D Chbib
email: dyaa.chbib@cpt.univ-mrs.fr

R Triay
email: triay@cpt.univ-mrs.fr

Inferences on cosmological expansion based on QSOs

Keywords: dark energy, cosmological parameters, quasars ArXiv ePrint: X.X Chapter 4 : Application to quasars data

By assuming that the evolutionary eects are negligible, SDSS QSO data favour a Friedmann-Lemaître model with a positive scalar curvature. Such a result is obtained by using a null correlation technique, which has the advantage of being independent of the luminosity function of sources and free from the Malmquist bias, what interprets as a robustness criteria in estimating the cosmological paramaters.

Résumé n sièle près le modèle d9univers de priedmnnEvemîtreD les oservtions le onfortent ve une onstnte osmologique Λ et une omposnte de mtière somre @noireA sns pression @poussièreA et froide dominnt elle ryoniqueD que l9on désigne pr modèle Λghw ou enore modèle stndrdF v9élértion de l9expnsion de l9nivers on(rmée pr le digrmme de rule des suE pernove en IWWV impose une vleur stritement positive à l onstnte osE mologiqueF wes trvux de thèse se folisent sur l9estimtion des vleurs de prmètres osmologiques du modèle stndrd en utilisnt l tehnique de orréltion nulleF gette pprohe présente l9vntge d9être plus rouste que les tehE niques usuellesF in prtiulierD il n9est ps requis de préiser l fontion de luminositéD elleEi est déduite pr ette méthodeF he plusD elle prend en ompte le iis de wlmquist due à l limittion en mgnitude pprentF ge trvil onsisté ussi à modéliser des éhntillons de l9événement qusr insi que l9événement supernovD une extrpoltion dptée du premierF ge qui permis de générer des éhntillons onformes ux hypothèses des modèlesD (n de vlider les pprohes sttistiquesF xous vons exploité les données du lon higitl ky urvey @hA pour les qusrsD et elles du uperxov vegy urvey @xvA et du hEss pour les supernoveF ves inférenes sttistiques ont onduit à un univers sptilement fermé et une présene de mtière noire plus fileF hns le dre d9une prohine pplition de ette tehniqueD elle ser utilisée pour ontrindre les modèles d9énergie noireF he mêmeD l9utilistion des ms de glxies oservées grâe à l9e'et de unyev el9dovihD servir d9éhntillon osmologiqueF ne telle étude pourr ontriuer à pporter un élément de réponse à l vlidité du rôle supposé des neutrinos mssifs dns l formtion des ms dns l9ère primordile de l9niversF wots glés X gosmologieD wtière noireD Énergie noireD prmètres osE mologiquesD sttistiquesD simultionD qusrsD supernovaeD digrmme de ruE leF Q Abstract e entury fter the niverse model of priedmnnEvemîtreD the oservtions omfort it with osmologil onstnt Λ nd drk mtter omponent withE out pressure @dustA nd old dominting the ryoni oneD whih is denoted y Λghw model or stndrd modelF he elertion of the expnsion of the niverse on(rmed y the rule digrm of the supernove in IWWV imposes stritly positive vlue on the osmologil onstntF wy thesis work fouses on the estimtion of the osmologil prmeters vlues of the stndrd model using the null orreltion tehniqueF his pE proh hs the dvntge of eing more roust thn the usul tehniquesF sn prtiulrD it is not neessry to speify the luminosity funtionD it is derived from the dt nlysisF sn dditionD it ounts for the wlmquist is due to seletion e'ets on pprent mgnitudeF his work dels with modelling smE ples of the qusr event nd the supernov eventD whih enles us to generte smples in order to vlidte the sttistil pprohesF e used dt from the lon higitl ky urvey @hA for qusrsD nd the uperxov vegy urvey @xvA nd hEss for supernoveF he ttistil inferenes suggest niverse sptilly glosed nd weker presene of drk mtter thn tht in the tndrd modelF uh sttistil nlysis n e used to onstrin drk energy modelsF epplition of this tehnique might e useful for nlyzing of lusters of glxies oserved through the e'et of unyev el9dovihD in view of deriving the osmologil model nd provide n nswer to the question of the ontriE ution of mssive neutrinos in the formtion of lusters in the primordil er of the niverseF uey ordsX gosmologyD drk mtterD drk energyD osmologil prmeE tersD sttistisD simultionD qusrsD supernovaeD rule digrmF Remerciements te tiens à remerier mon direteur de thèse olnd riy pour m9voir proposé un sujet de thèse en osmologieF te souhite remerier tous les memres de jury de thèse pour voir epté de donner de leur temps pour évluer mes trvuxF in prtiulierD les deux rpporteursD ierre estier et wr vhièzeEeyD pour leurs retours sur le mnusritF te souhite ussi remerier hierry wrtinD direteur de l9unité de reherhe wUQQPD qui epté de m9ueillir dns leur unité pour e'etuer ette thèseF te souhite églement exprimer mes remeriements à l9nien responsle d9informtique du loD inent fyleD pour son ide d9instlltion de plusieurs modules u déut de m thèseF te remerie mes olos @xdimD térômeD stisemD wrieD wihelleA ve qui j9i pssé des elles nnées à wrseilleD je n9oulie ps les onnes soirées u vieux port près un long jour de trvilD et ussi les elles rndonnées dns les lnques pendnt les weekends3 te vous souhite l réussite dns vos vies personnelles et professionnellesF te remerie prtiulièrement m fmille pour son soutien pendnt toute l durée de mes études et surtout dns les derniers moments de m thèseF weri pour vos sentimentsD on(nesD onseilsD idesFFF À tout moment di0ile vous étiez là pour me soutenirD grnd meriF te souhite le meilleur pour hun de vous dns s rrière et s vieF our (nirD je remerie m (néeD qui étit toujours présente et prohe pour rendre m vie joyeuse et m9ider à dépsser toutes les di0ultésF weri pour l joie que tu fis dns m vie et pour les ons moments qu9on pssés ensemle 3 weri pour tous les projets qu9on rêvé à réliser dns notre future 3 S Contents 1 Basics of Observational Cosmology 3

IFI sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R IFP xewtonin osmology F F F F F F F F F F F F F F F F F F F F F F F F R IFQ iinstein9s osmology F F F F F F F F F F F F F F F F F F F F F F F F S IFR he priedmnnEvemîtreEqmow model F F F F F F F F F F F F F T IFRFI he priedmnn9s equtions F F F F F F F F F F F F F F F F F U IFRFP rimordil niverse F F F F F F F F F F F F F F F F F F F F F V IFS gontents of the niverse F F F F F F F F F F F F F F F F F F F F F F W IFT fryoni eousti ysilltions F F F F F F F F F F F F F F F F F F F IQ IFU hrk mtter F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IQ IFV he hrk inergy F F F F F F F F F F F F F F F F F F F F F F F F F F IR
IFVFI wodels with lr pields F F F F F F F F F F F F F F F F F IS IFVFP he gosmologil gonstnt F F F F F F F F F F F F F F F F IT IFW rule digrm nd mesure of distnes F F F F F F F F F F F F IT IFWFI he gomoving histne F F F F F F F F F F F F F F F F F F IV IFWFP he ege nd the gonforml ime F F F F F F F F F F F F F PH IFWFQ he vuminosity histne

F F F F F F F F F F F F F F F F F F PH IFIH he wgnitude ystems F F F F F F F F F F F F F F F F F F F F F F PI IFIHFI he epprent wgnitude F F F F F F F F F F F F F F F F F PI IFIHFP he eg ystem F F F F F F F F F F F F F F F F F F F F F F PI IFIHFQ he AB ystem F F F F F F F F F F F F F F F F F F F F F F F PI IFII gonstrin the osmologil prmeters F F F F F F F F F F F F F F PP IFIP gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PP 2 Quasars samples 23 PFI sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PQ PFP petrum of qusr F F F F F F F F F F F F F F F F F F F F F F F F F PR U CONTENTS PFQ ivents in the speEtime digrm F F F F F F F F F F F F F F F F F PR PFR he seletion e'et F F F F F F F F F F F F F F F F F F F F F F F F F PU PFS wodelling smple of qusrs F F F F F F F F F F F F F F F F F F PV PFSFI ttistil modelling F F F F F F F F F F F F F F F F F F F F PW PFSFP imultion tehnique F F F F F F F F F F F F F F F F F F F F PW PFSFQ he kEorretion F F F F F F F F F F F F F F F F F F F F F F QR PFT gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QS
The null correlation technique 37 6 Résumé en français TFI v osmologie moderne F F F F F F F F F F F F F F F F F F F F F F F IIS TFP imultion d9éhntillon de qusr F F F F F F F F F F F F F F F F F IIV TFQ v tehnique de orréltion nulle F F F F F F F F F F F F F F F F F IPH TFR ésultts ve les données qusrs F F F F F F F F F F F F F F F F F IPI TFS wodélistion d9éhntillon de

QFI sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QU QFP eighting ftors F F F F F F F F F F F F F F F F F F F F F F F F F F RH QFQ vuminosity funtion nd seletion funtion F F F F F F F F F F F F RR QFR uolmogorovEmirnov test F F F F F F F F F F F F F F F F F F F F F F RT QFS he G max test F F F F F F F F F F F F F F F F F F F F F F F F F F RU QFSFI glultion of the G max terms F F F F F F F F F F F F F RV QFT gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SH Application to quasars data 51 RFI sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SI RFP he lon higitl ky urvey dt F F F F F F F F F F F F F F F F SP RFPFI he kEorretion F F F F F F F F F F F F F F F F F F F F F F SQ RFQ esults F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SR RFQFI he null orreltion test F F F F F F F F F F F F F F F F F F SR RFQFP he G max test F F F F F F F F F F F F F F F F F F F F F F
F F F F F F F F F F F F F F F F F IHI SFRFI hesription of smples F F F F F F F F F F F F F F F F F F F IHP CONTENTS SFRFP esults F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHQ SFS gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHU

List of Figures

IFI his imge shows the gosmi wirowve fkground s seen y ie9s lnk stellite @upper right hlfA nd y its predeesE sorD xee9s ilkinson wirowve enisotropy roe @lower left hlfAF he olors represent the temperture )ututionsF ed @resF lueA orresponds to wrm @respF oldA regionsF smge tken from the wesite of ie nd the lnk gollortion F F W IFP he gwf power spetrum otined with lnk PHISF opX egreement of the )t stndrd model of osmology ΛCDM with the oservtion dt from the temperture relese of the lnk stelliteF esiduls with respet to this model re shown in the ottompnel lnk gollortion sss @PHISAF F F F F F F F F IH IFQ le t whih the hypothesis of sptil uniform distriution of grvittion soures eomes vlid is roughly ove IHH wpD ording to the distriution of lrge numer of glxies with Pdp surveyF he (gure shows the lrgeEsle struturesD whih re less visile with distne from the orserverF he wilky y is loted t the intersetion of the two slies of the oserved sky wF golless PHHQF F F F F F F F F F F F F F F F F F F F F F F F II IFR he sptil distriution of the qusrs of the fy tlog shows rdil strti(tion due to four distint phses of oE servtionsD nd in eh of suEsmples derese in the numer of ojets s the rdius inresesF fy tking into ount this seletion e'etD one n sfely guess tht this distriution is uniformF pigure reditX universetody wesiteF F F F F F F F F F F II IFS gon(dene ontours in the @Ω M D Ω Λ A pln otined y the uE pernov gosmology rojet with oservtions of RP xe s F erlmutter et lF IWWWF F F F F F F F F F F F F F F F F F F F F F F F IS II LIST OF FIGURES IFT ivolution of the densities of rdition Ω γ D of rdition Ω m D nd of the osmologil onstnt Ω Λ with respet to the expnsion ftorF he di'erent periods indited y the dshed lines orreE spond to the lnk sleD the eletrowek symmetry reking @iAD the fig fng nuleosynthesis @ffxA nd the present epoh @xowAF greditX F grroll PHHRF F F F F F F F F F F F F F IU IFU he rule digrm showing the expnsion of the niverseF he rdil veloity of strsD orreted for solr motion plotted ginst the distne estimted from the strs nd the verge of rightness of glxies in lusterF greditX iF rule IWPWF IV IFV he rule digrm of the upernove smples with the xvD hD r nd severl lowEz surveysF greditX fetouleD wF et

l PHIRF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IW
IFW prom top left to ottom rightX he omoving distneD the lumiE nosity distneD the distne modulus nd the omoving volume versus the redshift for six di'erent osmologil modelsF F F F F F PP PFI higrm unifying di'erent strophysil types of eqx @lzrsD qusrsD seyfert glxiesA depending on the inlintion ngle with respet to the line of sight of the dust torus surrounding the sme entrl engine @xoted in greenAF he vrious omE ponents of eqx @noted in whiteAX e luminous retion disk whih surrounds the entrl lk holeF frod emission lines whih re due to louds oriting ove the diskF e thik dusty torus whih osures the rodEline region from trnsverse linesE ofEsightF xrrow emission lines whih re originted from irrE dited louds so fr from the entrl soureF greditX gF wF rryD F dovniD IWWS F F F F F F F F F F F F F F F F F F F F F F PS PFP petrum of qusr t rest frme determined y the h surE veyF he dshed thin nd thik lines re the powerElw (ts to the estimted ontinuum )uxF greditX hF ferkD et l PHHIF F F PT PFQ petrum of qusr h IHRREHIPS t redshift SFVF he optil vymnEα emission @IPIS ÅA is shifted into ner snfrEed

∼ VPQH ÅF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PT
PFR he spetime digrm @onforml time η versus omoving speA X he red (lled irle stnds for our present positionF he world lines of soures re vertil @perennil E lkA nd @ephemerl E lueAD they ross the pst light one of the oserver for oserved ojetsF hese ones re hrterized y n solute mgnitude M nd omoving volume V for the perenil onesD nd lso y lifetime durtion ∆η for ephemerl onesF F F F F F F F F F F F F PU PFS imilr to pigFPFR showing the pperne @lueAGdispperne @greenA of newGexisting supernove due to the intersetionGout of intersetion of the light one with their limited lifetimeF F F F PV PFT he lue urve gives the mximum volume v max (M ) round the oserver up to whih n ojet with solute mgnitude M n e oserved y using devie tht detets up to limiting pprent mgnitude m lim F F F F F F F F F F F F F F F F F F F F F F QI PFU imultion of ys smple omplete up to limiting pprent mgnitudeF he lue urve rterizes the threshold on pprE ent mgnitudeF he distriution of simulted soures @redA is uniform with respet to omoving volume @V A nd qussin with respet to solute mgnitude @M AF F F F F F F F F F F F F F QP PFV e omplete smple up to redshift z max with (Ω • , λ • = 0.3, 0.7)F e uniform distriution ppers in vertil xis nd qussin distriution in the horizontl xisF F F F F F F F F F F F F F F F F F QQ PFW histriutions of omplete smple up to redshift z max nd limE iting mgnitude m lim D y disentngling the visile ojets @lueA from those tht re not visile @redAF eAX ristogrm of solute mgnitudesF fAX wE digrmF F F F F F F F F F F F F F F F F F F QR PFIH e typil qusr spetrum t redshift z aSF he y n e oserved in two nds thnks to the vyα emission line whih is in the i9 nd nd the vyα forest whih is long of the r9 ndF pigure reditX F hrp et l PHHIF F F F F F F F F F F F F F F F QR QFI he distriution of the smple in oth (x, y) nd (X, Y ) frmesF QV QFP erhing the zero vlue of the weighted ovrine of X nd Y F he eqution Γ(θ) = 0 shows two solutionsD the (rst one @θ = π 4 A mthes the one used for simultionF F F F F F F F F F F F QW QFQ ht distriution in the (M, V ) frme nd (M, m) frmeF F F F F RH QFR he inverse of the orreltion funtion versus redshift for given osmologil modelF F F F F F F F F F F F F F F F F F F F F F F F F F RI QFS he perentge of the lost of the sttistil informtion @leftX for L 1 D rightX for L 0 A with respet to the prmeter βF F F F F F F RP QFT he weighting ftor with respet to the redshift for given osmologil modelF F F F F F F F F F F F F F F F F F F F F F F F F F RP QFU veft to rightD top to ottomF he vlues of the weighted oE vrine with respet to osmologil modelF he null vlues of the xgg re found y horizontl nd vertil interpoltions in the ovrine gridF he xgg in the (Ω • , λ • ) plne retrieve the initil simultion model @the (lled red irleAF F F F F F F F F F F RQ QFV he shpe of the xgg of two simulted smples with di'erent sizes nd di'erent pprent mgnitude limitsF F F F F F F F F F F RQ IQ LIST OF FIGURES QFW he dependene of osmologil model on the xgg ppers in these urves when we use two smples with sme hrteristis ut with di'erent osmologil modelsF F F F F F F F F F F F F F F RQ QFIH he preision of the method is shown oviously y inresing the size of the smpleF prom left to right nd from top to otE tomD the size of the smple inreses from QS to QSH to PHHH respetivelyF e see tht the sttistil )ututions derese with the inresing of the numer of ojetsF F F F F F F F F F F F RR QFII ttistis of the luminosity funtion @upper pnelsA nd the seE letion funtion @lower pnelsA F F F F F F F F F F F F F F F F F F F RS QFIP imilr to pigFQFII ut with qumel distriution for the luE minosity funtionF F F F F F F F F F F F F F F F F F F F F F F F F F F RT QFIQ hree df testing uniform distriution with uolmogorovEmirnov testF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F RT QFIR he redshift rnge is not onneted for this model with positive urvture κ • = 0.4F he distne modulus ζ(z) of soure with solute mgnitude M hs most likely two solutions nd it nnot e oserved in the hthed reF F F F F F F F F F F F F RW QFIS uolmogorovE mirnov test @vikelihood of ndidtesAF erh for the minimum of D max @top leftA s funtion of λ • D nd @ottom rightA s funtion of Ω • D on the xgg @top rightAF F F F SH RFI he lon foundtion PFSm telesope itheyEghretien type loE ted t ephe oint yservtoryD in south est xew wexioF F SP RFP veftX he kEorretion of the hEhQ smple in the iEnd inluding oth the emissionEline nd ontinuum omponentsD normlized t z = 2 with (xed spetrl index α ν = -0.5F ightX he numeril derivtive of kEorretion with respet to the redshift zF F F F F F F F F F F F F F F F F F F F F F F F F F F F F SR RFQ he null orreltion urves orresponding to ihD hID hQD hU smples nd hQ susmplesF he urves re loted in the region with positive urvture κ • > 0 @ove the stright lineAF ell the urves give lose ndidte vlues for osmologil prmeters exepted the one of ih smple @red urveAF F F F F SS RFR The absolute magnitude-volume diagram of the EDR sample. The left panel with Ω • = 0.076, λ • = 1.211. The right panel with Ω • = 0.3, λ • = 0.7. The domains corresponding to limiting magnitudes i = 15, 19.1, 20.5 and 21.78 are delimited by the corresponding curves (yellow, blue, green and black). The second ordinate axis represents the redshift values. F SS RFS imilr to pigFRFRX the solute mgnitudeEvolume digrm of the hI smpleF F F F F F F F F F F F F F F F F F F F F F F F F F F SS RFT imilr to pigFRFRX the solute mgnitudeEvolume digrm of the hQ smpleF F F F F F F F F F F F F F F F F F F F F F F F F F F SS

LIST OF FIGURES

RFU imilr to pigFRFRX the solute mgnitudeEvolume digrm of the uniform sttistil susmple of the hQ @fhQAF F F F F SS RFV imilr to pigFRFRX the solute mgnitudeEvolume digrm of the hEhU smpleF F F F F F F F F F F F F F F F F F F F F F F F ST RFW The absolute magnitude-volume diagram of a simulated sample which consists of 46,420 objects with a Gaussian luminosity function with M 0 = -25, σ M = 1.2, a limiting magnitude m lim = 19.1, and a maximum redshift z max = 13.6 in the model Ω • = 0.076, λ • = 1.211. 3 . F F F F F F F F F F F F F F F F ST RFIP eighting ftor pplied to hQ smple nd simultion smE ple without @leftA nd with @rightA kEorretionF ed urve with @λ • = 1.275D Ω • = 0.132A nd lue with @λ • = 1.229D Ω • = 0.132AF he distured shpe of the urve is due to the ontriution of the emission linesF F F F F F F F F F F F F F F F F F SU RFIQ e represent the pprent mgnitude limits on the p digrm of the homogeneous sttistil susmpleD fhQD hosen to pply the V /V max X 18.65 < m < 19.1 F F F F F F F F F F F F F F F SW RFIR he df of V /V max ompred to uniform distriutionD for fhQ susmple with Ω • = 0.076 nd λ • = 1.211F F F F F F F SW RFIS histne etween the empiril df nd theoretil df @D max AD testing uniform distriution of V /V max vluesD for ndidtes osmologil models on the xgg orresponding to fhQF veft pnelX D max s funtion of λ • F ight pnelX D max s funtion of Ω • F sn oth pnelsD the vlues of D max on the ordinte xis is trnslted into signi(ne level on the right hnd side xisF F F F F F F F F F F F F F F F F F F F F F F F F F F F SW RFIT he df of V /V max ompred to uniform distriutionD for hU susmple with (Ω • = 0.0169 nd λ • = 1.079)F F F F F F F F SW RFIU imilr to pigFRFIS X D max with respet to λ • @leftA nd (Ω • @rightA s ndidtes given y the null orreltion urve of hUF SW RFIV gontriution of mesurement errors to xgg orresponding to hQ F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UW RFIW The luminosity function f (M ) of the subsample of SUBDR3 with (Ω • = 0.0305, λ • = 1.1180). F F F F F F F F F F F F F F F F F F F F F F F F F F UW RFPH The selection function φ(m) of the SUBDR3 subsample with (Ω • = 0.0305, λ • = 1.1180). The upper panel is an enlargement of the area in which the function is most constant. F F F F F F F F F F F F F F F F F F F F F F F UW IS LIST OF FIGURES SFI he supernov lssi(tion treeF F F F F F F F F F F F F F F F F F VP SFP he light urves of di'erent types of xe in the lue ndX the type sD sD ssEvD ssED nd x IWVUeF he gurve of the x s is righter thn ll other x typesF pigure ourtesyX heeler IWWHF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VQ SFQ he supernov spetrl ypesF he grvittionl supernove is the progenitor of the supernove type sD s nd ssF he progenE itor of x type s is thermonuler explosion of white dwrfsF pigure ourtesyX upernov gosmology rojet @gAF F F F F F VQ SFR en rtisti imge of white dwrf nd red gent strF pigure ourtesyX pgGhvid rrdyF F F F F F F F F F F F F F F F F F F F VQ SFS petr of three ype s supernoveD x IWVUhD x IWVUxD nd x IWWHxD out one week fter the mximum of their vgF pigure ourtesyX eF F pilippenko IWWU F F F F F F F F F F F F F VQ SFT fs light urves for x IWWVu @pigure ourtesyX xF fF untze' IWWWA F F F F F F F F F F F F F F F F F F F F F F F F F F F VR SFU vight urve stndrd otinedF he fEnd lightEurves for lowEredshift xe s from the glnEololo survey @pigure ourE tesyX rmuy et lF IWWTA F F F F F F F F F F F F F F F F F F F F F VS SFV gorreltion etween the solute mgnitude with the deline rte ∆m 15 of the lightEurve for lowEredshift x s from the glnEololo survey @pigure ourtesyX rmuy et lF IWWTA F F VS SFW veftX shemti of intrinsi light urve of xF ightX the intereption of the worldEline of the x with the pst light oneF VU SFIH vight urve of x xvEHRhQgxF his (gure shows light urve well smpled y oservtions whih help to hrterize its pE rmeters (x 0 , t * obs , x 1 , c) without isesF pigure ourtesyX estier F et lD PHHT F F F F F F F F F F F F F F F F F F F F F F F F F F F F WI SFII e representtive (gure of the x s light urves in the im to explin the seletion e'etsF he dshed urve represents the mster urve where the entire light urves must oinideF he highest lue urve represents the light urve of the x s righterD roderD slower nd luer thn the mster urveF he yn urve @the lowestA represents the light urve of the x s finterD nrrowerD fster nd less luer thn the mster urveF he t 1 nd t 2 re two oservtion dtes for whih ndidte of x is deteted y sutrting the imges tken t these dtesF WI SFIP he seletion of xe depends on two hrteristisX the mxE imum of luminosity m * nd the deline rte ∆m 15 F he lowest light urve is the threshold light urve hrterized y m * lim nd ∆m 15,lim whih re used s the seletion riteriF e x with deline rte ∆m 15 `∆m 15,lim is oserved s long s its pprent mgnitude t mximum luminosity m * `m * lim F F F F F WP

LIST OF FIGURES

SFIQ wo xe with the sme intrinsi hrteristis t di'erent redE shiftsF elthough the light urve of the x of the highest redshift @z = 1.5A hs smll deline rte ∆m 2 15 `∆m 15,lim due to the diltion of timeD this x will not e onsidered in the smple euse it hs not su0ient time to hve good smpling of its light urveF he seletion of x is done only if m * `m * lim F F WQ SFIR he righterEslower reltion @solute mgnitude versus shpe prmeterA

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WQ SFIS esolute mgnitudeEvolume digrm of simulted smples @eA @leftA nd @fA @rightAF F F F F F F F F F F F F F F F F F F F F F F F F WT SFIT rule digrm
of simulted smples @eA @leftA nd @fA @rightAF WU SFIU pour dimensionl digrms of hrteristis of smple @eAF F F F WU SFIV pour dimensionl digrms of hrteristis of smple @fAF F F F WU SFIW he xgg orresponding to @eA smple @leftA nd the one orE responding to @fA smple @rightAF F F F F F F F F F F F F F F F F F WW SFPH hpe of the xgg with respet to the hrteristis of smple nd osmologil modelF F F F F F F F F F F F F F F F F F F F F F F WW SFPI xgg of ten rndomly generted smples with the sme hrE teristis in the stndrd osmologil modelF F F F F F F F F F F F IHH SFPP erh for the minimum of u testD D max D @top leftA s funtion of λ • D nd @ottom rightA s funtion of

Ω • D on the xgg @top rightAF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHH SFPQ ttistis of the luminosity funtion @upper pnelsA nd the seE letion funtion @lower pnelsA F F F F F F F F F F F F F F F F F F F IHI SFPR imilr to pigFSFPQ F F F F F F F F F F F F F F F F F F F F F F F F F IHI SFPS he xggs of the SHH simulted smplesF F F F F F F F F F F F F F IHP SFPT et
of the est ndidtes of the osmologil model with the likelihood ontours @top leftAF he proility density funtions with respet to λ • @top rightA nd with respet to

Ω • @left otE tomAF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHP SFPU he p digrm of tve smple uild with the modelX Ω • = 0.
F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHQ SFPV he p digrm of r smple uild with the model @Ω • = 0.3, λ • = 0.7AF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHR SFPW he p digrm of lowEz smple uild with the model @Ω • = 0.3, λ • = 0.7AF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHS SFQH he p digrm of hEss smple uild with the modelX Ω • = 0.3, λ • = 0.7F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHT SFQI he p digrm of xvQ smple uild with the modelX Ω • = 0.3, λ • = 0.7F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHU IU
LIST OF FIGURES SFQP imilr to pigFSFIU for xvQ smple with lirtion of stnE drdiztion oe0ients with respet to the lowEz smpleF F F F F IHU SFQQ he null orreltion urve in the λ

• -Ω • digrm orresponding to xvQ smpleF F F F F F F F F F F F F F F F F F F F F F F F F F F IHV SFQR he null orreltion urves in the (Ω • , λ • ) digrm tht orreE
spond to y smples nd xvQ smple @the highestAF F F F F IHV SFQS D max with respet to λ • @leftA nd Ω • @rightA s ndidtes given orrespondnt ux éhntillons ihD hID hQD hU et ux sousEéhntillons hQF ves oures sont situées dns l région ve une ourure positive κ • > 0 @uEdessus de l ligne droiteAF outes les oures sont en ord et prohes l9une de l9utreD suf l oure rouge qui orrespond à l9éhntillon ihF F F F F IPI TFS ves oures de orréltion nulles dns le digrmme

y the xgg of xvQF F F F F F F F F F F F F F F F F F F F F F F F IHV SFQT he p digrm of hEss smple uild with the model (Ω • = 0.11, λ • = 1.32)F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHW SFQU he p digrm of xvQ smple uild with the model (Ω • = 0.11, λ • = 1.32)F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIH SFQV vuminosity funtion @left pnelA nd seletion funtion @right pnelA uilt with the model (Ω • = 0.11, λ • = 1.32)D orrespondE ing to xvQ smpleF F F F F F F F F F F F F F F F F F F F F F F F F IIH SFQW imilr to pigFSFPT F F F F F F F F F F F F F F F F F F F F F F F F F
(Ω • , λ • ) qui orrespondent ux éhntillons y et à l9éhntillon xv @le plus élevéAF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IPR

Introduction

he understnding of the niverse ws strted in the sth epoh when the polish gopernius suggested helioentri model of solr systemD leding to the ideX no speil ple for irth in the niverseF his revolution ws folE lowed y seond nd mjor one dted for ITVUD where ss xewton o'ers the (rst rel theory of grvittionF his theory explined for the (rst time the movement of the strs in the sky s well s flling odies on irthD giving desription of the grvittionl intertion s fore tht spreds instntlyF roweverD this theory ws not le to explin the nomlous preession of werE ury9s perihelion y RPFU rseonds per enturyF he questions tht formed the weknesses of xewton9s theoryD hve een nswered with the new theory of grvittion given y iinstein in IWHSEIWISF e new desription of grvity is given y the iinstein pield iqutionsD whih n e summed up in words s followsX mtter modify the urvture of speEtimeD nd the urvture of speEtime mps the movement of mtterF st is worth to mention tht iinstein dded to his equtions osmologil onstnt Λ serhing solution of stti osmi niverseF foth priedmnn @IWPPA nd vemître @IWPUAD independentlyD proved tht the iinstein9s theory hs osmologil solutions of n expndingD non stti niverseF st ws not until IWPWD the er of modern osmology egn with the disovery of rule of the expnsion of the niverse iF rule IWPWF his disovery mrked milestone nd mjor progress in osmologil studiesF yne generlly eE lieves tht this is the reson why iinstein withdrew the onstnt Λ liming tht its introdution ws gret mistkeF vterD this onstnt ws proposed s solution to explin the expnsion of the niverseF eentlyD the rule digrm of typeEs supernove grees with positive vlue of the osmologil onstntD wht interprets s n elertion of the osmologil expnsion eF qF iess et lF IWWVD F erlmutter et lF IWWWF I LIST OF FIGURES etullyD the usul sttistil pproh in the nlysis of the rule diE grm is fed to the systemti isesD ssuming hypotheses on the luminosity funtion nd the seletion funtionF yn the other hndD roust sttistil pproh requires weker numer of hypothesesF his thesisD introdues simultion methods of qusr nd supernov smplesD s well s roust sttistil tehnique with the im to onstrin forml vlues of the osmologil prmetersF sn ghpFID s present rief desription of the oservtionl osmology repE resented y the priedmnnEvemîtreEqmow modelF s present in ghpFP the proility lws whih desrie the hrteristis of qusr smplesD nd the simultion methods in whih my work is prtF he third hpter is devoted to desriing the roust sttistil methodF he pplition of this method on the qusr dt from h survey nd the results re exhiited in ghpFRF pinllyD e new modelling of supernov smple is presented in ghpFSF e dt proessing of supernove of hEss nd xvQ surveys is desried s well in this hpterF CHAPTER 1 wny enturies fter xewton @ITRQ E IUPUAD who introdued the (rst universl grvittionl theoryD the osmology follows xewton9s theories nd priniples 1 D serhing for wys to understnd the lws tht ontrol the niverse iF rrE rison PHHHF he niverseD during xewton9s timeD ws thought to onsist of only the soE lr system nd the strsF st ws unler how fr the ltter were from irthD sine no diret mesurement of the distnes ould e performed due to the qulity of the stronomil instrumenttion t tht timeF st ws not until the nineteenth enturyD tht the (rst diret mesurement of the distne of str ws performed y fessel F lotner PHHVF efter fessel9s prllx 2 in IVQUD it ws ovious tht the size of the solr system ws in(nitesiml ompred to tht of the niverseF vukilyD the spetrosopyD devie whih mesures the hemil omposition of strs nd estimtes their )uxes nd luminositiesD ws developed in IVRP y elexndre fequerel @IVPHEIVWIA sF eppenzeller PHIQ fy the end of tht enturyD the niverse ws reognized s n enormous glxyD the wilky y where we elongD with huge numer of strs vF qerrit IWQUF he strs re distriuted in hoti order in the skyD howeverD in some times they re umulted in lustersF he ltter re known s di'use neul or plnetry neul tF fFD uler IWUTF hese oservtions rise lot of quesE tions suh sX ht is the nture of these lusters nd how do they performc hose questions remined unnswered until the twentieth entury when in IWPSD idwin rule disovered gepheid in the rightest spirl neulD enE dromedF sing the periodEluminosity reltion of renriett vevitt @IWHVA for the lirted gepheids tF hF pernie IWTWD rule disovered the extrE glti nture of endromedF his neul is more thn P million light yers wy from our glxy sF is PHHSF ruleD thenD onluded tht the niE verse is not restrited to the wilky yD nd it is populted y millions of glxiesFFF 1.2 Newtonian cosmology hespite the suess of the xewtonin theory in explining the grvittionl lws on erth nd etween plnets nd their motion in the solr systemD it 1.3 Einstein's cosmology fils to desrie the niverse t lrge sleF he xewtonin theory predits non stle niverse wFF vongir PHIQF essuming two ojets of di'erent msses re seprted y distne dD these ojetsD in the xewtonin oneptD nnot remin in their positions inde(nitely onsidering their mutul ttrtionF eording to this theoryD these ojets undergo grvittionl fore whih leds to shorten the distne etween themF gonsequentlyD niverse whih surrounds enter of mss is doomed to ollpse on itselfF sn the xewtonin theoryD to otin suh osmologil model without enterD one must onsider loud (lling the niverse uniformly with onstnt densityF roweverD ording to rlley in IUPHD in n in(nite niverse ontining n in(nite numer of strsD the night sky should e s right s the dyF his on)it etween the drkness of the night sky nd the ssumption of n in(nite nd eternlly stti universe is the mjor prolem of the xewtonin osmologyD known y ylerts9 prdox iF F rrrison IWVWF his prdox hs een resolved lter through the modern osmology @desried in the next setionA whih shows tht the niverse is expndingF hue to this expnsionD the light emitted from distnt strs nd glxies is redshiftedF husD the emission spetr of those ojets pper to us s veering grdully to the light frequenies tht we nnot see @typilly in infrredAF his interprettion explins the drkness of our night skyF 1.3 Einstein's cosmology he modern osmology egn hundred yers go @IWISA where elert iinstein pulished his theory of generl reltivity @qAF his theory links the distriuE tion of the energy density tht (lls the niverse nd its geometry through the iinstein (eld equtions @ipiAX

Basics of Observational Cosmology

R µν - 1 2 Rg µν = 8πGT µν @IFIA
here µν is the ii tensorD g µν is the metri tensor used with the onvention signture (+, -, -, -)D is the ii slrD µν is the energyEmomentum tensor nd q is the xewtonin onstntF sn qD the units re so tht the speed of light in vuumD c = 13 F sn IWIUD iinstein introdued the osmologil onstnt Λ for otining stti niverseF iqFIFI eomesX

R µν - 1 2 Rg µν + Λg µν = 8πGT µν @IFPA
sn IWPPD the ussin meteorologist nd engineerD elexnder priedmnn proved tht the ipi lso llow dynmil worlds rF xussumer PHIRF roweverD it took iinstein few yers to reognize the relevne of priedmnn9s remrksF

Chapter 1 : Basics of Observational Cosmology sndependentlyD in IWPUD the felgin nonD qeorges vemître pulished pE per qF vemitre IWPU in whih he estlished the sme equtions s priedE mnn nd interpreted themF vemître lso predited the rule vw even efore the rule oservtions @getting orret estimtion for the ruE le onstnt of TPS km.s -1 .M pc -1 A4 F sn IWQID he pulished his theory of the primitive tom in whih he introdued the onept of the time zeroF st ws not until IWPW tht rule disovered the fmous liner reltionship etween the distne nd the veloity @etFIFWA of distnt glxies with the help of telesope of PFSR meters in dimeter pled t wount ilson iF rule IWPWF sn IWQID rule showed iinstein the oservtionl evidene of redshifted neulr spetr onvining iinstein tht the niverse is expndE ingF iinstein knowledgedD thereforeD tht there ws no reson to mintin his ide of (nite stti niverseF re ndoned the osmologil onstntD desriing it s the gretest mistke of his lifeF sn IWQPD iinstein pulished with de itter his euliden in(nite osmologil model of the niverse eF iinstein nd F de itter IWQPF he onept of n expnding niverse lter quired the termD fig fng theoryF his term ws oined y pred royleD one of the proponents of the sttionry modelD who introdued vemître to his ollegues in IWSH with the wordsX his is the fig fng mn3F fy the end of the UH9sD rF plihe 8 tF wF ouriu IWUW reonsidered the grvE ittionl equtions with the osmologil onstnt in their sttistil nlysis of the rule digrm of ysF his llowed to estimte positive vlue of ΛF vterD riyD F et lF IWWHD sttistil investigtion on the rightest luster led to similr results whih shows the elertion of the osmologil expnsionF iight yers lterD eF qF iess et lF IWWV nd F erlmutter et lF IWWW pointed out the evidene of osmi elertion y mesuring the urte distnes to distnt typeEs supernoveD used s stndrd ndlesF ine thenD the use of Λ eome the dilemm of the enturyF 1.4 The Friedmann-Lemaître-Gamow model he frmework of the modern osmology is sed on the osmologil prinE ipleF st supports the hypotheses of the homogeneity nd the isotropy of the universeF eording to the gopernin prinipleD there is no privileged ple in the niverseF purthermoreD ssuming the isotropy of this ltterD whih mens tht eh point in the niverse hs identil properties in di'erent diretionsD thusD the niverse would look homogeneous5 F es mtter of ftD oserving the universe t given length in di'erent diretions with the sme opening 1.4 The Friedmann-Lemaître-Gamow model solid ngle must led to n identil result independent of the hosen diretionF his hrteristi implies thtD on lrge sleD we n desrie the oservE le universe s eing sptilly homogeneous nd isotropiF his rings us to the desription of speEtime with the priedmnnEvemîtreEoertsonElker @pvA metriF hespite the inhomogeneity of the niverse t smll sles s shown in the mp of Pdf survey in pigFIFQD the pv metri desries the speEtime t lrge slesD more thn IHH wpD where the osmologil hypotheses re vlid s seen in pigFIFRF 1.4.1 The Friedmann's equations he pv metri illustrtes the osmologil priniple mthemtillyF st is de(ned in spheril oordintes @rD ΘD ϕA sX

ds 2 = dt 2 -a 2 (t)[ dr 2 1 -kr 2 + r 2 (dΘ 2 + sin 2 Θdϕ 2 )] @IFQA
where t is the osmi timeD a(t) is the @dimensionlessA expansion parameterD its present vlue is a • = a(t • ) = 1D nd k is the sign of urvture of spe whih determines the type of geometri topologyX k = -1 for n open niverseD k = 0 for )t niverse nd k = 1 for losed niverseF he derivtion of the pv metri is purely geometrilD sujet to the onstrints of homogeneityF en lterntive form of the metri is given y tking into ount the representtion of the (xed oordintesF e n write the lterntive pv metri sX

ds 2 = dt 2 -a 2 (t)[dχ 2 + S 2 k (χ)(dΘ 2 + sin 2 Θdϕ 2 )] @IFRA
where dχ redsX

dχ = dr √ 1 -kr 2 @IFSA nd S k (χ)X S k (χ) =          sin(χ) if k = 1 χ if k = 0 sinh(χ) if k = -1 @IFTA
sn homogeneous niverseD the energyEmomentum tensor T µν ounts for perfet )uid of density ρ nd pressure P F sing the T µν tensor with the metri in iqFIFQD one n derive the priedmnn equtionsX

H 2 = ȧ a 2 = 8πG 3 ρ - k a 2 + Λ 3 @IFUA U Chapter 1 : Basics of Observational Cosmology ä a = - 4πG 3 (ρ + 3P ) + Λ 3 @IFVA
where ȧ stnds for the time derivtive of @tAF H is the Hubble parameterD its present vlue is H • = 67.8 ± 0.9 km.s -1 .M pc -1 lnk gollortion s @PHISAF 1.4.2 Primordial Universe ntil the lte thirtiesD the formtion of elements in the niverse ws sujet of detesF sn IWRPD qeorge qmow mentioned for the (rst time the ide of primordil nuleosynthesisF sn IWRVD qmowD elpher nd rermn predited the osmi mirowve kgroundD working on model of hot fig fng sE suming primordil niverse omposed only of neutrons @whih susequently disintegrte into protonsA F tF iF eeles PHIRD F elpher PHIRF sn this model of fig fng xuleosynthesis @ffxAD the originl niverse ws mde up of n extremely hot nd dense plsm of photonsD fermions nd qurksF he niverse then expndedD nd its density nd temperture deresedD fostering primordil nuleosynthesis iFeF the formtion of the (rst wek nulei @hydroE genD heliumD deuteriumD lithiumAF hen the temperture eme low enough @∼ 3000 • KA for the (rst toms to form through proess lled reomintionF et tht timeD the ryons deoupled the photons nd the light egn to spred freelyF his rditionD whih ws simply ooled during the niverse evolutionD is known s the gosmi wirowve fkground @gwfAF he ryons nd old drk mtter @etFIFUA evolved in n tthed wy under the fore of grvityD nd formed the strutures tht we re urrently seeingD iFeF lustersD glxies nd strsF snterestinglyD two min oservtions on(rmed the ffx modelF he (rst one ws estlished in IWTSD when enzis nd ilson deteted the osmi mirowve kgroundF he nturl explntion for the presene of this rdiE tion is tht the niverse pssed through wrm nd dense phse efore the reomintion hppened nd the rdition ws emittedF his rdition hs lmost perfet lk ody spetrum with n verge temperture of 2.73 • K produed y the photons of the primordil niverse nd its spetrum peks in the mirowve frequeny rnge of ITHFP qrz orresponding to wvelength of IFW mmF he temperture of gwf is isotropi nd homogeneous down to vritions of IH -5 @see pigFIFIAF he mesures undne of the wek elements in the niverseD s seond proofD re in exellent greement with the preditions of the primordil nuleE osynthesis qF teigmn PHHRF his inludes the mss undne of relium reR predited etween PQ% nd QH%D in perfet greement with oservtions gF eF fertulniD et lF PHITF 1.5 Contents of the Universe pigure IFIX his imge shows the gosmi wirowve fkground s seen y ie9s lnk stellite @upper right hlfA nd y its predeessorD xee9s ilkinson wirowve enisotropy roe @lower left hlfAF he olors represent the temperture )ututionsF ed @resF lueA orresponds to wrm @respF oldA regionsF smge tken from the wesite of ie nd the lnk gollortion 1.5 Contents of the Universe he perfet )uid is hrterized y the properties of the energyEmomentum tensor with pressure P nd density of energy in the niverse ρF st stis(es n @empirilA eqution of stteX P = wρ @IFWA where w = -1 for osmologil onstnt @vuum energy densityA nd w < -1 3 for drk energy models suh s quintesseneF gomining iqsFIFU nd IFV we getX

ρ + 3(ρ + P ) ȧ a = 0 @IFIHA
prom the derivtion of iqsFIFU nd IFVD we otinX d(ρa 3 ) = -P da 3 @IFIIA using iqFIFIHD one n de(ne the evolution in time of the energy density

ρ(t) ∝ a(t) -3(1+w) @IFIPA W
Chapter 1 : Basics of Observational Cosmology pigure IFPX he gwf power spetrum otined with lnk PHISF opX egreeE ment of the )t stndrd model of osmology ΛCDM with the oservtion dt from the temperture relese of the lnk stelliteF esiduls with reE spet to this model re shown in the ottompnel lnk gollortion sss @PHISAF por the severl omponents of grvittionl soures in the niverseD this energy density is written sX

ρ i (t) ∝ a(t) -3(1+w i ) @IFIQA
with w γ = 1/3 for the rdition nd w m a H for the nonEreltivisti mtterF iquting iqFIFIQD we getX

ρ γ a 4 ∝ const. @IFIRA ρ m a 3 ∝ const. @IFISA
por reltivisti mtter suh s neutrinosD w γ is not onstnt nd vries dependE ing on the erD ut this omponent n e negleted for reent periodsF fy negleting this termD iqFIFU eomesX

H 2 = 8πG 3 (ρ m + ρ γ ) - k a 2 + Λ 3 @IFITA ith the following nottion X ρ k = -3k 8πGa 2 @IFIUA
1.5 Contents of the Universe pigure IFQX le t whih the hypothesis of sptil uniform distriution of grvittion soures eomes vlid is roughly ove IHH wpD ording to the distriution of lrge numer of glxies with Pdp surveyF he (gure shows the lrgeEsle struturesD whih re less visile with distne from the orserverF he wilky y is loted t the intersetion of the two slies of the oserved sky wF golless PHHQF we n express iqFIFIT s followsX

H 2 = 8πG 3 (ρ m + ρ γ + ρ Λ + ρ k ) @IFIVA
where

ρ c = 3H 2 8πG @IFIWA ith the dimensionless prmetersX Ω m = ρ m ρ c , Ω γ = ρ γ ρ c , Ω Λ = ρ Λ ρ c , Ω k = ρ k ρ c @IFPHA nmed osmologil prmetersD iqFIFIV eomesX H 2 H 2 • = ρ m ρ c,• + ρ γ ρ c,• + ρ Λ ρ c,• + ρ k ρ c,• @IFPIA sing iqsF IFIR nd IFISD we otinX H 2 H 2 • = ρ m ρ c,• a • a 3 + ρ γ ρ c,• a • a 4 + ρ Λ ρ c,• + ρ k ρ c,• a • a
a • = 1D thereforeX H 2 H 2 • = Ω m a -3 + Ω γ a -4 + Ω Λ + Ω k a -2 @IFPQA
where these dimensionless prmeters stis(yX

Ω m + Ω γ + Ω Λ + Ω k = 1 @IFPRA
he ove nottions of osmologil prmeters re the usul nottionsD ut they re sustituted y

Ω Λ ≡ λ • = Λ 3H 2 • D Ω k ≡ -κ • = k• H 2 • D Ω m ≡ Ω • = 8πGρm,• 3H 2 • nd Ω γ ≡ α • = 8π 3 G(kT•) 4 45 3 H 2 • ∝ 10 -5
whih re more pproprited euse these quntities show di'erent ehviors with timeF vet us de(ne the polynomilX

P (a) = λ • a 4 -κ • a 2 + Ω • a + α • @IFPSA
1.6 Baryonic Acoustic Oscillations

P (1) = 1; λ • -κ • + Ω • + α • = 1 @IFPTA
sn our sttistl nlysisD we estimte the prmeters λ • nd Ω • F 1.6 Baryonic Acoustic Oscillations he properties of the gwf gurntee the existene of sle from whih the universe eomes isotropi nd homogeneousD s well s the mesurement of the veloity of our glxy with respet to the referentil tht it de(nesF iven though the temperture of the gwf is extremely uniform throughout the skyD the gyfi stellite @gosmi fkground ixplorerA deteted tiny )ututions or temperture nisotropies in the gwf in IWWPF he we @ilkinson wiE rowve enisotropy roeA stelliteD lunhed in PHHID on(rmed the gyfi oservtion nd mpped the temperture )ututions with muh higher resE olutionF wore reentlyD the lnk stellite mpped the gwf nisotropy with even higher ury @see pigFIFIAF sn generl greement with the oservtionD the nisotropy in the gwf grew from the grvittionl tension of smll )utuE tions existent in the erly niverseF hese perturtions gve rise to ousti osilltions in the photonEryon )uid efore the reomintionF huring tht periodD the grvittionl ttrtion etween ryons tended to ollpse the system nd ompress the photonEryon )uid wheres the photon pressure proE vided n opposite restoring foreF his reted sound wves tht propgte in the )uid of the primordil niverseF et the time of reomintionD the photons were di'used freely nd these osilltions outlined in the gwfF e modelling of the dynmis of these strutures in the primordil plsm mkes it possile to interpret the ngulr )ututions of its tempertureD whih re oserved y mens of spetrl nlysisD whih provides estimtes of osmologil pE rmetersF woreoverD these fetures hve n imprint on ryoni strutures t every stge of the evolution of the niverse lled fryoni eousti ysilE ltions @feyAF st is elieved tht the high density res ssoited with the ousti wves in the gwf ondense to rete the urrent struturesF he size of struturesD n e used to tre k the osmologil prmeters F ru IWWUF pigFIFP shows the power spetrum of the gwf otined y lnk stelliteF 1.7 Dark matter hrk mtter is n d ho ingredient used in the modelling of grvittionl struture to interpret oservtionsF st does not intert eletromgnetilly nd therefore nnot e deteted diretly with telesopesF sts properties re deduted from grvittionl e'ets on visile mtterF st is elieved tht only IQ Chapter 1 : Basics of Observational Cosmology less thn IH% of the energy density of the niverse omes from ryoni nd in prinipleD visile mtterF iF rule IWQR proposed the (rst mesure of the density of @visileA mtter in the niverse sed on n verge of the mesured glti mss @T E VA × IH 8 w F he existene of suh mtter ws derived y tn yort in IWQP who studied the orits of strs in the glxy nd onluded tht the wilky y should ontin more mtter thn wht ws visile tF rF yort IWQPF oon fterD wiky mesured the dispersion of the rdil veloities of eight glxies in the gom superluster nd found surprisingly lrge vlue pF wiky IWQQF epplying the viril theoremD he found tht the totl mss of the superluster should e RHH times greter thn the mss of visile mtter pF wiky IWQUF re resolved the prolem y introduing hidden mss distriuted in the lusterF sn the IWUHsD er uin mesured the rottionl urves of spirl glxies nd disovered tht the rdil veloity of the glxy endromed @wQIA remins onstnt until t lest QH kp using the PI m line of rydrogen uin et lF IWUHF sn disgreement with the expeted keplerin rottionl urveD this study hs given rise n evidene of the existene of drk mtter in the glxiesF ine thenD the modelling of severl oservtions re in greement with the existene of drk mtter in the universe suh s the dynmis of glxies nd lustersD the grvittionl lensing nd the temperture distriution of hot gs in the glxy lusters s well s osmologil proes suh s the gwfD fey nd ype s upernoveF hree di'erent types of drk mtter hve een de(nedX old drk mtter @ghwAD wrm drk mtter @hwA nd hot drk mtter @rhwAF he neutrino is the est ndidte for hot mtter while its mss is not yet well onstrinedF gonerning the old drk mtterD the est ndidtes re the sws @ek snterting wssive rtilesAD whih re predited y mny supersymmetri theoriesD nd wegry @wssive gompt rlo estronomil yjetAF he deoupling dte of these prtiles intervenes in the formtion of lrge strutures in the niverseF e niverse dominted y hot drk mtter would hve topEdown senrio of struture formtionX the formtion of superlusters of glxies whih then rek up into lustersD then glxiesD nd so forthF gonverselyD niverse dominted y old drk mtter senrio would e ottomEupX formtion of glxies @from louds of gsA whih re grouped into lusters nd then into superlustersF xowdys oservtions gree with the theory of old drk mtter despite some luster oservtions suggest tht they hve formed previouslyF The coincidence problem he oserved energy densities of the vuum nd of mtter re of the sme order of mgnitudeD s seen in pigFIFSD whileD the energy densities of the omE ponents of the niverse @α • D Ω • D nd λ • A evolve di'erently s shown in pigFIFTF his (gure shows tht the vuum energy initilly hd negligile quntity with respet to the two other densitiesD then it eomes dominnt in the present timeF e lso see tht the trnsitions etween the dominnt omponents re 1.9 Hubble diagram and measure of distances firly shortD prtiulrly etween mtter nd energy of the vuumF st is inE triguing tht this trnsition hppens tody whih mkes the present dy privileged time in osmi historyF st isD thereforeD interesting to sk whether the oserved drk energy is form of energy hving stte eqution vrile with timeF reneD new model hs een introdued y dding slr (eld pigure IFTX ivolution of the densities of rdition Ω γ D of rdition Ω m D nd of the osmologil onstnt Ω Λ with respet to the expnsion ftorF he di'erE ent periods indited y the dshed lines orrespond to the lnk sleD the eletrowek symmetry reking @iAD the fig fng nuleosynthesis @ffxA nd the present epoh @xowAF greditX F grroll PHHRF slowly vrying with time @see etIFVFIAF 1.9 Hubble diagram and measure of distances ht rule did tht led to the disovery of his lw ws the mesure of oth the distne nd the veloity of glxies in the nery niverse @see pigFIFUAF re showed tht glxies reede with veloity tht inreses proportionlly with their distneD wht gives the rule9s lwX

v = H • d @IFQHA
where the reession veloity of glxies v nd their distne dF he onstnt of proportionlity H • is lled the rule onstntF ine its disoveryD the IU Chapter 1 : Basics of Observational Cosmology rule digrm eme mjor tool for retrieving of osmologil informE tionF he distriution of typeEs supernove in this digrm is used with @lrge smples of typeEs supernoveA deteted from surveys suh s the xvD hD r nd lowEz surveys @pigFIFVAF he spetr of soure is seen redshifted pigure IFUX he rule digrm showing the expnsion of the niverseF he rdil veloity of strsD orreted for solr motion plotted ginst the distne estimted from the strs nd the verge of rightness of glxies in lusterF greditX iF rule IWPWF y n oserverD due to the expnsion of the niverseF he redshift z is de(ned s followsX

1 + z = λ o λ e = a(t o ) a(t e ) = 1 a(t) @IFQIA
where o a IF hereforeD the formul of the expnsion ftor t time t is expressed sX

a(z) = 1 1 + z @IFQPA
τ * (z) = to te dt a(t) @IFQQA = a• ae da H • √ λ • a 4 -κ • a 2 + Ω • a + α • @IFQRA = 1 H • 1 1 1+z 1 P (a)
da @IFQSA rerefterD we use the dimensionless omoving distneX τ (z) = he dimensionless omoving volume V (z) is the volume of sphere of rdius τ (z) entered on the oserver9s lotionF st is de(ned y the following formulX

V (z) =            π κ 3 2 • (2τ -sin(2τ )) if κ • > 0 4π 3 τ 3 if κ • = 0 π |κ•| 3 2 (sinh(2τ ) -2τ ) if κ • < 0 @IFQUA where τ = τ | κ • | is the ngulr distneF 1.9.2
The Age and the Conformal Time he lookk time n e mesured from the oserver t z = 0 @time t • A k to redshift z @time tA6 F st is given yX

t(z) = 1 H • 1 1 1+z ada P (a) @IFQVA por )t osmologil model @λ • = 0.7D Ω • = 0.3A nd H • = 70 km.s -1 .M pc -
η * (z) = 1 H • η (z) @IFRIA
where η is the dimensionless onforml time @see iqFIFQTAF 1.9.3 The Luminosity Distance he emitted light of soure is reeived y the oserver s he is on surfe element of sphere of rdius d L D the luminosity distneD entered on the light soureF he reeived light per unit of time represents the )ux nd the pprent 1.10 The Magnitude Systems rightness of this soure whih dereses with the squre of the distneD y the known inverse squre lwX

f = L 4πd 2 L @IFRPA
where L is the intrinsi luminosity of n ojetF st turns out tht the luminosity distne is relted to the omoving distne nd n e written s followsX

d L = c H • (1 + z).T (τ (z) ) @IFRQA ithX T (τ (z) ) =            sin(τ ) √ κ 0 if κ 0 > 0 τ if κ 0 = 0 sinh(τ ) √ |κ 0 | if κ 0 < 0 @IFRRA
1.10 The Magnitude Systems sn stronomyD the mgnitude system is frequently used to desrie the rightE ness of n ojetF he mgnitude is sle used to lssify the strs depending on their rightnessF he rightest strs hve the lowest mgnitude for whih one degree of mgnitude orresponds to di'erene of PFSI times in rightnessF 1.10.1 The Apparent Magnitude his mgnitude orresponds to the pereption of str9s luminosity in the eyes of n oserverF ereption through the humn eye is nonEliner nd is sensitive to reltive di'erenes of luminosity etween strsD therefore the oserved mgnitude is presented in logrithmi sleF he pprent mgnitude m of n ojetD is de(ned s its )uxD f D mesured in given oservtionl (lterD nd the one of referene ojet @for exmple eg str whih hs null mgnitudeAD f 0 D in the sme (lterF m = -2.5 log 10 f f 0 @IFRSA he mgnitude of the referene str is null wht de(nes the origin of mgnitude y de(ning the zero point notion @A sX

m AB = -2.5 log 10 d(log ν)f ν T ν d(log ν)T ν -48.6 @IFRVA
with f ν the )ux in units of frequenyD of unit ergFs -1 Fm -2 Frz -1 D nd T ν is the trnsmission of the (lter used @dimensionlessAF he onstnt RVFT is hosen so tht AB a V for n ojet whose spetrum is )tF his referene system is sed on four dwrf strs of type pF 1.11 Constrain the cosmological parameters sing the de(nition of pprent mgnitudeD we n introdue the solute mgnitude M of n ojetF his ltter is mesure of the intrinsi rightness of the ojet nd is de(ned s it would pper to hypothetil oserver t distne of IHpF he pprent mgnitude for n ojet t redshift z is relted to the solute mgnitude @following iqFIFRPAD we n writeX

m(z) = -2.5 log 10 L 4π(10pc) 2 + 5 log 10 d L (z) 10pc @IFRWA m(z) = M + 5

Introduction

usrs were disovered in the lte IWSH9s using the rdio telesopes with no oservtion of orresponding visile ojetsF he (rst one ws tied to n opE til ojet ws the rdio soureF urprisinglyD the spetrum of this ojet hd mny unidenti(ed rod emission linesF oon fterD it ws disovered y the huth stronomer wrten hmidt tht these lines were not so strngeD s they were just redshiftedF felieving he ws oserving strD hmidt disE overed the rdio qusrD Qg PUQD in IWTQ nd identi(ed its emission lines of hydrogen nd showed tht they were shifted to lrger wvelengths wF hmidt IWTQF st resemled strD euse it ws seen s point soure through the PS Chapter 2 : Quasars samples telesopeF fut unlike the strsD the spetrum of this point soure hd strong rdio emissionF wore ojets of the sme type were soon oservedF sn the sene of the explntion of the strophysil nture of these ojetsD they were lled qusiEstellr rdio souresD the nme whih hs een susequently revited s qusrsF usrs re onsidered suset of the popultion of tive glti nulei @eqxA @eFgF see F entonui IWWQAF pigFPFI represents eqx whih onsists of superEmssive lk hole in the enter surrounded y n retion diskF st is formed due to the surrounding mterils whih fll under the huge grvity of the lk holeF hile fllingD these mterils relese their potentil energy s result of the frition generted etween themF hrough di'erent turulent phenomenD these mterils generte high energy prtiles hnneled y the mgneti (eld lines of the eqxF hese prtiles re lter relesed in the form of two ollimted jets from the mgneti polesF he retion disk is surrounded y thik torus of dustF en eqx whose jet of prtiles ollimtes in our line of sight is lled flzrF sf the jet tips in nother diretion nd the host glxy of the lk hole is visileD the eqx is eyfert glxyD otherwiseD the eqx is qusrF he qusr @y for qusi stellr ojetsA is the rightest lss of eqxsF 2.2 Spectrum of quasar ys re identi(ed minly y their emission linesF vymnEα emission line @λ Ly-α a IPIS ÅA hrterizes prtiulrly the qusrs @see pigF PFPAF edE ditionl emission lines re lso visile in their spetrumD suh xD isD gsD wgssD mong other ones less importntD see petrl vines used in h tleF sn generl vymn lph forest is present in most of ys spetrD depresE sion of the ontinuum in the spetrum used y suession of sorption lines t smller wvelength thn the vymnE α emission lineF st is used y the the presene of neutrl hydrogen loted long the line of sight of the qusrF his is the âqunnEeterson e'etâ tF qunn nd fF eterson IWTSF pigFPFQ represents y from the h survey t redshift of SFVF he prt of the spetrum etween the vymnEβ nd the vymnEγ emission lines is lled the forest vymnEβD etFFF his depression of the ontinuum due to the superpoE sition of sorption lines is used y hydrogen louds nd other sorers in front of the yF 2.3 Events in the space-time diagram ith the motivtion in mind to use events @suh s light emissions from sE tronomil ojetsA to proe the geometry of speEtimeD one hs to speify 2.3 Events in the space-time diagram pigure PFIX higrm unifying di'erent strophysil types of eqx @lzrsD qusrsD seyfert glxiesA depending on the inlintion ngle with respet to the line of sight of the dust torus surrounding the sme entrl engine @xoted in greenAF he vrious omponents of eqx @noted in whiteAX e luminous retion disk whih surrounds the entrl lk holeF frod emission lines whih re due to louds oriting ove the diskF e thik dusty torus whih osures the rodEline region from trnsverse linesEofEsightF xrrow emission lines whih re originted from irrdited louds so fr from the entrl soureF greditX gF wF rryD F dovniD IWWS their intrinsi properties tht hrterize single fmilyF hey re ssumed to e not very di'erent from one nother in terms of their intrinsi luminosity @stndrd ndlesAF woreoverD we ssume tht they do not show evolutionry e'ets nd tht they re uniformly distriuted in speF hese hrteristis stnd for rndom vriles with proility densities who re de(ned y speE i( working hypothesesF he dt re fe to seletion e'ets in oservtionD nd seletion funtion is used t this purposeF e hek the e0ieny of our sttistil method on simultion smples s representtion of the rel dtF he min seletion e'etD it hs een desried y wlmquist @see etFPFRAD depends only on the pprent mgnitudeF hereforeD seletion funtion must e used in the proility density in order to tke it into ount mong other seletion e'ets in oservtionF PU Chapter 2 : Quasars samples pigure PFPX petrum of qusr t rest frme determined y the h surveyF he dshed thin nd thik lines re the powerElw (ts to the esE timted ontinuum )uxF greditX hF ferkD et l PHHIF pigure PFQX petrum of qusr h IHRREHIPS t redshift SFVF he optil vymnEα emission @IPIS ÅA is shifted into ner snfrEed ∼ VPQH ÅF eording to working hypothesesD the ojets hve (xed position on the omoving spe V 3 nd they re expeted to e uniformly distriutedF sn the speEtime digrm @pigsFPFR 8 PFSA s desried y the diret produt of the onforml time with V 3 D perennil 1 soures show vertil world lines @lkA while ephemerl ones @lue nd greenA stnd for prts of those onesF fy ssuming tht they re right enoughD they n e oserved solely when the light pst one of the oserver rosses tht gives dte @iFeF redshiftAF wo dtes @the eginningA η 1 nd @the endA η 2 re displyed for ephemerl eventsF imilrlyD the durtion of oservtionD eing in(nitesiml ompred to osmologil sleD is mgni(ed on the speEtime digrmF yn these (gE uresD the intersetion of the world lines with the pst light one of the oserver @the two rossed lines t the red irleA ensures the ojet to e visileD if it is right enough with respet to oservtion devieF he ojets @qusrD glxyDFFFA re hrterized intrinsilly y the omoving distne τ nd the solute mgnitude M y ssuming tht it is onstnt in timeF yn the other hndD the ephemerl ojets re hrterized y luminosity fontion @light urveA insted of onstnt solute mgnitudeF 2. 4 The selection eect pigure PFRX he spetime digrm @onforml time η versus omoving speA X he red (lled irle stnds for our present positionF he world lines of soures re vertil @perennil E lkA nd @ephemerl E lueAD they ross the pst light one of the oserver for oserved ojetsF hese ones re hrterized y n solute mgnitude M nd omoving volume V for the perenil onesD nd lso y lifetime durtion ∆η for ephemerl onesF 2. 4 The selection eect ith limiting mgnitude in oservtionD sine the pprent luminosity deE reses with distneD the smple verge of solute mgnitudes does not proE vide us the men solute mgnitude of the popultionD see qunnr wlmquist @IWPPAF his is rises due to the limited sensitivity of instruments nd teleE sopesD using the overErepresenttion of luminous ojets in mgnitudeE limited smpleF hereforeD the verge of solute mgnitude of the trunted smple will e righter thn the one of the omplete smple finney 8 werE ri(eldD IWWVF his limittion in the pprent mgnitude is the min seletion e'etF PW Chapter 2 : Quasars samples pigure PFSX imilr to pigFPFR showing the pperne @lueAGdispperne @greenA of newGexisting supernove due to the intersetionGout of intersetion of the light one with their limited lifetimeF 2.5 Modelling a sample of quasars qiven tht the qusrs re deteted t high redshiftD therefore the use of the rule digrm with this type of ojets will e good tool to otin the osmologil informtionF he frthest qusr oserved up to now is the qusr ve tIIPHHIFRVCHTRIPRFQ deteted t redshift z = 7.085 iF womjin et l PHIQF es usulD non evolution of qusrs is ssumed on interpreting the nonE liner rule digrmF sn the followingD we ssume H • = 70 Km.M pc -1 .s -1 hF xF pergel et l PHIQF

Statistical modelling

sn this setionD we fous on uilding simultion smple of ys with n pproh whih is di'erent thn tht used in the wgwg vewis et l PHHPF es we hve noted in etFPFQD the ys re hrterized intrinsilly y n solute mgnitude M nd uniform distriution in the omoving speF sf 2.5 Modelling a sample of quasars the solute mgnitude M is distriuted ording to the luminosity funtion f (M )D then the y9s smple is desried with the produt of two independent proility densities s followX

dP th ∝ f (M )dM.dV @PFIA
essuming tht the seletion e'ets depend solely on mD we use seletion funtion φ(m) for desriing the seletion proess in oservtionF henD the proility density of the oservle vriles re desried yX

dP obs = φ(m)dP th P th (φ) @PFPA
where P th (φ) is normliztion ftorX

P th (φ) = φdP th @PFQA
e de(ne the umultive distriution funtion @dfA of the rndom vrile M sX

F (x) = x -∞ dP obs = 1 P th (φ) x -∞ f (M )dM V (M ) @PFRA
where V (M ) is de(ned s followsX

V (M ) = φ(ζ(z) + M )dV (z) @PFSA F (x)
is uniform rndom vrile etween 0 nd 1F he solute mgnitudeEvolume digrm is grphi representtion of dt tht enles one to hek visully ny rtift in the distriutionF por the true vlues of osmologil prmetersD they re distriuted uniformly long the Exis nd ording to the luminosity distriution funtion long the wExisF his digrm ws proposed y plihe nd ouriu rF plihe 8 tF wF ouriu IWUWD @herefter nmed p digrmAF

m lim = ζ( z max ) + M @PFUA
reneD let us de(ne the funtion v max (M ) tht oinides with V (M ) is this prtiulr seX

V (M ) = v max (M ) = v( z max ) @PFVA
rving de(ned orrespondene tle of the umultive distriution funtion F (x)D we n dedue y interpoltion its inverse funtionF reneD we generte rndom uniform smple F X Fk ∈ [0, 1] nd we retrieve the orresponding solute mgnitude M k y solving the inverse mpping X

M k = F -1 ( Fk ) @PFWA nd hene the volume v max (M k ) following iqFPFV @or iqFPFS if φ(m) is not reviside funtionAF e generte uniform rndom vrile Vk etween [0, v max (M k )]D nd then we determine the redshiftX z k = V -1 ( Vk ) @PFIHA
where V -1 stnds for the inverse fontion of the omoving volume iqFIFQUF pigFPFT shows the digrm v max versus M for given pprent mgnitude m lim F he red symol plus is reliztion of uniform rndom vrile distriuted etween H nd v max (M k )F his step is done for N numer of ojetsF pinllyD we ompute the pprent mgnitude of eh ojet ording to iqFIFSHD to otin simulted smple of ysX {(z k , m k )} k=1..N @presented in pigFPFUA2 F e generted severl smples tht llowed us to vlidte the tehniques used herefterF e hose the stndrd osmologil model @λ • = 0.7D Ω • = 0.3A nd qussin luminosity funtion of men M 0 = -20 nd stndrd devition σ M = 0.3D with limiting pprent mgnitude m lim = 26F

An example of biased sample generator elthough not solutely neessryD it is not uninteresting to desrie method of smple simultionD whih n e found in the literture to estimte the orE retion of the is of wlmquistD whih is unfortuntely erroneousF o generte smple omplete up to n pprent limiting mgnitude m lim D it onsists on hoosing d ho redshift vlue domin so to minimize the numer of ojets 2.5 Modelling a sample of quasars pigure PFTX he lue urve gives the mximum volume v max (M ) round the oserver up to whih n ojet with solute mgnitude M n e oserved y using devie tht detets up to limiting pprent mgnitude m lim F disrded euse of the pprent mgnitude thresholdF vet the z max e suh vlueD the orresponding volume reds V max s given y iqFIFQUF reneD one genertes uniform distriution of N ojets etween 0 nd V max nd y ssuming qussin distriution of the solute mgnitude with proilE ity density funtion g G (M ; M 0 , σ M ) whih gives the smpleD {M k D V k } k=1..N F hereforeD the redshift is determined y solving the inverse funtion of volumeD V -1 (V k )D nd the pprent mgnitude following iqFIFSHF es resultD one oE tins the oserved quntities {(z k , m k )} k=1..N F uh smple is shown in the M -V digrm in pigFPFVF e see lerly tht the ojets with solute mgE nitude M re distriuted uniformly until mximum volume is rehedF o represent the seletion e'etsD we pply threshold m lim to the resulting smE ple exluding ll events tht hve lrger vlue of pprent mgnitude thn m lim Y this uto' is trnslted y reviside funtion θ(m limm)F e uto' t z max is present @in the simultionAD s desried y reviside funtion θ(z maxz)F hereforeD the proility density of the events redsX

dP obs = 1 P th (θ m θ z ) θ(m lim -m)θ(z max -z)g G (M ; M 0 , σ M )dM dV @PFIIA QQ
Chapter 2 : Quasars samples pigure PFUX imultion of ys smple omplete up to limiting pprent mgnitudeF he lue urve rterizes the threshold on pprent mgnitudeF he distriution of simulted soures @redA is uniform with respet to omoving volume @V A nd qussin with respet to solute mgnitude @M AF where P th (θ m θ z ) is normliztion ftorX

P th (θ m θ z ) = θ(m lim -m)θ(z max -z)g G (M ; M 0 , σ M )dM dV @PFIPA
he omprison etween pigFPFV nd pigFPFWEf shows up the soures @redA tht re not right enough to e visile t limiting mgnitude m lim F he histogrm in pigFPFWEe shows this distriution of solute mgnitudes of omplete smple up to redshift z max D the red domin orresponds to soures not enough right to e visile with limiting pprent mgnitude m lim F he verge of the oserved smple @lue dominA is found to e righter thn the men tht hrterizes the popultion @ M = M 0 -δM AD ppering notieE ly on the histogrmF his is the e'et of the wlmquist isF he numer de(it orresponds to ≈ 20% of the smpleF gontrry to the tehnique used oveD determining mximum redshift of the smple efore strting the simE ultion is unlikely sine we would lose lrge numer of ojetsD s n e oserved in the empty irle in pigFPFWEfF sn other wordsD we pplied uto' on the redshiftD in ddition to the uto' pplied on the pprent mgnitudeD might e the use of rising the systemti errorsF ithout this uto'D the irle must ontin suh numer of ojetsF woreoverD seletion funtion pigure PFVX e omplete smple up to redshift z max with (Ω • , λ • = 0.3, 0.7)F e uniform distriution ppers in vertil xis nd qussin distriution in the horizontl xisF depending on the redshift put n dditionl orreltion etween the vrilesD whih is inonsistent with the hypotheses of work of the sttistil test deE sried in ghpFQF his is the min di'erene etween this method nd the simultion method desried in etFPFSFID whih mke this lst the reomE mended method of simultionF 2.5.3 The k-correction es we hve seen in etsFIFW 8 PFPD the osmologil expnsion of the universe shifts the spetrum of y towrd the red wvelength @pigFPFIHAF hereforeD in order to hve n homogeneous dt smpleD one needs k-orretion @ rumsonD wyllD 8 ndge IWSTY yke 8 ndge IWTVAF xmelyD one uses the solute mgnitude M t the sme wvelength for ll ojets t ny redshift nd iqFIFSH eomesX

m X = M Y + M X -M Y + ζ(z) @PFIQA m X = M Y + k XY (z) + ζ(z) @PFIRA

QS

Chapter 2 : Quasars samples pigure PFWX histriutions of omplete smple up to redshift z max nd limiting mgnitude m lim D y disentngling the visile ojets @lueA from those tht re not visile @redAF eAX ristogrm of solute mgnitudesF fAX wE digrmF where m X is the pprent mgnitude in the oserved ndD M Y is the solute mgnitude of the soure in the restEfrme nd nd k XY is the kEorretion etween the nd ndsF k XY is lulted through the rte of the oserved )ux in nd nd the one mesured in (tive (lter in the rest frme roggD hF F PHHPX

k XY (z) = -2.5log 10 (1 + z) -2.5log 10 Φ(λ)X(λ)dλ Φ( λ 1+z )Y (λ)dλ Φ 0 (λ)Y (λ)dλ Φ 0 (λ)X(λ)dλ @PFISA
where Φ(λ) is the spetrum of the soure nd Φ 0 (λ) is the spetrum of the referene ojetF feuseD qusrs spetrum shows vrious rod emission linesD their kE orretion ounts for omponent due to the underlying ontinuumD uontD nd nother omponent due to the emission linesD uem oss xF F PHIQF e de(ne the redshift9s termX

ζ(z) = 5 log 10 d L (z) 10pc + k(z) @PFITA husD iqFPFIR is written s followsX m = M + ζ(z) @PFIUA
feuse of the presene of the emission lines in the spetrum of yD the kE orretion is not monotonous funtionF reneD the redshift9s term in iqFPFIU is no longer n invertile funtionF 36

Conclusion

pigure PFIHX e typil qusr spetrum t redshift z aSF he y n e oserved in two nds thnks to the vyα emission line whih is in the i9 nd nd the vyα forest whih is long of the r9 ndF pigure reditX F hrp et l PHHIF

Conclusion

sn this hpter we hve desried how to model y smple from theoretE il to n oserved proility lws tht desries the eventsF e presented n e0ient simultion method for providing smples in greement with the working hypothesesF QU CHAPTER 3

The null correlation technique he orretion of ised sttistis requires to know the proility lw of hrteristis desriing the smple usedD suh s the luminosity funtion nd the seletion e'etsF rerefterD we use roust sttistil method whih does not depend on the luminosity funtion nd the seletion funtionF hese fetures re derived s result of this pprohF he innovtive spet of this pproh is the use of the weighting ftorF

dP ∝ g G (x)dx f (y)dy @QFIA
where f (y) is reviside funtionX

f (y) = 1 if 0 ≤ y < 1 0 otherwise @QFPA
ith rottion of ngle π 4 the distriution ppers s shown in pigFQFIF xowD let us suppose tht the oserver is fed to nlyze suh distriutionD whih is desried with proility lw in the (X, Y ) frmeX

dP ∝ G(X, Y )dX dY @QFQA
where q stnds for the prolity density funtion of nd D whih works s orreltion funtion etween the rndom vriles

X nd Y X G(X, Y ) = g G (X cos(θ) -Y sin(θ)) h(X, Y ) @QFRA with h(X, Y ) = 1 if 0 ≤ X sin(θ) + Y cos(θ) < 1 0 otherwise @QFSA
he im is to estimte the vlue of θ y ssuming liner orreltion etween these rndom vrilesF ith the usul proedure to estimte suh vlue y men of lest squre (ttingD one is fed to the ritrry hoie of the distne @horizontl or vertilAF e wy to overome suh prolem is to trnslte this optimiztion into (nding wy to mke the rndom vriles independentsF iqFQFQ shows tht the orreltion etween X nd Y is given y the orreltion funtion ρ ∝ h(X, Y ) e cos(θ) sin(θ) X Y s prt of the proility density funtionF o remove suh orreltionD one n use weighting ftor

w i = 1 ρ F he weighted ovrine redsX Γ(θ) = N i=1 w i (X i -X )(Y i -Y ) @QFTA whereX X = N i=1 w i X i nd Y = N i=1 w i Y i @QFUA
is expeted to e null for the true vlue of thetD Γ(θ) = 0D see pigFQFPF sul sttistil pproh onsists to (t the dt to the theoretil urve y mens of lestEsqure methodD so tht the vlues of the osmologil pE rmeters minimize the dispersionF o know if the seletion e'ets is the estimtes of these prmetersD we must mke ssumptions to desrie them in the proility distriution of these dtF he null orreltion tehnique does not require to mke suh dditionl hyE pothesesD ut solely tht there is no seletion e'ets on redshiftF eording to the formul PFPD simulted smple under the model @λ • = 0.7D Ω • = 0.3A is performedD we used qussin luminosity funtionY this smple is used in this setion to disuss on the sttistil tehniqueF he oserved dt re desied in the (M, m) frme y the following proE ility densityX

dP (M,m) = ρ(z)φ(m)f (M )dM dm; ρ(z) = ∂V ∂m @QFVA
where ρ(z) is orreltion funtion etween M nd mD whih orresponds to the toin of the trnsformed vriles (M, V ) → (M, m)F he orreltion of the pprent mgnitude nd the solute mgnitude is shown in pigFQFQF et (rst glneD for the true vlues of the osmologil prmeters (Ω • , λ • )D the orreltion etween the rndom vriles M nd m vnish y dividing eh event y the orreltion funtion ρ(z)F his orreltion is given y the weighted ovrine of M nd mD it depends on osmologil prmeters nd redsX

Γ(Ω • , λ • ) = N k=1 ω k (M k -M )(m k -m ) @QFWA
where ω k is weighting ftor @detils in etFQFPA whih is inversely proporE tionl to ρ(z k )F he weighted verges M nd m re de(ned s followsX

M = N k=1 ω k M k @QFIHA m = N k=1 ω k m k @QFIIA he solutions of eqution Γ(Ω • , λ • ) = 0 re ndidte vlues for Ω • nd λ • D they orrespond to urve on the (Ω • , λ • ) plneF

Weighting factors

he orreltion funtion of the proility density given y iqFQFVD depends on the priedmnnEvemître modelX

w(z) = 1 ρ(z) = 1 ∂V ∂m @QFIPA 3.2 Weighting factors pigure QFQX ht distriution in the (M, V ) frme nd (M, m) frmeF
For the case of the close Universe κ 0 > 0:

e ompute ∂V ∂m a ∂V ∂ζ a ∂V ∂z G ∂ζ ∂z @the detils of lultion re in the ppendix ??AF hen we otinX ithout the kEorretion termX

∂V ∂ζ = ϑ (1 -cos(2κ 1 2 0 τ )) (1 + z) P ( 1 1+z ) + κ 1 2 0 cot(κ 1 2 0 τ ) @QFIQA ith the kEorretion termX ∂V ∂ζ = ϑ (1 -cos(2κ 1 2 0 τ )) (1 + z) P ( 1 1+z ) + κ 1 2 0 cot(κ 1 2 0 τ ) + (1 + z) 2 P ( 1 1+z ) ∂k(z)
∂V ∂ζ = ϑ (cosh(2|κ 0 | 1 2 τ ) -1) (1 + z) P ( 1 1+z ) + |κ 0 | 1 2 coth(|κ 0 | 1 2 τ ) @QFISA
ith the kEorretion termX

∂V ∂ζ = ϑ (cosh(2|κ 0 | 1 2 τ ) -1) (1 + z) P ( 1 1+z ) + |κ 0 | 1 2 coth(|κ 0 | 1 2 τ ) + (1 + z) 2 P ( 1 1+z ) ∂k(z)
∂z @QFITA where ϑ = 2π ln [START_REF] Souriau | Mécanique statistique, groupes de Lie et cosmologie[END_REF] 5|κ 0 | F For the case of the at Universe κ 0 = 0:

ithout the kEorretion termX

∂V ∂ζ = ϑ τ 3 τ (1 + z) P ( 1 1+z ) + 1 @QFIUA ith the kEorretion termX ∂V ∂ζ = ϑ τ 3 τ (1 + z) P ( 1 1+z ) + 1 + τ (1 + z) 2 P ( 1 1+z ) ∂k(z) ∂z @QFIVA
where ϑ = 4π ln [START_REF] Souriau | Mécanique statistique, groupes de Lie et cosmologie[END_REF] 5 F he weighting ftor redsX

ω k = w(z k ) N i=1 w(z i ) @QFIWA
he prie to py for using the weighting ftors is the prtil loss of the sttistil informtion riy et l IWWID mostly t high redshift s we show in pigFQFRF st n e evluted y mesuring of the perentge of the de(it s followsX @QFPIA where β is hosen so tht weighting ftors ω k 1 re s lose s possileF sts vlue is determined y minimizing the funtion over the smpleX

L 1 = 1 + 1 ln N N k=1 ω k ln ω k @QFPHA
L 0 = M ax{w β (z)} -M in{w β (z)} @QFPPA
pigFQFS shows the de(it of sttitil informtion versus β @left pnelA tht n e ompred to the rnge of weighting ftorsD tht n e ompred to the dispersion of weighting ftors vlues @right pnelAF he weker of L 1 isD the more relile the resultsF e smll di'erene on the vlues of β otined y the two methods does not hve signi(nt impt on the estimtion of the osmologil prmetersF yn the other hndD the simultions show tht it is preferle to perform the minimiztion of L 0 rther thn L 1 F he ltter is used loss estimtorF he weighting ftor versus redshift s it is shown in pigFQFTF he use of these weighting ftors fvors the soures t low nd high redshift ompred to otherD whih is t the ene(t to reover osmologil informE tionD see pigFQFRF e seek the vlues of osmologil prmeters in the ) on this xgg to simulte nother smple with the sme hrteristis @luminosity funE tionD size nd limited pprent mgnitudeAF he orresponding xggD whih inludes the vlues of the osmologil prmeters used to perform the simuE ltionD ut swithes round the ndidte model with respet to the previous one @lue urve in pigFQFWAF he ury of the null orreltion pproh is pigure QFVX he shpe of the xgg of two simulted smples with di'erent sizes nd di'erent pprent mgnitude limitsF sertined y the study of the sttistil )ututionsF reneD one hundred rndom omplete smples were generted ording to the density de(ned in iqFPFPF hese smples were simulted with the sme osmologil model @)t stndrd modelAD qussin luminosity funtionD size nd limiting pprent mgnitudeF hese smples provided us with one hundred xggs for omputE ing the sttistis of iqFQFWF he nlysis ws performed y vrying the numer of ojetsF st turns out tht the sttistil estimtes re more urte with lrger numer of ojetsF es shown in pigFQFIHD utter)y shpe is otined y the hundredth xggsF pigFQFIH @top pnelD leftA is otined for smples of QS ojetsD @top pnelD rightA for smples of QSH ojets nd @ottom pnelA for smples of PHHH ojetsF sn the (rst (gureD the shpe of the utter)y is wideD only QH% of the urves retrieved the initil modelF por QSH ojetsD most @WV7A of the urves retrieves the initil simulted modelF sn the ottom pnel we see lerly tht the model is retrieved for ll smples with high uryF pigure QFWX he dependene of osmologil model on the xgg ppers in these urves when we use two smples with sme hrteristis ut with di'erent osmologil modelsF

(Ω • D 1 ω k = w β (z k ) N i=1 w β (zi)

RS

Luminosity function and selection function

he null orreltion pproh llows us to determine the shpes of the lumiE nosity funtion f (M ) nd the seletion funtion φ(m)F por the true vlues of the osmologil prmetersD the smple is desried with the following proility densityX

dP w = f w (M )dM φ w (m)dm @QFPQA
where the funtions f w (M ) nd φ w (m) re de(ned s followsX

f w (M ) ∝ 10 -β 5 M f (M ) @QFPRA φ w (m) ∝ 10 β 5 m φ(m) @QFPSA
hey re evluted from the umultive distriution funtionsX

F w (M ) = M -∞ f w (M )dM @QFPTA Φ w (m) = m -∞ φ w (m)dm @QFPUA RW
F w (M ) ≈ F st w (M ) = N k=1 ω k θ(M -M k ) @QFPVA Φ w (m) ≈ Φ st w (m) = N k=1 ω k θ(m -m k ) @QFPWA
where θ is the reviside funtionF o otin the funtions f (M ) nd φ(m)D we ompute f w (M ) nd φ w (m) y determining the derivtives of F w (M ) nd Φ w (m) nd use iqsFQFPR nd QFPSF sn order to void the ritrry hoie of 50 3.4 Kolmogorov-Smirnov test the (tD the derivtives of smooth pproximtions of F st w (M ) nd Φ st w (m) re determined y the liner interpoltionsF henD the luminosity funtion f (M ) nd the tehnologil funtion φ(m) re determined s followsX

   f (M ) ∝ F st w (M +∆rM )-F st w (M -∆ l M ) ∆rM +∆ l M 10 β 5 M φ(m) ∝ Φ st w (m+∆rm)-Φ st w (m-∆ l m) ∆rm+∆ l m 10 -β 5 m @QFQHA
where ∆ r M nd ∆ l M re the right nd left intervls of the vrile M in whih ontin

√ N 2 vlues of M k > M nd M k < M respetivelyF he intervls ∆ r m nd ∆ l m lso ontin √ N 2 vlues of m k > m nd m k < m respetivelyF es result of the use of liner interpoltion tehniqueD the @ √ N
2 A points on the left side nd the @ √ N

2 A points on the right side re isolted nd thereforeD must e disrded in the estimtion of the funtions f (M ) nd φ(m)F pigsFQFII nd QFIP show the result otined for f (M ) nd φ(m) with simulted smples in whih f (M ) is qussin funtion with n verge M 0 = -20D stndrd devition σ M = 0.15D qumel funtion with mode M 0 = -20 nd sle prmeter γ M = 1.09 @see ppendix eFPAF he φ(m) funtion is reviside funtionF he upper pnels of these (gures show the sttistis of the lumiE nosity funtionF he @left pnelA illustrtes the empiril df of the evluted luminosity funtion represented y the green urveD it is ompred to the theoE retil df of the luminosity funtion represented y the red urveF he @right pnelA shows the luminosity funtion with respet to the solute mgnitudeF he red urve is the luminosity funtion used for the simultion of the smple nd the green plus symols represent the luminosity funtion evluted y the interpoltion tehniqueF he lower pnels show the sttistis of the seleE tion funtionF he @left pnelA represents the empiril df of the evluted seletion funtion represented y the green urve nd the theoretil df of the reviside funtion represented y the red urveF he @right pnelA shows the tehnologil funtion with respet to the pprent mgnitude mF he funE tion ϕ(m) ≡ φ(m) @the green plus symolsA otined is the sme s the one used in the simultionD it is evluted s reviside funtion @in redAF hese (gures show tht the initil luminosity nd tehnologil funtions used for the simultion re retrieved perfetly y this tehnique with M = -20.005 nd σ M = 0.158F

Kolmogorov-Smirnov test

he uolmogorovEmirnov @uA test is nonprmetri test tht llows one to hek whether devitions etween the theoretil df nd the empiril @stepA df n e onsidered s rndom )ututionsF he u is sed on the proility lw of the mximum distne de(ned s followsX

D max = M ax(|cdf T heoretical -cdf Empirical |) @QFQIA

SI

Chapter 3 : The null correlation technique pigure QFIIX ttistis of the luminosity funtion @upper pnelsA nd the seE letion funtion @lower pnelsA he result of D max is ompred to the ritil vlueD D max,α D @for signi(E ne level αA otined from the u tle with respet to the size of the smpleF st llows one to deide if the null hypothesis is epted @D max ≤ D max,α A or rejeted @D max ≥ D max,α A for given α @y defultD α = 5%AF pigFQFIQ shows the df of three distriutions with inresing smple size @xA @from left to rightA tht n e ompred to the theoretil df of uniform distriution @solid lineAF st shows how the empiril nd the theoretil df get loser when x growsF he u test enles one to hek whether spei( funtions f (M ) nd φ(m) (t the dty ompring F w (M ) with F st w (M ) nd Φ w (m) with Φ st w (m)F por this purpose we strted with the se of the luminosity funtionF sf f (M ) hs qussin form @eppFeFIAD iqFQFPR shows tht f w (M ) is still qussin with men M de(ned in iqFQFII nd stndrd devition σ M de(ned s the followingX -1) @QFQSA his lst llows us to otin the mximum devitions with respet to the empiril df

σ 2 M ≈ N N -1 N k=1 ω k (M k -M ) 2 @QFQPA 52
Φ st w (m)X D max = M ax(|Φ w (m) -Φ st w (m)|) @QFQTA
his test llows us to deide whether these devitions re sttistil )utuE tionsF 3.5 The V/V max test he V /V max test is well known sttistil pprohes tht hve een used in osmologyF he rndom vrile V /V max hs uniform distriution etween H to IF rereinD we pply this test s desried y qF figot 8 F riy IWWIF e de(ne the uniform rndom vrileX

h(z, m) = V 0 (z) V m (M ) @QFQUA
where V 0 (z) is the omoving volume t redshift z nd V m (M ) is the omoving volume tht the soure of solute mgnitude M remins visile up to redE shift z max t limiting pprent mgnitude m lim of the surveyD it orresponds to V (z max ) de(ned in iqFPFV hmidt wFD IWTVF he proility density of omplete smple redsX

dP obs = 1 A θ(m lim -m)θ(z)θ(z f orm -z)f (M )dM dV @QFQVA
where e denotes the normliztion ftorD z f orm the redshift of the formtion epoh nd θ the reviside funtionF e rnge of redshift is de(ned y the produt of the reviside funtions in the proility density in whih soure of solute mgnitude M is oservedF pollowing iqFPFIU we de(ned this rnge y Π

M (z)X Π M (z) = θ(m lim -M -ζ(z))θ(z)θ(z f orm -z) @QFQWA
e little lger shows tht the proility density of M nd h redsX

dP ∝ V m (M ) f (M )dM dh @QFRHA
efter otining set of ndidte models @the xggAD we pply the u test to test to these ndidte modelsF ht requires the lultion ofX the soE lute mgnitude M k D the mximum of redshift z max @whih orresponds to the limited pprent mgnitude m lim AD the mximum of volume V (z max ) nd the volume V 0 (z k ) ording to iqsFPFIUD PFUD PFV nd IFQUD respetivelyF 3.5 The V/V max test 3.5.1 Calculation of the V/V max terms he lultion of V /V max is strightforwrdD ut for positive urvture pE rmeterD needs one pys ttention to its lultionF sn suh seD the redE shift rnge Π M (z) might onsist in severl intervlsD s in models with positive urvtureF sn tht sesD the funtion ζ(z) is not monotoni nd the inverse funtion ζ -1 (m lim -M ) does not existF hereforeD to ompute the terms V 0 (z) nd V m (M ) ndt determine the redshift domin Π M (z)D iFeF the regions where the ojet with luminosity M is oservleF vet z † e the redshift for whih ζ rehes its mximumF he domin Π M (z) is not onneted if z † `zform F sn suh seD iqFPFIU hs two rootsX z 1 nd z 2 F vet us de(ne the following mgnitudesX

M † = m lim -ζ(z † ) @QFRIA M f orm = m lim -ζ(z f orm ) @QFRPA
he soures with solute mgnitude M whih stisfy the ondition M † < M < M f orm hs visiility redshift window de(ned y union of disjoint redE shift intervlsX [0, z 1 ] ∪ [z 2 , z f orm ] @see pigFQFIRAF o summrizeD the lultion of h(z, m) in se of positive urvture model distinguished in two sesF pirstlyD if iqFPFIU hs single root then we hveX

V 0 (z) = V (z) @QFRQA V m (M ) = V (min{z f orm , z max }) @QFRRA
eondlyD if iqFPFIU hs two roots then h(z, m) is lulted with the following termsX

V 0 (z) =      V (z) if z < z 1 V (z 1 ) + V (z) -V (z 2 ) if z > z 2 @QFRSA V m (M ) = V (z 1 ) + V (z f orm ) -V (z 2 )
@QFRTA e similr issue ours lso in the presene of the kEorretion term in iqFPFIU @euse the emission lines mke this funtion solely invertile y prtAF e note tht the V /V max test n e pplied on susmples de(ned with riteriX

m min < m < m max nd z min < z < z max F hereforeD h(z, m) is lulted s followsX h(z, m) = V 0 (z) -max{ 1 (M ), V (z min )} min{ 2 (M ), V (z max )} -max{ 1 (M ), V (z min )} @QFRUA

SS

Chapter 3 : The null correlation technique where j (M ) re given y iqsFQFRR nd QFRT sustituting m lim into iqFPFIUD y m min for j a ID or y m max for jaPF yne the terms of vrile h(z, m) re determinedD we ompute its umultive distriution funtion s followsX

F h (x) = 1 N N k=1 θ(x -h(z k , m k )) @QFRVA
nd thereforeD we ompre it to the df of theoretil uniform distriution etween H nd I @the linerityA using the u test s followsX he lon higitl ky urvey is one of the well known surveys speilized in the identi(tion of the ys nd glxies for mpping the niverseF ine PHHH the h hs evolved in two editionsX hEs @up to PHHSAD hEss @PHHSEPHHVA followed y the hEsss in PHHV whih minly detets the hrE teristi sle imprinted y ryon ousti osilltions @feyA in the erly niverse through the h9s fryon ysilltion petrosopi urvey rojet @fyAF he h telesope with dimeter PFSm is loted t ephe oint yservtoryD in outhest xew wexio @eAD using multiEnd with phoE tometri system of (ve (lters @rD iD uD zD nd g sed on the ef mgnitude system s explined in etFIFIHFQA nd overing out IIDTHH deg 2 of the sky in xorthern remisphereF st is lso ssoited with pir of spetrogrphs with optil (er feedD whih mesure the spetr of seleted ojetsF e hve used severl tlogs to pply the null orreltion testF e foused pigure RFIX he lon foundtion PFSm telesope itheyEghretien type loted t ephe oint yservtoryD in south est xew wexioF on the smples from hEs nd hEssF ine our investigtion requires stE tistil properties suh tht the homogeneity of the smpleD we were unle to perform the test on the hEsss smplesF his is due to the seletion of ojets whih ws dependent on the redshift @more spei(lly for the nineth dt relese hWEfyAF sn the hW smpleD the ys were seleted in order to mesure the fey sle in the vymnEα forest t redshift ≈ 2.5F husD most of the ys in this smple spn rnge of redshiftsX 2.5 < z < 3.5 âris et l PHIHF por the reson of this seletion sed on redshiftD s seen in the solute mgnitudeEvolume digrm of hW @see etFRFREQFI for more desription nd nlysis of this digrmAD we did not use this smple for the pplition of the null orreltion testF roweverD the two editions sGss grow inresingly with huge numer of qusrs @ysA nd goodness of the qulity of the seletion

D max = M ax(|F h (x) -x|) @QFRWA
Ω • = 0.3D λ • = 0.7 nd H • = 70 Km.M pc -1 .
s -1 AD s well s with full width t hlf mxE imum @prwA of lines from the rod line region greter thn 1000 km.s -1 F st is worth to mention tht pplying limit on the solute mgnitude @whih is vrile depends on the osmologil prmetersA is prohiited euse it mkes is on the estimtion of the prmetersF roweverD the vlue M = -22 orresponds to z ≤ 0.06D then its e'et is not signi(nt nd thereE fore ignoredF TI Chapter 4 : Application to quasars data

The k-correction

es mentioned in etFPFSFQD the kEorretion mkes the solute mgnitude M entered t the sme wvelength for ll ojets t ny redshiftF he dependene of the kEorretion on the overll qusr petrl inergy histriution @ihA neessittes the study of its omponentD usefully y seprting etween the ontinuum @K cont A nd the emission lines @K em AF he qusr ih n e represented y ssuming power lw distriution f ν ∝ ν αν F henD the K cont is givenD s trditionllyD t redshift zero y the followingX

K cont = -2.5(1 + α ν )log 10 (1 + z) @RFIA
eording to ihrds et l PHHT pperD the ontinuum slopes pper orreE lted with the oserved reltive gi olorD while the true orreltion is not found yetD notly for high redshiftF henD it is useful to hnge the restE frme9s redshift whih voids the mentioned orreltionF hereforeD redshift lose to the medin redshift of the hQ smple is hosen for the K cont with onstnt ontinuum slope α ν = -0.5D iFeD the ontinuum kEorretions re zero t z = 2D whih redues the systemti error @his error inurred when using (xed spetrl index for ll the luest nd reddest ojets t high redshiftAF rnsferring the zero point of the kEorretion to z = 2D signi(ntly redues the systemti error inurred y extrpolting the wrong spetrl index to high redshift @see ihrds et l PHHT for more detilsAF kEorreting the prt of emission linesD K em D is estlished with seletion of the )ux limitF his limit hs een settled in the i nd t z = 2 due to the ft tht the iEnd is reltively free of strongly peked emission lines t this redshiftF xote thtD the vyα forest does not inlude the i nd until muh higher redshiftF he ontriution of host glxies is likely to e smll in this nlysisF pigFRFP shows the totl kEorretionD K cont nd K em D in the i ndF 4.3 Results

The null correlation test

he null orreltion urves found with the smples nd susmples desried ove re very lose to eh otherD nd they oinide in the region of the low density mtter in the (Ω • , λ • ) digrm @see pig RFQAF hese results re in good greement with the results of rF plihe 8 tF wF ouriu IWUW who (rst pplied this test on the ysF roweverD it must e noted tht the xgg of the ih smple @red urve in pig RFQ A shows deresing shpe whih gives di'erent resultsF his issue is due to the d qulity of the seletion dt of ih smpleF sndeedD it hs een noti(ed tht originl version of the qusr trget seletion lgorithm used in the ih dt did prtiulrly poor jo of seleting qusrs with redshifts lose to z = 3.5 ihrds et l PHHTF

Results

pigure RFPX veftX he kEorretion of the hEhQ smple in the iEnd inluding oth the emissionEline nd ontinuum omponentsD normlized t z = 2 with (xed spetrl index α ν = -0.5F ightX he numeril derivtive of kEorretion with respet to the redshift zF hereforeD this urve is exludedF he model with zero urvture @de(nes )t universeD Ω • = 0.6A is not good ndidte @see etFRFREQFPAF pigure RFQX he null orreltion urves orresponding to ihD hID hQD hU smples nd hQ susmplesF he urves re loted in the region with positive urvture κ • > 0 @ove the stright lineAF ell the urves give lose ndidte vlues for osmologil prmeters exepted the one of ih smple @red urveAF he V /V max test provide us with more likely vlues of the osmologil prmeters in greement with the working hypothesesF sts pplition suggest TQ the most likely ndidte de(ned y Ω • = 0.076D λ • = 1.211F rerefterD we use those vlues in the digrms nd grphis for the disussion of resultsF efE terwrdD we plot the solute mgnitudeEvolume digrm tht will e nmed fterwrds p digrm @for plihe nd ouriu digrmAF st is worth to note tht two limiting pprent mgnitudeD i = 19.1 nd i = 20.5D re present in this digrm tht orrespond to two distint set of oservtionsF sn pigsF RFRD RFSD RFTD RFU nd RFV we show the di'erene etween the p digrms uilt y determined ndidte model @leftA nd the model of the stndrd )t universe @rightA for the smples ihD hID hQD the homogeneous susmE ple of hQ nd hUF st is ler tht the p digrm of ih smple su'ers from the wekness of the size with respet to other tlogsF por the other tlogsD we notied tht ojets re missing in the p digrms t redshifts z = 2.7 nd z = 3.5F he ovious reson of this seletion e'et is euse the h olors of qusrs t these redshifts re similr to the olors of their strs @host glxiesAD whih mkes di0ult to distinguish qusrs from strsF purthermoreD we remrk lk of right ys in the ottomEleft of the p diE grm @white zoneD whih egn more signi(nt with )t universeA ompred to the simulted digrm in pigFRFW @uilt with model Ω • = 0.076D λ • = 1.211Y the volumeEredshift nd distne modulusEredshift digrms re represented in pigsRFIH nd RFII respetivelyD this lst enles one to understnd the deE resing of the luminosity t high redshift in the simultion smpleAF his is due to the seletion of right ys iFeF limit on the pprent mgnitude hs een pplied to selet the rightest ojetsD i = 15D s mentioned oveD in ddition to rrity of ys t low redshift @detils re given in etFRFREQFPAF pigFRFIP @right pnelA shows how the kEorretion intervenes the weighting ftor nd its dependene on the osmologil model with the dt @the red urvesA nd with simultion @the lue urvesA where vrition etween the urves is due to the hnges of λ • F he negtive vlues @see left pnel of pigFRFIP A re rtefts tht pper when the distneEmodulus is no longer invertile t high redshiftD whih onern solely IH ojets @HFHP7 of the smE ple sizeAF TU

4.3.2

The V/V max test feuse the V /V max test requires the most possile homogeneous dt setD we fous on the hQ nd we use the suset of ys tht were seleted uniformly s desried in ihrds et l PHHTF his susmpleD noted fhQD onE tins ISQRQ ysD where WH7 of ys re finter thn the notly limit of the pprent mgnitudeD i = 19.1F ntil this pointD we hve used only smE ples provided y their uthors ut the pplition of the V /V max test requires omplete smples qF figot 8 F riy IWWID see detils in etFQFSF he p digrm enles one to hek whether smple stis(es suh propertyF es visile on piguresD they pper not homogenousD whih fores us to idenE tify omplete susmplesF he theory llows one to pply uto' on the pprent mgnitude @m min < m < m max A s well s uto' on the redshift @z min < z < z max AF e hve pplied this test with severl susmples hosen y seleting domins on redshift nd pprent mgnitude to ndidte models s given y the xggF he method is itertiveD it onsists in delimiting domins for whih the susmple ppers homogeneousF henD y mens of the seleE tion funtion determined y iqFQFQHD one identi(es the pprent mgnitude rnge for whih the susmple shows to e ompleteF reneD the u test is pplied to the orresponding susmpleD whih provides us with the likelihood level of the osmologil prmetersF pigure RFIPX eighting ftor pplied to hQ smple nd simultion smple without @leftA nd with @rightA kEorretionF ed urve with @λ he pplition of V /V max test suggests @Ω • = 0.0169D λ • = 1.079A t SW7 signi(ne level @pigsFRFIT nd RFIUF hese seletion riteri hve een hoE sen to ensure the smple ompleteness within the lrgest mgnitude rngeD ordingly to pplition requirements of the testF his is on(rmed with the seletion funtion φ(m) whih is roughly onstnt in this intervl nd lose to I @pig RFPH E see next setionAF Error on the cosmological paramaters he mesurement errors on the pprent mgnitude re ville only for the hQ smple @RTRIW ysAD this smple ws used to mesure the )ututions of the null orreltion urve ording to these errorsF pigFRFIV shows two urves tht on(ne the estimtes of the osmologil prmeters orresponding to the tking into ount of these errorsF his llows us to estimte the error on the osmologil prmtersF e found for Ω 2.9

• = 1.275D Ω • = 0.132A nd lue with @λ • = 1.229D Ω • = 0.132AF he distured
• = 0.132D λ • = 1.229X δΩ • = ±0.003D δλ • = ±0.0025F
The methodology when using the null correlation method 9

3 Application to QSOs samples 9

3.1

The Sloan Digital Sky Survey (SDSS) data 9

3.2

The null correlation test 11

3.3

Rening the cosmological parameter's estimates 14

Conclusion 15 1 Introduction

In the 80's, because the estimation of cosmological parameters is more reliable when the sources are remote, one had expected that the use of the Hubble diagram was the most promising approach by using QSOs. However, such a viewpoint has been abandoned because of the large dispersion in their luminosity distribution and possible evolutionary eects of these sources. Nevertheless, despite of their weak capacity of being standard candles, an innovative statistical approach, known as the null correlation technique, provided us with the earliest estimates of the cosmological constant that accounts for the acceleration of the cosmological expansion. This approach has been introduced by Fliche and Souriau [11, 12] (FS) and later adapted with additional features to brightest cluster galaxies [1719]. Our aim is to perform such an approach to QSOs data sets from the Sloan Digital Sky Survey (SDSS), in view of inferring cosmological informations.. In the following sub-sections, we remind the geometry of the standard model with the cosmological constant (no dark energy) and dene quantities used herein, with their notations.

The null correlation technique

We show how to probe the geometry of spacetime through a statistical modelling of the advantageous algorithm that provides complete samples up to a given apparent magnitude is described.This chapter concludes with the methodology of the null correlation test, that we use in the next section.

The Friedman-Lemaître-Gamov model

In Friedman-Lemaître-Gamov (FLG) model [10], the space-time is described by a RW metric

ds 2 = dt 2 -a 2 (t)dσ 2 (2.1)
where t is the cosmic time, a(t) is the (dimensionless) expansion parameter and dσ 2 is the metric element on an homogeneous 3-dimensional manifold V 3 (namely, the comoving space).

It corresponds to the space-time events from where the CMB is observed as an isotropic Black-

λ • = 1 3 ΛH -2 • , κ • = K • H -2 • = λ • + Ω • + α • -1, Ω • = 8 3 πGρ • H -2 • > 0 (2.2) α • = 8 45 π 3 Gk B T 4 • -3 H -2 • ≈ 2.5 10 -5 h -2 , h = H • /(100 km s -1 Mpc -1 ) (2.3)
These are dimensionless quantities that stand respectively for the present values of the reduced versions of the cosmological constant Λ, the scalar curvature K of the comoving space, the matter density, ρ being the specic density of massive particles (dark matter included), G is the Newton's constant, and the radiation energy density 2 , which accounts of CMB photons as sources of gravity, k B is the Boltzmann constant, and is the Planck constant; the deceleration parameter reads q = α + Ω 2λ. These notations, for time dependent quantities of a dierent origins, are more appropriated than the usual ones

Ω Λ ≡ λ • , Ω K ≡ -κ • , Ω M ≡ Ω • and Ω γ ≡ α • ,
for preserving their own meaning (in particular, for the sign of the scalar curvature).

The dimensionless expressions of the geometrical quantities used hereafter are :

• the comoving distance 3 of a source at redshift z,

τ (z) = 1 (1+z) -1 da P (a) , P (a) = λ • a 4 -κ • a 2 + Ω • a + α • , P (1) = 1
(2.4)

1
The V3 homogeneity being related to the distribution of the gravitational sources, because of the presence of large scale structures in the universe, one agrees that this model describes an anamorphosis of the galaxies distribution that is smoothed at scale ≈ 100 Mpc. The Sun is moving at speed 369 km s -1 from such an event.

2 Although α• is negligible today, it provides us with a better description of the Universe at recombination epoch than describing the radiation-dominated era and the matter-dominated era separately. [START_REF] Schneider | The sloan digital sky survey quasar catalog. II. First Data Release[END_REF] The comoving distance reads τ /H• 2 Chapter 4 : Application to quasars data

• the comoving volume of a sphere of radius τ = τ (z),

V (τ ) = π ×    (2τ -sin(2τ )) κ -3/2 • if κ • > 0 4 3 τ 3 if κ • = 0 (sinh(2τ ) -2τ ) (-κ • ) -3/2 if κ • < 0 (2.5) where τ = τ |κ • | for (κ • = 0).

Statistical modeling of data

A class of luminous objects, described intrinsically by an absolute magnitude M and a position in the comoving space, can be used to probe the space-time geometry, with even more precision than magnitude dispersion is small (standard candles). For those that can be observed, their world lines intercept the observer's past lightcone, one measures their redshift z and their apparent magnitude

m = M + ζ(z) (2.6)
where the distance modulus ζ(z) reads in term of cosmological parameters. For broad-band photometric measurements expressed in magnitudes, the estimation of rest frame luminosities requires a K-correction term K(z). Hence, the distance modulus reads

ζ(z) = 5 log d L 10pc + K(z) + A, d L = (1 + z)    sin τ / √ κ • if κ • > 0 τ if κ • = 0 sinh τ / √ -κ • if κ • < 0 (2.7)
where d L is the luminosity distance and A quanties the foreground extinction.

According to FLG model, the spatial distribution of gravitational sources being uniform in V 3 , their corresponding volume V = V (τ ), of a sphere of comoving radius τ centered at the observer location, stands for a uniform random variable. If (one assumes that) these objects are permanent, the redshit z of those within the even horizon can be measured, (the visible part of ) the comoving space is representatively sampled 4 by mean of their comoving distance τ = τ (z). If the related sample shows a luminosity function f (M ) then the probability density of the random variables (M, V ) reads [START_REF] Schneider | The sloan digital sky survey catalog. III. Third data release[END_REF] dP th ∝ f (M )dM dV

(2.8)
If the selection eects depend solely on m, the probability density of observed objects reads

dP = 1 P th (φ) φ(m)f (M )dM dV, P th (φ) = φ(m)f (M )dM dV (2.9)
where φ is a selection function that describes the diculties to identify/collect the sources and/or the selection criteria, it works as a lter response (i.e., 0 ≤ φ(m) ≤ 1). According to eqs. (2.4),(2.5),(2.6),(2.7), it works as a correlation function of the random variables (M, V ). 5 The absence of evolutionary eects is a requirement for being considered as standard candles.

ω k = ω β (z k ) k=1,N ω β (z k ) (2.12)
where the value of the parameter β is chosen to minimize the dierence 6 between the weights

ω min = min k=1,N {ω β (z k )} , ω max = max k=1,N {ω β (z k )} (2.13)
and hence to diminish the lose of information. Figure 1 shows the weighting factors (scaled by the factor N ) for the SDSS-DR3 sample calculated by assuming (Ω

• = 0.132, λ • = 1.275) (bottom-left panel) and (Ω • = 0.3, λ • = 0.7) (bottom-right panel).
Let us emphasise that the K-correction term, which intervenes through the distance modulus ζ, plays an important rule in the weighting. Indeed, the comparison between the upper panels (without K-correction term) to the bottom ones of gure 1, shows that its eect in the weighting is not negligible with respect to that of cosmological parameters. It takes into account that a source is more easily detectable when an emission line of its spectrum overlaps with the observed wavelength window of the lter. Ignoring this eect necessarily produces biased estimates of cosmological parameters.

Therefore, the (weighted) correlation coecient of random variables (M, m), which reads

Γ(Ω • , λ • ) = k=1,N ω k (M k -M ω ) (m k -m ω ) σ ω (M )σ ω (m) , M k = m k -ζ(z k ) (2.14)
where the weighted statistics of the mean and variance are dened by substituting X by M or m in the following equations According to working hypothesess, the null correlation test can be applied to any subsample selected in terms of the apparent magnitude 8 . It must be also noted that this method does not require to specify the selection function φ(m) and the luminosity function f (M ) of sources, which advantageously characterizes the robustness of our estimates.

X ω = k=1,N ω k X k , σ ω (X) 2 = N N -1 k=1,N ω k (X k -X ω ) 2

Luminosity function and Selection function

For the candidate model, the weighted probability density that describes the sample reads

dP ω ∝ f ω (M )dM φ ω (m)dm
(2.17) 8 e.g., the sources whose apparent magnitude stand within the interval [mmin, mmax], or any sampling, as long as it is free from selection eects on redshift.

f ω (M ) ∝ f (M )10 -β 5 M , φ ω (m) ∝ φ(m)10 β 5 m (2.18)
For large N , their cumulative distributions functions can be approched as follow

F ω (M ) ≈ Fw (M ) = N k=1 ω k θ(M -M k ), Φ ω (m) ≈ Φw (m) = N k=1 ω k θ(m -m k ) (2.19)
where θ stands for the Heaviside step function. Hence, from discrete derivatives of statitstics, one obtains estimates of the luminosity function and the selection function

f (M ) ∝ ∆ Fw (M ) ∆M 10 β 5 M , φ(m) ∝ ∆ Φw (m) ∆m 10 -β 5 m
(2.20)

The FS diagram

A graphic representation of the data in the (M, V ) diagram, hereafter named FS diagram [11], enables one to detect inconsistencies with the working hypotheses as artifacts in the distribution. A complete sample up to a limiting apparent magnitude m lim contains all the sources in the eld whose absolute magnitude M < M max (z) at redshift z, where that characterizes the threshold at the apparent magnitude m lim . For the true values of cosmological parameters, they are distributed uniformly along the V -axis and according to the luminosity distribution function along the M -axis.

M max (z) = m lim -ζ(z) (2.21) is either a decreasing function (if κ • ≤ 0) or has a minimum for τ (z) = π 2 (if κ • > 0), in
It is obvious that one can have alterations of this idealist picture, such as a smoothed cut o at m lim instead of a sharp one, which might be a closer description to the real situation.

Moreover, we must also expect the existence of structures characterizing the presence of several surveys, bumps on the curve C which are related to emission lines in the spectrum, and others alterations that depend on the identication process of sources. These structures, related to the selection eects on the apparent magnitude, do not contradict the working hypotheses, they are characterized by shapes that are parallel to C. Any other structure that cannot be described by the selection function φ(m) biases the null correlation test. With this in mind, and according to working hypotheses, the presence of any structure suggesting evolution with redshift invalidates the candidate model. Thanks to the power of the discernment of the eye, a simple visual analysis of the FS diagram makes it possible to verify whether the working hypotheses are satised and also to discern the artefacts from real structures.

Rening the determination

If the sample is (statiscally) complete up to a limiting apparente magnitude, then the results obtained by the null correlation test can be rened by using the V /V max test. Indeed, for the candidate model, the data should be distributed uniformly along the V -axis. If the sample For brevity, let us describe a case that does not account of coincidences of the lter with the emission lines of sources [START_REF] Pâris | The sloan digital sky survey catalog: ninth data release[END_REF] and of the case κ • > 0 for which the sample contains objects with redshift z > z , that is a threshold at which the apparent magnitude increases with redshift 10 . Hence, the selection eects being described by a cuto at a limiting apparent magnitude m lim , the selection function reads

φ(m) = θ(m lim -m) (2.23)
A source with absolute magnitude M is visible up to a redshift z lim , as dened by

m lim = M + ζ(z lim ) (2.24)
For the candidate model, namely the values of cosmological parameters that satisfy eq. (2.16), the random variable

v = V V max (M ) ∈ [0, 1[, V max (M ) = V (z lim ) (2.
v k = v(z k , m k ) = V (z k ) V max (m k -ζ(z k )) , (k = 1, N ) (2.27)
If the working hypotheses are correct, the distance between the empirical F (x) and the theoretical F (x) cumulative distribution functions, which is dened as follows

δ = sup x∈[0,1] | F (x) -x|, F (x) = 1 N N k=1 θ(x -v k ) (2.28)
is a random variable with the KS cumulative distribution function Prob N ( δ < δ) 12 . 9 In that case, according that the K-correction term takes emission lines into account, see eq. (2.7), the distance modulus is not an invertible function, and the formulation of the null correlation test becomes slightly dierent. Indeed, if the spectrum shows p emission lines, then equation Mmax(z) = M shows a rst root ẑ0 in addition (eventually) of p ≤ p pairs of roots {ẑ2j-1, ẑ2j} j=1,p , ranked by increasing values, that correspond to redshift ranges where the souce becomes visible. Hence, the uniform random variable as give in eq. (2.27)

transforms as follows v = 1 Vmax(M ) V (z) - p j=0 θ (z -ẑ2j+1) (V (ẑ2j+1) -V (ẑ2j)) , Vmax(M ) = V (ẑ0)+ p j=1 (V (ẑ2j) -V (ẑ2j-1))
10 The redshift dependance of the distance modulus, excepted that through the K-correction term, is given by log (1 + z) + log sin τ (z), see eq. (2.7). Hence, if κ• > 0, the distance modulus decreases with redshit above a given value z > τ -1 ( π 2 ); what characterizes a spatially closed universe (i.e., the comoving space V3

stands for the 3-sphere).

11 Indeed, its cumulative distribution function reads

F (x) = Prob(v < x) = Prob(V < xVmax(M )) = x P th (φ) Vmax(M )f (M )dM = x
(2.26) [START_REF] Fliche | Evualuation des paramètres cosmologiques à l'aide des propriétés optiques des quasars. Fluctuations des modèles de Firedmann-Lemaître[END_REF] In short, the value ProbN ( δ ≥ δ), the probability to obtain statistical uctuations larger than δ (the one measured), stands for the likelihood that the working hypotheses are correct. If the selection function φ is known, or given by eq. (2.20), then the V /V max test can be adapted for using the entire sample by substituting V and V max in eq. (2.27) by Ṽ and Ṽmax dened as follows

Ṽ = V 0 φ (M + ζ(z)) dV, Ṽmax (M ) = Vmax(M ) 0 φ (M + ζ(z)) dV (z) (2.29)
where V max (M ) is given by eq. (2.24).

A comparison with the Hubble diagram

In both approaches, the data sampled the cosmological expansion throughout cosmic time, and along the observer's past light cone. The sources are assumed to be permanent and not evolving (number density, luminosity, spectral). Hence, a K-correction is used (unless using The null correlation method could have being introduced solely as a search for the roots of the statistics dened in eq. (2.16) but the presentation in terms of weighted probabilities allows to highlight a property akin to a lessening of the statistical information contained in the sample. This one can be quantied as a (sample) number decit N L dened by means of the Shannon's entropy H(ω) as follows

L = 1 + 1 ln N H(ω), H(ω) = k=1,N ω k ln ω k (2.30)

Simulations techniques

Let us focus on the case of a sample complete up to an apparent magnitude m lim that are uniformly distributed in space and with a luminosity function f (M ), in a given cosmological model (Ω • , λ • ). It is clear that to simulate such a sample {(z k , m k )} k=1,N with a trial and error method, that keeps those (V k , M k ) in agreement with selection eects, is extremely time consuming. In order to avoid such a diculty, one needs to proceed in a dierent way.

Chapter 4 : Application to quasars data 84

The probability density is given in eq. (2.9), the selection eects in observation are solely described by a cuto in apparent magnitude, as described by eq.(2.23). Hence, the cumulative distribution function 13 reads

G(x) = 1 P th (φ) x -∞ V max (M )f (M )dM
M k = G -1 (g k ), V k = v k V max (M k ), z k = V -1 (V k ) and m k = M k + ζ(z k ).

The methodology when using the null correlation method

Since only the selection eects allowed in the data are the ones on the apparent magnitude, one has to verify that there is no of other kinds, and in particular those on the redshift. For this purpose, a visual inspection on the data distribution in the FS diagram enables us to check whether anomalies at constant V are present. At this point, let us emphasize that the overlapping of emission lines of the spectrum of a source with the observed wavelength window of the lter interprets as a selection eect on the apparent magnitude and not on redshift.

Among the roots of eq. 2.16, that dene the null correlation curve, one can rene the estimate of cosmological parameters if (and only if ) the sample is complete up to a given limiting apparent magnitude. Indeed, the most likely values that account for the data is the one for which the spatial distribution of (the observed) sources appears to be the most uniform in the comoving space. The completeness criteria of the data can be checked by means of the statistic given in eq. (2.20), which provides us with an estimate of the selection function φ(m); it must coincide with a Heaviside step function at a threshold m lim . In general, the samples do not show a sharp cuto at the (apparent) threshold m lim but a smoother one. In that case, a brighter limiting magnitude m lim < m lim has to be chosen to perform the V /V max test. It is clear that such a step diminishes the sample size but to the benet of an unbiased estimation of the Kolmogorov-Smirnov test on the uniformity of the spatial distribution of sources. If necessary, other subsampling criteria can be chosen, to test the distribution on specic part of the FS-diagram, as described in section 2.6.

3 Application to QSOs samples

The Sloan Digital Sky Survey (SDSS) data

The data releases of Sloan Digital Sky Survey (SDSS) (http://www.sdss.org/) contain a large number of QSOs data, whose the size and accuracy of samples increase with their release numbers. However, we have been forced to limit our investigation on the rst ones, because of the required statistical properties for our investigation.

For example, the FS diagram of DR9-BOSS on gure 2 shows that the SDSS-III samples suer from an inhomogeneous parcelling due to data processing. For a homogenous sample, one should have a regular distribution of dots situated at the left side of a curve (or below it) representing the limiting apparent magnitude. Such a border is present, although its sharpness is diminished because of an additional and sparse distribution of dots beyond it. [START_REF] Bigot | A determination of q • by a technique of null correlation[END_REF] If φ cannot not be identied to a Heaviside step function then Vmax in eq. ( 2.31) has to be substituted by Ṽmax as dened in the footnote of sec. 2.6. Table 1. QSOs samples At the left side, and parrallel to it, two similar structures are also present. They probably account for surveys with brighter limiting magnitudes. Such features characterize surveys at dierent limiting apparent magnitudes. They can be treated globally by the null correlation technique, as long as selection eects on redshift are absent, which is not the case. Indeed, four prominent structures are present, the zones of higher density of dots with horizontal borders, and a sparse region (with a rhombus shape) at the bottom of the distribution, that characterize, as a matter of fact, selection eects on redshift. One of the reasons of such an inhomogeneity is that these QSOs have been selected with the aim to measure the BAO scale in the Lyman-α forest at redshift ≈ 2.5. Most of them span a range of redshift: 2.5 < z < 3.5 [9].

Consequently, our choice has dropped to the SDSS Legacy Survey (i.e., SDSS-I and SDSS-II) which provides us with an uniform and well-calibrated map of the Universe. These samples are described in table 1, characterized by their (number) size N and their range on the redshift ∆z and the i magnitude ∆i-mag.

Because of successive improvements of the data processing, these samples were made in dierent ways. For example, the dierences between the DR1 and DR3 samples are due to several modications in the selection criteria of candidates (on the eciency of the S/N ratio to determine the redshift, on the constraint of the line width of the spectrum, new techniques are used for the photometric measurement and the data imaging.). Although it is recommended not to use previous versions to DR3 (see, http://classic.sdss.org/dr1/), we have performed the null correlation test to each of these samples, what enabled us to test the stability of this method versus measurement errors (and improvements in measurement).

The K-correction term for the i-band used herein is described in [5, 6], it takes into account the continuum and the emission lines, see gure 3.

The null correlation test

The null correlation test has been applied to these four samples and the results are shown in the Ω We use the FS diagram of the samples DR3 and DR7 for evaluating the relevance of such a result, see gure 5. As representant, among these candidate models, we choose the one with the smallest scalar curvature (Ω • = 0.076, λ • = 1.211) (left panels), since it is common to these samples; their information decit is L = 18. 10 -4 for DR3 and L = 17. 10 -4 for DR7, see eq. (2.30). These diagrams are compared to their versions with (Ω • = 0.3, λ • = 0.7) (right panels); their information decit is L = 43. 10 -4 for DR3 and L = 50. 10 -4 for DR7.

For purposes of the analysis, the graduation on the ordinate is translated into redshift on the right hand side axis. The curves, they correspond to the apparent i-magnitudes m = {15, 19.1, 20.5, 21.78}, give ranges on the magnitude. Let us keep in mind that, while the distribution along the vertical axis is expected to be uniform for a complete sample up to a limiting apparent magnitude, the presence of selection eects in apparent magnitude alters this characteristics, in addition of possible horizontal alterations due to selection eects by the redshift. With gure 2, one can note that the homogeneity of these samples has improved with respect to that of DR9 sample.

• FS diagram of DR3 sample :

On the left panel, let us identify rst the obvious structures which are produced by selection eects, they appear as discontinuities in the distribution of dots. It is clear that the low density region (hole) in the range 19.1 ≤ m ≤ 20.5 and 2.7 ≤ z ≤ 3.1 that extends (slightly) at brighter magnitudes at z ≈ 3.38, is due to selection eects on the redshift. This attests of diculties on identifying QSOs in the data processing. Mainly because the colors of quasars and the stars of their host galaxies are similar at these redshifts.

Moreover, the distribution, which extends up to magnitude of m = 21.78 is very sparse down to i-magnitude m = 20.5 that connes more clearly the sample. A similar feature is also present at i-magnitude m ≈ 19 characterizing a distinct survey. The region within 15 ≤ m ≤ 19.1 could be splitted in two parts by a curve at constant i-magnitude, so to detach a sparse region of nearby sources. These low density regions on either side of the distribution are due to selection eects on apparent magnitude, while it is not the case for the hole.

Hence, because of this inconsistency with the working hypothesis, these results can be questioned, unless the hole is (interpreted as) a statistical uctuation.

On the right panel, for an unbiased comparison with the left panel, the analysis of structures has to take in account their relative sizes with respect to the extend of the sample. With this in mind, the most prominent dierences are that the size of the hole z and vanishes at z ≈ 3.2, which characterizes a selection eect on the redshift. An additional and sparse distribution is also present, that extends beyond m = 21.78.

On the right panel, the spatial distribution of bright QSOs appears less uniform compared to the the left panel, and the number of missing objects at low redshifts increases (statistically), which stands for as a signicant issue.

Rening the cosmological parameter's estimates

The candidates models as obtained from DR3 and DR7 samples are not so dierents from each other, that is the related null correlation curves coincide almost, see gure 4. With the aim to rene the cosmological parameter's estimates by mean of the V /V max test we need complete subsamples.

The only comparison between the FS diagrams of DR3 and DR7 samples with a simulation on gure 6 shows that they are not complete, what is the required statistical property which enable us to rene these estimates of cosmological parameters by means of the V /V max test.

We start by applying the test to subsamples using the corresponding NCC. The results for each subsample are given as follows:

4 Conclusion

The application of the null correlation and V /V max tests to QSO samples of the SDSS survey allowed us to discriminate the candidates values for the cosmological parameters that agree with observations. 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ojets in the universe disE overed until todyF st is the result of the explosion of mssive strs t the end of its lifeF sts luminosity n mth the one of its host glxy in some sesD nd they re oserved t osmologil distnesF he xe re lssi(ed into groupsD one of whih n e used s stndrd ndles for proing osmologil distnes with high uryF he spetrl nd photometri studies of x points out two tegories @winkowski IWRHA ording to their propertiesD minly sed on the presene @type ssA or not @type sA of rydrogen elements in their spetrF he spetrumD is relted to the photosphere phse @ few instnts fter explosionAF he fst expnsion of its envelope mkes the medium opque to rditionD using the sorption lines to pper in the spetrumF vterD during the neulr phse @efter few weeks of growthA emission lines rise in the spetrum sine the medium eomes trnsprent to rditionF por type sD the strong sorption feture t wvelength ≈ TISH Å due to silion ss @issA hrterizes the type sF sn sene of issD the x is lssi(ed depending on the preseneGsene of relium re element s type sGsF his lssi(tion is summrized in pigFSFIF he photometri lssi(tion is sed on the omprison of the evolution of luminosity with time iFeF the light urve of the xF pigFSFP showsD the light urves of type ss xe whih deline slowly in the f ndF wo dditionl lsses re distinguished from the type ssnD the x type ssEv tht displys liner lightEurveD nd the x type ssE whih displys plteu lightEurveF roweverD the x type s dereses fster thn the type ssD it represents simE ilr lightEurve to tht of x sF his lst group is the rightest mong the other types of xeF eording to gppellro et l IWWWD the x type sD sD 96 5.2 Type Ia Supernovae as standard candles pigure SFIX he supernov lssi(tion treeF pigure SFPX he light urves of di'erent types of xe in the lue ndX the type sD sD ssEvD ssED nd x IWVUeF he gurve of the x s is righter thn ll other x typesF pigure ourtesyX heeler IWWHF nd ss re the result of the explosion of mssive strs in the frmework of their oreEollpseF hese types re formed in the rms of spirl glxies nd they never hd the possiility to e in the elliptil glxiesD where the rte of old strs is highF gontrstinglyD the typeEs x n e in ny type of glxyD s WU Chapter 5 : Application to supernovae type Ia sample they re the result of the thermonuler explosions of white dwrfs @pigsFSFQ 8 SFRAF he lowEmss strs @< 8M A 1 evolve s their hydrogen omusts nd pigure SFQX he supernov spetrl ypesF he grvittionl supernove is the progenitor of the supernove type sD s nd ssF he progenitor of x type s is thermonuler explosion of white dwrfsF pigure ourtesyX upernov gosmology rojet @gAF trnsforms into reliumD whih is explodes with inresing the temperture to trnsform into ron nd oxygenF he most wellEknown model tht deE sries the progenitor of the typeEs x s lowEmss str is tht the one whih eomes white dwrf t the end of its lifeF his ltter is ompnied with nother str s inry prtner system @pigFSFRA reting mtter on the white dwrfD nd then inresing its mss until rehing the ghndrsekhr mss limit @≈ 1.4M AF gonsequentlyD the white dwrf explodes to produe the 1 M is the solar mass 98 5.2 Type Ia Supernovae as standard candles typeEs supernoveF he min hrteristi of this lss is the homogeneity of the x spetr s shown in pigFSFSF pigure SFRX en rtisti imge of white dwrf nd red gent strF pigE ure ourtesyX pgGhvid rrdyF pigure SFSX petr of three ype s supernoveD x IWVUhD x IWVUxD nd x IWWHxD out one week fE ter the mximum of their vgF pigure ourtesyX eF F pilippenko IWWU 5.2.1 The lightcurve of type-Ia SN he light urve @vgA represents the temporl evolution of the luminosity during the lifetime of xF por ojets tht hnge their rightnessD the vg is mjor soure of informtion nd vlule tool for studying these eventsF pigFSFT shows the x IWWVu with lmost similr or di'erent vg in optil nds @referring to fessel ndsAY eFgFD note tht two umps pper in the s nd nds with dely of the seond ump y PI to QH dys from the mximum in the f ndF st is worth mentioning tht mong ll types of xD the vg in the f nd of the typeEs x is lmost the sme for ll of them @pigFSFU top pnelAF his feture mkes them the most homogeneous fmily mong the x types with respet to vgF he deline of x s light urves fter the mximum of luminosity orreltes with the rdiotive dey of 56 N i to 56 Co for three to four weeks t the rte of 0.11mag.day -1 nd with the rdiotive dey 56 Co to 56 F e t the rte of 0.01mag.day -1 more thn one monthF

Standardisation of type-Ia SN

sn view to determine osmologil prmetersD it is generlly dmitted tht the vg of typeEs xe re good ndidtes to e used s stndrd ndlesD euse the dispersion of luminosity t the mximum seems to e smllD whih is n WW Chapter 5 : Application to supernovae type Ia sample pigure SFTX fs light urves for x IWWVu @pigure ourtesyX xF fF untze' IWWWA ovious improvement ompred to ysF wF wF hillips IWWQY rmuy et lF IWWSD IWWTY iess et l IWWSD IWWTY hve shown up orreltion etween the solute mgnitudeD M B D t the mximum of luminosity nd the deline rte of luminosity fter IS dysD noted ∆m 15 @pigFSFVAF his orrelE tion enles one to estimte the solute mgnitude nd hene the distne modulusF everl models hve een pplied to stndrdize their vgs in orE der to redue the dispersion t the mximum of luminosity in the f ndF emong these modelsD one hs the wvg @wultiolor vight gurve hpeAD the wvgPkPD the treth @strething of the vgAD the ipyD the ev @petrl edptive vighturve emplteA nd the evP models whih re the most widely usedF hese models stndrdize the xe withX the pprent mgniE tude t mximum of luminosity @free from the kEorretionAD the @fEA olor t this mximum of luminosity nd shpe prmeter for desriing the vgF A brief description of SALT2 Model he evP model @for the seond version of petrl edptive vighturve emplteA is n empiril model tht determines the )ux of typeEs xe s 100 5.2 Type Ia Supernovae as standard candles pigure SFUX vight urve stndrd otinedF he fEnd lightEurves for lowEredshift xe s from the glnEololo survey @pigure ourtesyX rmuy et lF IWWTA funtion of p @phseA nd λ @wvelenghthAD where p is de(ned s followsX

p = t obs -t * obs 1 + z @SFIA
where t obs stnds for the dte of oservtion nd t * obs t mximum of lumiE nosity in rest frmeF imilrlyD t * stnds s the emission dte t mximum of luminosityF his model is prmeterized yX the normliztion ftorD the shpe nd the olor prmetersF st is sed on modelling the spetrl energy IHI @SFPA where x 0 is the glol normliztion in )uxD x 1 is the shpe prmeter @equivE lent to @s E IA ftor in treth method of stndrdiztionA whih is used s n estimtor of the solute mgnitudeD nd the olor of xF he S k re the spetrl sequenesX S 0 stnds for the men spetrl sequeneD S 1 stnds for the (rst order devition round S 0 D etFFFY fEsplines funtions re used to de(ne themF gv stnds for the phseEindependent olorElwF he evP output provides us with X the pprent mgnitude in f nd free from @foregroundA ltertions m * c = -2.5 log 10 (x 0 ) + ZP D the dte t mximum of luminosity t * obs D the di'erene of olor etween f nd nds nd the shpe prmeter x 1 quy et l PHIHD fetouleD wF et l PHIRF 5.2.3 Modelling of the Supernova event he x s event trnsltes to powerful )sh whose intrinsi luminosity vries with timeF he light urve is hrterized y the dte t * t whih the lumiE nosity rehes its mximum nd y the orresponding solute mgnitude M * F 102 5.2 Type Ia Supernovae as standard candles he evolution of the solute mgnitude with time is prmetrized y shpe prmeter x 1 D tht ts for diltionGontrtion of pttern urveF he lifetime of x is shown to e linerly orrelted with M * @pigFSFV desries the pprent orreltion etween the solute wgnitude nd the deline rte ∆m 15 AX

M * = -αx 1 + M 0 @SFQA
where M 0 stnds for the solute mgnitude of the pttern urve t mximum luminosity nd α ≥ 0F he solute mgnitude t dte t redsX

M (t) = -2.5 log 10 (F (t; t * , M * , x 1 )/F 0 ) @SFRA
where F 0 stnds for the )ux of the referene str nd F (t; t * , M * , x 1 ) the inE trinsi )uxY it is desried y evP model in the referene frme of the supernovF he evolution of the pprent mgnitude with time is given y the followingX

m(t obs ) = M (t) + ζ(z) @SFSA
he solute nd the pprent mgnitude t mximum luminosity stisfyX

m * = M * + ζ(z) @SFTA
he neessry property whih is required for x to e servle is tht its worldEline interepts the oserver9s pst light oneD s desried in pigsFPFR nd SFWF rereinD the x event is de(ned y (M * , t * , x 1 , V ) with proility density given yX

dP th = g(M * , x 1 )dM * dx 1 dt * dV @SFUA
whih desries uniform distriution s muh in @omovingA spe s in @osmiA timeY where V stnds for the omoving volume of sphere entered t the oserver lotionD nd the omoving distne of the soure s rdiusF usequentlyD we will limit ourselves to onsidering the rndom vrile t * s prmeter required in the evP proedureD s usulF x ndidte re (rstly identi(ed mong vrile light soures y ompring two imges of the sme (eld tken t di'erent timesF henD spetrosopi monitoring llows to identify the ndidte nd determine their redshiftsF piE nllyD if it turns out to e typeEs x @sed on the presene of identi(ed sorption linesAD photometri monitoring is performedD tht privides us with light urveF he drwk in serhing of typeEs xe @or ny nonEperennil ojetA is the loss of mny ojets due to the d wether onditions or dely in spetrosopi nd photometri oservtionsF sn the im to inrese the time nd the (eld of oservtionD the rolling serh method @developed in the frmework of the uperxov vegy urvey @xvA projetA is followed to oserve this type of eventsF his method onsists of four oservtion (elds during severl regulr time intervls nd in di'erent (ltersF his tehnique enles one to disover nd monitor severl xe simultneously in four nds @g D r D i D z AY it is used in hD referring to the mith tlog @PHHPA nd sed on the ef system of mgnitude 2 F sing multiple nds n over wide wvelength rngeD whih mkes possile the oservtion of x s t high redshiftsF en other dvntge of using severl (lters is tht the )ux in the referene of the x in the f nd n e estimted y n interpoltion from the olor of the x mesured in two other nds @t lestAF ih (eld is oserved during QER dys nd the suessive imges re omE pred y sutrting their pixels rithet PHHRF sn the se of signi(nt di'erene etween the pixelsD photometri mesurement is reordedF his tehnique llows to otin good smpling of the light urveF sn idel sesD x with lifetime ∼ VH dys @the verge for the xe lifetimeA might hve ∼ PH photometri mesurementsD whih provides us with n urte (t of the x light urveF por instneD pigFSFIH shows the x xvEHRhQgx light urve in the i nd smpled with IU photometri mesurementsF he smpling of x urves needs photometri mesurements in prtiulr phses pD s desried in quy et l PHIHF xmelyD the xe re seleted ording to the following requirementsX they must show t lestX IF four mesurements t di'erent epohs rnging within EIH `p CQS dysF PF one mesurement in erly epoh rnging within EIH `p `CS dysF QF one mesurement rnging within CS `p `CPH dysF RF two ndsD with one mesurement or more in the rnge EV `p CIH dys to ensure genuine estimtion of luminosity pek nd olorF e x with light urve poorly smpled is disrded from the smpleF eordE ing to quy et l PHIHD this e'et is not signi(nt for xv smple @xe with high redshiftsAD while the x hEss smple @xe with low redshiftsA is 'eted signi(ntly fetouleD wF et l PHIRF pigFSFII shows light urves tht help to understnd the stndrdiztion methodD see F erlmutter et lF IWWWF hey re desried y referring to pttern urve @dshedD the mster light urveA with streth s p D olor c p nd n solute mgnitude t mximum luminosity M p F he xe with light urves lying ove the mster urve re hrterized y X s > s p nd c < c p F hey re righter @ long lifetimeAD showing slow dereses nd luer @f B > f V A 3 F gonverselyD the ones elow the mster urve re hrterized y X s < s p nd c > c p F hey re finter @ short lifetimeAD showing fst dereses nd less luer @f B < f V AF e note tht the olor of x t mximum of its luminosity is more negtive @respFD positiveA for the xe with light urves ove @respFD elowA the one of the mster urveF he opportunity to detet supernov requires tht the event lies in the pst light one of the oserver nd tht its pprent luminosity is righter thn given threshold f lim D let m lim e the orresponding limiting pprent mgnitude tht hrterizes devieD see the red line in pigFSFIIF feuse the righter the ojet the longer the lifetimeD s (rst pproximtionD single seletion riteri on the mgnitude ounts for othF woreoverD the x light urve requires to show su0ient time to ful(ll the seletion riteri sed on the phseF reneD the threshold of seletion hs to e righter thn m lim F he light urve must show signi(nt gp etween m lim nd m * D nd smll ∆m 15 to hve good smpling in ordne with the detetion tehnique desried oveF ith this in mindD we might idelize 4threshold4 light urve in term of m * lim nd ∆m 15,lim whih ful(lls the forementioned riteri @see pigFSFIPAF st is ler tht seletion on m * D whih tkes into ount the lifetimeD lso desries wlmquist is @see pigFSFIQAF henD the proility density of oserved event redsX

∝ φ B (m * )dP th @SFVA
where φ B (m * ) stnds for seletion funtion on the pek of pprent mgE nitude in the f ndF elthough suh representtion of the seletion e'ets is the most ommonly eptedD it is ler tht n improvement tking into ount the lifetime of the events is neessryF roweverD euse the longer the lifetime the righter the supernovD the orretion to this representtion n e expeted to e smllF e work is in progress to enle us to ensure this quntittivelyF he olor stnds for orretion to derese the stter of the distriution of the solute mgnitude t mximum luminosity M * F he orreted solute mgnitude M * c stis(es the reltionX F hepending on the method usedD s given y iqsFSFIQ nd SFISD the smple n e desried y the orresponding proility densitiesX where the normliztion ftor reds X

M * = M * c -βc; β ≥ 0 @SFWA
dP obs = 1 P th (φ B ) φ B (m * )g M * (x 1 )dx 1 g G (c; 0, σ c )dcdt * e γζ dζ × g G ( ; 0, σ )d @SFIVA dP obs = 1 P th (φ B ) φ B (m * )g x 1 (M * )dM * g G (c; 0, σ c )
P th (φ B ) = φ B (ax 1 + b + ζ)g M * (x 1 )dx 1 e γζ dζ × dt * g G (c; 0, σ c )dcg G ( ; 0, σ )d @SFPHA P th (φ B ) ∝ φ B (ax 1 + b + ζ)g M * (x 1
)dx 1 e γζ dζ @SFPIA he sttistis of prmeters D nd σ re determined y following the order of their rting s followsX for the hf reltion X

a DBS = Cov(x 1 , M * ) σ 2 x 1 @SFPPA IHW
* m * lim F σ DBS = σ M * 1 -ρ 2 (x 1 , M * ) @SFPQA where ρ(x 1 , M * ) is the orreltion oe0ientF b DBS = M * -a DBS x 1 + γσ DBS 2 @SFPRA
for the sf reltionX

a IBS = σ 2 x 1 Cov(x 1 , M * ) @SFPSA b IBS = M * -a IBS x 1 @SFPTA 110 5.
2 Type Ia Supernovae as standard candles pigure SFIRX he righterEslower reltion @solute mgnitude versus shpe prmeterA

σ IBS = σ M * 1 ρ 2 (x 1 , M * ) -1 @SFPUA
st is worth to mention tht the existene of the ftor γσ DBS 2 in the estimtion of DBS prmeter stnds for is orretionF sn the followingD we use the sf lirtion methodD euse it is more roust @with respet to seletion e'etsA thn the hf one riy et l IWWRF feuse the lirtion shem turns out to e identil to the orreltion @M * c D AD one otins similr equtionsD ut this step requires the hf method to estimte the solute mgnitudeF 5.2.6 Simulation e proeed similrly s for the simultions of qusrs smplesD see etFPFSFPD ut y inluding the olor orretion with the proility density given in iqFSFIH nd the sf reltionD given in iqFSFITF henD one hs the following proility densityX

∝ φ B (m * )g x 1 (M * )dM * g G (c; 0, σ c )dc g G ( ; 0, σ )d dt * dV @SFPVA
where t * is required in the evP routineF rereD we ssume tht φ B reds s reviside funtion θ(m * limm * ) hrterizing omplete smples up to limiting pprent mgnitude m * lim F III Chapter 5 : Application to supernovae type Ia sample he pprent mgnitude in restEfrme f nd otined with stndrdiztion of xe @free from the kEorretionA is given yX

m * c = M * c + ζ(z) @SFPWA
where

M * c = ax 1 + b + βc if M a stellar < 10 10 M ax 1 + b + βc + δ otherwise @SFQHA
δ stnds for orretion tht depends on the host stellr mss M a stellar ulE livnD wF et l PHIHD gonley et l PHIIF sts vlue hs een evluted in fetouleD wF et l PHIR δ = -0.070 ± 0.023 y using the joint lightEurve nlysis smple of URH xeF st is used in this investigtionF o simulte smple free from @foregroundA ltertionsD we used M * c insted of M * D then iqFSFPV redsX

∝ φ B (m * ) f (M * c )dM * c g G (c; 0, σ c )dc dV @SFQIA
where f (M * c ) is the distriution funtion of M * c F vet us de(ne the umultive distriution funtion of the rndom vrile

M * c s follows Ĝ ≡ G(x) = 1 P th (φ B ) x -∞ f (M * c ) Q(M * c ) dM * c @SFQPA
where

P th (φ B ) = φ B (m * ) f (M * c )dM * c g G (c; 0, σ c )dc dV @SFQQA nd Q(M * c ) = θ(m * lim -M * c + βc -ζ(z))g G (c; 0, σ c )dc dV @SFQRA
por given vlue of M * c D the proility density of rndom vriles nd is de(ned s follows dP [START_REF] Wisotzki | Quasar spectra and the K correction[END_REF] = 1

P (1) th (φ B ) θ(m * lim -M * c + βc -ζ(z))g G (c; 0, σ c )dc dV @SFQSA
where

P (1) th (φ B ) = θ(m * lim -M * c + βc -ζ(z))g G (c; 0, σ c )dc dV @SFQTA
he df of the rndom vrile reds

Ĉ ≡ C(x) = 1 P (1) th (φ B ) x -∞ θ(m * lim -M * c + βc -ζ(z))g G (c; 0, σ c )dc dV @SFQUA 112
5.2 Type Ia Supernovae as standard candles he simultion sheme onsists in the following trils

M * c -→ c -→ V @SFQVA vet us remind tht Ĝ ∈ [0, 1] nd Ĉ ∈ [0, 1] re uniform rndom vrilesF ih generted rndom vlues for Ĝ nd ĈD provide M * c = G -1 ( Ĝ) nd c = C -1 (
Ĉ) from the inverse funtionsD see iqFSFQPD SFQUF henD with M * = M * c -βcD one simultes vlue of the uniform rndom vrile V ∈HD max (M * )]D see iqFPFVF he redshift is given y z = V -1 ( V ) nd the pprent mgnitude t mximum of luminosity m * c y iqFSFPWF he shpe prmeter x 1 is determined either with iqFSFIT for the lirtion prmeters a IBS nd b IBS or with iqFSFIR for a DBS nd b DBS D nd vlue generted y white noiseD in ordne with working hypothesesF vet us nlyze two smples of PRH ojets whih re simulted with the following hrteristis nd ording the models of white noise s desried in the lirtion proessesX 04D mkes lerly the ound of the distriutionF feuse of suh fetureD lest squre (tting method requires horizontl estimtes of distnes to the expeted urveF he hrteristis of two smples @eA nd @fA re desried in four dimensionl digrms in pigsFSFIUD SFIVF he likelihood of working hypotheses on the olor nd the shpe prmeter n e esily heked on these digrmsF ht re the orreltion etween M * c nd @middle left pnelAD lso etween M * nd x 1 @top right pnelAF IIQ Chapter 5 : Application to supernovae type Ia sample pigure SFISX esolute mgnitudeEvolume digrm of simulted smples @eA @leftA nd @fA @rightAF pigure SFITX rule digrm of simulted smples @eA @leftA nd @fA @rightAF 5.3 The null correlation test on the supernova sample rerefterD we dpt the null orreltion pproh to x smples for determinE ing the osmologil prmetersF he x event is desried in the (M * c , m * c ) frme y the following proility densityX

@eAX            Ω • = 0.3, λ • = 0.7 σ M * c = 0.3, σ c = 0.085 β = 3.14, σ = 0.09 a DBS = -0.14, b DBS = -19 M * c,0 = -19, m * lim = 25.04 @fAX            Ω • = 0.3, λ • = 0.7 σ M * c = 0.
∝ ρ φ B (m * ) g G (c; 0, σ c )dc f (M * c )dM * c dm * c @SFRHA where ρ = ρ(z) = ∂V ∂m * c
stnds s orreltion funtion whih depends on the osmologil prmetersD see iqsFQFIQD QFIS nd QFIUF he weighting ftor is de(ned y iqFQFPID ut where the distne modulus is given yX

ζ est = m * c -M * c @SFRIA
where the solute mgnitude M * c eing estimted y x 1 nd F his is the mjor di'erene with the null orreltion proedure ove used with qusrsF where µ is determined y minimizing the quntity X

∆ w = max{w µ (z)} -min{w µ (z)} max{w µ (z)} + min{w µ (z)} @SFRQA
st must e noted tht the weighted orreltion oe0ient redsX

Γ(Ω • , λ • ) = N k=1 ω k (M * c,k -M * c )(m * c,k -m * c ) Σ M * c Σ m * c @SFRRA
where M * c is given y iqFSFQH he sttistis of stndrd devitions σ M * c nd σ m * c re de(ned yX o nlyse the dependene of the shpe of the xgg on the osmologil model nd the smple hrteristisD thirty smples re simulted similr to @fA ut with di'erent limiting pprent mgnitude @m * lim = 25.04D m * lim = 23A nd di'erent smple sizes xF o illustrte the resultsD we hve hosen two repreE senttive smples llowing us to desrie them lerlyD see pigFSFPHF he shpe depends strongly on the smple hrteristis @left pnelAF st is mtter of ft tht the xgg retrieve the osmologil prmeters used to simulte the smplesD whih shows the roustness of this pprohF he degenery of the rule lw is nlyzed with two simulted smples hving the sme hrE teristis of @fA s followsX the (rst one is de(ned y @Ω • = 0.3D λ • = 0.7A nd the seond one is hosen on the xgg of the (rst smpleD @Ω • = 0.4D λ • = 1.03AF he orresponding xggs devite quite slightly from eh other @right pnelA whih shows the need to use n dditionl sttisti to disentngle the true modelF he ury on the determintion of the osmologil pE 116 5.3 The null correlation test on the supernova sample rmeters whih form the xgg n e estimted from sttistil )ututions spei( to ten rndomly generted smples with the sme hrteristis of @fAD see pigFSFPIF e n hve hint of this preision from squre tht overs the edges of this distriution of urvesD nd we otin ∆Ω • ≈ 0.02 nd ∆λ • ≈ 0.125D tht orrespond roughly to 3σ errorF pigure SFIWX he xgg orresponding to @eA smple @leftA nd the one orreE sponding to @fA smple @rightAF pigure SFPHX hpe of the xgg with respet to the hrteristis of smple nd osmologil modelF he smple is desriedD fter the determintion of the olorD s followsX

Σ 2 M * c = N N -1 N k=1 ω k (M * c,k -M * c ) 2 @SFRSA Σ 2 m * c = N N -1 N k=1 ω k (m * c,k -m * c ) 2 @SFRTA IIS
dP ∝ θ(m * lim -m * )f (M * c )dM * c dV @SFRUA
reneD we pplied the V /V max testD see eFQFSFIF he model used for the simultion shows WU7 signi(ne level s it is shown in pigFSFPPF

f ω (M * c ) ∝ f (M * c )10 -µ 5 M * c , φ B,ω (m * c -βc) ∝ φ B (m * c -βc)10 µ 5 m * c @SFRWA
por lrge N D their umultive distriutions funtions n e pproximted y the stepEfuntionsX st is generlly knowledged tht the typeEs x stnds for good stndrd ndle to onstrin the osmologil prmetersF xeverthelessD the nowdys tlogs su'er from their smll sizeF he joint lightEurve nlysis @tveA4 smpleD pulished in fetouleD wF et l PHIRD onsists of URH typeEs xe on(rmed spetrosopillyF st inludes two min smplesX the hEss @QUR xeA nd xvQ @PQW xeA smples on whih the present nlysis is performedD see tle SFIF 5.4.1 Description of samples he p digrm of tve smple is desried in pigFSFPUF vet us remind thtD ording to the working hypothesesD the Edistriution should e uniform nd limited y urve relted to limiting pprent mgnitudeF st is ler tht this distriution does not meet this expettionF sndeedD the three points t V ≥ 3 re signi(ntly dethed from the rest of the distriutionF hey elong to the r smpleD whih shows inhomogeneities in seleting ojetsD see pigSFPVF rene it must e exludedD in ordne with the requirement of the null orreltion pprohF he red distriution in the p digrm of tve represents the hEss smple @ojets with redshift less thn HFRAF his onentrtion is due to the di'erene etween the oservtion onditions with other surveysD minly the xvQ smple @the lue pointsAD hEss hs limiting pprent mgnitude smller thn the one used for xvQF por this resonD we deided to use the susmples of tve seprtelyF he p digrm of lowEz smple is represented in pigFSFPW where the threshold on pprent mgnitude ppers oviouslyF he p digrm of hEss supernov smple is shown in pigFSFQH D suh struture is deteted in this digrm within the volume intervl [0, 0.04] for M * c < -19F sn this regionD there is onentrtion of ojetsF pper to this loud of pointsD we (nd retngulr white speF his shows tht the hEss smple is not uniformly distriuted in volume xis nd then it is not onsistent with the working hypothesesF pigFSFQI represents the p digrm of xvQ smple whih ontins ojets up to high redshift @z = 1.06AF his smple shows uniform distriution nd does not su'er from signi(nt seletion e'etsD prt from the limittion in pprent mgnitudeF gonsequentlyD this smple is more suitle to e used in our pprohF

F ω (M * c ) ≈ Fω (M * c ) = N k=1 ω k θ(M * c -M * c,k ) @SFSHA Φ B,ω (m * c -βc) ≈ ΦB,ω (m * c -βc) = N k=1 ω k θ(m * c -βc -(m * c,k -βc k )) @SFSIA

Results

he lirtion of dt is performed with respet to the steps detiled in etFSFPFSF ine the lirtion method suggests to use osmologil model yne the solute mgnitude is estimted y using iqFSFQHD the null orreE ltion test is pplied on oth xvQ nd hEss smplesF he xgg ws not found for the ltterF sn ordne with the ove nlysis on the p digrm @pigFSFQHAD this smple turns out indeed not omptile with the working hyE e studied the typeEs supernove events to infer the osmologil prmeters y mens of their light urves whih re good ndidtes to e used s stnE drd ndlesF uh feture tht provides us n estimtion of the solute mgnitude t mximum of luminosity thnks to its orreltion with the shpe prmeter of the light urveF his is n dvntge over the y investigtionF he seletion of xe does not depend solely on the oservtion of the ojet t given dteD ut lso on the smpling of the light urveD whih mkes the seletion of ojets more di0ultF por this purposeD we nlyzed the seletion e'et nd we determined seletion riterion @m * lim A tht is hrteristi of (tive limiting light urveF sn (rst stepD we simulted supernove smples tking into ount the orE reltion etween the solute mgnitude nd the shpe prmeterF he null orreltion test ws dpted to these smplesF sn seond stepD we studied the tve smple whih onsists of URH xeF nE fortuntelyD we found tht this smple is not onsistent with our pprohF hereforeD we pplied the null orreltion test to its susmplesD hhEss nd xvQF he xvQ smple provided null orreltion urve lose to the ones found y using the y smplesF his result fvors model with positive slr urvture with wek presene of drk mtterF 130 5.5 Conclusion smportnt improvement n e onsidered to extend this workF en imE provement in the desription of the seletion funtion tht tkes into ount the lifetime of events n provide more urte resultsF sn the futureD this work will e strong inititive to study the qmm y fursts @qfsAD the most powerful explosive events ever oserved in the niverseF he qfs re extrglti ojets distriuted in wide redshift rnge rehing z = 8.3 @qf HWHRPQD nvirD xF F et l PHHWAF he study of qfs using the sttistil pproh proposed in this thesis will llow to infer the osmologil informtion more urtelyF IQI Résumé en français n sièle près le modèle d9univers de priedmnnEvemîtreD les oservE tions le onfortent ve une onstnte osmologique Λ et une omposnte de mtière somre @noireA sns pression @poussièreA et froide dominnt elle ryE oniqueD ommunément ppelé wodèle vmd gold hrk wtter @ΛghwA ou enore modèle stndrdF v9élértion de l9expnsion de l9nivers on(rE mée pr le digrmme de rule des supernove en IWWV impose une vleur positive à l onstnte osmologiqueF wes trvux de thèse se folisent sur l9estimtion des vleurs de prmètres osmologiques du modèle stndrd en utilisnt l tehnique de orréltion nulleF gette pprohe présente l9vntge d9être plus rouste que les tehE niques usuellesF in prtiulierD il n9est ps requis de préiser l fontion de luminositéD elleEi est déduite pr ette méthodeF he plusD elle prend en ompte le iis de wlmquist due à l limittion en mgnitude pprentF ge trvil onsisté ussi à modéliser des éhntillons de l9événement qusr insi que l9événement supernovD une extrpoltion dptée du premierF ge qui permis de générer des éhntillons onformes ux hypothèses des modèlesD (n IQQ Chapter 6 : Résumé en français de vlider les pprohes sttistiquesF our e qui onerne les qusrsD nous vons utilisé le lon higitl ky urvey @hA d9une prtD et pour les suE pernove le uperxov vegy urvey @xvA et hEss d9utre prtF ves inférenes sttistiques de es données ont onduit à un univers sptilement fermé et une présene de mtière noire plus fileF hns le dre d9une prohine pplition de ette tehniqueD elle pourr être utilisée pour ontrindre les modèles d9énergie noireF he mêmeD l9utilistion des ms de glxies oservées grâe à l9e'et de unyev el9dovihD servir nturellement omme éhntillon osmologiqueF ne telle étude pourr onE triuer à pporter un élément de réponse à l vlidité du rôle supposé des neuE trinos mssifs dns l formtion des ms dns l9ère primordile de l9niversF wots glés X gosmologieD wtière noireD Énergie noireD prmètres osE mologiquesD sttistiquesD simultionD qusrsD supernoveD digrmme de ruE leF 134 6.1 La cosmologie moderne 6.1 La cosmologie moderne v osmologie moderne est sé sur le prinipe osmologiqueD qui fvorise les hypothèses de l9homogénéité et de l9isotropie de l9universF in e'etD le prinipe operniien proposé qu9il n9y ps de ple privilégiée dns l9universF in outreD l9hypothèse de l9isotropie de e dernier signi(e que hque point de l9nivers possède des propriétés identiques dns des diretions di'érentesD lors l9nivers semle homogène @l9isotropie de hque point implique l9homogénéitéAF v9oservtion dns des di'érentes diretions ve l même ngle solide ont onduit à un résultt identiqueF gei implique queD à grnde éhelleD nous pouvons dérire l9univers oservle omme étnt sptilement homogène et isotropeF gel nous mène à l desription de l9espeEtemps ve l métrique priedmnnEvemîtreEoertsonElker @pvAX

ds 2 = dt 2 -a 2 (t)dσ 2
@TFIA où t est le temps osmiqueD a(t) est le paramètre d'expansion @sns dimensionA et dσ 2 est l9élément métrique de l9espe homogène tridimensionnel V 3 @nommé prD espace comobileAF wlgré l9inhomogénéité de l9nivers à petites éhellesD l métrique pv dérit l9espeEtemps à grndes éhellesD à prtir de plus de IHH wpD où les hypothèses osmologiques sont vlles omme les montré l distriution des qusrs à grnde éhelle ve le sondge fryon ysilltion petrosopi urvey @fyAF in IWRVD qmowD elpher et rermn ont prédit le fond di'us osmologiqueD ils ont proposé un modèle de fig fng hud en supposnt un univers primorE dil omposé uniquement de neutrons @qui se désintègrent ensuite en protonsA F tF iF eeles PHIRD F elpher PHIRF hns e modèle @onnu sous le nom de ffx pour fig fng xuleosynthesisAD l9nivers primordil étit omE posé d9un plsm de photonsD fermions et qurks extrêmement hud et denseF v9nivers ensuite montré une expnsion où s densitéD insi s tempérture ont diminuéD fvorisnt l nuléosynthèse primitiveD 9estEàEdire l formtion des premiers noyux files @rydrogèneD héliumD deutériumD lithiumAF invE iron QVH HHH ns près le fig fngD l tempérture est devenue ssez file @∼ 3000 • KA pour que les premiers tomes se forment à trvers un proessus ppelé reominison @ien que les tomes n9ient jmis été ominés vntAF à ette époqueD les photons ont déouplé de ryons et l lumière ommené à se propger lirementF ge ryonnementD qui été simplement refroidi penE dnt l9évolution de l9universD est onnu sous le nom de fond di'us osmologique @gwf pour gosmi wirowve fkground en nglisAF ves oservtions ont on(rmé le modèle ffxD une fois lorsque enzis et ilson ont déteté le gwfF ge ryonnement presque un spetre de orps noir prfit ve une IQS Chapter 6 : Résumé en français tempérture moyenne de 2.73 • K produit pr les photons de l9nivers primorE dilF v tempérture de gwf est isotrope et homogène à des vritions de l9ordre de IH -5 F it une utre fois pr les mesures de l9ondne des éléments légers dns l9universD en tnt que deuxième preuveD qui sont en exellent ord ve les préditions de l nuléosynthèse primordile qF teigmn PHHRF gel inlut l9ondne mssive d9rélium reR prédite entre PQ% et QH%D ve un ord prfit ve les oservtions gF eF fertulniD et lF PHITF ve ontenu de l9univers est onsidéré omme un )uide prfit qui se rtérise pr les proE priétés du tenseur énergieEimpulsion ve l pression et l densité d9énergie dns l9univers ρF in e'etD les équtions de priedmnn nous permettent de dé(nir le polynôme suivntX

P (a) = λ • a 4 -κ • a 2 + Ω • a + α • ; P (1) = 1
@TFPA où les oe0ients sont des prmètres sns dimensionD et ils sont dé(ni omme suivntX

λ • = 1 3 ΛH -2 • , κ • = K • H -2 • = λ • + Ω • + α • -1, Ω • = 8 3 πGρ • H -2 • > 0@TFQA α • = 8 45 π 3 Gk B T 4 • -3 H -2 • ≈ 2.5 10 -5 h -2 , h = H • /(100 km s -1 Mpc -1 )@TFRA
ge sont des quntités qui se situent respetivement pour les vleurs tuelles des versions réduites de l onstnte osmologique ΛD l ourure slire K de l9espeD l densité de l mtièreD ρ étnt l densité spéi(que des prtiules mssives @wtière noire inluseAD G est l onstnte de xewtonD l densité d9énergie du ryonnement et l onstnte de ruleF ges prmètres stisfont l formuleX

λ • -κ • + Ω • + α • = 1 @TFSA r l suiteD nous estimons les prmètresX λ • et Ω • F
ge que rule mené à l déouverte de s loi élère étit l mesure de l distne et de l vitesse des glxies dns l9nivers proheF sl déouvert que les glxies semlient reuler à une vitesse qui ugmentit proportionE nellement ve leur distneF ge phénomène se re)ète dns v loi de ruleX

v = H • d @TFTA
eve les oordonnées omoile des glxiesD on peut dé(nir l distance comobile τ entre l9oservteur et l soure de lumière à redshift zX

τ (z) = 1 1 1+z da P (a)
@TFUA 136 6.1 La cosmologie moderne v distne de luminositéD d L D est dé(ni omme le ryon d9une sphère entrée sur l soure lumineuse dont son énergie lumineuse est distriué sur l surfe de ette sphèreF v lumière reçue pr unité de temps représente le )ux @l luminosité pprenteA de ette soure qui diminue ve le rré de l distneD pr l loi d9inverse rré X

f = L 4πd 2 L @TFVA
où L est l luminosité intrinsèque de l9ojetF sl s9vère que l distne de luminosité est liée à l distne omoile et elle s9érit omme suitX

d L = c H • (1 + z)            sin(τ ) √ κ 0 if κ 0 > 0 τ if κ 0 = 0 sinh(τ ) √ |κ 0 | if κ 0 < 0 @TFWA où τ = τ |κ 0 | est l distne ngulireF
in stronomieD le système de mgnitude est fréquemment utilisé pour dérire l luminosité d9un ojetF v mgnitude est une éhelle utilisée pour lsser les étoiles en fontion de leur luminositéF ves étoiles les plus rillntes ont l plus file mplitude pour lesquelles un degré de mgnitude orrespond à une di'érene de PDSI fois de luminositéF in utilisnt l dé(nition de l mgnitude pprente mD nous pouvons introduire l mgnitude solue M d9un ojetF ge dernier est une mesure de l luminosité intrinsèque de l9ojet et est dé(ni omme il pprîtrit à un oservteur hypothétique à une distne de IHpF v mgnitude pprente d9un ojet à redshift z est liée à l mgnitude solue @suivnt iqFTFVAD on peut érireX gonformément ux hypothèses de l9homogénéité et de l9isotropie de l9universD les ojets dns l9espe sont ensés d9être uniformément réprtisF ves qusrs @ysD pour usr stellr yjets en nglisA supposés d9être des ojets permnents pour sonder l géométrie de l9espeEtempsF sls sont rtérisés intrinsèquement d9une mgnitude solue M et une distne omoile τ @le ryon de volume V entré à l position de l9oservteur ve une réprtition uniforme dns l9espe omoileAF i l mgnitude solue M est réprtie selon l fontion de luminosité f@wAD l9éhntillon de y est dérit ve le produit de deux densités de proilité indépendntes omme suitX

m = M + ζ(z) @TFIHA où ζ(z)
dP th ∝ f (M )dM.dV @TFIIA
in supposnt que les e'ets de séletion dépendent uniquement de l mgnitude pprente mD nous utilisons une fontion de séletion φ(m) dns l densité de proilité dns le ut de prendre en ompte le proessus de séletion en oservtionF r onséquentD l densité de proilité des vriles oservles sont dérites prX dP obs = φ(m)dP th P th (φ) @TFIPA où P th est un fteur de normlistionF xous dé(nissons l fontion de réE prtition en terme de l vrile létoire de mgnitude solue M omme 138 6.3 La technique de corrélation nulle l9intégrle de l9éqution TFIPF v fontion de réprtition est une vrile léE toire uniforme entre H et IF v9éhntillon simulé est rélisé pr l reherhe de l solution de l9inverse de l fontion de réprtition en déterminnt les deux quntités intrinsèqueD l mgnitude solue M et le volume V F insuiteD le redshift z est déterminé pr l solution de l9inverse de olume V -1 F in(nD l mgnitude pprente est lulé en utilisnt l9iqFTFIHF ve digrmme de mgnitude solueEvolume (M -V ) est représenté omme une dérivtion du digrmme de ruleF elors que e digrmme ne montre ps les e'ets de séletionD le digrmme M -V donne une illustrtion lire des e'ets de séletionF pigure TFPX ve digrmme mgnitude solueEvolume de l9éhntillon simulé présenté en rougeF v oure leue est le volume en fontion de l mgnitude solue à une mgnitude pprente limite donnéeF get éhntillon été généré ve un univers plt (Ω • , λ • = 0.3, 0.7)F ne réprtition uniforme pprît sur l9xe vertil et une distriution gussienne sur l9xe horizontl tel qu9ils ont été hoisi initilement dns l simultionF ges simultions nous permettent d9otenir des éhntillons de y simiE lires ux données réellesD qui peuvent être utilisés ve le test de orréltion nulleF 6.3 La technique de corrélation nulle v méthode sttistique hituelleD u moyen de l9justement du moindre rréD onsiste à dpter les données de l9éhntillon oservle u modèle théorique supposéF in d9utres termesD les prmètres osmologiques sont dé(nis de telle sorte que les données montrent l dispersion minimle du omportement IQW Chapter 6 : Résumé en français théoriqueF gependntD le iis de wlmquist empêhe à estimer orretement l mgnitude d9une telle dispersion à moins qu9on onnisse l fontion de luE minosité et les e'ets de séletion de l9oservtionF eve l tehnique de orréltion nulleD nous sommes exemptés de fournir es rtéristiquesF xénmoinsD les e'ets de séletion ne doivent ps dépenE dre du redshift mis uniquement de l mgnitude pprenteF elon l formule TFIPD un éhntillon simulé sous le modèle (Ω • , λ • = 0.3, 0.7) est e'etuéD nous vons utilisé une fontion de luminosité gussienneF get éhntillon été utilE isé omme référene pour véri(er l performne de l tehnique sttistiqueF v tehnique de orréltion nulle est une méthode sttistique rouste qui n9exige ps l onnissne de l fontion de luminosité prioriF he plusD ette tehnique est exempte du iis de wlmquistF v tehnique de orrélE tion nulle nous permettre d9otenir un ensemle de modèles qui sont onsidéré omme les meilleurs ndidts qui s9ordent ve les donnéesF v9éhntillon oservé est représenté dns le pln @wD mA pr l densité de proilité suivnteX

dP (M,m) = ρ(z)φ(m)f (M )dM.dm; ρ(z) = ∂V ∂m @TFIQA
où ρ(z) est une fontion de orréltion entre M et mF our les vries vleurs des prmètres osmologiques (Ω • , λ • )D l orréltion entre les vriles létoires M et m disprissent en divisnt hque événement pr l fontion de orréltion ρ(z)F gette orréltion est donnée pr l ovrine pondérée de M et mD elle dépend des prmètres osmologiques et s9érit omme suivntX

Γ(Ω • , λ • ) = N k=1 ω k (M k -M )(m k -m ) @TFIRA
où ω k est le fteur de pondértion qui est inversement proportionnel à ρ(z)F ves moyennes pondérées de M et m sont notées M et m respetivementD elles sont dé(nies pr les formules suivntesX v9extitude de ette tehnique été montré en étudint les )utution sttistiques qui diminuent ve l9ugmenttion du nomre d9ojetsF h9utre prtD l9pprohe de orréltion nulle nous permis de déterminer l fontion de luminosité et l fontion de séletionF n deuxième test été étliD u moyen du test V /V max D ve une petite modi(tion de l dé(nition hituelleF xous vons 0né les résultts en ppliqunt e test sur l oure de orréltion nulle dns le ut d9otenir les meilleurs ndidts des prmètres osmologiquesF 6.4 Résultats avec les données quasars xous vons ppliqué le test de orréltion nulle et elui de V /V max sur plusieurs éhntillons et sousEéhntillons de y fournis pr les données du sondge hF xous vons otenu plusieurs oures de orréltion nulles qui nous ont permis de disriminer les meilleurs ndidts qui s9ordent ve les hyE pothèses de trvilF in outreD nous vons 0né les résultts en véri(nt l réprtition sptile uniforme des y en utilisnt le test V /V max D en prtiE ulier sur le sousEéhntillon le plus homogèneF ges deux tests nous ont permis de déterminer une onstnte osmologique positive ve une file vleur de densité de mtière pour une ourure positive @pigTFRAF qrâe ux tehniques sttistiques utilisées dns e trvilD les ys ont des inférenes de l9élértion osmique ien qu9elles ne soient ps lssées omme des ougies stndrdF IRI Chapter 6 : Résumé en français pigure TFRX ves oures de orréltion nulles dns le digrmme(Ω • , λ • ) orE respondnt ux éhntillons ihD hID hQD hU et ux sousEéhntillons hQF ves oures sont situées dns l région ve une ourure positive κ • > 0 @uEdessus de l ligne droiteAF outes les oures sont en ord et prohes l9une de l9utreD suf l oure rouge qui orrespond à l9éhntillon ihF 6.5 Modélisation d'échantillon de supernova et résultats sur les données de SDSS-II/SNLS3 v supernov @xA est l9un des ojets les plus lumineux dns l9univers déE ouvert jusqu9ujourdâhuiF ge type d9ojet est le résultt de l9explosion des étoiles mssives ve une intensité phénoménleF gette explosion est le signe du hngement d9étt d9une étoile à l (n de s vieF v x est un ojet très lumineux @gltique et extrgltiqueA ve une luminosité qui peut orreE spondre à elle de s glxie hôte dns ertins sF qrâe à leurs luminosités intrinsèquesD les xe sont oservés à des distnes osmologiquesF illes ont été lssées et distinguées en groupesD dont l9une est onsidérée omme une fmille de ougies stndrdF uisque l dispersion de l luminosité u mximum de l oure de luE mière @gvA des xe du type s est suseptile d9être fileD elles sont onE sidérées omme de ons ndidts des ougies stndrdD e qui en fit les meilleurs outils pour estimer les distnes osmologiques et ensuite ontrinE dre les prmètres osmologiques ve une grnde préisionF wF wF hillips IWWQD rmuy et lF IWWSD IWWTD iess et l IWWS et IWWT ont démontré l orréltion entre un prmètre de l gv et l mgnitude solueD M B D u mximum de luminositéF ge prmètre est le tux de délin de l gvD noté 142 6.5 Modélisation d'échantillon de supernova et résultats sur les données de SDSS-II/SNLS3 ∆m 15 ou x 1 D shnt que l nottion et l signi(tion du prmètre de forme dépendent de l méthode de stndrdistion hoisiF yn mentionne spéileE mentD x 1 utilisé ve le modèle de stndrdistion evPF v desription des événements ve une durée de vie (nie 1 D omme 9est le s pour les xe de type sD néessite une détermintion sensile de leurs quntités @vrilesA intrinsèquesF v9événement x s est un )sh puissnt rtérisé pr s oure de lumièreD 9estEàEdire l9évolution de s luminosité intrinsèque ve le tempsF v oure de lumière est identi(ée pr l dte u mximum de luminositéD notée t * D et pr l mgnitude solue qui orreE spond à e mximumD noté M * F ges deux vriles intrinsèques rtérisent l9évolution de l mgnitude solue ve le tempsD w@tAD qui est églement prmétré pr un prmètre de forme x 1 D 9estEàEdire l dilttion ou l onE trtion de l durée de vie pr rpport à un oure ptronF ve prmètre de forme x 1 représente l durée de vie de hque x et il est supposé d9être linéirement orrélé ve M * F our prendre en ompte les e'ets de séletionD nous vons étudié le omE portement des oures de lumière d9un éhntillon de x donné (n de souligner un ritère générl qui peut être utilisé pour distinguer les xe etD pr onE séquentD éliminer les xe ve des oures de lumière de muvis éhntillonE ngeF n supernov une oure de lumière ien éhntillonnée si elle eu su0smment du temps pour montrer une évolution de s luminosité uEdessus du )ux limite f lim qui orrespond u )ux de l mgnitude pprente limite m lim F sl fut rppeler que l oure de lumière oservée se rtérise pr l mgnitude pprente u mximum de luminosité m * et pr le tux de déE lin ∆m 15 @de plus ∆m 15 est grndD de plus l diminution de l oure de lumière est viteAF v oure de lumière doit présenter un ért signi(tif entre l mgnitude pprente limite de m lim et son pi de mgnitude pE prente m * d9une prtD et d9utre prt un petit ∆m 15 pour voir un temps su0snt pour éhntillonner l oure de lumièreF hns ette visionD les tehE niques d9oservtion susmentionnées permettent de déterminer une oure de lumièreD dé(nie omme une oure de lumière limiteD pour lquelle les exiE genes minimles sont ssuréesF gette oure limite dépend fortement de l pité des tehniques d9oservtion de hque enquête et elle se rtérise pr m * lim et ∆m 15,lim F in onlusionD ette oure de lumière limite est utilisée pour juger l séletion des xe en fontion du pi de l mgnitude pprente m * D où une x est oservée ve suès et séletionnée si son m * `m * lim F eve e ritère de séletionD nous inluons non seulement le iis de wlmquistD mis [START_REF] Wisotzki | Quasar spectra and the K correction[END_REF] Un objet non pérenne IRQ Chapter 6 : Résumé en français ussi les iis qui dues u muvis éhntillonnge des oures de lumièreD en prtiulier pour les xe ve des ourtes durées de vieF eve ette desription des e'ets de séletionD nous vons introduit dns l densité de proilité des ojets oservésD une fontion tehnologique φ B (m * ) qui dépend du pi de l mgnitude pprente dns l nde fX dP obs = φ B (m * )g(M * , x 1 )dM * dx 1 dt * dV @TFIUA v ouleur est utilisée pour rendre l distriution de l mgnitude solue u mximum de luminosité plus étroiteF insuiteD nous notons M * c pour l vleur orrigée de l mgnitude solueX

M = N k=1 ω k M k @TFISA m = N k=1 ω k m k @TFITA v solution de l9éqution Γ(Ω • , λ • ) =
M * = M * c -βc @TFIVA
où β est un oe0ient orretif positifF r onséquentD l ouleur doit être prise en ompte dns l loi sur l densité de proilité de l9événementF elorsD iqFTFIU est érisX

dP obs = φ B (m * )g(M * , x 1 )dM * dx 1 g G (c; 0, σ c )dcdt * dV @TFIWA où M * c s9érisX M * c = ax 1 + b -βc if M a stellar <

•

[2κ

1 2 • ∂τ ∂z -2κ 1 2
• ∂τ ∂z cos(2κ

1 2 • τ )] = 2π κ • ∂τ ∂z [1 -cos(2κ 1 2 • τ )] = 2π κ •   1 -cos(2κ 1 2 • τ ) (1 + z) 2 P ( 1 1+z )   @fFIA ∂ζ ∂z = 5 ln(10) ∂((1+z) sin(κ 1 2 • τ )) ∂z (1 + z) sin(κ 1 2 • τ ) = 5 ln(10)   1 (1 + z) + κ 1 2
• cot(κ

1 2 • τ ) (1 + z) 2 P ( 1 1+z )   = 5 ln(10)   (1 + z) P ( 1 1+z ) + κ 1 2
• cot(κ

1 2 • τ ) (1 + z) 2 P ( 1 1+z )   @fFPA
henD we otinX ISI Chapter B : Calculation of the weighting factor

∂V ∂ζ = 2π ln(10) 5κ • (1 -cos(2κ 1 2 • τ )) [(1 + z) P ( 1 1+z ) + κ 1 2
• cot(κ 1 2

• τ )] @fFQA ith the similr lultion for the negtive κ • D we otinX

∂V ∂ζ = 2π ln(10) 5|κ • | (cosh(2|κ • | 1 2 τ ) -1) [(1 + z) P ( 1 1+z ) + |κ • | 1 2 coth(|κ • | 1 2 τ )] @fFRA APPENDIX C Calculation of V(M)
rereD we detil the determintion of the urve of the eqution m(M, V ) = m lim F sn other wordsD the volume s funtion of w t limiting pprent mgnitude m lim F e note tht the solute mgnitude w is repled y the redued solute mgnitude for resons of simpliity in the lultionF por κ • a HX

τ 3 = 1 (1 + z lim ) 3 10 3(m lim -M ) 5
@gFIA where

z lim = ζ -1 (m lim -M ) @gFPA
the soure of solute mgnitude w remins visile up to redshift z lim under given limited pprent mgnitude m lim D thereforeD using iqsFIFQU nd gFID the volume eomesX

V (M ; m lim , z lim ) = 4π 3 1 (1 + z lim ) 3 10 3(m lim -M ) 5 @gFQA por κ • b HX V (z) = π κ 3/2 • (2τ κ 1/2 • -sin(2τ κ 1/2 • )) @gFRA d l = (1 + z) sin(τ κ 1/2 • ) κ 1/2 • @gFSA ISQ Chapter C : Calculation of V(M) thereforeD one otinsX sin(τ κ 1/2 • ) = κ 1/2 • (1 + z lim ) 10 (m lim -M ) 5 @gFTA τ κ 1/2 • = arcsin κ 1/2 • (1 + z lim ) 10
(m lim -M ) 5
@gFUA sine from the trigonometri reltions we hveD sin@PαA a Psin@αAcos@αA nd cos@arcsin@αAA a sin@arccos@αAA a √ 1α 2 where α ∈ EIDID soX

V (M ; m lim , z lim ) = π κ 3/2 • 2τ κ 1/2 • -2 sin(τ κ 1/2 • ) cos(τ κ 1/2 • ) @gFVA V (M ; m lim , z lim ) = π κ 3/2 • 2 arcsin(α l ) -2α l 1 -α 2 l @gFWA whereX α l ≡ α(M ; m lim , z lim ) = κ 1/2 • (1 + z lim ) 10
(m lim -M ) 5 @gFIHA por κ • `HX V (z) = π |κ 3/2 • | (sinh(2τ |κ • | 1/2 ) -2τ |κ • | 1/2 ) @gFIIA d l = (1 + z) |κ • | 1/2 sinh(2τ |κ • | 1/2 ) = 10 (m lim -M ) 5 @gFIPA sinh(τ |κ • | 1/2 ) = |κ • | 1/2 (1 + z lim ) 10 (m lim -M ) 5 @gFIQA τ |κ • | 1/2 = |κ • | 1/2 (1 + z lim ) 10
(m lim -M ) 5 @gFIRA α l ≡ α(M ; m lim , z lim ) = |κ • | 1/2 (1 + z lim ) 10
(m lim -M ) 5
@gFISA the trigonometri reltions llow us to writeX sinh@PαA a Psinh@αAcosh@αA nd cosh((α)) a ----------------------- ) ) * ( ( 1 . + s e l f . z ) * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) ) ) + ( 1 . / tau ) + s e l f . d_kcorr * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) ) ) * ( 1 . + s e l f . z ) * * 2 . ) e l i f s e l f . param . kappa_0 > 0 . 0 : e p s i l o n = 2* tau * s q r t ( s e l f . param . kappa_0 ) jac_1 = ( ( 1 . + s e l f . z ) * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) ) ) + ( s q r t ( s e l f . param . kappa_0 ) / ( ( s i n ( e p s i l o n / 2 . 0 ) ) / cos ( e p s i l o n / 2 . 0 ) ) ) + ( ( ( ( 1 . + s e l f . z ) * * 2 . ) * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) ) ) ) * s e l f . d_kcorr ) ) / ( 1 . -cos ( e p s i l o n ) ) e l i f s e l f . param . kappa_0 < 0 . 0 : e p s i l o n = 2 . * tau * s q r t ( abs ( s e l f . param . kappa_0 ) ) jac_1 = ( ( 1 . + s e l f . z ) * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) )

√ 1 + α 2 D thenX V (M ; m lim , z lim ) = π |κ • | 3/2 2α l 1 + α 2 l -2(α l ) @

ISU

Chapter D : Code Python ) + ( s q r t ( abs ( s e l f . param . kappa_0 ) ) / tanh ( e p s i l o n / 2 . 0 ) ) + ( ( ( ( 1 . + s e l f . z ) * * 2 . ) * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) ) ) ) * s e l f . d_kcorr ) ) /( cosh ( e p s i l o n ) -1 . ) e l s e : pass e l i f s e l f . K c o r r e c t i o n == ' Kcorrection_NO ' : tau = s e l f . c o m o b i l e D i s t a n c e i f s e l f . param . kappa_0 == 0 . 0 : jac_1 = ( 1 . / ( tau * * 3 . ) ) * ( tau * ( 1 . + s e l f . z ) * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) ) ) + 1 . ) e l i f s e l f . param . kappa_0 > 0 . 0 : e p s i l o n = 2* tau * s q r t ( s e l f . param . kappa_0 ) jac_1 = ( ( 1 . + s e l f . z ) * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) ) ) + ( s q r t ( s e l f . param . kappa_0 ) / ( ( s i n ( e p s i l o n / 2 . 0 ) ) / cos ( e p s i l o n / 2 . 0 ) ) ) ) / ( 1 . -cos ( e p s i l o n ) ) e l i f s e l f . param . kappa_0 < 0 . 0 : e p s i l o n = 2 . * tau * s q r t ( abs ( s e l f . param . kappa_0 ) ) jac_1 = ( ( 1 . + s e l f . z ) * s q r t ( s e l f . param . P ( 1 . / ( 1 . + s e l f . z ) ) ) + ( s q r t ( abs ( s e l f . param . e l i f s e l f . param . kappa_0 > 0 . 0 : e p s i l o n = tau * s q r t ( s e l f . param . kappa_0 ) weighing = s e l f . Jacobian ( z ) * ( ( ( 1 . + z ) * abs ( s i n ( e p s i l o n ) ) * 1 0 . * * ( s e l f . k c o r r ) / s q r t ( s e l f . param . kappa_0 ) ) ** beta ) e l i f s e l f . param . kappa_0 < 0 . 0 : e p s i l o n = tau * s q r t ( abs ( s e l f . param . kappa_0 ) ) weighing = s e l f . Jacobian ( z ) * ( ( ( 1 . + z ) * s i n h ( e p s i l o n ) * 1 0 . * * ( s e l f . k c o r r ) / s q r t ( abs ( s e l f . param . kappa_0 ) ) ) ** beta ) e l s e : pass e l i f s e l f . K c o r r e c t i o n == ' Kcorrection_NO ' : i f s e l f . param . kappa_0 == 0 . 0 : weighing = s e l f . Jacobian ( z ) * ( ( ( 1 . + z ) * tau ) ** beta ) e l i f s e l f . param . kappa_0 > 0 . 0 : e p s i l o n = tau * s q r t ( s e l f . param . kappa_0 ) weighing = s e l f . Jacobian ( z ) * ( ( ( 1 . + z ) * s i n ( e p s i l o n ) / s q r t ( s e l f . param . kappa_0 ) ) ** beta ) e l i f s e l f . param . kappa_0 < 0 . 0 : e p s i l o n = tau * s q r t ( abs ( s e l f . param . kappa_0 ) ) weighing = s e l f . Jacobian ( z ) * ( ( ( 1 . + z ) * s i n h ( e p s i l o n ) / s q r t ( abs ( s e l f . param . a l f a _ l = ( s q r t ( s e l f . param . kappa_0 ) /(1 + z_l ) ) * ( 1 0 * * ( (m_lmus e l f . k c o r r ) / 5 . ) ) e l i f s e l f . K c o r r e c t i o n == ' Kcorrection_NO ' : a l f a _ l = ( s q r t ( s e l f . param . kappa_0 ) /(1 + z_l ) ) * ( 1 0 * * ( (m_lmu) / 5 . ) ) e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" V = ( p i / s e l f . param . kappa_0 * * ( 3 . / 2 ) ) * (2* a r c s i n ( a l f a _ l ) -2* a l f a _ l * s q r t (1 -a l f a _ l * * 2 . ) ) e l i f s e l f . param . kappa_0 < 0 . 0 : mu = M + 5* l o g 1 0 ( ( ( L i g h t V e l o c i t y * 1 0 * * 6 . 0 ) / ( 1 0 . *H_0) ) ) a l f a _ l = ( s q r t ( abs ( s e l f . param . kappa_0 ) ) /(1 + z_l ) ) * ( 1 0 * * ( ( m_l -mu) / 5 . ) ) i f s e l f . K c o r r e c t i o n == ' Kcorrection_OK ' : a l f a _ l = ( s q r t ( abs ( s e l f . param . kappa_0 ) ) /(1 + z_l ) ) * ( 1 0 * * ( ( m_l -mus e l f . k c o r r ) / 5 . ) ) e l i f s e l f . K c o r r e c t i o n == ' Kcorrection_NO ' : a l f a _ l = ( s q r t ( abs ( s e l f . param . [ 1 ] ) . c u r v a t u r e ( z2 , a [ i ] [ 0 ] , a [ i ] [ 1 ] , 0 . 0 ) [ 1 ] p l t . p l o t ( z2 , V) """ r e t u r n aa Cov_All_Models = func ( All_models , r e d s h i f t , Beta ) #p r i n t "Cov_All_Models = " , Cov_All_Models , "\n" p r i n t " l e n ( Cov_All_Models ) = " , l e n ( Cov_All_Models ) , "\n" p r i n t " l e n ( All_models ) = " , l e n ( All_models ) , "\n" # make the array as matrix to s e a r c h the n u l l c o r r e l a t i o n curve on the g r i d ! Cov_All_Models = Cov_All_Models . reshape ( i n t ( s q r t ( l e n ( All_models [ : , 1 ] ) ) ) , i n t ( s q r t ( l e n ( All_models [ : , 0 ] ) ) ) ) All_modelslambda = All_models [ : , 0 ] . reshape ( i n t ( s q r t ( l e n ( All_models [ : , 0 ] ) ) ) , i n t ( s q r t ( l e n ( All_models [ : , 0 ] ) ) ) ) # For matrix o f lambda All_modelsomega = All_models [ : , 1 ] . reshape ( i n t ( s q r t ( l e n ( All_models [ : , 1 ] ) ) ) , i n t ( s q r t ( l e n ( All_models [ : , 0 ] ) ) ) ) # For """ R e g i s t r a t i o n o f a l l l i s t s and a r r a y s i n one f i l e """ d e f __init__ ( s e l f , l i s t o f a r r a y s , s t r i n g l i s t o f a r r a y s , f i l e n a m e ) : s e l f . l i s t o f a r r a y s = l i s t o f a r r a y s s e l f . s t r i n g l i s t o f a r r a y s = s t r i n g l i s t o f a r r a y s s e l f . f i l e n a m e = f i l e n a m e d e f ReadWrite ( s e l f , header , private_pc ) : path = s t r ( s e l f . f i l e n a m e ) f = open ( path , 'w ' ) i f header == True : f o r l i n range ( l e n ( s e l f . s t r i n g l i s t o f a r r a y s ) ) : l i s t n a m e = s e l f . s t r i n g l i s t o f a r r a y s [ l ] l i s t n a m e = s t r ( l i s t n a m e ) f . w r i t e ( l i s t n a m e ) f . w r i t e ( " " ) f . w r i t e ( "\n" ) e l i f header == F a l s e : pass 188 l e n g t h = [ l e n ( a ) f o r a i n s e l f . l i s t o f a r r a y s ] f o r i i n range (max( l e n g t h ) ) :

c o n d i t i o n = ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ]== D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) i f c o n d i t i o n == F
c o n d i t i o n 1 = ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ]== D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) c o n d i t i o n 2 = ( abs ( f l o a t ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ] ) -f l o a t ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) )>= 0 . 0 6 ) i f c o n d i t i o n 1 == F a l
c o n d i t i o n 1 = ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ]== D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) c o n d i t i o n 2 = ( abs ( f l o a t ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ] ) -f l o a t ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) )>= 0 . 0 6 ) i f c o n d i t i o n 1 ==
c o n d i t i o n = ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ]== D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) i f c o n d i t i o n == F a l s e : z_l_l . append ( z_l_jj [ i ] ) e l s e : pass D i s t a n g u i s h _ o f _ s o l u t i o n s = [ s t r ( gg ) [ 0 : 4 ] f o r gg i n z_l_l ] z_l = [ ] z_l . append ( z_l_l [ 0 ] ) f o r i i n range ( 1 , l e n ( z_l_l ) ) : c o n d i t i o n 1 = ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ]== D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) c o n d i t i o n 2 = ( abs ( f l o a t ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ] ) -f l o a t ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) )>= 0 . 0 6 ) i f c o n d i t i o n 1 == F a l
f o r l i n range ( l e n ( s e l f . l i s t o f a r r a y s ) ) : i f i < l e n ( s e l f . l i s t o f a r r a y s [ l ] ) : element = s e l f . l i s t o f a r r a y s [ l ] [ i ] element = s t r ( element ) f . w r i t e ( element ) f . w r i t e ( " " ) e l s e :

f . w r i t e ( "Nan" ) f . w r i t e ( " " )

f . w r i t e ( "\n" ) f . c l o s e ( ) p r i n t path , "\n"

r

  SU RFQFQ istimtion of the luminosity funtion nd the seletion funtionF F F F F F F F F F F F F F F F F F F F F F F F F F F TH RFR snferenes on osmologil expnsion sed on ys F F F F F F TI RFS gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UW Application to supernovae type Ia sample 81 SFI sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VI SFP ype s upernove s stndrd ndles F F F F F F F F F F F F F VP SFPFI he lighturve of typeEs x F F F F F F F F F F F F F F F F VQ SFPFP tndrdistion of typeEs x F F F F F F F F F F F F F F F VR SFPFQ wodelling of the upernov event F F F F F F F F F F F F F VT SFPFR mpling the light urve of typeEs x nd seletion e'ets VV SFPFS glirtion sttistis F F F F F F F F F F F F F F F F F F F F WH SFPFT imultion F F F F F F F F F F F F F F F F F F F F F F F F F FWR SFQ he null orreltion test on the supernov smple F F F F F F F F WV SFQFI vuminosity funtion nd eletion funtion F F F F F F F F IHH SFQFP reision nd error F F F F F F F F F F F F F F F F F F F F F IHI SFR epplition to the tve smple F

  supernov et résultts sur les donE nées de hEssGxvQ F F F F F F F F F F F F F F F F F F F F F F IPP Appendices A Probability density functions B Calculation of the weighting factor C Calculation of V(M) D Code Python W CONTENTS

  F F F F F F F F F F F F ST RFIH The volume-redshift diagram in the model Ω • = 0.076, λ • = 1.211. The graduation on the ordinate is translated into angular distance on the right hand side axis. F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ST RFII The distance modulus ζ(z)-redshift diagram in the model Ω • = 0.076, λ • = 1.211, showing the angular distance on the right hand side axis. The maximum of ζ(z) corresponds to τ = 2π

  IIH TFI ve digrmme de rule des éhntillons de upernove ve xvD hD r et plusieurs sondges lowEzF pigure tirée de fetouleD wF et l PHIR F F F F F F F F F F F F F F F F F F F F F F IIV TFP ve digrmme mgnitude solueEvolume de l9éhntillon simulé présenté en rougeF v oure leue est le volume en fonE tion de l mgnitude solue à une mgnitude pprente limE ite donnéeF get éhntillon été généré ve un univers plt (Ω • , λ • = 0.3, 0.7)F ne réprtition uniforme pprît sur l9xe vertil et une distriution gussienne sur l9xe horizontl tel qu9ils ont été hoisi initilement dns l simultionF F F F F F F F IIW TFQ v oure de orréltion nulle dns le pln (Ω • , λ • )F v oure retrouve le modèle osmologiques utilisé pour e'etuer l simuE ltion de l9éhntillonF F F F F F F F F F F F F F F F F F F F F F F F IPI TFR ves oures de orréltion nulles dns le digrmme(Ω • , λ • )

Contents 1 . 1 8 1. 5

 1185 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Newtonian cosmology . . . . . . . . . . . . . . . . . . 4 1.3 Einstein's cosmology . . . . . . . . . . . . . . . . . . . 5 1.4 The Friedmann-Lemaître-Gamow model . . . . . . . 6 1.4.1 The Friedmann's equations . . . . . . . . . . . . . . 7 1.4.2 Primordial Universe . . . . . . . . . . . . . . . . . . Contents of the Universe . . . . . . . . . . . . . . . . 9 1.6 Baryonic Acoustic Oscillations . . . . . . . . . . . . . 13 1.7 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . 13 1.8 The Dark Energy . . . . . . . . . . . . . . . . . . . . 14 1.1 Introduction

1 :

 1 Basics of Observational Cosmology pigure IFRX he sptil distriution of the qusrs of the fy tlog shows rdil strti(tion due to four distint phses of oservtionsD nd in eh of suEsmples derese in the numer of ojets s the rdius inresesF fy tking into ount this seletion e'etD one n sfely guess tht this distriution is uniformF pigure reditX universetody wesiteF ine

1. 9 . 1

 91 The Comoving Distance sn the omoving speD the glxies hve (xed oordintes nd the comoving distance τ * etween the oserver nd the soure of light t redshift zF sing 1.9 Hubble diagram and measure of distances pigure IFVX he rule digrm of the upernove smples with the xvD hD r nd severl lowEz surveysF greditX fetouleD wF et l PHIRF iqs IFUD IFPQ nd IFPSD one otinsX

1 :

 1 Basics of Observational Cosmology

2. 5 . 2

 52 Simulation technique e simultion tht desries smple of ys in osmologil model @λ • D Ω • AD depends on pprent mgnitude m lim nd luminosity funtion f (M ) @usully qussin funtionD see ppendix eFIF sf the smple is omplete up to pprent mgnitude m lim D then the seletion funtion redsX φ(m) = θ(m limm) @PFTA QI Chapter 2 : Quasars samples where θ stnds for reviside funtionF por given threshold m lim D soure with solute mgnitude M is visile up to redshift z max D vlue tht stis(es the following equtionX

Contents 3 . 1

 31 Introduction . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Weighting factors . . . . . . . . . . . . . . . . . . . . 40 3.3 Luminosity function and selection function . . . . . 44 3.4 Kolmogorov-Smirnov test . . . . . . . . . . . . . . . . 46 3.5 The V/V max test . . . . . . . . . . . . . . . . . . . . . 47 3.5.1 Calculation of the V/V max terms . . . . . . . . . . . 48 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 50

3. 1

 1 Introduction he null orreltion tehnique ws formulted y rF plihe 8 tF wF ouriu IWUW for qusr sttistis nd lter dpted for the rightest luster glxies y riyD F et lF IWWHF o illustrte this tehniqueD let us desrie simple exmple of liner orreltion with uniform distriution on the yExisD nd qussin distriution on the xExis with men 0 nd stndrd devition 0.5F QW Chapter 3 : The null correlation technique he proility density funtion of the qussin distriutionX

3. 1

 1 Introduction pigure QFIX he distriution of the smple in oth (x, y) nd (X, Y ) frmesF pigure QFPX erhing the zero vlue of the weighted ovrine of X nd Y F he eqution Γ(θ) = 0 shows two solutionsD the (rst one @θ = π 4 A mthes the one used for simultionF he rule digrm of soures t osmologil distnes shows orreltion etween the two oservlesD the redshift z nd pprent mgnitude mD whih re not linerly dependentF RI Chapter 3 : The null correlation technique

F 3 :

 3 For the case of the open Universe κ 0 < 0: he sme lultion for negtive κ 0 D e otinX RQ Chapter The null correlation technique ithout the kEorretion termX

  pinllyD with the im of ensuring the estimtion of the λ • nd Ω • without depending on prtiulr hoie of the mgnitudeD the weighting ftor is multiplied y ritrry funtions of M nd mF hese funtions void the onE entrtion of ll the weight on the ner ojetsF e hve hosen the following pigure QFRX he inverse of the orreltion funtion versus redshift for given osmologil modelF funtion in prtiulrX 10 β 5 [m-M ] F hereforeD the weighting ftor is lulted y the following formulX w β (z

Chapter 3 :

 3 The null correlation technique pigure QFSX he perentge of the lost of the sttistil informtion @leftX for L 1 D rightX for L 0 A with respet to the prmeter βF pigure QFTX he weighting ftor with respet to the redshift for given osE mologil modelF λ • ) digrm with zero weighted ovrineY ouned models re expliitly exludedF he sttistis depend on dt (m k , z k ) nd re lulted in eh vertex of grid (Ω i • , λ j • ) i,j=0..n ording to iqsFIFSHD QFPI nd QFWF uh proedure seprtes the domin in two prtsD positive ovrine nd negtive ovrine on the (Ω • , λ • ) digrmD see pigFQFUF he roots orresponding to zero weighted ovrineD loted etween the negtive nd positive vluesD re determined y men of horizontl nd vertil proli interpoltionF he set of solutions de(nes the null orreltion urve @xggAX the candidate values of the cosmological parametersF pigFQFU shows the xgg when using the simuE 3.2 Weighting factors ltion smple with the stndrd modelF xote tht the red irle on the urve orresponds to this modelD wht ensures the vlidity of this methodF pigure QFUX veft to rightD top to ottomF he vlues of the weighted ovrine with respet to osmologil modelF he null vlues of the xgg re found y horizontl nd vertil interpoltions in the ovrine gridF he xgg in the (Ω • , λ • ) plne retrieve the initil simultion model @the (lled red irleAF e me y simultions to the onlusion tht the xgg depends on the osE mologil model nd the smple feturesF imultion smples were performed y ssuming the stndrd model (Ω • = 0.3, λ • = 0.7) nd qussin lumiE nosity funtion g G (M ; M 0 = -20, σ M = 0.6) with di'erent limiting pprent mgnitude @m lim = 26D m lim = 24A nd di'erent smple sizes N F pigFQFV shows two xggs whih ontin the vlues of osmologil prmeters used to simE ulte the smplesF eny ndidte model hosen on xgg whih is di'erent thn the one used to simulte the relted smpleD hs di'erent xggF he shpe of the xgg depends strongly on the hosen smpleF pigFQFW shows the RU Chapter 3 : The null correlation technique xgg @red urveA orresponding to simulted smple in the model (Ω • = 0.1, λ • = 1.2)F reneD we hose the model (Ω • = 0.19, λ • = 1.4

Chapter 3 :

 3 The null correlation technique pigure QFIHX he preision of the method is shown oviously y inresing the size of the smpleF prom left to right nd from top to ottomD the size of the smple inreses from QS to QSH to PHHH respetivelyF e see tht the sttistil )ututions derese with the inresing of the numer of ojetsF por su0iently lrge numer of ojetsD F w (M ) nd Φ w (m) n e pproxiE mted y stepEfuntions F st w (M ) nd Φ st w (m) s followsX

3. 4 2 M

 42 Kolmogorov-Smirnov test pigure QFIPX imilr to pigFQFII ut with qumel distriution for the lumiE nosity funtionF pigure QFIQX hree df testing uniform distriution with uolmogorovEmirnov testF efterwordsD we ompute nlytilly the df F w (M ) de(ned in iqFQFPTF he mximum devitions etween the ontinuous df F w (M ) nd the empiril df F st w (M ) re omputed s followsX D max = M ax(|F w (M ) -F st w (M )|) @QFQQA sf qussin hypothesis for f (M ) is likely iFe the null hypothesis is epted y the u test for signi(ne levelD one n estimte the hrteristis of the qussin shpe of f (M )X he stndrd devition σ M whih is de(ned in iqFQFQP nd the men M 0 X M 0 = M + β ln (10) 5 σ @QFQRA he ssumption of the ompleteness of the smple up to the limiting pprent mgnitude m lim leds to tehnologil funtion φ(m) like reviside funtion SQ Chapter 3 : The null correlation technique θ(m limm)F he lultion of iqFQFPU using iqFQFPS gives us the following theoretil df of φ w (m)X Φ w (m) = 1 + θ(m limm)(10 -β(m lim -m) 5

Chapter 4 :

 4 Application to quasars data pigure RFRX The absolute magnitude-volume diagram of the EDR sample. The left panel with Ω • = 0.076, λ • = 1.211. The right panel with Ω • = 0.3, λ • = 0.7. The domains corresponding to limiting magnitudes i = 15, 19.1, 20.5 and 21.78 are delimited by the corresponding curves (yellow, blue, green and black). The second ordinate axis represents the redshift values.

644. 3

 3 Results pigure RFSX imilr to pigFRFRX the solute mgnitudeEvolume digrm of the hI smpleF pigure RFTX imilr to pigFRFRX the solute mgnitudeEvolume digrm of the hQ smpleF TS Chapter 4 : Application to quasars data pigure RFUX imilr to pigFRFRX the solute mgnitudeEvolume digrm of the uniform sttistil susmple of the hQ @fhQAF pigure RFVX imilr to pigFRFRX the solute mgnitudeEvolume digrm of the hEhU smpleF 66 4.3 Results pigure RFWX The absolute magnitude-volume diagram of a simulated sample which consists of 46,420 objects with a Gaussian luminosity function with M 0 = -25, σ M = 1.2, a limiting magnitude m lim = 19.1, and a maximum redshift z max = 13.6 in the model Ω • = 0.076, λ • = 1.211.

Chapter 4 :

 4 Application to quasars data pigure RFIHX The volume-redshift diagram in the model Ω • = 0.076, λ • = 1.211. The graduation on the ordinate is translated into angular distance on the right hand side axis.

684. 3

 3 Results pigure RFIIX The distance modulus ζ(z)-redshift diagram in the model Ω • = 0.076, λ • = 1.211, showing the angular distance on the right hand side axis. The maximum of ζ(z) corresponds to τ = 2π 3 .

  shpe of the urve is due to the ontriution of the emission linesF e strt y pplying the test to fhQ smple using the orresponding xgg @the yellow urve in pigFRFQAF he results for eh susmple re given TW Chapter 4 : Application to quasars data s followsX 18.65 ≤ m ≤ 19.1, 0.04 ≤ z ≤ 2.45 : Ω • = 0.132, λ • = 1.275D t QS7 signi(ne levelF 18.65 ≤ m ≤ 19.1, 1.8 ≤ z ≤ 2.4 : Ω • = 0.076, λ • = 1.211D t UH7 signi(ne levelF hey re shown in pigsFRFIR nd RFISF he overdensity in the top of the p digrm of fhQ t redshift ≈ 3 is notiele in pigFRFIQF sndeedD this my n upset the result of the null orreltion test nd then the V /V max testF por thisD we selet susmple of fhQ limited t n pprent mgnitude 19.1D whih ontins IQVPV ysD getting rid the pkge in topF hereforeD we pply the null orreltion test on this susmple whih provids us xgg @in yn olorA in pigFRFQF reneD we perform the V /V max test on susmple limited yX 18 ≤ m ≤ 19.1, 1.8 ≤ z ≤ 2.4 : Ω • = 0.0305, λ • = 1.1180D t UU7 signi(ne levelF detils out these results is presented in etFRFREQFQF e selet susmple of hU in the following dominX m min = 18.65D m max = 19.1D z min = 1.8D z max = 2.4F

4. 3 4 :

 34 Resultspigure RFIQX e represent the pprent mgnitude limits on the p digrm of the homogeneous sttistil susmpleD fhQD hosen to pply the V /V max X 18.65 < m < 19.1 pigure RFIRX he df of V /V max ompred to uniform distriutionD for fhQ susmple with Ω • = 0.076 nd λ • = 1.211F UI Chapter 4 : Application to quasars data pigure RFISX histne etween the empiril df nd theoretil df @D max AD testing uniform distriution of V /V max vluesD for ndidtes osmologil models on the xgg orresponding to fhQF veft pnelX D max s funtion of λ • F ight pnelX D max s funtion of Ω • F sn oth pnelsD the vlues of D max on the ordinte xis is trnslted into signi(ne level on the right hnd side xisF pigure RFITX he df of V /V max ompred to uniform distriutionD for hU susmple with (Ω • = 0.0169 nd λ • = 1.079)F 4.3.3 Estimation of the luminosity function and the selection function he null orreltion pproh enles us to evlute the luminosity funtion f (M ) nd the seletion funtion φ(m) @for detils of lultion see etFQFQAF he osmologil prmeters used for these estimtions re @Ω • = 0.0305D λ • = 1.1180AF he luminosity funtion shows pproximtely qussin for the 4.3 Results pigure RFIUX imilr to pigFRFIS X D max with respet to λ • @leftA nd (Ω • @rightA s ndidtes given y the null orreltion urve of hUF pigure RFIVX gontriution of mesurement errors to xgg orresponding to hQ rightest ojetsD whilst it eomes onstnt for the fintest ojets @pigFRFIW represents f (M ) of the susmple of fhQ limited t 19.1 of pprent mgnitudeF por the other smplesD see the etFRFREQFRAF he ompleteness of the smple t limited pprent mgnitude is estimted in the simultion with WS7 signi(neF es point of ftD this is not suitle for the entire hosen smpleF e found tht the funtion φ(m) otined from the h dt inreses s n exponentil funtion for the rightest ojets @qusrs re very rre t low redshiftAD while it lusters round φ(m) = 1.4 from IVFTS to IWFI of pprent mgnitudesF sn this rnge of mgnitudeD the ys re esily identi(ed in the i ndD thnks to the vyα emission line whih is more likely to e oserved in this ndF eove the (rst limit iaIWFID the funtion φ(m) UQ Chapter Application to quasars data dereses quikly s the de(ieny of ys is importntF futD φ(m) eomes onstnt until rehing the seond pprent mgnitude limit i = 20.2 @pigFRFPH represents φ(m) of the susmple desried oveF por the other smplesD see the etFRFREQFRAF he shpe of the φ(m) suggests tht the h smples re onstituted of t lest two pprent mgnitude limitsD nd of regions where the emission lines of ys re esily identi(ed in the i ndF he ottom pnel represents φ(m) with n inresing shpe until iaIVFTSF his funtion eomes onstnt nd lose to φ(m) = 1.4 up to the limit i = 19.1 where it strts to derese t this limitF he top pnel represents the zoom on φ(m) etween the rnge from 17.9 to 19.1 where the funtion egin to e lose to IF his trend is due to the rre right ys @the nery ysA nd to the esy identi(tion of ys in this rngeF pigure RFIWX The luminosity function f (M ) of the subsample of SUBDR3 with (Ω • = 0.0305, λ • = 1.1180).

  Statistical analysis in observational cosmology is faced to the problem of selection eects, which makes the derivation of unbiased inferences a major challenge, e.g. see the Malmquist bias [1416, 20], among others. One of the interesting features of the null correlation technique is indeed to contribute in solving such problems in the determination of cosmological parameters.

1 4. 4

 14 Inferences on cosmological expansion based on QSOs UU Hubble law by means of weighting factors. Then, we remind the denitions of statistics which provide us with estimates of the luminosity function and of the selection function, for cosmological models that agree with the working hypotheses, and how to test them by means of the FS diagram. Then, we show the appropriate use of the V /V max test for rening the estimation of cosmological parameters. A comparison to the usual tting technique in Hubble diagram, shows up the benets of this approach. With respect to simulation techniques, an

4

  When reasoning in the conformal spacetime diagram, this property becomes obvious since the past light cone of the observer intercept the worldlines of objects at the same angle of 45 • .

3 4. 4 2 . 3

 3423 Inferences on cosmological expansion based on QSOs UW Probing the geometry of spacetime with weighting factors According to eq.(2.9), the probability density of random variables (M, m) can be written as the random variables (M, m) are correlated, because of the correlation fonction ∂V ∂m . Hence, by weighting the data with the function ω β (z) = ∂V ∂m variables (M, m) become independent for the true values of cosmological parameters, which stands for the principle of the null correlation test. For a sample {(z k , m k )} k=1,N , one denes the normalized weighting factors as follows

(2. 15 ) 7 Γ( 2 . 16 ) 6

 1572166 has a vanishing expectation for the true values of cosmological parameters. Hence, the candidate model is dened by those values that satisfy the equation (Ω • , λ • ) = 0 It can be also the relative dierence ∆ω = ωmax-ω min ωmax+ω min .

Figure 1 .

 1 Figure 1. The weighting factors (scaled values by a factor ×N ) for the SDSS-DR3 sample without K-correction (upper panels) and with K-correction (bottom panels), by assuming (Ω • = 0.132, λ • = 1.275) (left panels) and (Ω • = 0.3, λ • = 0.7) (right panels).

5 4. 4

 54 Inferences on cosmological expansion based on QSOs VI as a product of two probability density functions, namely

  addition of bumps related to the K-correction term, see gure 3. In the FS diagram, these data are located to the left hand side of the curve C : (M max (z), V (z))(2.22)

Chapter 4 :

 4 Application to quasars data 82 shows such a property solely within a redshift range of values, this test applies to the related subsample.

  25) is uniformly distributed 11 within the range [0, 1]. The likehood of this assumption can estimated by the Kolmogorov-Smirnov (KS) test to the sample of values

7 4. 4

 74 Inferences on cosmological expansion based on QSOs VQ One can easily prove that the V /V max test can be applied to sub-domains dened in apparent magnitude m ∈ [m min , m max ] and/or in redshift z ∈ [z min , z max ]. Moreover, to avoid making inferences on the formation epoch of sources, one can use z max corresponding to the largest redshift value in the sample, which is particularly useful if κ • > 0.

  bolometric magnitude), what requires a composite spectrum of sources. Unlike the FS diagram, the Hubble diagram (m, z) the coordinates do not depend on cosmological parameters, which facilitates the identication of selection eects in observation. The data are expected to lie around a model dependent curve that gives the apparent magnitude of a standard candle as a function of redshift. The goal is to t that curve to data by means of a least squares method, for estimating the values of cosmological parameters. To perform such an approach, one is faced to the arbitrary choice of a distance between the data to their the predicted values, which often results from heuristic method with subjective reasoning. Moreover, in order to perform properly the Malmquist bias correction, one has to specify the selection function φ(m) and the luminosity function f (M ). Because of these requirements, the null correlation technique appears then more robust in testing the working hypotheses. Moreover, the uniformity on the spatial distribution of sources, as required by FLG models, is also tested. The crucial dierence with the Hubble diagram is that the eq. (2.16) does not have necessarely a solution if the data do not fulll the required properties that are induced by the working hypotheses.

( 2 .

 2 31)For each object, two random values g k and v k are generated, both uniformly distributed within the interval [0, 1]. By inversing the functions given in eqs.(2.31) and (2.5), one obtains successively

9 4. 4 Figure 2 .

 942 Figure 2. The FS diagram for the SDSS-III:DR9 sample by assuming Ω • = 0.3 and λ • = 0.7.

Figure 3 .

 3 Figure 3. The K-correction of the SDSS-DR3 sample in the i-band, including both the emission-line and continuum components, normalized at z = 2 with a xed spectral index α ν = -0.5.

Figure 4 .

 4 Figure 4. The null correlation curves in the Ω • -λ • diagram corresponding to samples : EDR, DR1, DR3 and DR7. They provide us with the candidate values of cosmological parameters which agree with the working hypotheses. The spatially closed cosmological models κ • > 0 are located above the straight line, that corresponds to κ • = 0, where the standard solution is represented by a yellow dot. The bottom curve corresponds to the EDR sample, these models are in complete disagreement with those derived from the other samples.

Figure 5 . 13 4. 4 Figure 6 .

 51346 Figure 5. The FS diagram for the DR3 sample (upper panels) and the DR7 sample (bottom panels), by assuming (Ω • = 0.076, λ • = 1.211) (left panels) and (Ω • = 0.3, λ • = 0.7) (right panels). The curves correspond to the apparent i-magnitudes 15 (yellow), 19.1 (blue), 20.5 (green) and 21.78 (black). The graduation on the ordinate is translated into redshift on the right hand side axis.

Chapter 4 :4. 5 4. 5 5 Application to supernovae type Ia sample Contents 5 . 1

 455551 Application to quasars data 92 Conclusion pigure RFPHX The selection function φ(m) of the SUBDR3 subsample with (Ω • = 0.0305, λ • = 1.1180). The upper panel is an enlargement of the area in which the function is most constant. Conclusion he pplition of the null orreltion nd V /V max tests to y smples of the h survey llowed us to disriminte the ndidtes vlues for the osE mologil prmeters tht gree with oservtionsF e found model with positive slr urvture with wek presene of drk mtterF WQ CHAPTER Introduction . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 Type Ia Supernovae as standard candles . . . . . . . 82 5.2.1 The lightcurve of type-Ia SN . . . . . . . . . . . . . 83 5.2.2 Standardisation of type-Ia SN . . . . . . . . . . . . . 84 5.2.3 Modelling of the Supernova event . . . . . . . . . . . 86 5.2.4 Sampling the light curve of type-Ia SN and selection eects . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2.5 Calibration statistics . . . . . . . . . . . . . . . . . . 90 5.2.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 The null correlation test on the supernova sample . 98 5.3.1 Luminosity function and Selection function . . . . . 100 5.3.2 Precision and error . . . . . . . . . . . . . . . . . . . 101 5.4 Application to the JLA sample . . . . . . . . . . . . 101 5.4.1 Description of samples . . . . . . . . . . . . . . . . . 102 5.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.1 Introduction ristoril reords of supernove hve een found over the pst two thousnd yersF he (rst x in our own glxy ws oserved nd reorded y the ghiE WS Chapter 5 : Application to supernovae type Ia sample

Chapter 5 :

 5 Application to supernovae type Ia sample pigure SFVX gorreltion etween the solute mgnitude with the deline rte ∆m 15 of the lightEurve for lowEredshift x s from the glnEololo survey @pigure ourtesyX rmuy et lF IWWTA distriutionX F SN (p, λ) = x 0 × [S 0 (p, λ) + x 1 S 1 (p, λ) + ...] × e [c×CL(λ)]

5. 2 . 4 5 :

 245 Sampling the light curve of type-Ia SN and selection eects elthough the xe re known to e used s stndrd ndlesD they ould e 'eted y lrge unertinties due to the limittions in x oservtionF imE ilrly to ysD the seletion e'ets must e tken into ount in deriving inferenesD eFgF the one responsile of the wlmquist isF roweverD in this IHQ Chapter Application to supernovae type Ia sample pigure SFWX veftX shemti of intrinsi light urve of xF ightX the intereption of the worldEline of the x with the pst light oneF respet x di'ers from yD minly euse it is not perenilF

5 :

 5 whih mkes righter the ojet the luerF reneD ording to iqFSFVD the proility density of oserved x trnsformsX ∝ φ B (m * ) g G (c; 0, σ c )dc dP th @SFIHA 5.2.5 Calibration statistics he lirtion of the frighterElower @fA reltion ssumed s liner orreE ltion etween M * nd x 1 s desried y the proility density tht redsX ∝ g(M * , x 1 )dM * dx 1 @SFIIA e ssume tht the error in estimting M * from x 1 D s given yXM * = ax 1 + b + @SFIPAis rndom vrile with qussin distriution funtion g G ( ; 0, σ )D see pigFSFIRF he estimtion of prmeters a nd b @lirtionAD n e performed either X 106 5.2 Type Ia Supernovae as standard candles pigure SFIHX vight urve of x xvEHRhQgxF his (gure shows light urve well smpled y oservtions whih help to hrterize its prmeters (x 0 , t * obs , x 1 , c) without isesF pigure ourtesyX estier F et lD PHHT with the diret righterEslower reltion @hfAX g(M * , x 1 )dM * dx 1 ≈ g M * (x 1 )dx 1 g G ( ; 0, σ DBS )d @SFIQA where X M * = a DBS x 1 + b DBS + @SFIRA or with the indiret righterEslower reltion @sfAX g(M * , x 1 )dM * dx 1 ≈ g x 1 (M * )dM * g G ( ; 0, σ IBS )d @SFISA where X x 1 = a IBS M * + b IBS + @SFITA he estimtes of orresponding prmeters re denoted a DBS D b DBS D σ DBS D a IBS D b IBS nd σ IBS F es usulD we proeed with the mximum likelihood @wvA tehnique s deE sried in riy et l IWWRF ith nery ojets used for lirtionD we n sfely ssume uniform diestriution in n eulidin speF reneD the proility density tht desries the lirtion smple redsX ∝ φ B (m * )g(M * , x 1 )dM * dx 1 g G (c; 0, σ c )dc dt * e γζ dζ @SFIUA IHU Chapter Application to supernovae type Ia sample pigure SFIIX e representtive (gure of the x s light urves in the im to explin the seletion e'etsF he dshed urve represents the mster urve where the entire light urves must oinideF he highest lue urve represents the light urve of the x s righterD roderD slower nd luer thn the mster urveF he yn urve @the lowestA represents the light urve of the x s finterD nrrowerD fster nd less luer thn the mster urveF he t 1 nd t 2 re two oservtion dtes for whih ndidte of x is deteted y sutrting the imges tken t these dtesF where γ = 3ln(10) 5

  dc dt * e γζ dζ × g G ( ; 0, σ )d @SFIWA pigure SFIPX he seletion of xe depends on two hrteristisX the mxiE mum of luminosity m * nd the deline rte ∆m 15 F he lowest light urve is the threshold light urve hrterized y m * lim nd ∆m 15,lim whih re used s the seletion riteriF e x with deline rte ∆m 15 `∆m 15,lim is oserved s long s its pprent mgnitude t mximum luminosity m * `m * lim F

Chapter 5 :

 5 Application to supernovae type Ia sample pigure SFIQX wo xe with the sme intrinsi hrteristis t di'erent redE shiftsF elthough the light urve of the x of the highest redshift @z = 1.5A hs smll deline rte ∆m 2 15 `∆m 15,lim due to the diltion of timeD this x will not e onsidered in the smple euse it hs not su0ient time to hve good smpling of its light urveF he seletion of x is done only if m

3 ,

 3 σ c = 0.085 β = 3.14, σ = 0.6 a IBS = -5, b IBS = -95 M * c,0 = -19, m * lim = 25.04 @SFQWA es expetedD the p digrms in pigFSFIS show tht the distriutions re upper ounded y the urve V max (M * )D hrterizing the limiting pprent mgniE tude m * lim D uniformly distriuted with respet to V D nd qussin with respet to M * c F st n e noted tht righter ojets hve more lue olorF he ruE le digrms s shown in pigFSFIT provide us with dt distriution out the expeted reltion for stndrd ndel with M * c = -19 @color = 0AF he seE letion e'et t the limiting pprent mgnitudeD m * lim = 25.

5. 3

 3 The null correlation test on the supernova sample pigure SFIUX pour dimensionl digrms of hrteristis of smple @eAF reneD the weighting ftor redsX w µ (z

Chapter 5 :

 5 Application to supernovae type Ia sample pigure SFIVX pour dimensionl digrms of hrteristis of smple @fAF pigFSFIW shows the xggs of the two simulted smples desried oveF es expetedD the vlues of the osmologil prmeters used to generte the smE ples re roots of the sttisti Γ(Ω • , λ • ) = 0F st will e seen tht the di'erenes in shpe @eFgFD the urvtureA of the xggs orresponding to these smplesD otined ording to di'erent lirtionsD re of rndom nture nd do not re)et the lirtion typeF

Chapter 5 :

 5 Application to supernovae type Ia sample pigure SFPIX xgg of ten rndomly generted smples with the sme hrE teristis in the stndrd osmologil modelF pigure SFPPX erh for the minimum of u testD D max D @top leftA s funtion of λ • D nd @ottom rightA s funtion of Ω • D on the xgg @top rightAF 5.3.1 Luminosity function and Selection function yne of the dvntges of the null orreltion pproh is to estimte the luE minosity funtion nd the seletion funtionF ine the olor is determined y mens of evP routineD the smple n e desried y the weighted proility density tht redsX dP ω ∝ f ω (M * c )dM * c φ B,ω (m * )dm * c @SFRVA vet us emphsize tht φ B (m * ) stnds s orreltion funtion of c nd m * c D sine m * = m * c -βcF he proility density funtions reds s produt of two independent funtionsD the olor orretion eing performed

  where θ is the reviside step funtionF hereforeD the derivtives of Fω (M * c ) nd ΦB,ω (m * cβc) llow to estimte the luminosity funtion nd the seletion funtion f (M * c ) SFPR show the estimtion of the funtions f (M * c ) nd φ B (m * c -βc) with simulted smple in whih f (M * c ) is qussin funtion with n verge M * c,0 = -19 nd stndrd devition σ M * c = 0.25F hese sttistis hve een vlidted y tens simulted smplesD the (gures show tht the ssumed pro(les of funtions re retrievedD they hve not een smoothed from sttistil )uE tutions nd those due to the numeril derivte methodD humn eyes eing more e0ient estimtors thn utomti methodsF 5.3.2 Precision and error sn order to estimte the error udget round the est model of the osmologil prmetersD we performed simultion of SHH smples using the rteristis IIW Chapter 5 : Application to supernovae type Ia sample pigure SFPQX ttistis of the luminosity funtion @upper pnelsA nd the seE letion funtion @lower pnelsA pigure SFPRX imilr to pigFSFPQ listed in the @fA se of iqFSFQWF ih smple onsists of SHH x eventsF pirstlyD we pplied the null orreltion test to the SHH smples nd we determined their xggsD see pigFSFPSF sn this (gureD we n see the e'et of the sttistil )ututions for the determintion of the true osmologil modelF henD we performed the V /V max test whih provided us set of models onsidered s est estimtes t given signi(ne level @etween VT7 nd WU7AF hereforeD we split the re of models in squres @IH x IHA nd we determined the density of points in eh squreF his llowed us to determine the proility density funtion on the Ω • nd λ • xesD see pigFSFPTF es resultD we otinedX Ω • = 0.3 ± 0.03D λ • = 0.7 ± 0.01 t TV7 on(dene levelF 120 5.3 The null correlation test on the supernova sample pigure SFPSX he xggs of the SHH simulted smplesF pigure SFPTX et of the est ndidtes of the osmologil model with the likelihood ontours @top leftAF he proility density funtions with respet to λ • @top rightA nd with respet to Ω • @left ottomAF IPI Chapter 5 : Application to supernovae type Ia sample 5.4 Application to the JLA sample

5. 4

 4 Application to the JLA sample pigure SFPUX he p digrm of tve smple uild with the modelX Ω • = 0.3, λ • = 0.7F he red dots represent the hEss dtD nd the lue ones represent those of xvQF he rest is represented with multiolorF with zero slr urvture @we used Ω • = 0.3, λ • = 0.7AD it will e more resonE le to use smple t low redshift where the e'et of osmologil prmeters n e ignoredF por thisD we rried out the lirtion for the lowEz smple whih ontins ojets up to z a HFHVF e otinedX a = -0.138D b = -19.043 nd β = 2.641F he orreltion etween the vriles for xvQ smple is shown in pigSFQPF

Chapter 5 : 5 . 4 5 . 4

 55454 Application to supernovae type Ia sample pigure SFPVX he p digrm of r smple uild with the model @Ω • = 0.3, λ • = 0.7AF pigure SFPWX he p digrm of lowEz smple uild with the model @Ω • = 0.3, λ • = 0.7AF 124 Application to the JLA sample pigure SFQHX he p digrm of hEss smple uild with the modelX Ω • = 0.3, λ • = 0.7F pigure SFQIX he p digrm of xvQ smple uild with the modelX Ω • = 0.3, λ • = 0.7F IPS Chapter 5 : Application to supernovae type Ia sample pigure SFQPX imilr to pigFSFIU for xvQ smple with lirtion of stnE drdiztion oe0ients with respet to the lowEz smpleF pothesesF yn the other hndD the xgg hs een determined with the xvQ smpleD s shown in pigFSFQQF pigure SFQQX he null orreltion urve in the λ • -Ω • digrm orresponding to xvQ smpleF his xgg is oviously oherent with the results otined with the hEss y smplesD whih fvor osmologil model with positive urvture @see pigFSFQRAF 126 Application to the JLA sample pigure SFQRX he null orreltion urves in the (Ω • , λ • ) digrm tht orreE spond to y smples nd xvQ smple @the highestAF he pplition of the V /V max test on the xgg provides us with osmoE logil prmeters estimtes (Ω • = 0.11, λ • = 1.32) t RH7 signi(ne level @pigFSFQSAF st must e noted tht euse of the wek smple sizeD we hve not seleted susmples omplete to limiting pprent mgnitude for performE ing properly the V /V max testD ut we did to the entire smpleD whih redues inevitly the signi(ne levelF pigure SFQSX D max with respet to λ • @leftA nd Ω • @rightA s ndidtes given y the xgg of xvQF IPU Chapter 5 : Application to supernovae type Ia sample pigsFSFQTD SFQU show the p digrms for hEss nd xvQ smplesDy ssuming these vluesF st is ler tht the xvQ smple ppers more uniE formly distriuted in Exis thn the stndrd oneD see pigFSFQI with this osmologil model (Ω • = 0.3, λ • = 0.7)F pigure SFQTX he p digrm of hEss smple uild with the model (Ω • = 0.11, λ • = 1.32)F fy using the estimtion method for the luminosity funtion nd seletion funtionD desried in etFSFQFID we otined lmost qussin luminosity funtion for the xvQ smpleD see pigFSFQV @left pnelAF he verge nd stndrd devition σ M * c is estimted using iqFSFRSF his yields n verge M * c,0 ≈ -19.139 nd σ M * c ≈ 0.19F pigFSFQV @right pnelA shows tht the seleE tion funtion φ B (m * ) is omptile with reviside funtion up to mgE nitude of PR nd deresesF his result gives evidene of the di0ulties in smpling the light urve of fint mgnitude ojetsF e determined the error on the the osmologil prmeters y perfomE ing perturtion of the pprent mgnitude with qussin white noise hrterized y stndrd devition of HFHQF hereforeD we found the orreE sponding xggs nd we pplied the V /V max testF e performed this proess for PHH timesF he otined ndidtes re determined within QS7 nd RQ7 signi(ne levelF pigSFQW shows the results of the tests within their error inE 128 5.4 Application to the JLA sample pigure SFQUX he p digrm of xvQ smple uild with the model (Ω • = 0.11, λ • = 1.32)F pigure SFQVX vuminosity funtion @left pnelA nd seletion funtion @right pnelA uilt with the model (Ω • = 0.11, λ • = 1.32)D orresponding to xvQ smpleF tervls with the proility density funtionsF his work rought the resultX Ω • = 0.13 ± 0.02D λ • = 1.34 ± 0.01F IPW Chapter 5 : Application to supernovae type Ia sample pigure SFQWX imilr to pigFSFPT 5.5 Conclusion

1 Chapter 5 : 1

 151 Application to supernovae type Ia sample La cosmologie moderne . . . . . . . . . . . . . . . . . 115 6.2 Simulation d'échantillon de quasar . . . . . . . . . . 118 6.3 La technique de corrélation nulle . . . . . . . . . . . 120 6.4 Résultats avec les données quasars . . . . . . . . . . 121 6.5 Modélisation d'échantillon de supernova et résultats sur les données de SDSS-II/SNLS3 . . . . . . . 122

6 :

 6 est le module de distne qui dépend des prmètres osmologiquesF v meilleure mesure ou estimtion de ette dernière distne représente l lé pour ontrindre les perles de digrmme de ruleD FàFd les prmètres osmologiques λ • et Ω • F IQU Chapter Résumé en français pigure TFIX ve digrmme de rule des éhntillons de upernove ve xvD hD r et plusieurs sondges lowEzF pigure tirée de fetouleD wF et l PHIR 6.2 Simulation d'échantillon de quasar

  0 donne les vleurs ndidtes de Ω • et λ • D u moyen de l oure de orréltion nulle dns le pln (Ω • , λ • ) @pigFTFQAF 6.4 Résultats avec les données quasars pigure TFQX v oure de orréltion nulle dns le pln (Ω • , λ • )F v oure retrouve le modèle osmologiques utilisé pour e'etuer l simultion de l9éhntillonF

10 10 M ax 1 + 2 x 1 @ 6 . 5

 12165 bβc + δ otherwise @TFPHA où δ est le oe0ient de orréltion entre l mgnitude solue et l msse stelE lire de l glxie hôte M a stellar ullivnD wF et l PHIHD gonley et l PHIIF ve oe0ient de stndrdistion hôteEmsse est évlué dns fetouleD wF et l PHIR pour l vleur δ = -0.070 ± 0.023 en utilisnt l9nlyse de l9éhntillon tve @pour toint vightEurve enlysisA qui se ompose de URH xeF ves prmètres D et σ sont lulés selon les formules suivntesXa = Cov(x 1 , M * ) σ TFPIA σ = σ M * 1ρ 2 (x 1 , M * ) @TFPPA où ρ(x 1 , M * ) estle oe0ient de orreltionF b = M *a x 1 + γσ 2 @TFPQA Modélisation d'échantillon de supernova et résultats sur les données de SDSS-II/SNLS3 ves mêmes étpes de lirtion sont e'etuées pour déterminer les prmètres de orréltion linéire dns le pln @M * c D AF xous vons dpté le test de orréltion nulle pour un éhntillon de superE novF xous vons ppliqué e test sur les éhntillons de hEss et xvF yn trouvé que l9éhntillon de x de hEss n9est ps pproprié ve les hyE pothèses de trvilF eutrementEditD le digrmme mgnitude solueEvolume ne montre ps une distriution uniforme sur l9xe de volumeD on remrqué un struture et une onentrtion sur e digrmmeF r onséquentD le test de orréltion nulle ve et éhntillon n9 ps trouvé des ndidts pour les prmètres osmologiques @FàFd on n9 ps trouvé l oure de orréltion nulleAF in revnheD l9pplition de e test à l9éhntillon de xv nous permis d9otenir un ensemle de ndidts de modèles osmologiques où les données s9ordent ve les hypothèses de trvilF v oure de orréltion nulle otenu ve l9éhntillon de xv onforte les résultts otenus ve les éhntillons de ys de h @pigFTFSAF pigure TFSX ves oures de orréltion nulles dns le digrmme (Ω • , λ • ) qui orrespondent ux éhntillons y et à l9éhntillon xv @le plus élevéAF hns e trvilD nous vons e'etué une nouvelle modélistion d9un éhnE tillon de supernov du type s en tennt ompte de l orréltion entre l mgE nitude solue et le prmètre de forme de l oure de lumièreF v9pplition du test de orréltion nulle à l9éhntillon de xv on(rmé un modèle osmologique ve une ourure positive et une présene file de l mtière noireF IRS APPENDIX B Calculation of the weighting factor ith iqFIFQUD one omputes the weighting ftors depending of the sign of the redued urvtureD nd the intermedite results re given s follows por the se of the lose niverse κ • b HX

H_0 = 7

 7 0 . 0 #Km. s -1.Mpc-1 L i g h t V e l o c i t y = 2 . 9 9 7 9 2 4 5 8 * 1 0 * * 5 . 0 # Km. s-1 c l a s s ApparentMagnitude : ISS Chapter D : Code Python d e f __init__ ( s e l f , z , lambda_0 , omega_0 , M, Kcorrection , redsh_k , k_z ) : s e l f . K c o r r e c t i o n = K c o r r e c t i o n s e l f . redsh_k = redsh_k s e l f . k_z = k_z s e l f . param = Functions ( lambda_0 , omega_0 ) s e l f . z = z s e l f .M = M s e l f . cosmoModel = Model ( lambda_0 , omega_0 ) s e l f . d_L = s e l f . cosmoModel . c u r v a t u r e ( s e l f . z , lambda_0 , omega_0 , M) [ 2 ] s e l f . c o m o b i l e D i s t a n c e = s e l f . cosmoModel . c u r v a t u r e ( s e l f . z , lambda_0 , omega_0 , M) [ 0 ] #---------K-c o r r e c t i o n ---i f s e l f . K c o r r e c t i o n == ' Kcorrection_OK ' : dkcorr = [ ] f o r i i n range ( l e n ( redsh_k ) ) : i f i==0 or i ==(l e n ( redsh_k ) -1) : dkcorr . append ( k_z [ i ] ) e l s e : dkcor = ( k_z [ i +1] -k_z [ i -1]) /( redsh_k [ i +1] -redsh_k [ i -1]) dkcorr . append ( dkcor ) dkcorr = array ( dkcorr ) d_kcorr = [ ] k c o r r = [ ] f o r i i n range ( l e n ( s e l f . z ) ) : k_cor = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n (k_z , redsh_k , s e l f . z [ i ] ) k c o r r . append ( k_cor ) f o r i i n range ( l e n ( s e l f . z ) ) : d_k_cor = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( dkcorr , redsh_k , s e l f . z [ i ] ) d_kcorr . append ( d_k_cor ) #alfa_nu = -0.5 #s e l f . k c o r r = kcorr_con + kcorr_em s e l f . k c o r r = array ( k c o r r ) s e l f . d_kcorr = array ( d_kcorr ) #-

  e l i f s e l f . K c o r r e c t i o n == ' Kcorrection_NO ' : pass e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" #------------------------de f m_theor ( s e l f , z , lambda_0 , omega_0 , M) : i f s e l f . K c o r r e c t i o n == ' Kcorrection_OK ' : m_theorie = s e l f .M + 5 . * np . l o g 1 0 ( s e l f . d_L/ 1 0 . 0 ) + s e l f . k c o r r e l i f s e l f . K c o r r e c t i o n == ' Kcorrection_NO ' : m_theorie = s e l f .M + 5 . * np . l o g 1 0 ( s e l f . d_L/ 1 0 . 0 ) e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" r e t u r n m_theorie #===========Jacobian_1============ d e f Jacobian ( s e l f , z ) : i f s e l f . K c o r r e c t i o n == ' Kcorrection_OK ' : tau = s e l f . c o m o b i l e D i s t a n c e i f s e l f . param . kappa_0 == 0 . 0 : jac_1 = ( 1 . / ( tau * * 2 .

  kappa_0 ) ) / tanh ( e p s i l o n / 2 . 0 ) ) ) /( cosh ( e p s i l o n ) -1 . ) e l s e : pass e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" r e t u r n jac_1 #=======Weighting f a c t o r========== d e f W e i g h t i n g f a c t o r ( s e l f , z , beta ) : tau = s e l f . c o m o b i l e D i s t a n c e i f s e l f . K c o r r e c t i o n == ' Kcorrection_OK ' : i f s e l f . param . kappa_0 == 0 . 0 : weighing = s e l f . Jacobian ( z ) * ( ( ( 1 . + z ) * tau * 1 0 . * * ( s e l f . k c o r r ) ) ** beta )

  kappa_0 ) ) ) ** beta ) e l s e : pass e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" r e t u r n weighing d e f VOLUME( s e l f , M, zz , ZETA, m_l) : z_l = s e l f . param . N o n l i n e a r _ I n t e r p o l a t i o n ( zz , ZETA, m_l -M) i f s e l f . param . kappa_0 == 0 . 0 : mu = M + 5* l o g 1 0 ( ( ( L i g h t V e l o c i t y * 1 0 * * 6 . 0 ) / ( 1 0 . *H_0) ) ) i f s e l f . K c o r r e c t i o n == ' Kcorrection_OK ' : V = ( 4 . * p i / 3 . ) * ( ( 1 / ( 1 . + z_l ) ) * * 3 . ) * 1 0 * * ( 3 . * (m_l -mus e l f . k c o r r ) / 5 . ) ISW Chapter D : Code Python e l i f s e l f . K c o r r e c t i o n == ' Kcorrection_NO ' : V = ( 4 . * p i / 3 . ) * ( ( 1 / ( 1 . + z_l ) ) * * 3 . ) * 1 0 * * ( 3 . * (m_l -mu) / 5 . ) e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" e l i f s e l f . param . kappa_0 > 0 . 0 : mu = M + 5* l o g 1 0 ( ( ( L i g h t V e l o c i t y * 1 0 * * 6 . 0 ) / ( 1 0 . *H_0) ) ) i f s e l f . K c o r r e c t i o n == ' Kcorrection_OK ' :

  kappa_0 ) ) /(1 + z_l ) ) * ( 1 0 * * ( ( m_l -mu) / 5 . ) ) e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" V = ( p i / abs ( s e l f . param . kappa_0 ) * * ( 3 . / 2 ) ) * (2* a l f a _ l * s q r t (1 + a l f a _ l * * 2 . ) -2* a r c s i n h ( a l f a _ l ) ) e l s e : pass r e t u r n V c l a s s Model : d e f __init__ ( s e l f , lambda_0 , omega_0 ) : s e l f . param = Functions ( lambda_0 , omega_0 ) d e f c u r v a t u r e ( s e l f , z , lambda_0 , omega_0 , M) : zz = s e l f . param . LowerBound ( z ) c o m o b i l e D i s t a n c e = s e l f . param . Trapeze ( s e l f . param . F u n c t i o n t o I n t e g r a t e , zz , 1 . 0 ) lookback_time = s e l f . param . Trapeze ( s e l f . param . F u n c t i o n t o I n t e g r a t e 2 , zz , 1 . 0 ) i f s e l f . param . kappa_0 > 0 . 0 : e p s i l o n = c o m o b i l e D i s t a n c e * s q r t ( s e l f . param . kappa_0 ) volume = p i * ( 2 . 0 * e p s i l o ns i n ( 2 . 0 * e p s i l o n ) ) /( s e l f . param . kappa_0 * * ( 3 . / 2 . ) ) d_L = ( L i g h t V e l o c i t y * 1 0 * * 6 . 0 ) * ( 1 . + z ) * ( s i n ( e p s i l o n ) ) /(H_0* s q r t ( s e l f . param . kappa_0 ) ) e l i f s e l f . param . kappa_0 < 0 . 0 : e p s i l o n = c o m o b i l e D i s t a n c e * s q r t (-s e l f . param . kappa_0 ) volume = p i * ( s i n h ( 2 . 0 * e p s i l o n ) -2 . 0 * e p s i l o n ) /((-s e l f . param . kappa_0 ) * * ( 3 . / 2 . ) ) d_L = ( L i g h t V e l o c i t y * 1 0 * * 6 . 0 ) * ( 1 . + z ) * s i n h ( e p s i l o n ) /(H_0* s q r t (-s e l f . param . kappa_0 ) ) e l i f s e l f . param . kappa_0 == 0 . 0 : volume = 4 . * p i * ( c o m o b i l e D i s t a n c e * * 3 . 0 ) / 3 . d_L = ( L i g h t V e l o c i t y * 1 0 * * 6 . 0 ) * ( 1 . + z ) * c o m o b i l e D i s t a n c e /H_0 e l s e : pass r e t u r n comobileDistance , volume , d_L, lookback_time c l a s s SIMULATED: d e f __init__ ( s e l f , lambda_0 , omega_0 ) : ITI Chapter D : Code Python s e l f . param = Functions ( lambda_0 , omega_0 ) d e f Sample ( s e l f , M_0, sigma , m_lim , zform , z_max , z_min , S i z e o f s a m p l e , NumbOfSamples , Kcorrection , redsh_k , k_z ) : lambda0 , omega0 = s e l f . param . lambda_0 , s e l f . param . omega_0 kappa0 = s e l f . param . kappa_0 p r i n t "kappa0 = " , kappa0 , "\n" p r i n t Functions ( lambda0 , omega0 ) . NoBigBang ( omega0 , ' p e r m i s s i o n ' ) B e t a l i s t = l i n s p a c e ( np . f l o a t 6 4 ( 0 . 0 1 ) , np . f l o a t 6 4 ( 6 ) , 600) Beta = array ( [ array ( [ a ] ) f o r a i n B e t a l i s t ] ) #=========K-c o r r e c t i o n====== t = asc . read ( ' /SDSS-DR3/K-c o r r e c t i o n . dat ' , g u e s s=F a l s e ) #redsh_k , k_z = t [ ' redsh ' ] , t [ ' k c o r r e ' ] dkcorr = [ ] f o r i i n range ( l e n ( redsh_k ) ) : i f i==0 or i ==(l e n ( redsh_k ) -1) : dkcorr . append ( k_z [ i ] ) e l s e : dkcor = ( k_z [ i +1] -k_z [ i -1]) /( redsh_k [ i +1] -redsh_k [ i -1]) dkcorr . append ( dkcor ) dkcorr = array ( dkcorr ) zz = np . l i n s p a c e ( z_min , zform , 10**4) i f K c o r r e c t i o n == ' Kcorrection_NO ' : redsh_k_l , k_z_l = [ ] , [ ] pass e l i f K c o r r e c t i o n == ' Kcorrection_OK ' : List_of_Poly = [ ] f o r i i n range ( i n t ( l e n ( t [ ' k c o r r e ' ] ) /5) ) : j = i *5 f i t 4 = np . p o l y f i t ( t [ ' redsh ' ] [ j : j +5+1] , t [ ' k c o r r e ' ] [ j : j +5+1] , i n t ( l e n ( t [ ' k c o r r e ' ] [ j : j +5+1]) ) )f i t _ f n = np . poly1d ( f i t 4 ) List_of_Poly . append ( ( min ( t [ ' redsh ' ] [ j : j +5+1]) , max( t [ ' redsh ' ] [ j : j +5+1]) , f i t _ f n ) )d e f Fit_of_Kcorrection ( List_of_Poly , z ) : i = 0 Condition = z>List_of_Poly [ i ] [ 1 ] w h i l e Condition == True and i<=l e n ( List_of_Poly ) -2: i i n range ( l e n ( zz ) ) : i f i==0 or i ==(l e n ( zz ) -1) : redsh_k_l . append ( Fit_of_Kcorrection ( List_of_Poly , zz [ i ] ) ) e l s e : redsh_k_l . append ( ( Fit_of_Kcorrection ( List_of_Poly , zz [ i +1])-Fit_of_Kcorrection ( List_of_Poly , zz [ i -1]) ) /( zz [ i +1]-zz [ i -1]) ) redsh_k_l = array ( redsh_k_l ) k_z_l = [ ] f o r i i n range ( l e n ( zz ) ) : k_z_l . append ( Fit_of_Kcorrection ( List_of_Poly , zz [ i ] ) ) k_z_l = array ( k_z_l ) e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" #========================== ZETA = ApparentMagnitude ( zz , lambda0 , omega0 , 0 . 0 , Kcorrection , redsh_k_l , k_z_l ) . m_theor ( zz , lambda0 , omega0 , 0 . 0 ) paths_of_samples = [ ] f o r bb i n range ( 0 , NumbOfSamples ) : Number_of_simulation = bb m_app = [ ] r e d s h i f t = [ ] r ed r ed = [ ] Mlist , Volume , Volume_l , r e d s h i f t _ l = [ ] , [ ] , [ ] , [ ] HH = np . random . uniform ( low =0.0 , high =1, s i z e =( S i z e o f s a m p l e ) ) M l i s t 1 = l i n s p a c e ( -30 , -22. , 2* S i z e o f s a m p l e ) F_M = [ ] F_th = quad ( Functions ( lambda0 , omega0 ) . Pro_DensityFunction , -70, 70 , a r g s =(z_min , m_lim , M0, sigma ) ) [ 0 ] f o r i i n range (2* S i z e o f s a m p l e ) : ITQ Chapter D : Code Python z_l = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( zz , ZETA, m_lim -M l i s t 1 [ i ] ) f_M = quad ( Functions ( lambda0 , omega0 ) . Pro_DensityFunction , -70, M l i s t 1 [ i ] , a r g s =(z_l , m_lim , M0, sigma ) ) [ 0 ] F_M. append (f_M) F_M = array (F_M) /F_th #F_M = array (F_M) /max(F_M) f o r i i n range ( S i z e o f s a m p l e ) : F_cap = np . random . uniform ( low =0.0 , high =1, s i z e =(1) ) [ 0 ] M_k = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( Mlist1 , F_M, F_cap ) M l i s t . append (M_k) M l i s t = array ( M l i s t ) p l t . h i s t ( Mlist , b i n s =50, h i s t t y p e= ' s t e p ' , normed=False , c o l o r= ' r ' ) p l t . show ( ) #=============== i f K c o r r e c t i o n == ' Kcorrection_NO ' : i f kappa0 > 0 : Angular_distance = Model ( lambda0 , omega0 ) . c u r v a t u r e ( np . l i n s p a c e ( z_min , 1100 , 10**4) , lambda0 , omega0 , 0 . 0 ) [ 0 ] * s q r t ( kappa0 ) z_pi = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( np . l i n s p a c e ( z_min , 1100 , 10**4) , Angular_distance , p i ) zz = np . l i n s p a c e ( z_min , z_pi , 10**4) ZETA_pi = ApparentMagnitude ( zz , lambda0 , omega0 , 0 , Kcorrection , 0 , 0) . m_theor ( zz , lambda0 , omega0 , 0) M_star = m_lim -max(ZETA_pi) ZETA_form = ApparentMagnitude ( z_pi , lambda0 , omega0 , 0 . 0 , Kcorrection , 0 , 0) . m_theor ( z_pi , lambda0 , omega0 , 0 . 0 ) M_form = m_lim -ZETA_form f o r i i i n range ( S i z e o f s a m p l e ) : M_k = M l i s t [ i i ] mu_lim = m_lim -M_k i f M_k>M_star and M_k<M_form : ZETA_jj = ZETA* 1 . 0 z z _ j j = zz * 1 . 0 z_l_jj = [ ] f o r j j i n range ( 1 0 ) : z _ l j j = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( zz_jj , ZETA_jj , mu_lim) z_l_jj . append ( z _ l j j ) index_of_mu_lim = l i s t ( abs (ZETA_jj -mu_lim) ) . index ( min ( abs (ZETA_jj -mu_lim) ) ) ZETA_jj = l i s t (ZETA_jj*1) z z _ j j = l i s t ( z z _ j j * 1) ZETA_jj . remove (ZETA_jj [ index_of_mu_lim ] ) z z _ j j . remove ( z z _ j j [ index_of_mu_lim ] ) ZETA_jj = array (ZETA_jj*1) z z _ j j = array ( z z _ j j * 1) z_l_jj = array ( z_l_jj ) z_l_jj . s o r t ( ) D i s t a n g u i s h _ o f _ s o l u t i o n s = [ s t r ( gg ) [ 0 : 5 ] f o r gg i n z_l_jj ] z_l_l = [ ] z_l_l . append ( z_l_jj [ 0 ] ) f o r i i n range ( 1 , l e n ( z_l_jj ) -1) :

  a l s e : z_l_l . append ( z_l_jj [ i ] ) e l s e : pass D i s t a n g u i s h _ o f _ s o l u t i o n s = [ s t r ( gg ) [ 0 : 4 ] f o r gg i n z_l_l ] z_l = [ ] z_l . append ( z_l_l [ 0 ] ) f o r i i n range ( 1 , l e n ( z_l_l ) ) : c o n d i t i o n 1 = ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ]== D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) c o n d i t i o n 2 = ( abs ( f l o a t ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ] ) -f l o a t ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) )>= 0 . 0 6 ) i f c o n d i t i o n 1 == F a l s e and c o n d i t i o n 2 == True : z_l . append ( z_l_l [ i ] ) e l s e : pass i f ( l e n ( z_l ) % 2 == 0) : #even p r i n t " l e n ( z_l ) " , l e n ( z_l ) z_l . append ( zform ) z_l . s o r t ( ) z_l = array ( z_l )Volume_list = l i s t ( Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 )[ 1 ] )V_l = Volume_list [ 0 ] RandomChoiceList_of_Volume = [ ] f o r i i n range ( 1 , i n t ( ( l e n ( Volume_list ) -1) /2) +1) : V_l += Volume_list [ 2 * i ] -Volume_list [ 2 * i -1]ITSChapter D : Code Python Volume_l . append (V_l) ZoneProhibited_of_Volume = [ ] Z o n e P r o h i b i t e d _ o f _ r e d s h i f t = [ ] f o r i i n range ( i n t ( l e n ( Volume_list ) /2) ) : ZoneProhibited_of_Volume . append ( ( Volume_list [ 2 * i ] , Volume_list [ 2 * i +1]) ) Z o n e P r o h i b i t e d _ o f _ r e d s h i f t . append ( ( z_l [ 2 * i ] , z_l [ 2 * i +1]) ) HH_list = [ ] condition_on_HH = ( l e n ( HH_list ) == 0) i j = -1 V_z_list = [ ] condition_on_V_z = ( l e n ( V_z_list ) == 0) w h i l e condition_on_V_z == True and condition_on_HH == True : i j = i j + 1 i f i j > l e n (HH) -1: break e l s e : V_K = HH[ i j ] * V_l V_z = V_K RandomChoiceList_of_Volume . append (V_z) f o r i i n range ( l e n ( Volume_list ) -1) : i f ( i % 2 == 0) : V_z += -Volume_list [ i ] e l s e : V_z += Volume_list [ i ] RandomChoiceList_of_Volume . append (V_z) f o r dd i n range ( l e n ( RandomChoiceList_of_Volume ) ) : V_z = RandomChoiceList_of_Volume [ dd ] f o r i i n range ( l e n ( ZoneProhibited_of_Volume ) ) : c o n d i t i o n = (V_z > min ( ZoneProhibited_of_Volume [ i ] ) and V_z < max( ZoneProhibited_of_Volume [ i ] ) ) i f c o n d i t i o n == True : s t l i s t . append ( M l i s t [ i i ] ) e l s e : #odd z_l . s o r t ( ) z_l = array ( z_l )Volume_list = l i s t ( Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 )[ 1 ] )V_l = Volume_list [ 0 ] RandomChoiceList_of_Volume = [ ] f o r i i n range ( 1 , i n t ( ( l e n ( Volume_list ) -1) /2) +1) : V_l += Volume_list [ 2 * i ] -Volume_list [ 2 * i -1] Volume_l . append (V_l) ZoneProhibited_of_Volume = [ ] Z o n e P r o h i b i t e d _ o f _ r e d s h i f t = [ ] f o r i i n range ( i n t ( l e n ( Volume_list ) /2) ) : ZoneProhibited_of_Volume . append ( ( Volume_list [ 2 * i ] , Volume_list [ 2 * i +1]) ) Z on e P r o h i b i t e d _ o f _ r e d s h i f t . append ( ( z_l [ 2 * i ] , z_l [ 2 * i +1]) ) HH_list = [ ] condition_on_HH = ( l e n ( HH_list ) == 0) i j = -1 V_z_list = [ ] condition_on_V_z = ( l e n ( V_z_list ) == 0) w h i l e condition_on_V_z == True and condition_on_HH == True : i j = i j + 1 i f i j > l e n (HH) -1: dd i n range ( l e n ( RandomChoiceList_of_Volume ) ) : V_z = RandomChoiceList_of_Volume [ dd ] f o r i i n range ( l e n ( ZoneProhibited_of_Volume ) ) : c o n d i t i o n = (V_z > min ( ZoneProhibited_of_Volume [ i ] ) and V_z < max( ZoneProhibited_of_Volume [ i ] ) ) i f c o n d i t i o n == True : c o n t i n u e e l s e : i f V_z > max( ZoneProhibited_of_Volume [ i ] ) : c o n t i n u e e l s e : V_z_list . append (V_z) HH_list . append (HH[ i j ] ) HH. remove (HH[ i j ] ) break condition_on_HH = ( l e n ( HH_list ) == 0) condition_on_V_z = ( l e n ( V_z_list ) == 0) i f condition_on_V_z == True and condition_on_HH == True : c o n t i n u e e l s e : pass V_z = V_z_list [ 0 ] Volume . append (V_z) M l i s t l i s t . append ( M l i s t [ i i ] ) e l s e : z_l = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( zz , ZETA, mu_lim) r e d s h i f t _ l . append ( z_l ) V_l = Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 ) [ 1 ] Volume_l . append (V_l) V_K = HH[ i i ] * V_l Volume . append (V_K) M l i s t l i s t . append ( M l i s t [ i i ] ) e l s e : f o r i i i n range ( S i z e o f s a m p l e ) : M_k = M l i s t [ i i ] mu_lim = m_lim -M_k z_l = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( zz , ZETA, mu_lim) r e d s h i f t _ l . append ( z_l ) V_l = Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 ) [ 1 ] Volume_l . append (V_l) V_K = HH[ i i ] * V_l Volume . append (V_K) M l i s t l i s t . append ( M l i s t [ i i ] ) e l i f K c o r r e c t i o n == ' Kcorrection_OK ' : i f kappa0 > 0 : Angular_distance = Model ( lambda0 , omega0 ) . c u r v a t u r e ( np . l i n s p a c e ( z_min , 1100 , 10**4) , lambda0 , omega0 , 0 . 0 ) [ 0 ] * s q r t ( kappa0 ) z_pi = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( np . l i n s p a c e ( z_min , 1100 , 10**4) , Angular_distance , p i ) M_star = m_lim -max(ZETA) k_z_zform = Fit_of_Kcorrection ( List_of_Poly , zform ) ZETA_form = ApparentMagnitude ( zform , lambda0 , omega0 , 0 . 0 , Kcorrection , 0 , k_z_zform ) . m_theor ( zform , lambda0 , omega0 , 0 . 0 ) M_form = m_lim -ZETA_form f o r i i i n range ( S i z e o f s a m p l e ) : M_k = M l i s t [ i i ] mu_lim = m_lim -M_k i f M_k>M_star and M_k<M_form : ZETA_jj = ZETA* 1 . 0 z z _ j j = zz * 1 . 0 z_l_jj = [ ] f o r j j i n range ( 2 0 ) : z _ l j j = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( zz_jj , ZETA_jj , mu_lim) z_l_jj . append ( z _ l j j ) index_of_mu_lim = l i s t ( abs (ZETA_jj -mu_lim) ) . index ( min ( abs (ZETA_jj -mu_lim) ) ) ZETA_jj = l i s t (ZETA_jj*1) z z _ j j = l i s t ( z z _ j j * 1) ZETA_jj . remove (ZETA_jj [ index_of_mu_lim ] ) z z _ j j . remove ( z z _ j j [ index_of_mu_lim ] ) ZETA_jj = array (ZETA_jj*1) z z _ j j = array ( z z _ j j * 1) z_l_jj = array ( z_l_jj ) z_l_jj . s o r t ( ) D i s t a n g u i s h _ o f _ s o l u t i o n s = [ s t r ( gg ) [ 0 : 5 ] f o r gg i n z_l_jj ] ITW Chapter D : Code Python z_l_l = [ ] z_l_l . append ( z_l_jj [ 0 ] ) f o r i i n range ( 1 , l e n ( z_l_jj ) -1) : c o n d i t i o n = ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ]== D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) i f c o n d i t i o n == F a l s e : z_l_l . append ( z_l_jj [ i ] ) e l s e : pass D i s t a n g u i s h _ o f _ s o l u t i o n s = [ s t r ( gg ) [ 0 : 4 ] f o r gg i n z_l_l ] z_l = [ ] z_l . append ( z_l_l [ 0 ] ) f o r i i n range ( 1 , l e n ( z_l_l ) ) :

  s e and c o n d i t i o n 2 == True : z_l . append ( z_l_l [ i ] ) e l s e : pass i f ( l e n ( z_l ) % 2 == 0) : #even p r i n t " l e n ( z_l ) " , l e n ( z_l ) z_l . append ( zform ) z_l . s o r t ( ) z_l = array ( z_l )Volume_list = l i s t ( Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 )[ 1 ] )V_l = Volume_list [ 0 ] RandomChoiceList_of_Volume = [ ] f o r i i n range ( 1 , i n t ( ( l e n ( Volume_list ) -1) /2) +1) : V_l += Volume_list [ 2 * i ] -Volume_list [ 2 * i -1] Volume_l . append (V_l) ZoneProhibited_of_Volume = [ ] Z o n e P r o h i b i t e d _ o f _ r e d s h i f t = [ ] f o r i i n range ( i n t ( l e n ( Volume_list ) /2) ) : ZoneProhibited_of_Volume . append ( ( Volume_list [ 2 * i ] , Volume_list [ 2 * i +1]) ) Z o n e P r o h i b i t e d _ o f _ r e d s h i f t . append ( ( z_l [ 2 * i ] , z_l [ i i n range ( l e n ( Volume_list ) -1) : i f ( i % 2 == 0) : V_z += -Volume_list [ i ]e l s e : V_z += Volume_list [ i ] RandomChoiceList_of_Volume . append (V_z) f o r dd i n range ( l e n ( RandomChoiceList_of_Volume ) ) : V_z = RandomChoiceList_of_Volume [ dd ] f o r i i n range ( l e n ( ZoneProhibited_of_Volume ) ) : c o n d i t i o n = (V_z > min ( ZoneProhibited_of_Volume [ i ] ) and V_z < max( ZoneProhibited_of_Volume [ i ] ) ) i f c o n d i t i o n == True : c o n t i n u e e l s e : i f V_z > max( ZoneProhibited_of_Volume [ i ] ) : c o n t i n u e e l s e : V_z_list . append (V_z) HH_list . append (HH[ i j ] ) HH. remove (HH[ i j ] ) break condition_on_HH = ( l e n ( HH_list ) == 0) condition_on_V_z = ( l e n ( V_z_list ) == 0) i f condition_on_V_z == True and condition_on_HH == True : s t l i s t . append ( M l i s t [ i i ] ) e l s e : #odd p r i n t " l e n ( z_l ) " , l e n ( z_l ) z_l . s o r t ( ) IUI Chapter D : Code Python z_l = z_l [ 0 ] r e d s h i f t _ l . append ( z_l ) V_l = Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 ) [ 1 ] Volume_l . append (V_l) V_K = HH[ i i ] * V_l Volume . append (V_K) M l i s t l i s t . append ( M l i s t [ i i ] ) e l s e : ZETA_jj = ZETA* 1 . 0 z z _ j j = zz * 1 . 0 z_l_jj = [ ] f o r j j i n range ( 2 0 ) : z _ l j j = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( zz_jj , ZETA_jj , mu_lim) z_l_jj . append ( z _ l j j ) index_of_mu_lim = l i s t ( abs (ZETA_jj -mu_lim) ) . index ( min ( abs (ZETA_jj -mu_lim) ) ) ZETA_jj = l i s t (ZETA_jj*1) z z _ j j = l i s t ( z z _ j j * 1) ZETA_jj . remove (ZETA_jj [ index_of_mu_lim ] ) z z _ j j . remove ( z z _ j j [ index_of_mu_lim ] ) ZETA_jj = array (ZETA_jj*1) z z _ j j = array ( z z _ j j * 1) z_l_jj = array ( z_l_jj ) z_l_jj . s o r t ( ) D i s t a n g u i s h _ o f _ s o l u t i o n s = [ s t r ( gg ) [ 0 : 5 ] f o r gg i n z_l_jj ] z_l_l = [ ] z_l_l . append ( z_l_jj [ 0 ] ) f o r i i n range ( 1 , l e n ( z_l_jj ) -1) : c o n d i t i o n = ( D i s t a n g u i s h _ o f _ s o l u t i o n s [ i ]== D i s t a n g u i s h _ o f _ s o l u t i o n s [ i -1]) i f c o n d i t i o n == F a l s e : z_l_l . append ( z_l_jj [ i ] ) e l s e : pass D i s t a n g u i s h _ o f _ s o l u t i o n s = [ s t r ( gg ) [ 0 : 4 ] f o r gg i n z_l_l ] z_l = [ ] z_l . append ( z_l_l [ 0 ] ) f o r i i n range ( 1 , l e n ( z_l_l ) ) :

  F a l s e and c o n d i t i o n 2 == True : z_l . append ( z_l_l [ i ] ) e l s e : pass i f ( l e n ( z_l ) % 2 == 0) : #even z_l = z_l [ 0 ] r e d s h i f t _ l . append ( z_l ) V_l = Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 ) [ 1 ] Volume_l . append (V_l) V_K = HH[ i i ] * V_l Volume . append (V_K) M l i s t l i s t . append ( M l i s t [ i i ] ) e l s e : #odd z_l . s o r t ( ) z_l = array ( z_l )Volume_list = l i s t ( Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 )[ 1 ] )V_l = Volume_list [ 0 ] RandomChoiceList_of_Volume = [ ] f o r i i n range ( 1 , i n t ( ( l e n ( Volume_list ) -1) /2) +1) : V_l += Volume_list [ 2 * i ] -Volume_list [ 2 * i -1] Volume_l . append (V_l) ZoneProhibited_of_Volume = [ ] Z o n e P r o h i b i t e d _ o f _ r e d s h i f t = [ ] f o r i i n range ( i n t ( l e n ( Volume_list ) /2) ) : ZoneProhibited_of_Volume . append ( ( Volume_list [ 2 * i ] , Volume_list [ 2 * i +1]) ) Z on e P r o h i b i t e d _ o f _ r e d s h i f t . append ( ( z_l [ 2 * i ] , z_l [ 2 * i +1]) ) HH_list = [ ] condition_on_HH = ( l e n ( HH_list ) == 0) i j = -1 V_z_list = [ ] condition_on_V_z = ( l e n ( V_z_list ) == 0) w h i l e condition_on_V_z == True and condition_on_HH == True : i j = i j + 1 i f i j > l e n (HH) dd i n range ( l e n ( RandomChoiceList_of_Volume ) ) : V_z = RandomChoiceList_of_Volume [ dd ] f o r i i n range ( l e n ( ZoneProhibited_of_Volume ) ) : c o n d i t i o n = (V_z > min ( ZoneProhibited_of_Volume [ i ] ) and V_z < max( ZoneProhibited_of_Volume [ i ] ) ) i f c o n d i t i o n == True : c o n t i n u e e l s e : i f V_z > max( ZoneProhibited_of_Volume [ i ] ) : c o n t i n u e e l s e : V_z_list . append (V_z) HH_list . append (HH[ i j ] ) HH. remove (HH[ i j ] ) break condition_on_HH = ( l e n ( HH_list ) == 0) condition_on_V_z = ( l e n ( V_z_list ) == 0) i f condition_on_V_z == True and condition_on_HH == True : c o n t i n u e e l s e : pass V_z = V_z_list [ 0 ] Volume . append (V_z) M l i s t l i s t . append ( M l i s t [ i i ] ) e l s e : f o r i i i n range ( S i z e o f s a m p l e ) : M_k = M l i s t [ i i ] mu_lim = m_lim -M_k ZETA_jj = ZETA* 1 . 0 z z _ j j = zz * 1 . 0 z_l_jj = [ ] f o r j j i n range ( 2 0 ) : z _ l j j = Functions ( lambda0 , omega0 ) . N o n l i n e a r _ I n t e r p o l a t i o n ( zz_jj , ZETA_jj , mu_lim) z_l_jj . append ( z _ l j j ) index_of_mu_lim = l i s t ( abs (ZETA_jj -mu_lim) ) . index ( min ( abs (ZETA_jj -mu_lim) ) ) ZETA_jj = l i s t (ZETA_jj*1) z z _ j j = l i s t ( z z _ j j * 1) ZETA_jj . remove (ZETA_jj [ index_of_mu_lim ] ) z z _ j j . remove ( z z _ j j [ index_of_mu_lim ] ) ZETA_jj = array (ZETA_jj*1) z z _ j j = array ( z z _ j j * 1) z_l_jj = array ( z_l_jj ) z_l_jj . s o r t ( ) D i s t a n g u i s h _ o f _ s o l u t i o n s = [ s t r ( gg ) [ 0 : 5 ] f o r gg i n z_l_jj ] z_l_l = [ ] z_l_l . append ( z_l_jj [ 0 ] ) f o r i i n range ( 1 , l e n ( z_l_jj ) -1) :

  s e and c o n d i t i o n 2 == True : z_l . append ( z_l_l [ i ] ) e l s e : pass i f ( l e n ( z_l ) % 2 == 0) : #even z_l = z_l [ 0 ] r e d s h i f t _ l . append ( z_l )V_l = Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 )[ 1 ] Volume_l . append (V_l)V_K = HH[ i i ] * V_l Volume . append (V_K) M l i s t l i s t . append ( M l i s t [ i i ] )e l s e : #odd z_l . s o r t ( ) z_l = array ( z_l ) Volume_list = l i s t ( Model ( lambda0 , omega0 ) . c u r v a t u r e ( z_l , lambda0 , omega0 , 0 . 0 )[ 1 ] )V_l = Volume_list [ 0 ] RandomChoiceList_of_Volume = [ ] f o r i i n range ( 1 , i n t ( ( l e n ( Volume_list ) -1) /2) +1) : V_l += Volume_list [ 2 * i ] -Volume_list [ 2 * i -1] e P ro h i b i t e d _ o f _ r e d s h i f t = [ ] f o r i i n range ( i n t ( l e n ( Volume_list ) /2) ) : ZoneProhibited_of_Volume . append ( ( Volume_list [ 2 * i ] , Volume_list [ 2 * i +1]) ) Z o n e P r o h i b i t e d _ o f _ r e d s h i f t . append ( ( z_l [ 2 * i ] , z_l [ 2 * i +1]) ) HH_list = [ ] condition_on_HH = ( l e n ( HH_list ) == 0) i j = -1 V_z_list = [ ] condition_on_V_z = ( l e n ( V_z_list ) == 0) w h i l e condition_on_V_z == True and condition_on_HH == True : i j = i j + 1 i f i j > l e n (HH) -1: break e l s e : V_K = HH[ i j ] * V_l V_z = V_K RandomChoiceList_of_Volume . append (V_z) f o r i i n range ( l e n ( Volume_list ) -1) : i f ( i % 2 == 0) : #even V_z += -Volume_list [ i ] e l s e : #odd V_z += Volume_list [ i ] RandomChoiceList_of_Volume . append (V_z) f o r dd i n range ( l e n ( RandomChoiceList_of_Volume ) ) : V_z = RandomChoiceList_of_Volume [ dd ] f o r i i n range ( l e n ( ZoneProhibited_of_Volume ) ) : c o n d i t i o n = (V_z > min ( ZoneProhibited_of_Volume [ i ] ) and V_z < max( ZoneProhibited_of_Volume [ i ] ) ) i f c o n d i t i o n == True : (HH[ i j ] ) HH. remove (HH[ i j ] ) break condition_on_HH = ( l e n ( HH_list ) == 0) condition_on_V_z = ( l e n ( V_z_list ) == 0) p l t . p l o t ( a [ i ] [ 1 ] , L_1 , marker = '+ ' , c o l o r =' r ' ) p l t . f i g u r e ( 6 ) V = Model ( a [ i ] [ 0 ] , a [ i ]

  All_models = np . a s a r r a y ( All_models ) Cov_All_Models = np . a s a r r a y ( Cov_All_Models ) #*********** g r i d o f c o v a r i a n c e ***************** # f i g u r e o f 3D( s u r f a c e ) #All_models [ : , 0 ] i s a l i s t e o f lamda #All_models [ : , 1 ] i s a l i s t e o f omega f i g = p l t . f i g u r e ( 7 ) ax = f i g . add_subplot ( 1 1 1 ) c o l = p l t . s c a t t e r ( All_models [ : , 1 ] , All_models [ : , 0 ] , marker= ' . ' , s =150 , c=Cov_All_Models , l i n e w i d t h s =0.3 , cmap=p l t . cm . Spectral_r ) #f o r vv , ww, dd i n z i p ( All_models [ : , 1 ] , All_models [ : , 0 ] , Cov_All_Models ) : # p l . t e x t ( vv , ww, '%.2 f ' % dd , ha=' c e n t e r ' , va='bottom ' ) p l t . p l o t ( [ omega0_Model ] , [ lambda0_Model ] , marker= ' o ' , c o l o r= 'm' ) # Add a c o l o u r b a r . #cax = f i g . c o l o r b a r ( col , o r i e n t a t i o n =' v e r t i c a l ' , format ='%.30 f ' ) cax = f i g . c o l o r b a r ( col , o r i e n t a t i o n= ' v e r t i c a l ' , format= ' %.2 f ' ) cax . s e t _ l a b e l ( ' Covariance ' ) p l t . x l a b e l ( ' $\\Omega_{\\ c i r c }$ ' , f o n t s i z e =14) p l t . y l a b e l ( ' $\\lambda_{\\ c i r c }$ ' , f o n t s i z e =14) #===========For the save o f the data================ OmegaOfGrid = All_models [ : , 1 ] * 1 . 0 LambdaOfGrid = All_models [ : , 0 ] * 1 . 0 IVI Chapter D : Code Python C o r r C o e f f i c i e n t = Cov_All_Models * 1 . 0 #=================================================== #p l t . f i g u r e ( 8 ) #f o r i i n range ( l e n ( Cov_All_Models ) ) : # p l t . p l o t ( Cov_All_Models [ i ] , All_models [ i ] [ 0 ] , marker='o ' , c o l o r =' r ' ) #n , bins , patches = p l t . h i s t ( Cov_All_Models , 4 0 , normed='True ' ) #( mu_hist , sigma_hist ) = norm . f i t ( array ( Cov_All_Models ) ) #y = mlab . normpdf ( bins , mu_hist , sigma_hist ) #p l t . p l o t ( bins , y , ' r --', l i n e w i d t h =2) #p l t . t i t l e ( r ' $\rho_{0}=%.7 f , \ \sigma_{0}=%.3 f $ ' %(mu_hist , sigma_hist ) , c o l o r =' r ' ) #p l t . x l a b e l ( ' C o r r e l a t i o n c o e f f i c i e n t ' , f o n t s i z e =14) #**************************************

  matrix o f omega #To make the c o n t o u r s o f c o n f i d e n c e l e v e l s #CS = p l . contour ( All_modelsomega , All_modelslambda , abs ( Cov_All_Models ) , 3 , l i n e w i d t h s=np . arange ( . 5 , 4 , . 5 ) , c o l o r s =( ' r ' , ' green ' , ' blue ' , ( 1 , 1 , 0 ) , '# a f e e e e ' , ' 0 . 5 ' ) ) #p l t . c l a b e l (CS , i n l i n e =1, f o n t s i z e =10) #p l t . show ( ) #s e a r c h o f c o v a r i a n c e z e r o by HORIZONTAL i n t e r p o l a t i o n f o r i i n range ( l e n ( o m e g a l i s t ) ) : #----------------""" f o r i i n range ( l e n ( X_Fit3 ) ) : X. append ( min ( X_Fit3 ) ) Y. append ( Y_Fit3 [ X_Fit3 . index ( min ( X_Fit3 ) ) ] ) Y_Fit3 . remove ( Y_Fit3 [ X_Fit3 . index ( min ( X_Fit3 ) ) ] ) X_Fit3 . remove ( min ( X_Fit3 ) ) """ #-----p r i n t " l e n (X) = " , l e n (X) p r i n t " l e n (Y) = " , l e n (Y) p r i n t "X = " , X p r i n t "Y = " , Y i f omega0_Model < min (X) : o m e g a l i s t 2 = l i n s p a c e ( omega0_Model -0.02 , max(X) , 200) e l i f omega0_Model > max(X) : o m e g a l i s t 2 = l i n s p a c e ( min (X) , omega0_Model +0.02 , 200) e l s e : o m e g a l i s t 2 = l i n s p a c e ( min (X) , max(X) , 200) IVU Chapter D : Code Python p l t . p l o t (X, Y, c o l o r=c o u l e u r ) Lamdalimit = [] o m e g a l i s t = l i n s p a c e ( min ( o m e g a l i s t ) , max( o m e g a l i s t ) , 200) f o r i i n range ( l e n ( o m e g a l i s t ) ) : Lamdalimit . append ( Functions ( 0 . 7 , 0 . 3 ) . NoBigBang ( o m e g a l i s t [ i ] , ' l i m i t ' ) ) p l t . f i l l _ b e t w e e n ( o m eg al i st , Lamdalimit , max( Lamdalimit ) , c o l o r = ( 0 . 5 , 0 . 5 , 0 . 5 ) ) p l t . t e x t ( 0 . 0 0 1 , 1 . 4 6 8 , 'No Big Bang ' , c o l o r= 'w ' ) p l t . t e x t ( 0 . 7 , 1 . 3 5 , ' ( $ \\Omega_{\\ c i r c }$ , $ \\lambda_{\\ c i r c }$ ) = ( '+s t r ( omega0_Model )+ ' , '+s t r ( lambda0_Model )+ ' ) ' , c o l o r= ' r ' ) p l t . p l o t ( [ omega0_Model ] , [ lambda0_Model ] , marker= ' o ' , c o l o r= ' r ' ) p l t . x l a b e l ( ' $\\Omega_{\\ c i r c }$ ' , f o n t s i z e =16) p l t . y l a b e l ( ' $\\lambda_{\\ c i r c }$ ' , f o n t s i z e =16) g r i d ( True ) r e t u r n OmegaOfGrid , LambdaOfGrid , C o r r C o e f f i c i e n t , X, Y c l a s s Save :

  @gA F erlmutter et lF IWWW nd the righE supernov eF qF iess et lF IWWVF eording to pvq modelD suh n elertion results from either positive zero osmologil onstnt or the n unknown grvittionl soureD n d ho lterntive motivted y the osmlogil onstnt prolem in righ inergy hysisF hese two interprettions eing integrted in single one nmed 4hrk inergy4F pigFIFS presents the on(dene ontours of Ω m nd Ω Λ @see etFIFSA otined y g using RP xe sF his setion outlines the two frequent pprohes used to explin this elertionF pigure IFSX gon(dene ontours in the @Ω M D Ω Λ A pln otined y the uperE nov gosmology rojet with oservtions of RP xe s F erlmutter et lF IWWWF

	1.8 The Dark Energy	Chapter 1 : Basics of Observational Cosmology
	quintessene models fF tr IWVVD gF etterih IWVV in whih the drk
	energy is desried y slr (eld in slow rolling plnD in wy quite
	similr to in)tionF he dynmis of drk energy y tht of slr (eld
	whih is hrterized y potentil tht hs to (t to oservtionl onstrintsF
	he quintessene models pper interesting when onsidering tht he min
	issue with these models is the determintion of the potentil to otin vlue
	of drk energy density lose to the mesured energy densityF o solve this
	prolemD other slr (elds hve een proposed s phntom (eld F gldwell
	PHHPD PHHQ nd uEessene model F ghi et lF PHHHD gF ermendrizEion
	PHHHD PHHIF fut these models su'er from similr prolemsF
	1.8.2 The Cosmological Constant
	hile in qD the osmologil onstnt stnds for universl onstntF prom
	the point of view of prtile physisD it would e the energy density of quntum
	)ututions of vuumF reneD two mjor ostles pperX the osmologil
	onstnt prolem nd the prolem of oinideneF
	The Cosmological constant problem
	eording to the prtile physisD the vlue of the energy density of the vuum
	is estimtedX
	ρ vacuum Λ	∼ (10 18 GeV ) 4 ∼ 2.10 110 erg/cm 3	@IFPUA
	yn the other hndD oservtions of typeEs xe nd gwf )ututionsD the
	oserved energy density is of orderX
	ρ obs Λ ∼ (10 -12 GeV ) 4 ∼ 2.10 -10 erg/cm 3	@IFPVA
	hereforeD the expeted vlue is ∼ IPH order of mgnitude lrger thn the oserved energy density F einerg IWVWD F grroll PHHIX
		ρ vacuum Λ	∼ 10 120 ρ obs Λ	@IFPWA
	his disgreement is known s the gosmologil onstnt prolemF
	1.8 The Dark Energy 1.8.1 Models with Scalar Fields
	ine the end of WHsD elertion of osmi expnsion hs een deteted y o sketh the pproh to this issueD we limit to desrie models whih re
	using rule digrm of typeEs supernove thnks to upernov gosmology ehind severl drk energy modelsF he (rst models of this lss re the
		IS	14

rojet

pigure IFWX prom top left to ottom rightX he omoving distneD the luE minosity distneD the distne modulus nd the omoving volume versus the

  est mesurement or estimtion of this ltter represents the lue of onE strining the rule digrm perlsD the osmologil prmeters λ • nd Ω • F

	1.12 Conclusion
			CHAPTER 2
			Quasars samples
	Contents
	2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 23
	2.2 Spectrum of quasar . . . . . . . . . . . . . . . . . . . 24
	2.3 Events in the space-time diagram . . . . . . . . . . . 24
	2.4 The selection eect . . . . . . . . . . . . . . . . . . . 27
	2.5 Modelling a sample of quasars . . . . . . . . . . . . 28
	2.5.1 Statistical modelling . . . . . . . . . . . . . . . . . . 29
	2.5.2 Simulation technique . . . . . . . . . . . . . . . . . . 29
	2.5.3 The k-correction . . . . . . . . . . . . . . . . . . . . 34
	2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 35
	redshift for six di'erent osmologil modelsF
	1.12 Conclusion	log 10	d L (z) 10pc	@IFSHA
	he di'erene etween the two mgnitudes is lled distance modulus ζ(z) sn this ghpterD we hve presented the stndrd osmologil model ΛCDM
	@pigFIFWA nd is de(ned sX sed on the priedmnnEvemîtreEqmow modelF sn the frmeworkD we will
	ζ(z) = 5 log 10 seek the more pproprited vlues of the osmologil prmeters using stE d L (z) 10pc @IFSIA tistil tehniquesF
	PQ

he

  por eh model on the xggD we otin vlue of D max F he lower the D max vlueD the more the greement etween the two distriutionsF husD the most likely vlues of osmologil prmeters orrespond to the minimum of D max over the xggF he V /V max test is ppliedD on the xgg in pigFQFIS with Ω • = 0.2, nd λ • = 0.5F he vlues of D max is given for eh model on the urveF he yellow urve gives D max s funtion of λ

	Chapter 4 : Application to quasars data
	4.2 The Sloan Digital Sky Survey data
	CHAPTER 4
	Application to quasars data
	79
	4.1 Introduction
	he lrgest homogeneous smple of ys ever foundD re provided y puli
	dt releses of hF rerefterD we proeed to sttistil nlysis of these
	smples with the im to pply the null orreltion method for inferring osE
	mologil informtionF por suh purposeD we use the solute mgnitude
	digrm nd the V /V max testF
	SU SW

• @pigFQFIS in top right nd leftAF he minimum is rehed t vlue of λ • hosen for the simulted smpleF imilrlyD the mgent urve gives D max s funtion of Ω • D nd we otin the sme resultD the vlue of Ω • hosen for the simultion @pigFQFIS in top right nd in ottom rightAF e mesure tht the likelihood of this result is estimted t t WW7 signi(ne levelF 56 3.6 Conclusion pigure QFIRX he redshift rnge is not onneted for this model with positive urvture κ • = 0.4F he distne modulus ζ(z) of soure with solute mgnitude M hs most likely two solutions nd it nnot e oserved in the hthed reF

3.6 Conclusion

e hd desriedD n innovtive sttistil method tht does not require the knowledge of the luminosity funtion nd whih is free from the wlmquist isF st onsists on (rst step whih uses the null orreltion tehnique to otin set of models whih gree with dtF he ury of estimtes imE proves with the inresing of numer of ojetsF woreoverD one estimtes the luminosity nd the seletion funtionsF e seond step uses V /V max testD with little modi(tion of the usul de(niE tionF st provides us with etter estimtes of osmologil prmetersF Contents 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 The Sloan Digital Sky Survey data . . . . . . . . . . 52 4.2.1 The k-correction . . . . . . . . . . . . . . . . . . . . 53 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3.1 The null correlation test . . . . . . . . . . . . . . . . 54 4.3.2 The V/V max test . . . . . . . . . . . . . . . . . . . . 57 4.3.3 Estimation of the luminosity function and the selection function . . . . . . . . . . . . . . . . . . . . . . 60 4.4 Inferences on cosmological expansion based on QSOs 61 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .

  4.2 The Sloan Digital Sky Survey data trget ndidtesF pirstlyD we pplied the test to the (rst edition of h qusr tlogD the irly ht elese @ihAF st onsists of QDVIR ys with redshifts rnging from HFIS to SFHQF he survey rehes )ux limit of i ≈ 19 for ojets with uv < 1.5F his limit eomes i ≈ 20 for ojets with uv > 1.5 hneider et l PHHPF eondlyD we hose from hEssD the (rst nd third dt releses @hI nd hQ respetivelyA where the ltter overs ll the re of the former with more thn three times of qusrsF pinllyD we used smple where the numer of ys is muh higherF st is the ltest smple of hEssD the seventh dt relese hUD nd is two times more thn the hQ ited oveF rereD s present the di'erent hrteristis of eh smple @hID hQ nd hUAF he min di'erene etween hI nd hQD is the numer of qusrs seleted in ehF he hI tlog ontins ITDUIQ ys with redshift from HFHV up to SFRID nd spns rnge of pprent i mgnitude etween 15.15 nd 21.79F hile the hQ tlog onsists of RTDRPH ys @three times more qusrs thn hIA overing the full re of hID n re of RIVV deg 2 F he hQ smple is distinguished from hI due to the severl modi(tions in the riteri of the seletion ndidtesF snterestinglyD due to new tehniques used for this smpleD the hQ hs more ury thn the previous dt releses in

determining the signlGnoise rtio used to determine the redshiftF his llows to improve the loliztion of the seleted qusrs with more preisionF woreE overD good improvement of the photometri mesurement nd the imging dt re lso relesedF he seleted ys in hQ hve redshift rnge from z = 0.078 to 5.414 where SPH qusrs hve redshifts greter thn RD of whih IU re t redshifts greter thn SF his smple extends from i = 15 to 21.78 @i = 19.1 is the (rst limit on the pprent mgnitude t z < 3 nd i = 20.2 for z > 3AF emong this smpleD only ITH ys hve rightness finter thn PHFS in the i ndD for whih only (ve ys hve i > 21.0 hneider et l PHHSF he hU is the (nl dt relese for hEssD it ontins IHSDUVQ spetrosopiE lly on(rmed ys whih represent the entire set of ys from hEs nd hEss usr surveyF he tlog overs n re of WQVH deg 2 F he rnge of redshift in this tlog is from HFHTS to SFRTI with pprent mgnitude from i = 14.86 to 22.36F his smple hs IPRV ys with redshifts lrger thn RD of whih ST hve redshifts ove S hneider et l PHIHF ell of the ys in these tlogs re seleted with luminosities lrger thn M = -22 in the i nd @lulted with the osmologil modelX

  complete disagreement with those of other samples. As a matter of fact, . . . , the original version of the quasar target selection algorithm used in the EDR data did a particularly poor job of selecting quasars with redshifts close to z = 3.5. (loc. cit., see[6]).Hence, let us focus on the other null correlation curves. They slightly dier from each other, which is probably due to improved estimates of data and of dierent sample sizes. These candidate values lie rather above a straight line dened by (Ω • ≈ 0.04, λ • ≈ 1.1) and (Ω • ≈ 0.5, λ • ≈ 1.5), whose slope ≈ 0.52 indicates that estimates of λ • are twice as much accurated as those of Ω • . The unsettling result is that λ • > 1.1 for all these candidate values, what does not match with the standard value of λ • ≈ 0.7; furthermore one obtains λ • ≈ 1.3 by choosing Ω • ≈ 0.3.

• -λ • diagram, see gure 4. Each sample provides us with a null correlation curve that corresponds to candidate values in agreement with the working hypotheses. The obvious result is that the candidate values obtained with the EDR sample are in 4.4 Inferences on cosmological expansion based on QSOs VU

  e t u r n path visting hFIX ython exmple IVW Bibliography vF qerrit IWQU qerrit vF ershuurD snterstellr wttersD issys on guriosity nd estronomil hisoveryD fowieD wh PHUISD eD virry of gongress gtlogingEinEulition htD IWQU xF storello PHIQ xFD storelloD wFD rziD wFD gppellriD iFD imsellemD et lFD he plnetry neule popultion in the nuler regions of wQIX the eyx viewD wonthly xoties of the oyl estronomil oietyD olume RQHD PHIQ ehEPHIQwxeFRQHFIPIW tF fFD uler IWUT tF fFD ulerD e tlog of reltive emission line intensities oserved in plnetry nd di'use neuleD estrophysil tournl uppleE ment eriesD volF QID IWUT ehEIWUTeptFFFQIFFSIUu sF is PHHS sF isD gF tordiD pF ilrdellD et lFD pirst hetermintion of the histne nd pundmentl roperties of n ilipsing finry in he endromed qlxyD estrophysFtFTQSXvQUEvRHDPHHS rxivEHSIIHRS wFF vongir PHIQ wlolm F vongirD pringer iene fusiness wediD pge RD IR mrs PHIQ iF F rrrison IWVW rrrisonD iF FD he qlti nd ixtrglti fkE ground ditionF roeedings of the IQWthF ymposium of the snternE tionl estronomil nionD held in reidelergD pqD tune IPEITD IWVWF ehEIWWHseFFIQWFFFFQr F lotner PHHV F lotnerD he xight ky gompnionD e erly quide to kyEthing PHHVEPHHWD PHHV iF rrrison PHHH idwrd rrrisonD gosmologyX he iene of the niverseD gmridge niversity ressD pge QQQD IT mrs PHHH IWI BIBLIOGRAPHY sF eppenzeller PHIQ sF eppenzellerD sntrodution to estronomil peE trosopyD gmridge university pressD PHIQ tF hF pernie IWTW pernieD tF hFD he periodEluminosity reltionX e historil reviewD ulitions of the estronomil oiety of the i(D olF VID xoF RVQD pFUHUD IWTW ehEIWTWeFFFVIFFUHUp rF xussumer PHIR rrry xussumerD iinstein9s onversion from his stti to n expnding universeD olF QWD pF QUETPD iuropen hysis tournlE ristoryD PHIR rxivEIQIIFPUTQ eF iinstein nd F de itter IWQP eF iinstein nd F de itterD yn the eE ltion etween the ixpnsion nd the wen hensity of the niverseD roE eedings of the xtionl edemy of ienes of the nited ttes of emerE iD IVD PIQEPIR @IWQPA qF vemitre IWPU qF vemitreD n nivers homogene de msse onstnte et de ryon roissnt rendnt ompte de l vitesse rdile des neuleuses extrE gltiquesD ennles de l oietee ienti(que de fruxellesD eRUD pF RWESWD IWPU ehEIWPUefFFFRUFFFRWv iF rule IWPW iF ruleD e eltion etween histne nd dil eloity mong ixtrEqlti xeuleD roeedings of the xtionl edemy of ienes of the nited ttes of emeriD IWPW ehEIWPWxeFFFISFFITVr oert wF ld IWVR oert wF ldFD qenerl eltivityF he niversity of ghigo ressD ghigo nd vondonD IWVR F elpher PHIR lph e elpherD qeorge entonovih qmowD nd the reE dition of the gosmi wirowve fkground ditionD olF PQD esin tourE nl of physisD PHIR rxivEIRIIFHIUP F tF iF eeles PHIR F tF iF eelesD hisovery of the hot fig fngX ht hppened in IWRVD physisFhistEphD PHIR rxivEIQIHFPIRT eF go PHIT eF go rimordil xuleosynthesisD xuler hysis in estroE physis s @xeTAD tournl of hysisX gonferene eries TTSD PHIT rxivE IPHVFRURV hFtF pixsen IWWT hFtF pixsenD iFF ghengD tFwF qlesD tFgF wtherD FeF hferD iFvF rightD he gosmi wirowve fkground petrum from the pull gyfiGpse ht etD IWWTD estrophysFtF RUQ rxivEWTHSHSR gF vF fennett PHIQ gF vF fennettD hF vrson et lFDxsxiEie svusxE yx wsgyei exsyy yfi @weA yfiesyxX psxev we exh ivD PHIQ 192 BIBLIOGRAPHY gF eF fertulniD et lF PHIT gF eF fertulni nd F ujinoDprontiers in xuE ler estrophysisD PHIT rxivEITHRFHQIWU qF teigmn PHHR qF isqwexD fig fng xuleosynthesisX roing the pirst PH winutesD grnegie yservtories estrophysis eriesD olF PX weE suring nd wodeling the niverseD edF F vF preedmn @gmridgeX gmE ridge nivF ressAD PHHR wF golless PHHQ wF gollessD et lFD he Pdp qlxy edshift urveyX pinl ht eleseD estrophysis @stroEphAD PHHQ rxivEHQHTSVI tF rF yort IWQP tF rF yortF he fore exerted y the stellr system in the diretion perpendiulr to the glti plne nd some relted prolemsF fullF estronF snstF xetherlndsD T XPRWD eugust IWQP pF wiky IWQU pF wikyF yn the wsses of xeule nd of glusters of xeE uleF estrophysil tournlD olF VTD pFPIUD IWQU iF rule IWQR iF ruleF he distriution of extrEglti xeuleF esE trophysil tournlD volF UWD pFVD IWQR ehEIWQReptFFFFUWFFFFVr pF wiky IWQQ pF wikyF epulition ofX he redshift of extrglti neuleD qenerl eltivity nd qrvittionD olume RID sssue ID ppFPHUE PPRD @originl pulitionX relveti hysi etD olF TD pF IIHEIPUD IWQQA ehEPHHWqeqrFFRIFFPHUi lnk gollortion s @PHISA lnk ollortionD lnk PHIS resultsF sF yverview of produts nd sienti( resultsD estronomy estrophysisD euE gust IID PHIS rxivEISHPFHISVPvP lnk gollortion sss @PHISA lnk ollortionD lnk PHIS resultsF sssF gosmologil prmetersD estronomy estrophysisD tune PHD PHIT rxivEISHPFHISVW eF qF iess et lF IWWV eF qF iess et lFD yservtionl ividene from uE pernove for n eelerting niverse nd gosmologil gonstntF etD IIT X IHHWEIHQVD IWWV rxivEWVHSPHIvI F grroll PHHI en wF grrolFD he gosmologil gonstntF viving eviews in eltivityD R XID PHHI F grroll PHHR grrollD F wFD hy is the niverse eelerting c PQSEC ges of preedmnD F vF @edAD estronomil tournlD PHHR wesuring nd wodeling the niverseF F einerg IWVW einergD F IWVWF he osmologil onstnt prolemF eviews of wodern hysisD TI @tnF IWVWAD IEPQ IWQ BIBLIOGRAPHY F erlmutter et lF IWWW F erlmutter et lF nd upernov gosmology rojetF wesurements of ymeg nd vmd from RP righEedshift uE pernoveF eptD SIU XSTSESVTD tune IWWW fF tr IWVV trD fFD nd eelesD F tF iFD gosmologil gonseE quenes of olling romogeneous lr pieldD hysF evFD IWVV ehE IWVVhvhFFQUFQRHT gF etterih IWVV etterihD gFD gosmology nd the pte of hilttion ymE metryD xulF hysFD IWVV ehEIWVVxuhfFQHPFFTTV F gldwell PHHP gldwellD FD e phntom menec osmologil onseE quenes of drk energy omponent with superEnegtive eqution of stteD hysFvettF fSRSD PHHP PHHQ gldwellD F FD umionkowskiD wFD nd einergD xF xFD hntom energy nd osmi doomsdyD hysFevFvettF WID PHHQ rxivEHQHPSHT F ghi et lF PHHH ghiD FD ykeD FD nd mguhiD wFD uinetilly driven quintesseneD hysFevFh TPD PHHH gF ermendrizEion PHHH ermendrizEionD gFD et lFD e dynmil soluE tion to the prolem of smll osmologil onstnt nd lteEtime osmi elertionD hysFevFvettF VSD PHHH PHHI ermendrizEionD gFD wukhnovD FD nd teinhrdtD F tFD issentils of kEesseneD hysFevFh TQD PHHI F ru IWWU ru FD ugiym xF & ilk tFD he hysis of wirowve fkE ground enisotropiesD xtureD QVTD IWWU rxivEWTHRITTvI tohnson nd worgn IWSQ tohnsonD rF vF & worgn F FD pondmentl stellr photometry for stndrds of spetrl type on the revised system of the erkes spetrl etlsD eptD volume IIUD IWSQ ehEIWSQeptFFFIIUFFQIQt fF tF ylor IWVT ylotD fF tFD rnsformtion equtions nd other ids for s photometryD eptD THD SUUESWWD IWVT ehEIWVTeptFFFTHFFSUU hF FD ryes et lF IWVS ryesD hF FD sinettiD vF iFD hilipD eF qF hFD glE irtion of fundmentl stellr quntitiesD roeedings of the ymposiumD gomoD stlyD IWVS ehEIWVSseFFIIIFFFFFr fohlin et lF PHHR fohlinD F gFD qillilndD F vFD rule pe elesope esolute petrophotometry of eg from the prEltrviolet to the snE frredD etD IPUD PHHR ehEPHHRetFFFFIPUFQSHVf 194 BIBLIOGRAPHY pukugit et lF IWWT pukugitD wFD shikwD FD qunnD tF iFD hoiD wFD hiE mskuD uFD hneiderD hF FD he lon higitl ky urvey hotometri ystemD etD vFIIID IWWT ehEIWWTetFFFFIIIFIURVp tF fF yke et lF IWVQ ykeD tF fFD qunnD tF iFD eondry stndrd strs for solute spetrophotometryD etD vF PTTD IWVQ ehEIWVQeptFFFPTTFFUIQy uin et lF IWUH uinD er gFD pordD F uentD trFD ottion of the enE dromed xeul from petrosopi urvey of imission egionsD estroE physil tournlD volF ISWD pFQUWD IWUH ehEIWUHeptFFFISWFFQUW riyD F et lF IWWH qF figotD F riyD e null orreltion tehnique to deterE mine q • D in he quest for the fundmentl onstnts in osmologyD roF of the PRth enontre de woriond @Wth woriond estrophysis weetingAD @IWWHA IUU uF urisiuns IWWQ urisiunsD uevinD vook fk imeD the ege of the niE verseD nd the gse for ositive gosmologil gonstntD tF oyF estronF oF gndD volume VUD pges PPQD IWWQ wesuring nd wodeling the niE verseF rxivEWQHTHHP fetouleD wF et l PHIR fetouleD wF et lD smproved osmologil onstrints from joint nlysis of the hEss nd xv supernov smplesD estronF estrophysD volume STVD pges 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volF RIQD noF PD pF vIHSEvIHVD IWWQ ehEHQHITHQ rumsonD wyllD 8 ndge IWST rumson wF vFD wyll xF FD ndge eF FD edshifts nd mgnitudes of extrglti neuleD estronomil tournlD olF TID pF WUEITPD IWST ehEIWSTetFFFFFTIFFFWUr yke 8 ndge IWTV yke tF fFD ndge eFD inergy histriutionsD u gorE retionsD nd the teinsEhitford i'et for qint illiptil qlxiesD estrophysil tournlD volF ISRD pFPID IHEIWTV ehEIWTVeptFFFISRFFFPIy roggD hF F PHHP roggD hF FD he u orretionD stroEphGHPIHQWRvID PHHP rxivEHPIHQWRvI oss xF F PHIQ oss xF FD he hEsss fryon ysilltion petrosopi urveyX he usr vuminosity puntion from ht elese xineD estroE physil tournlD olume UUQD xumer ID PHIQ finney 8 werri(eldD IWWV finney tFD werri(eld wD qlti estronomyD rineton niversity ressD pge IIID IWWV golless wFD PHHS golless wFD he xew gosmologyD orld ienti(D pge PHD PV tun PHHS estier F et lD PHHT estier F et lD he upernov vegy urveyX mesureE ment of Ω M D Ω Λ nd w from the (rst yer dtD estronomy 8 estrophysE il tournlD olume RRUD pFQI E RVD xumer ID perury sss PHHT ehE PTeFFFRRUFFFQIe rF plihe 8 tF wF ouriu IWUW rFrF pliheD tFwF ouriuD usrs et osE mologieD estronF 8 estrophysF 78 @IWUWA VU heeler IWWH heeler tgFD heeler et l ppF IEWQD IWWH gF wF rryD F dovniD IWWS rryD gF wF nd dovniD FD ni(ed hemes for dioEvoud etive qlti xuleiD ulitions of the estroE nomil oiety of the i(D vFIHUD pFVHQD IWWS ehEIWWSeFFIHUFFVHQ wF hmidt IWTQ hmidtD wFD Qg PUQ X e trEvike yjet with vrge edE hiftF xture IWUD IWTQ ehEIWTQxturFIWUFIHRH F entonui IWWQ entonuiD FD ni(ed models for tive glti nulei nd qusrsD e&e QID RUQESPID IWWQ tF qunn nd fF eterson IWTS qunnD tF iF nd etersonD fF eFD yn the henE sity of xeutrl rydrogen in snterglti peF ept IRPD ITQQâITRID IWTS ehEIWTSeptFFFIRPFITQQq 196 BIBLIOGRAPHY iF womjin et l PHIQ womjinD iF nd grilliD gF vF nd lterD pF nd enemnsD fFD he righest edshift usr t z = 7.085X e dio uiet oureF estronF tF IRUD pFTD PHIQ rxivEIQIHFUWTHvI hF xF pergel et l PHIQ pergel hF xF et lD hreeEer ilkinson wiE rowve enisotropy roe @weA yservtionsX smplitions for gosE mologyF estrophysil tournlD olume IUHD pFQUUD PHHU hF ferkD et l PHHI pergel hF et lD gomposite usr petr from the lon higitl ky urveyF estronomil tournlD olume IPPD pFSRWD PHHI ehEPHHIetFFFFIPPFFSRW ient tF wrtinez et l ient tF wrtinezD inn rD inrique wrtinez qonzlesD wri tesus onsEforderiD ht enlysis in gosmologyD PHHW F hrp et l PHHI hrp FD rodgkin FD wwhon FD srwin wFD righ edshift usrs in the sxq ide pield urveyF he sxq newsletterD xoF RD pFIQ wrh PHHI rmuy et lF IWWS rmuyD wrioY hillipsD wF wFY wzD toseY untze'D xihols fFY hommerD F eFY evilesD F e0litionX e rule digrm of distnt type s supernoveD estronomil tournlD volF IHWD noF ITTWD pF IEIQD IWWS ehEIWWSetFFFFIHWFFFFIr IWWT rmuyD wFD hillipsD wF wFD untze'D xF fFD hommerD F eFD wzD tFD evilesD FD he esolute vuminosities of the glnGololo type s uE pernoveD estronomil tournlD volume IIPD pFPQWID IWWT rxivEWTHWHSW iess et l IWWS iessD edm qFY ressD illim rFY uirshnerD oert FD sing 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Principle of inertia, fundamental principle of dynamics and principle of reciprocal actions

Due to the source or observer movement, two dierent lines of sight illustrate two positions of the source with an angle that allows to measure distance using geometry.

c = 1, i.e. time can be measured in unit of length, 1s = 2.999792458 ×10 8 m S

Mpc: 10 6 × pc. A parsec (pc) is a unit of measurement of distance outside the solar system and it is used to express the measuring of the cosmological distances. It is equivalent to 3.26 light years: 1pc = 3.08 × 10 16 m.

Isotropy from every point implies homogeneity [Robert M.Wald 1984]

In a common language (poor), it is the time taken by the emitted light to travel from an object at redshift z to the observer [K.Krisciunas 1993].

A perennial object is an object which emits a stream of light and has been doing so since the inception of the observable universe, since the decoupling epoch, and for all time, such as galaxies, quasars and clusters.

This simulation sample is dierent than the one obtained by the MCMC method where the major drawback of this latter is that it is extremely time consuming [Vicent J. Martinez et al ] (see COSMOMC based on MCMC approach)

pigure QFISX uolmogorovE mirnov test @vikelihood of ndidtesAF erh for the minimum of D max @top leftA s funtion of λ • D nd @ottom rightA s funtion of Ω • D on the xgg @top rightAF

Note that if one limits solely to obtain these roots one can use the (weighted) covariance instead.[START_REF] Abazajian | The Third Data Release of the Sloan Digital Sky Survey[END_REF] Chapter 4 : Application to quasars data

Which can be converted to Johnson-Cousins bands system (B, V, R, I) referring to the Landolt catalog (1992) based on the Vega system of magnitude.

c = B -V = -2.5log 10 ( f B f V ) IHS

http://supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html

@eFIA he proility density funtion for the qumel @nmed for qermn mthemtiin imil tulius qumelA distriution isX p(x; x 0 , γ x ) = e -(x-x 0 )/γx γ x e -e -(x-x 0 )/γx , @eFPA where x 0 is the modeD lotion prmeterD nd γ x is the sle prmeterF he pdf hs men of x 0 + 0.57721γ x nd vrine of π Volume . append (V_z) M l i s t l i s t . append ( M l i s t [ i i ] ) e l s e : p r i n t "You must determine the c o n d i t i o n o f the K c o r r e c t i o n as ' Kcorrection_OK ' or ' Kcorrection_NO ' . " , "\n" M l i s t l i s t = np . a s a r r a y ( M l i s t l i s t ) M l i s t = M l i s t l i s t * 1 Weighting = ApparentMagnitude ( r e d s h i f t , lambda0 , omega0 , 0 . 0 , Kcorrection , redsh_k , k_z ) . W e i g h t i n g f a c t o r ( r e d s h i f t , beta ) SumWeighting = sum( Weighting ) f a c t o r W e i g h t i n g = array ( Weighting ) /SumWeighting #======================== #=======s a v i n g data===== f i l e n a m e = ' SimulationData_ '+s t r ( lambda0 )+ '-'+s t r ( omega0 )+ ' _NbOfObjects '+s t r ( S i z e o f s a m p l e )+ ' _limitMag '+s t r (m_lim)+ ' _NbOfSimulation '+s t r ( Number_of_simulation )+ ' Beta '+s t r ( beta )+ ' . dat ' l i s t o f a r r a y s = [ Mlist , m_app, r e d s h i f t , r e d s h i f t _ l , Volume , Volume_l , Weighting , factorWeighting , redred , z1 , redsh_k , k_z ] s t r i n g l i s t o f a r r a y s = [ ' M l i s t ' , 'm_app ' , ' r e d s h i f t ' , ' r e d s h i f t _ l ' , ' Volume ' , ' Volume_l ' , ' Weighting ' , ' f a c t o r W e i g h t i n g ' , ' re d re d ' , ' z1 ' , ' redsh_k ' , ' k_z ' ] Save ( l i s t o f a r r a y s , s t r i n g l i s t o f a r r a y s , f i l e n a m e ) . ReadWrite ( True , ' work ' ) path = Save ( l i s t o f a r r a y s , s t r i n g l i s t o f a r r a y s , f i l e n a m e ) . ReadWrite ( True , ' work ' ) paths_of_samples . append ( path ) #====================== p r i n t " Number_of_simulation = " , Number_of_simulation , "\n" p r i n t " path = " , path , "\n" ---------------#s e a r c h o f c o v a r i a n c #---------------- ---------------#