Abstract

prom numeril simultionsD oupling the growth of lrge sle struture nd of drk mtter hlos with gs physisD we now understnd theoretilly tht glxy evolution involves in)ows of old gs from the osmi weF qlxies in these old retion models hve lrge mounts of ryons ompred to oserved glxiesF o overome this issueD theorists fous on mking str formtion ine0ient y lowing ryons out of strE forming disks through mssive gseous out)owsF sn my thesisD s explore di'erent rodD investigting proesses tht my moderte gs retion onto disksF wy methodology is sed on multiphse gs physis honed through studies of the interstellr medium @swA in our own nd nery glxies over the pst dedes using nlytil modelling nd numeril simultions to follow the gs thermodynmis nd struturtion from osmi we (lments to glxy hlosF sn ghpter PD we present phenomenologil senrio where gs retion )ows ! if it is shoked s it enters the hlo s we ssume ! eome iphsi ndD s resultD turulentF e onsider ollimted strem of wrm gs tht )ows into hlo from n over dense (lment of the osmi weF he postEshok streming gs expnds euse it hs higher pressure thn the mient hlo gsD nd frgments s it oolsF he frgmented strem forms two phse mediumX wrm loudy phse emedded in hot postEshok gsF e rgue tht the hot phse sustins the retion shokF huring frgmenttionD frtion of the initil kineti energy of the inflling gs is onverted into turulene mong nd within the wrm loudsF he thermodynmi evolution of the postEshok gs is lrgely determined y the reltive timesles of severl proessesF hese ompeting timesles hrterize the oolingD the expnsion of the postEshok gsD the mount of turulene in the loudsD nd the dynmil time of the hloF e expet the gs to eome multiphse when the gs ooling nd dynmil times re of the sme orderEofE mgnitudeF sn this frmeworkD we show tht this ours in the importnt mss rnge of w halo ∼ 10 11 -10 13 M D where the ulk of strs hve formed in glxiesF hue to expnsion nd turuleneD gs reting long osmi we (lments my eventully lose oherene nd mix with the mient hlo gsF hrough oth the phse seprtion nd disruption of the stremD the retion e0ieny onto glxy in hlo dynmil time is loweredF heEollimting )ows mke the diret intertion etween glxy feedk nd retion strems more likelyD therey further reduing the overll retion e0ienyF es we disussD moderting the gs retion e0ieny through these mehnisms my help to llevite numer of signi(nt hllenges in theoretil glxy formtionF o estlish the likely initil onditions for retion )ows into hlosD using the dptive meshEre(nement ode RamsesD s performed zoomEin simultion of spei( hlo nd extrted the results for prtiulr reting interglti (lment into ∼ 310 11 M t z ∼ 2F his prt of my work is presented in ghpter QF efter hving presented the need for simultionD s rie)y desrie the ode RamsesF s investigte the gs thermodynmis nd struturtionD long nd ross the (lmentD with respet to drk mtterF s study severl key quntities s they evolve long the (lment nd derive re(ned prdigm to study (lmentsD s well s onsequenes regrding their fte fter v entering hloF s (nlly mke use of these results to extrpolte gs proesses tht the simultion my not hve ptured urtelyF sn ghpter RD s o'er some onlusions nd perspetives on how to extend our pheE nomenologil model presented in ghpter P nd how the results of simultions from ghpter Q relte to our phenomenologil results generllyF ith this thesis s strted exploring nd understnding the importne of nonEliner gs physis within the ontext of lrge sle struture nd glol hloEglxy evolutionF s introdued some of the physis of the multiphse gsD proesses whih re well known in the swD ut not wellEmodeled within osmi we (lmentsF s see my work s (rst step towrds dedited numeril studies for understnding the impt of gs physis on glxy evolutionF vi Résumé ves simultions numériquesD prennt en ompte le ouplge des strutures d9éhelle osE mique à l physique des gzD nous ont ppris que l9évolution des glxies doit impliquer un pport de gz froid provennt de l toile osmiqueF ves modèles en tennt ompte onduisent à des glxies plus rihes en ryons que elles oservéesF our surmonter e prolèmeD les théoriiens oisissent de rendre l formtion stellire ine0e grâe à une éjetion mssive de gz pr les disques en trin de former leurs étoilesF hns ette thèseD j9explore une utre voieD en étudint les proessus qui pourrient tténuer l9rétion de gz sur les disquesF w méthode s9ppuie sur l physique des gz multiphses onnue à trvers les études du milieu interstellire de notre glxie et de ses prohes voisines u ours des dernières déenniesD à l9ide d9une modélistion nlytique et de simulE tions numériquesD pour suivre l9évolution thermodynmique et struturelle du gzD des (lments osmiques ux hlos gltiquesF ve hpitre P expose un sénrio phénoménologique dns lequel l9rétion du gzD s9il y un ho virielD devient iphsique etD en onséqueneD turulentF xous omE mençons pr onsidérer un ournt ollimté de gz tiède qui s9éoule dns un hloD à l9emouhure d9un (lment osmiqueF v9éoulement postEho s9étend puisqu9il une pression supérieure à elle du gz environnnt dns le hlo et se frgmente u ours de son refroidissementF ve milieu omporte lors deux phses X un phse nugeuse et tiède ignnt dns le gs resté hu'é pr le hoF xous expliquons en quoi ette phse hude supporte le hoF eu ours de l frgmenttionD une frtion de l9énergie inéE tique est onvertie en une turulene prmi et u sein des nuges tièdesF v9évolution thermodynmique est lors en grnde prtie déterminée pr les rpports d9éhelles temE porelles de di'érents proessusD à svoir le refroidissementD l9expnsionD l turulene et l dynmique propre u hloF yn s9ttend à e que le gz devienne multiphse qund les temps de refroidissement et de l dynmique du hlo sont omprlesF hns e sénrioD nous montrons que el rrive dns l gmme de msses de hlo w halo ∼ 10 11 -10 13 M D pour lesquelles l mjorité des étoiles est déjà formée dns les glxiesF v9expnsion et l turulene peut fire perdre à terme leur ohérene ux (lments et entrîner leur dissoE lution dns le gz du hloF v séprtion des phses et l disruption de l9éoulement onduit lors à moindrir l9e0ité d9rétion sur l glxie en un temps ynmique du hloF v déollimtion roît de même l proilité d9intertion entre le ournt d9rétion et les éjetions gltiquesD diminunt d9utnt plus l9e0ité d9rétionF xous disutons en(n de l9e'et de ette diminution en tnt qu9elle permet de résoudre un ertin nomre de prolèmes théoriques onernnt l formtion des glxiesF e(n d9étlir les onditions initiles propres ux éoulements d9rétion sur les hlos et qui soient rélistesD nous vons utilisé le ode à r0nement dpttif de millge Ramses pour simuler de fçon ilée un (lment intergltique rétnt sur un hlo de msse ∼ 3 10 11 M à z ∼ 2F gette prtie de mon trvil est exposée dns le hpitre QF eprès voir étli l néessité d9une simultion numériqueD les propriétés du ode luiE même sont présentéesF t9i ensuite étudié l thermodynmique et l struturtion du gzD le long et à trvers le (lmentD orréltivement à l distriution de mtière noireF vii t9i suivi l9évolution de quntités importntes le long du (lment et i dérivé un dre plus judiieux pour l9étude des (lmentsD insi que les onséquenes ttendues sur leur destin près voir pénétré dns le hloF t9i en(n extrpoléD à prtir de es résulttsD ertins proessus que l simultion pouvit n9voir ps pturé ve préisionF hns le hpitre RD je propose des onlusions générles insi que les développements possiles de notre modèle phénoménologique présenté u hpitre PD insi qu9une mise en rpport plus générle des résultts du modèle et de l simultionF eu ours de ette thèseD j9i ommené à explorer et à ironsrire le rôle joué pr l physique nonElinéire du gz dns le ontexte des strutures osmiques et de l9évolution glole des ouples hloEglxieF t9i pu introduire une prt de l physique des gz multiphseD ien onnue en e qui onerne le milieu interstellireD mis enore imprfitement modélisée dns le dre des (lments osmiquesF ge trvil onstitue une premièr étpe vers une série d9études numériques spéilement dédiées à l ompréhension de l9impt de l physique des gz sur l9évolution des glxiesF viii ix Foreword At the end of the writing of this thesis, I wondered: how is this work in particular related to science in general? Here is my view on science, and on how my particular experience relates to it, introducing the outline of this manuscript.

A lot has been written on science, but what results from it, is that there is no clear and simple formulation circumscribing all of what science is. Is it only a search for truth? Is it particular because of its usage of mathematics? Is it dened by a back and forth motion between experiment and theory? Has it emerged because of its relative ability to predict? Can it be reduced to the hypotheticodeductive model? All of these limiting assertions are way too simple and too naive.

I will propose here a very crude description of how science works.

Science is the product of social activities of nding correlations in nature, constrained by what the agents recorded from nature and by self-consistency, of nding explanations suciently convincing to improve the overall consistency, and of keeping a memory of itself.

I intentionally omitted any term like aiming at or trying to.

It is a social activity. Science is made in a collective and organized way. Scientists have to think collectively, then share their ndings and convince of the reliability thereof. Finding alone and keeping it to oneself is not doing science.

Publishing is neither an end to a scientic process, but rather a step in its course.

Convincing the community is fundamental. It makes science subject to individual and sociological internal conicts of course. More, since scientists also live in a broader society, their interactions outside of their eld matter, especially when metaphysical, ethical or technological questions arise therefrom. This is why science is essentially social. But rather than making it awed, this essence makes it dynamic.

Its internal functioning is also formally, even if not always explicitly, constrained. The constraints come rst from observed data, that is from nature. But these do not constitute an ultimate guide. One needs to know that no observation can be categorized as direct or indirect. It is always indirect, mediated by several physical transformations and observational biases. What are the measured 1 quantities? What do they trace? What is their accuracy? Previously established models are often needed to simply interpret what is seen in nature, setting up loops of trust that can be one day broken. Not only is the trust in measurements dicult to establish, but the way they are represented has much to do with their acceptation by a scientic community. To nd correlations, certain combinations of observed quantities are to be tested and this choice may already be a source of distrust. Is the correlation robust? Is there enough data? What are the odds that it be coincidental?

After having recognized a new correlation as universal, or at least suciently general, scientists are eager to provide an explanation. This has nothing to do with a quest for causality. Explaining correlations consists in forging a symbolic structure. The most technical way of explaining involves modelling to some degree.

Modelling is an abstract construction of reality which needs:

• to conceive, that is gathering a minimal set of relevant concepts and detailing relationships therebetween,

• to formalize, that is gathering a minimal set of symbolic forms and ordering them to translate the conceptual links.

Mathematics provide us with a considerably powerful symbolic formalism. And even if it leaves, in general, no room to doubt when used rigorously, it is rarely the essence of conviction. The choices lie rather mainly in the conceptualization, and both requirements of a minimal set and of its relevance are often the heart of the issue.

A given personal scientic work can also become a game 1 . So many observations were made through exploration, guided by a more or less sharp intuition, sometimes even luck! I can remember a past experimental work I was involved into, where by playing with the knobs of instruments I found new regimes of a physical instability. And this exploration is not limited to observation and experiment.

Exploring by means of models may be quite vertiginous and exciting. Given the existing models, one can extend their results to what has not yet been observed. We may refer to this as prediction. When scientists extrapolate from already accepted models, they address the invisible, the unknown. Instead of fabricating a magical belief related to a mystic world, they keep following the aforementioned constraints while stating a consistent hypothesis about unknown or unsure situations.

This is what we have done here. Few information from observational instruments indicate the existence of intergalactic laments. Given the expected central role of laments in shaping the evolution of galaxies, which we will set forth in 1 I here mean a game with no competition and sometimes with strict rules, sometimes very free. In Roger Caillois' words, it would be more alea than agôn, more ilinx than mimicry, and oscillating between ludus and paidia.

Chapter 1, we chose to address what is not yet known, namely their descriptive and dynamical properties.

Current knowledge about gas physics at galactic and cosmological scales allowed to consistently envision what could happen to the gaseous laments when approaching galaxies. We stated that, in common circumstances, microphysics alone would play such a role that accreting lamentary streams could lose their identity. The expected timescales of interaction with galaxies and their close environments are then reassessed. This work is presented in Chapter 2.

A combination of physical, fundamental models with conditions that are currently considered realistic allows for a mathematical estimation of plausible solutions. Given the complexity of the said conditions, a numerical computation was used to investigate the state of resulting intergalactic laments. The analyses of the simulation revealed a possibly complex structuration of lamentary streams before their entry in a galactic halo, product of an interplay between dynamics ascribed by cosmic structures and physics specic to a condensing gas. We report our ndings in Chapter 3.

Both consistently arose from current knowledge but gure new results, whose in turn are to be tested observationally. This is what would be enjoined by the empiricism. However, as said before, science can not be reduced to a simple cycle of hypothesis and falsication. We may later nd that our results are not as general as we thought, or new developments may stem from them. Science is about acceptance of a nding by a community. I hope that this work will be a convincing, building brick to erect one new piece of the scientic edice.

Chapter 1

The role of dark matter and gaseous laments in galaxy evolution Abstract Over the last decade, it has been realized that galaxies may acquire much of their dark matter and baryonic mass through accretion along lamentary structures.These structures grow gravitationally from small uctuations in the density eld in the early Universe. Why are intergalactic laments so important to the eld of galaxy evolution? Such accretion ows shape many of the essential properties of galaxies, like their shape, their colors, their mass, their gaseous and stellar contents, and so on. We can only understand galaxy evolution as a balance or competition between gas accretion, through lamentary structures driven by the growth of large scale structure, star formation, and the outows intense star formation or active galactic nuclei may produce. But the fuel for all this activity and galaxy growth are all connected to the way matter has fallen, and is falling, onto galaxies, in combination with the way they react to the condensation processes and outows. Let us revise now the dierent theoretical elements needed to understand the role played by intergalactic laments (IGFs) in galaxy evolution.

Cosmological framework

The most fundamental principles governing the characteristics of the Universe, its expansion rate, mix of baryonic to dark matter, dark energy, its initial primoridial uctuation spectrum, are all of prime importance for understanding of processes that shaped the objects in the universe over the last ∼13 Gyrs, since the time of the Big Bang. The growth and evolution of galaxies is no exception. The cosmological conditions are the rst building blocks of any pertinent theory aiming to explain the history of one given galaxy, and of course, that of the ensemble of observed galaxies. The most important of these are the rate and rate of change of the cosmic expansion and the dominance of dark matter over (ordinary) baryonic matter. These are most consistently explained through variants of the cold dark matter with a non-zero cosmological constant, the Λ-CDM, model.

Cosmic expansion

The further the luminous bodies, the longer it took the light to travel from the source to the observer. But that distance also correlates with a widening of the wavelengths, making visible light become redderthis is the redshift :

z = λ -λ 0 λ 0 (1.1)
that is, the light initially emitted at a wavelength λ 0 is observed at a wavelength λ and the redshift z measures, for any wavelength, the light travelling time.

The most satisfying and convincing explanation for redshift is the cosmic expansion. This led to the introduction of a new energy source term Λ in Einstein's eld equation of general relativity, called the cosmological constant. The corresponding fraction of mass-energy density to the one needed to stop the ination (i.e. to make the Universe at) is Ω Λ and is estimated from Planck measurements (Planck Collaboration et al., 2014, 2016) of the CMB to be about 0.69. The linear term in the rate of expansion of the Universe is the Hubble's constant, H, and its current value is estimated by the Planck collaboration to be H 0 ≈ 67.3 km.s -1 .Mpc -1 .

The density required for a at Universe is called the critical density and is dened by:

ρ crit = 3H 2 8πG (1.2)
where G is the gravitational constant. This also means that a given redshift z also corresponds to a recessional velocity. The further the object, the faster it apparently recedes from the observer. This phenomenon impacts not only electromagnetic waves, but more generally all quantities whose dimension is proportional to a length at some non-zero power. Three simple examples reveal the ubiquity of the notion, when writing physical models: densities, sizes and velocities.

1.1.2 Dark and baryonic matter Several peculiarities arose from observations. Rotation curves of galaxies, if one only considered the distribution of luminous baryonic matter, show unexpected behaviour at large radii: they are almost at. It was expected from Newtonian dynamics applied to the mass of detected stars and gas that the orbital speed should decrease inversely to the distance. More, general relativity allows for light rays to be bent by masses, resulting in the so-called lensing eect. From the deformation of a luminous background, we can derive the value of a mass lensing it. Surprisingly, lensing galaxy clusters appeared to have a much higher mass than that computed from its stellar content 1 . Finally, the study of the cosmic microwave background led to an oddity: its spatial power spectrum has peaks, signatures of primordial acoustic oscillations, but their amplitudes were incompatible with models that only considered luminous matter.

A solution to the three problems is now widely accepted: there exists in the Universe a massive matter that does not produce radiation, and its mass density exceeds that of light-emitting matter. The latter is referred to as baryonic matter 2 , the former being dark matter (DM), and whose fundamental nature is still unclear.

Neglecting the density of mass-energy of radiation and assuming a at Universe, the fraction owing to matter is Ω m ≈ 0.31 and, still from Planck measurements (Planck Collaboration et al., 2014, 2016), the fraction for baryons is Ω b ≈ 0.05, leaving a fraction of DM Ω DM ≈ 0.26. The ratio f b = Ω b /Ω m ≈ 0.16 is called the baryonic fraction and this cosmic value is considered constant 3 . This enables us to write the redshift-dependent Hubble constant:

H(z) 2 = H 2 0 Ω m (1 + z) 3 + (1 -Ω m -Ω Λ )(1 + z) 2 + Ω Λ (1.3)
and, for a at Universe, the middle term of the parenthesis vanishes.

Dark and baryonic matter are thought to follow the same initial distribution but with dierent amplitudes, in proportion of their cosmic fractions. This is referred to as coupling of DM and baryons. The dark matter particles must have had initial velocities negligible compared to the speed of light to let the models t with observational data (see below about the large-scale structures) and this is 1 We could also add the cases of bound galaxies in superclusters whose dynamics suggested a higher mass than that derived from total luminous mass in galaxies.

2 The term is not the most accurate one: electrons are indeed not baryons, i.e. they are not sensitive to strong interaction. A more general term could be fermionic matter, since baryons are composed of quarks, which are fermions like the electrons.

3 Neither DM or baryons are produced, destroyed, or converted one into the other.

why the currently leading theoretical ground on which astrophysics stands now is called Λ-CDM, for non-zero cosmological constant, Λ, cold dark matter.

Initial density uctuations

Theory suggests that, in the young Universe, the distribution of matter was in a state of a slightly perturbed but generally homogeneous state. Such a smooth eld with small uctuations may be considered Gaussian in rst approximation its Fourier modes have independent imaginary and real parts, which simplied calculations of its evolution. The phases of these modes are however of prime importance in determining later growth of structure with the density eld. Statistical properties of those uctuations are derived from the knowledge gained in the last decades about the cosmic microwave background, notably the power spectrum.

Research on non-Gaussianities is still currently at the frontier of what is known or knowable. Gravity then drives matter towards regions of highest densities. This amplication of density peaks has been studied theoretically by means of (early) linear and (more recent) non-linear analysis.

1.2 Gravity in action

Large scale structures

The consideration of a gravitational and small scale uctuations within the cosmological density eld in three dimensions led to the prediction that matter should grow an organized characterizable structure. Dark matter halos sitting at the nexus of elongated, overdense regions (laments), themselves circumscribing slightly overdense patches (sheets or walls), themselves enclosing under-dense compartments (voids). We can see it as, at rst, an amplication of distinct features of a bidimensional density eld: low density valleys get deeper, intermediate density crests get sharper, and high density summits get higher.

Early views of halo formation

The modelling of the evolving density eld has given a more solid ground to the idea that density peaks, or halos, grow by acquiring mass from their environment. After the setup of an ideal, isotropic formulation for a self-similar spherical collapse, the mathematical developments of tridimensional, anisotropic growth allowed for ellipsoidal (triaxial collapse), rather than perfectly spherical halos. This more realistic depiction would then support some fundamental explanations for the strong asymmetry of their observed central luminous components, namely the galaxies that lie within these dark matter halos. However, even very elongated halos can not t the proles of observed laments, nor can very oblate halos t the observed sheets.

Detection

The easiest method to detect these particular spatial arrangements of dark matter was to look at the way galaxies and clusters were distributed in three dimensions (angular position and redshift (1985) studied the structures to nd that the Λ-CDM paradigm was particularly adapted to numerically reproducing observations, a work that opened the way to many others.

Fig. 1.1 shows a galaxy distribution slice of a plane in the sky computed from the galaxies the Sloan Digital Sky Survey (SDSS). In this slice following the northern equatorial plane, lamentary distributions of galaxies (here plotted in each galaxy's g-r color) are visible. Fig. 1.2 is a snapshot of the Millennium Simulation (Springel et al., 2005). Represented is the dark matter distribution, also organized in large-scale structures. Even visually, it is evident that galaxies and dark matter are both distributed in halos, laments, walls, and voids.

Often described as a honeycomb or form architecture, the pieces of that cosmic skeleton approximately draws a tessellation of the space, and they were entitled large-scale structures. In voids, matter ows as if expanding in all directions, as a negative pressure, until encounters a comparable ow from neighbouring voids, building panels, or sheets. In turn, matter from the sheets ows as if expanding on a surface, reaching other sheet-like ows and forming laments. Finally, matter expands in two directions along each laments, pulled to the regions where laments meet, namely the halos. This scenario for matter condensation is reminiscent of the chemical process of reticulation. From massive halos hosting galaxy clusters down to low-mass, dwarf galactic halos, the larger-scale structures should embed smaller-scale, similar ones. The model is hierarchical.

Dynamics

The formation and the ows associated with the growth of large-scale structure can be understood with a simple metaphor: rain, which occupies a tridimensional volume, falls on a sloping roof and forms a mostly bidimensional sheet, that in turn runs down to a gutter, giving rise to a thicker but mostly one-dimensional ow, and the water nally gets trapped into the sink that connects the gutters. Although useful, this picture proves inadequate, when describing the motion of the large-scale structures, and the associated deformation of the network. Indeed, dense structures not only give rise to a gravitational eld, but they also are subject to it. Individual structures, as they grow from the surrounding density eld, will eventually merge with any neighboring halos as they become gravitationally attracted. It is through this growth pattern where from the concept of hierarchical growth comes smaller halos grow and merge to form larger and larger halos. Halos growth from the small to larger scales, from numerous small low mass halos, into larger and larger halos.

In that way, halos, albeit pulling matter down their laments, are likely to ascend in the direction of their shortest feeding lament, pulled by the closest halo, at the other end. Depending on the mechanical conditions, gravitationally bound halos (for example, subhalos within a larger more massive one) may orbit each other. In the same way, laments can exhibit a woven pattern, with their axes transiently intertwined. Halos may undergo a merger, followed by some of their laments and nally some of their sheets.

Notwithstanding the cosmic expansion, these dynamics of the large-scale structure have a measurable impact: the two-point correlation functions should give dierent results depending on the redshift.

Halo growth

Merger mode

From the dynamics of large-scale structures, we have seen that the densest neighboring ones are attracted by each other. Because they travel until they merge with an similarly moving neighbor, and so on, halos follow a tree-like, or hierarchical storyline for their mass growth. The Press-Schechter formalism (Press & Schechter, 1974) is the earliest attempt to analytically predict the results. In a way, this paradigm for matter condensation is reminiscent of the process of coagulation and coalescence.

Accretion modes

As a corollary to the geometry of the large-scale structures and the associated ows, halos collect matter from laments, but also sheets in the corner of which they sit, and from voids in the vertices of which they sit. Similarly, laments accrete from sheets at the intersection of which they lie, but also from voids, Finally, sheets only receive mass ows from the voids they separate.

Smooth accretion, or (quasi-)spherical accretion corresponds to that coming from voids. It crosses a typical sphere around the halo with the widest angular sector but is also characterized by the lowest densities. Then comes the anisotropic accretion: from sheets (walls, or pancakes, representing the extremely oblate halos from the triaxial theory), and from laments. The lamentary accretion represents the densest ows (in both gas and dark matter) and cover the least angular sector on the spheres they cross.

Describing a halo

Halos, no matter how dark and baryonic matter constitute them, are commonly described by geometrical (a radius) and kinematic (a speed) parameters, that in turn are related to its mass M H and the redshift z. From a model of an initially isothermal gas exhibiting an isolated spherical over-density, Jeans derived a criterion to characterize the scales at which the system would be unstable to a gravitational collapse. Under the assumption that the system is initially virialized, we can derive the average over-density, compared to the critical density at that redshift. Overdense structures are gravitationally bound and will collapse. With the cosmology know cosmological parameters, yields an overdensity of ∆ = ρ H /ρ crit ≈ 100 (Bryan & Norman, 1998). But in this paradigm the Universe was thought to be matterdominated, the same calculation gave ∆ = 18π 2 ≈ 177.7 and most researchers continue to use this value of 200. Nevertheless, the idealized models being far from the reality, it is really only an arbitrary choice, chosen simply to make it easier to compare dierent and especially, older, studies.

Considering the sphere centered around the density peak of a halo and which, on average, has a density ∆ times the critical one, its radius is then set and denoted r ∆ . Its relation to the halo mass is simply:

M H = 4 3 π∆r 3 ∆ ρ crit (1.4)
and conversely, if the mass had been given rst:

r ∆ = 3M H 4π∆ρ crit 1/3 (1.5)
Using the denition of the critical density,

r ∆ = 2GM H π∆H(z) 2 1/3 (1.6)
Note that if the Universe was matter-dominated, we would get

r ∆ ∝ M 1/3 H (1 + z) -1 (1.7)
This means that, the halo mass being constant, the virial radius grows with time (i.e. with decreasing redshift). This eect may sometimes produce the illusion of growth (pseudo-growth), although the cosmic expansion is the sole responsible.

A typical order of magnitude for r ∆=200 is 100 kpc.

Having dened a radius and assuming a spherically symmetric distribution, it is possible to compute the speed at which a mass point would orbit precisely at that radius. It is called the virial velocity (for it may be derived in the simplest way from the virial theorem) and is dened by

v vir = GM H r ∆ 1/2 = π∆ 2 H(z)GM H 1/3 (1.8)
Note that if the Universe was matter-dominated, we would get

v vir ∝ M 1/3 H (1 + z) 1/2
(1.9)

A typical order of magnitude for v vir is 200 km. (2007), in collisional ionization equilibrium, for ve dierent metallicities (relative to solar).

.cm 3 ]) log(T [K]) Z = 10 -3 Z = 10 -2 Z = 10 -1 Z = 1 Z = 2 Figure 1.3: Cooling functions from Gnat & Sternberg
t dyn = r vir v vir (1.10)
which is close to a free-fall time from the virial radius to the center of the halo, if its mass was concentrated in its center, or alternatively an orbital time. It is remarkable that this time does not depend on the halo mass, and in a matterdominated Universe, it would be simply proportional to (1 + z) -3/2 .

1.3

The importance of gas physics

Between gas and dark matter, a decoupling may occur at some stage of the condensation process, due to (baryonic) gas physics emerging from microphysical interactions (e.g., collisions) and the transitions between and losses of dierent energy types (e.g., thermal, micro-kinetic, to electronic excitation through collisions leading to radiative cooling as the atoms recombine or change electronic states). The gas may either resist gravitational forces due to its own internal pressure, or endure a secondary condensation phase resulting from its cooling in pressurized environment and self-gravitation.

Radiative cooling

The interaction between the particles of gas may convert its internal energy into radiation. If the gas is largely transparent to this radiation, radiative losses represents a net cooling. The temperatures of interest in this work concern largely the circum-galactic and intergalactic media, and thus range from 10 4 to 10 8 (but only for the most extremely massive halos at low redshifts).

More precisely, for temperatures between 1 and 3 10 4 K, atoms of hydrogen get ionized, while helium atoms are collisionally excited. When ions recombine with an electron, they emit a photon. When atoms spontaneously decay (or de-excite), they also emit a photon. Around 10 5 K, helium is ionized. At temperature higher than 10 6 K, another phenomenon occurs: the plasma is almost fully ionized and the electrical interactions make fast electrons to deviate from their linear trajectories.

This braking is, again associated with radiation: this is the bremsstrahlung, which gets more intense with faster particles, so its intensity increases with increasing temperature. Immediately below 10 4 K, a gas of composition close to the primeval one displays a very inecient radiative cooling; this explains our temperature oor that we will assume subsequently in our phenomenological modeling. It also justies in many simulations why the cooling is truncated at 10,000 K.

Pristine gas being composed only of hydrogen and helium (with number density ratio n He /n H ≈ 1/12), Fig. 1.3 gives an example of the cooling power per unit of number density of such a very low metallicity (10 -3 Z ). These curves have been computed under the assumption that the gas is at all times in a collisional ionized equilibrium. Other higher metallicities are also represented and we can see the strong impact of metals on the cooling rate.

Given such tabulated values for the cooling eciency Λ(T, Z), it is possible to evaluate the temperature loss:

dT dt = - n e n H Λ(T, Z) (3/2 + s)(n e + n H )k B (1.11)
with n e and n H the number densities of electrons and hydrogen respectively, s is either 0 or 1 if the cooling is assumed to be isochoric or isobaric, respectively.

Although the eect of pressure may prevent the gas from following precisely the gravitational eld generated by the dark matter distribution as the dark matter grows into ever denser structures, the radiative cooling is often crucial for relieving that pressure and allowing gravity and external work from a still hot environment to compress the gas to an even larger extent than the dark matter would have allowed on its own in the absence of radiative cooling.

Shock due to condensation

When do compression eects occur to gas as it is gravitationally driven by dark matter? Let us reason with a theoretical, idealized formalism. When dark matter converges toward a dense structure, DM particles cross the system, because matter is transparent to them. After one crossing, and while the structure gains mass, they turn around, thus describing an orbital curve around the barycentric locusa plane for sheets, an axis for laments and a point for halos. Once a distribution of mass is settled, with a density decreasing with the distance to that locus, newly accreted shells (planar, cylindrical or spherical) will cross the distribution. The surfaces of shell-crossing are called caustics and their mathematical study is still ongoing (see for example Hidding et al., 2014). At this point, the accompanying gas shells collide with the previously condensed ones.

The shell crossing of dark matter is hence a source of the decoupling between dark and baryonic matter. The gaseous shell collision translates into an increase in pressure, that propagates to further shells. As can be seen from the scalings given above (Eqns. 1.2,1.9), densities are proportional to (1 + z) 3 and speeds to (1 + z) 1/2 , meaning that the higher the redshift, the stronger the collision. It is then expected that a so-called accretion shock should occur in the external regions of dense structures.

At high redshift, this ecient conversion of kinetic energy density (scaling like

(1 + z) 4 ) to thermodynamic and the associated entropy production must shape not only the spatial discrepancy between dark and baryonic density gradients, but also the history of the gas condensation. This motivated Birnboim & Dekel (2003) and later Birnboim et al. (2016) in determining the stability of such shocks against the pressure loss due to the ensuing radiative cooling.

From warm gas to stellar light

From halo growth to galaxy growth

The rst global theory for the formation of a dense, gaseous region at the center of dark matter halos, presumably then into forming realistic galaxies, came with Rees & Ostriker (1977); White & Rees (1978a);Fall & Efstathiou (1980a). They stated that while matter settles in isotropically converging ows forming halos, gas is prevented from condensing because of its collisional nature. The high pressure supports it, its temperature increases as the halo grows in mass, scaling as if the speed of sound was close to an orbital speed in the halo. More precisely, we call this temperature the virial one and dene it as:

T vir = µm p 3k B v 2 vir (1.12)
where µm p is the average particle mass (m p being the mass of a proton), and k B is the Boltzmann constant. Its typical values are around 3 10 5 -10 6 K.

The densest, innermost gas would then cool radiatively, up to a radius where the cooling time (density-dependent, so radius-dependent, as soon as a density prole is specied 4 ) is equal to the dynamical time of the halo. This radius where the two time scales are equal is called the cooling radius.

At the cooling radius, the gas loses its pressure, the central region further collapses and the larger shells of gas lack the necessary pressure support, collapsing as well, until a new pressure equilibrium is reached. This may never happen for very low-mass halos since neither the virial temperature is high nor the equilibrium can be reach quickly, leading to a rapid gas condensation. On the other hand, highmass halos (say, M H > 5 10 11 M ) exhibit a virial velocity higher than the speed of sound in the incoming gas (T 0 ≈ 104 K), thus possibly producing a spherical shock. The post-shock gas may then keep its pressure support for more than one dynamical time, and the rate of gas condensing in the center of the potential is strongly decreased.

During its cooling and compression, the gas, in this model, conserves its angular momentum which leads to a faster rotation. This has been one explanation for the formation of planar dense gaseous regions, precursor of (spiral) galaxies. We only showed that a substantial fraction of the mass of gas accreted by a galaxy came from thin, dense, almost unidimensional channels, with the peculiarity of being warm (∼ 10 4 K) even when the halo gas was much hotter. This lamentary accretion was found to be dominant over other modes, even mergers, above a redshift of about 1. While Birnboim & Dekel (2003) had provided a new explanation for the change of regime in smooth accretion, by reducing the virial shock stability to a simpler criterion based on an eective adibatic index, Dekel & Birnboim (2006) extended this study to the lamentary accretion. The involved densities being separated by orders of magnitude, the stability criterion was met so that even in Below ∼ 5 10 11 M , the smooth accretion cools faster than it is compressed and stimulates galactic star formation; above that limit, a stable shock prevents any ecient cooling. The cold-in-hot zone represents the masses and redshifts for which laments are most likely to undergo no stable shock, thus feeding the galaxy with warm gas. Dashed lines show the Press-Schechter 1-and 2-σ typical halo masses as functions of the redshift, to guide the eye through the process of halo growth. However, such a picture leads to galaxies drowning in baryons. To alleviate this, models then require very ecient outows to moderate the gas content of galaxies and to keep their growth inecient. We concern ourselves in this thesis with this supply of gas. Can micro-physical processes prevent or at least lengthen the time it takes for gas to accrete onto galaxies? Lowering the accretion eciency alleviates a number of signicant astrophysical challenges as we discuss in Chapter 2. the case when the spherical accretion is shocked, laments may endure no such shock and penetrate the halo, feeding the galaxy with fresh fuel to support the growth of the stellar populations within galaxies. As seen in Fig. 1.4, this cold mode accretion was found to occur at redshifts higher than 2 and for halo masses lower than a value which increases with redshift.

The work presented here is precisely an attempt to better understand the nature and the dynamics of this accretion mode. The next section explains why it is important to investigate this mode of accretion in more detail.

1.4

Challenges of galaxy evolution

Several pieces of observational information are intriguing. First of all, the overall average star formation rate (SFR) reaches a clear maximum up to a redshift of about 1.5 and then abruptly declines (e.g. Madau et al., 1996;Madau & Dickinson, 2014). Why would galaxies form more and more stars and then decline so precipitously? Moreover, as highlighted by Fig. 1.5, if one naïvely states that the stellar mass of galaxies should be proportional to their host halo mass, Λ-CDM would provide strong overestimates for low-mass and high-mass galaxies. So, what prevented low-mass and high-mass halos to grow a galaxy with a stellar component consistent with that of average L galaxies? This apparent lack of stars is connected to the broader question of the so-called missing baryon problem (less than 50% of the expected baryons are observed), the other side of the same coin being the over-cooling problem (numerical simulations tend to heavily overestimate the gas condensation by cooling). Why is galaxy formation so inecient? Two families of explanations were provided, and neither totally excludes the other.

Extrinsic explanations for inecient galaxy formation

The rst explanation may come by investigating processes for limiting the accretion of gas. As we said before, if a stable shock propagates outwards in the halo gas, baryons may remain at constant radius, supported by the hydrostatic pressure of the halo gas, for a time larger than one dynamical time, if for example, the circumgalactic medium is very dilute. This may be the case for the highest scale structures like inter-cluster laments or cluster halos. Hard X-ray emission is the signature of hot gas with a long cooling time. Within this context, missing baryons would lie in multi-phase gas retained in the halo as warm-hot intergalactic gas. It's emission would be extremely weak due to its rarity (low density).

Then, a new question arises: if strong mass inows of warm gas is allowed via a cold-mode accretion at high redshift, would not this mode stimulate the formation of baryon-rich, luminous galaxies? Without ruining the judicious reasoning that accounted for early massive galaxies, we need to investigate the precise fate of intergalactic laments close to the point where their role in supplying galaxies with gas shifts from leading to negligible. There may well be mechanisms to reduce their galaxy-feeding eciency earlier, at higher redshifts and in lower mass halos, than currently expected from theory.

Intrinsic explanations for inecient galaxy formation

The second possible type of explanation as to why galaxy formation is so inecient is illustrated in Fig. 1.5. These intrinsic explanations involve processes that are a result of the formation history of the galaxy itself. A galaxy is basically composed gaseous and a stellar components, and it exchanges mass and energy with its immediate environment, the circum-galactic medium, possibly the local inter-galactic medium. Internal evolution, correlated with mass growth and the evolution of its stellar populations or the growth of a supermassive black hole through gas accretion, could lie at the heart of galaxies evolving from a rapid accretion and vigorous outows mode of growth and evolution to a more quiescent growth of gas and stars with only limited outow rates.

On the low-mass end (∼ 10 8 M ) of the galaxy and dark matter mass distribution, the gravitational well of the dark matter halo is neither sucient to overcome the gas pressure up to a point where the cooling becomes ecient, nor, on the other hand, can it retain the gas expelled due to the combined energy output supernovae (if such energy is eectively thermalized). On the high-mass end (∼ 10 13 M ), mass accretion is accompanied by growth of a central (super)massive black hole. The more massive it gets, the more energy it extracts from the gas at constant accretion rate, which increases its bolometric luminosity and mechanical energy output in the form of intense radiation and powerful outows. This ejection, even if it is collimated and as long as it couples eectively to the ambient gas, disrupts and heats the interstellar medium. Since internal processes give rise to the shutdown of star formation, nominally by removing the gas, but also perhaps through the strong generation of turbulence, such processes are collectively called negative feedback.

Accretionfeedback interaction?

Feedback processes generally involve energy, mass, and momentum ejection from the galaxy. The thermal, turbulent, and kinetic energies, as well as the momentum must be deposited in the surrounding medium, that is the circum-galactic medium (CGM), possibly in to the IGM well beyond the virial radius. But these are also the very regions crossed by accreting material, themselves carriers of energy and momentum.

We expect the two families of ows to interact in some extent 5 . The result of such an interaction is dicult to predict: there exist many sets of conditions, allowing or preventing from condensing again and raining on the galaxy (see e.g. Voit et al., 2015b). In other words, the interaction between accreting and outowing gas could turn into either a positive or negative feedback loop. In fact, in reality, it is probably both. That's not the real question. The really poignant question is does feedback increase or decrease the gas accretion ecient overall.

With the current state of the art simulations, it appears that this interaction moderates the inow rates but it is not exactly clear how. In this context, it is essential to study the stability of the accreting IGFs, as a possible disruption would not only alter the accretion eciency, but also the way they interact with galactic feedback outows.

Chapter 2

Modeling the entry in the halo Abstract Simulations of cosmological lamentary accretion reveal ows (streams) of warm gas, T∼10 4 K, which are ecient in bringing gas into galaxies. We present a phenomenological scenario where gas in such ows if it is shocked as it enters the halo as we assume become biphasic and, as a result, turbulent. We consider a collimated stream of warm gas that ows into a halo from an over dense lament of the cosmic web. The post-shock streaming gas adjusts its pressure with the ambient halo gas, and fragments as it cools. The fragmented stream forms a two phase medium: a warm cloudy phase embedded in hot post-shock gas. We argue that the hot phase sustains the accretion shock. During fragmentation, a fraction of the initial kinetic energy of the infalling gas is converted into turbulence among and within the warm clouds. The thermodynamic evolution of the post-shock gas is largely determined by the relative, competing timescales of several processes: the cooling, the expansion of the post-shock gas, the amount of turbulence in the clouds, and the dynamical time of the halo. We expect the gas to become multiphase when the gas cooling and dynamical times are of the same order-ofmagnitude. In this framework, we show that this occurs in the important mass range of M halo ∼ 10 11 -10 13 M , where the bulk of stars have formed in galaxies.

Due to expansion and turbulence, gas accreting along cosmic web laments may eventually loose coherence and mix with the ambient halo gas. Through both the phase separation and disruption of the stream, the accretion eciency onto a galaxy in a halo dynamical time is lowered. De-collimating ows make the direct interaction between galaxy feedback and accretion streams more likely, thereby further reducing the overall accretion eciency. As we discuss, moderating the gas accretion eciency through these mechanisms may help to alleviate a number of signicant challenges in theoretical galaxy formation.

2.3

The beginnings of a phenomenological model for a turbulent and multiphase post-shock medium . . . . .

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Motivations

The physical state of the gas accreting onto a galactic halo is thought to largely determine the star formation history of the galaxy itself. When most of the accreting matter is still coming from intergalactic laments (IGFs), studying theoretically the physical conditions of the gas they carry into the halo is therefore of prime importance in understanding how galaxies grow and evolve. Two families of scenarii have been envisioned theoretically. The rst one, the gas is warm 1 , ∼ 10 4 K, as falls into the potential of the halo ultimately feeding the galaxy with fresh gas and supporting its star formation. In the other scenario the gas is hot, 5 10 5 K, and due to mixing generally leads to feeding the halo with hot gas. Unless the cooling time of the gas penetrating the halo is short relative to the dynamical time of the halo, it will simply feed the halo with gas, generally not the galaxy itself except perhaps over a very long timescale.

The lamentary accretion obviously shares common features with the spherical 2 accretion because the gas it involves is considered of the same general na- ture, but with a higher density and an expected lower temperature, and because its dynamics is also strongly driven by the potential well of the dark matter halo.

Nevertheless, the geometry of a lament, as opposed to spherical diuse accretion, has a stronger inuence on how the galaxy evolves. First, due to its collimated nature as its travel through the halo, its cross section is small enough that it may not encounter outows from the galaxy. The cross section of a stream is roughly two orders of magnitude less than the surface of a sphere encompassing the galaxy itself. More particularly, by conservation of angular momentum, infall trajectory of streams tends to align with the galactic plane, avoiding then possible powerful outows from an AGN or intense star formation occurring within the halo's central galaxy. Secondly, its possibly high overdensity compared to most of the halo gas provides it with enough inertia to penetrate deeply into the circum-galactic medium (CGM) of the galaxy. Because of their possible high eciency in reaching the galaxy proper, IGFs are expected to be major drivers of the gas supply within galaxies and hence support their star formation, when suitable conditions are met.

As with spherical accretion, the two scenarios rely on the existence of a shock propagating upwards into the cooler core of an IGF. When it exists, its stability, 1 This situation is often refered to as cold mode accretion. However, we will use the term warm, since this is the common nomenclature in studies of the interstellar medium for ionized and neutral gas at a few 1000 to 10 4 K such as the warm ionized or the warm neutral medium (WIM/WNM). The term cold is reserved for gas below these temperatures the cold neutral and the cold molecular media.

2 Spherical, or isotropic accretion are named this way because of idealized spherical halos.

It is of course rather unrealistic, as dark matter halos are expected to be ellipsoidal and irregular.

Some authors use the word smooth to refer to this kind of accretion.

its strength and the propensity to cool all determine whether the mode of accretion is hot or warm. If it is unstable, as predicted by Dekel & Birnboim (2006) at low redshifts (z 2 -3) and for high halo masses, the shock collapses down and no heating occurs. Its stability may be inferred from the radiative nature of the shock, as well as the compression due to the converging owlines and the pressure increase in the halo when falling in. If it is not strong enough (Mach numer M 5), the gas is heated up to only ∼ 10 5 K and the density is increased by a factor > 3.5. In such a case, the radiative cooling is ecient on a timescale shorter than the infall timescale (here the relevant dynamical time) and the stream ends up warm.

Rationale

Numerical simulations may be of great help to understand the fate of the gas driven along laments, especially their dynamics. Chapter 3 shows that a great deal of information about the physical conditions are revealed in zoom-in simulation. However, there are limitations that are inherent to numerical computations, such as the nite resolution or the nite number of processes modeled. A number of phenomena that stem from excluded (or of which scale is unreachable) microphysical processes, such as many hydro-and thermodynamic instabilities, may not develop in simulations of relatively low resolution. In particular, the turbulence because of the extended range of scales needed to cover the turbulent cascade(s), and the thermal fragmentationbecause of the small-scale uctuations needed to seed the rst fragments would not be computed. For example, observations of the ISM in the MW suggest that turbulence is dissipated on scales well below a pc (e.g., Miville-Deschênes et al. 2017).

We already noted the importance of the weak interactions between the halo gas and the galactic outows on the one hand, and the IGF material on the other hand, when accreting in warm mode. Allowing for the possibility to have both hot and warm gas leads to the possibly increased eciency of such interactions.

As far as this model allows it, we will then address at least the question of the eective cross-section of the lament when it penetrates into the virial sphere.

The redshift 2 will be the one we exemplify in specic illustrative calculations, but the model does not restrict to that value. There are two reasons for this. First, this is the epoch when, statistically, the co-moving density of star formation of the ensemble of galaxies is at a turning point. The star-formation rate (SFR) density evolution bends from increasing to decreasing around z ≈ 2. The prediction from Dekel & Birnboim (2006) that warm laments would stop feeding galaxies eciently in massive halos matches this value. Thus, learning more about the gas physics of lamentary accretion will help rening models that explain this evolution in the star-formation rate density with cosmic time.

Secondly, the redshift at which the dominant contribution to the growth of galactic halos shifts from the lamentary accretion to the halo mergers is estimated to be around z ∼ 1.5. Below this redshift, the study of IGF accretion may be somewhat less relevant within the context of galaxy growth, even if it would still be interesting to understand peculiarities of halo gas and CGM close to galaxies at all redshifts.

But as said, our model is more general and is aimed at determining the conditions under which a lamentary ow becomes turbulent and multiphase, and in what amount.

2.3

The beginnings of a phenomenological model for a turbulent and multiphase post-shock medium

In this section, we discuss our initial analysis of the formation of bi-phasic turbulent post-shock medium of the streams. We discuss the initial conditions of the stream and halo gas the stream is owing into. In this early analysis, our basic assumption was that the post-shock gas was in thermodynamic equilibrium. Thus, if the post-shock gas met the criterion for becoming thermally unstable (Field, 1965), then it would become bi-phasic. Turbulence would also be generated from the transformation of the bulk kinetic energy into random motions of the warm clouds and in turbulence within the clouds. However, subsequently, we realized that the assumption of thermodynamic equilibrium was not necessary, and perhaps even dicult to justify within the context of a phenomenological model. We developed a more general case where the stream develops inhomogeneities due to, for example, the shock being oblique, having a range of velocities due to shock curvature, and/or having an initially inhomogeneous stream (where the inhomogeneities are amplied in the post-shock gas). This section captures the development of our initial ideas and initial conditions and assumptions for a more fully developed model of gas accretion into galaxy halos.

Overview

The phenomenological and somewhat idealized models all considered an initially cylindrical ow of conned, warm gas and the possible shock was stationary, with a priori laminar, homogeneous post-shock ow, surrounded by a medium at hydrostatic equilibrium. We considered the case where:

• there is a stable, stationary shock;

• the IGM is at very low metallicity (Z < 10 -3 Z );

• turbulence is produced3 by the shock;

• the post-shock medium, being in thermodynamic equilibrium, heterogeneously cools to develop a multiphase medium leading to two phases: the warm one and the hot one;

• the relative motion of warm clouds has captured all of the turbulent motions.

Fixed parameters

Our model applies to any given halo mass M H at any given redshift z. Thus, the critical density of the Universe ρ crit , the virial radius r vir and the virial velocity v vir are considered known. A oor temperature is xed at T 0 = 10 4 K, below which the gas can not cool. This is a consequence of the abrupt fall of the cooling eciency at low metallicities. The initial temperature of the lament, dened as `warm', is set to T 0 .

Free parameters

A few parameters are kept free. The over-density of the lament compared to the surrounding IGM is denoted f and is considered free because simulations show that it may widely vary from one lament to another, and from one halo to another. The nal volume-averaged turbulent pressure over the initial ram pressure is labelled η and is free because it is extremely dicult to reliably quantify the energy conversion to random turbulent motions. We would indeed need, rst, a robust model for shock instabilities, and for compressive, non-isothermal, turbulence in radiating medium. Even with those, we would need to expand the number of assumptions and postulate realistic values for supplemental parameters. The volume lling factors of the warm and hot components are φ v,w and φ v,h = 1-φ v,w . The use of φ v,w as an input parameter is complementary to the use of φ m,w , the mass fraction of the gas falling into warm temperatures. It is a free parameter because only few may be known about the phase separation without the knowledge of power-spectra of thermodynamic quantities and other information related to microphysics. This, again, prevents us from making the model too intricate and too dubious due to an oversized set of assumptions. We note that ultimately, some of these parameters are not truly free but that microphysics within the gas must regulate the amount of energy, for example, it transformed into cloud-cloud motions.

Foundational denitions

Denition 1 The properties of the halo gas useful to our scenario are its density at the virial radius, ρ H , its temperature, T H , and the law for ideal gases let us also directly derive its pressure P H . The density of IGM/halo gas close to the virial radius, is the fundamental density of the model:

ρ 0 = ρ H .
Denition 2 The lament is cylindrical (perfectly collimated), homogeneous, its initial density is:

ρ 1 = f ρ 0 (2.1)
Denition 3 Its initial pressure is then:

p 1 = (k B /µm p ) ρ 1 T 0 (2.2)
Denition 4 The density of the warm and hot components are respectively designated as ρ w and ρ h .

Denition 5 The volume-averaged densities of the warm and hot components, of the overall medium are:

ρ w = φ v,w ρ w , ρ h = φ v,h ρ h (2.3)
which allows us to dene the overall average density as:

ρ 2 = ρ w + ρ h (2.4)
Denition 6 The mass fractions of each component are easily dened:

φ m,w = ρ w /ρ 2 , φ m,h = ρ h /ρ 2 (2.5)
Denition 7 The initial speed of sound in the lament is (squared):

c 2 s = γ (k B /µm p ) T 0 (2.6)
so that the initial Mach number is:

M 1 = v 1 /c s (2.7)
Denition 8 The nal post-expansion or contraction cross section of the ow is increased by a factor S. Note that it may take any real, positive value, meaning that S < 1 implies an actual contraction.

Hypotheses and their formal expressions

To allow for a suciently simple modelling, we posited a number of hypotheses that were plausible from what was known about cold streams, virial shocks, and gas physics in this kind of environment. We here present them, with their formal translations into mathematics, using the previous denitions.

Hypothesis 1 The lament comes in at the virial velocity:

v 1 = v vir (2.8)
This is equivalent to the case where gas is falling from innity toward an isolated halo. It is compatible with the ndings of Goerdt & Ceverino (2015a).

Hypothesis 2 The whole system (including the shock) is stationary.

The Rankine-Hugoniot jump relations apply easily to a stationary shock and that is why it is often easier to choose the reference frame of the discontinuity front. Would the shock propagate in the frame of the halo? Birnboim & Dekel (2003) numerically found that an accretion spherical shock would settle close to the virial radius along the cosmic expansion and the halo growth. Our intuition that the possible existence of a shock in the penetrating lament would occur at the virial radius and thus correspond to a virial shock led us to postulate that this shock would also be stationary. Assuming its existence also means that we assume that the conditions for its stability are met, implying that its post-shock medium would support it, even if dissipative processes occur afterward. To ensure this, the simplest hypothesis consists in the stationarity of the whole system.

Hypothesis 3 Halo is in hydrostatic equilibrium with an NFW prole. Halo is set to be at the virial temperature:

T H = T vir = γ -1 2 (µm p / k B )v 2 vir (2.9)
The isothermal sphere is a common assumption that ts to observations of cluster halos. Our goal being to model cases where the halos are massive, this is acceptable. The hydrostatic equilibrium is an idealized situation, as it is specifically applicable to isotropic and homogeneous gaseous halos. Using the virial temperature also means that the halo gas has itself been shocked, or at least compressed, without being able to radiate away its heat.

The density in the halo follows an NFW prole:

ρ 0 = f b ρnfw (r vir ) ≈ 37.0 × f b ρ crit (2.10)
The prole derived in Navarro et al. (1997) is known to t the observed density proles of halos. It is then appropriate to estimate the values of our initial conditions. However, the NFW prole applies to dark matter. By multiplying it with the cosmic baryonic fraction, we simply express the fact that before the entry in the halo, gas and dark matter are still coupled. This of course is an extreme assumption, but recent observational results seem to suggest that perhaps the CGM of galaxies has a signicant reservoir of gas (e.g., Werk et al. 2014).

The pressure close to the virial radius is then:

p H = (k B /µm p ) ρ 0 T H = γ -1 2 ρ 0 v 2 vir (2.11)
This directly results from the two previous choices. As we may see later,

this assumption is of great importance, as it sets the thermodynamic boundary condition of the ow.

Hypothesis 4 The system is impermeable, i.e. mass is conserved along the ow:

Sρ 2 v 2 = ρ 1 v 1 (2.12)
and there is no (immediate) mixing with the surrounding gas and such mixing only takes place when the hot stream gas has expanded and reached pressure equilibrium with the ambient halo gas.

If the gas originally coming from the IGF mixed at some point with the halo gas, the imaginary boundary of our system would be permeable and we could not rely anymore on the mass conservation relation. The mixing may occur wherever the surface separating the two uids is the site of a sheared ow. Giving rise to hydrodynamical instabilities such as that of Kelvin-Helmholtz (KHI), the growth rate thereof is to be evaluated. It appears to be, for the choices of densities we will make, of the order of the transverse sound-crossing time, which is in some cases shorter than other times typical of the evolution of our system. However, the rate may only be evaluated at the onset of the instability, where it is maximal, decreasing when entering the nonlinear regime. If the KHI develops, the resulting mixing layer would be restricted in width at its saturation and it would still be correct to apply then our model to a narrower system that does not undergo any rapid mixing.

Hypothesis 5 The ow system entirely reaches pressure equilibrium with the halo:

ρ w T w = ρ h T h = ρ H T H (2.13)
This assumption will be tested. The propagation of rarefaction waves from the interface with the halo gas to the innermost region of our ow should occur in a few times the transverse sound-crossing time. Computing this time and comparing it to other dynamical times has proved fundamental to our approach, as will be discussed subsequently.

Hypothesis 6

The system is adiabatically bounded (no heat exchange) and there is no momentum exchange, momentum is conserved:

S ρ 2 v 2 2 + ηρ 1 v 2 1 + p H = ρ 1 v 2 1 + p 1 (2.14)
which, in virtue of Hypothesis 3, can be written as:

Sρ 2 v 2 2 = 1 -S 1 3f + η ρ 1 v 2 1 + p 1 (2.15)
For a monatomic gas, the Prandtl number is Pr = 2/3. It relates the viscous term to the heat diusion term of the energetic equation, so that if viscosity is neglected at the interface with the halo gas, thermal diusion would also be. Like for the KHI, this layer may be omitted if needed, but viscous eects are very likely to be weak on the scales considered in our model: the Reynolds number is lower than a few hundreds, still close to a laminar regime.

Hypothesis 7 The warm component has radiated away its heat and ends at the oor temperature:

T w = T 0 (2.16)
The oor temperature is set by the cooling eciency of a metal-poor gas, which is very low below 10 4 K and also by possible heating sources of gas at this temperature such as the meta-galactic ux. The ability for the gas to cool down to that temperature is measured by the cooling time, dened later. Like the soundcrossing time, it is essential to compare its value relative to other times in order for our model to be valid.

Hypothesis 8 The ow rst endures a normal shock and the Rankine-Hugoniot equations apply at this stage:

ρ ps = γ+1 2 M 2 1 γ-1 2 M 2 1 + 1 ρ 1 ≈ γ + 1 γ -1 ρ 1 p ps = γ -1 γ + 1 2γ γ -1 M 2 1 -1 p 1 ≈ 2γ γ + 1 M 2 1 p 1 (2.17)
where the subscript ps stands for 'post-shock', and the rightmost members of each line correspond to the asymptotic case of a Mach number much greater than 1.

In fact, this is the strongest assumption we make. The post-shock conditions are indeed derived from it, and hence all the subsequent calculations. The existence of a shock inside the lament is debated. Numerical simulations may have found that it propagates up to a few virial radii but can not accurately resolve the discontinuity. Analytical arguments are not easy to develop either. In the next chapter we will hypothesize about possible ways of forming such a shock. The planarity and the orthogonality to the ow are chosen for the sake of simplicity.

Nevertheless, oblique shocks would not result in very dierent numbers and curved shocks imply many more parameters to be correctly dened. But ultimately, we will argue that oblique and curved shocks may be central to understanding how post-shock gas may fragment.

Hypothesis 9 If the phase separation occurred after having expanded, both component have the same bulk motion:

v w = v h = v 2 (2.18)
and if the phase separation occurred before the expansion, then:

v w = v ps , v h = v 2 (2.19)
In other terms, the mass of gas that will end up being warm must keep the velocity it has when the cooling begins, since both its mass and its momentum are constant, provided that warm clouds y freely into the hot component. In one asymptotic case, the warm component forms rapidly just after the shock, it keeps the post-shock velocity. In the other asymptotic case, it starts cooling after the pressure has adjusted to the halo and thus after the bulk velocity has evolved accordingly to the cross-section of the ow. This relation is needed to close the system of equations. But it depends on the ratio of the cooling to the soundcrossing time. As we will see, even if the intermediate situation is the most likely one, the two extreme cases (a ratio much greater or much lower than one) are of great help in understanding the evolution of the ow.

Hypothesis 10 The hot component has only adiabatically and reversibly, i.e., no heat transfer and no entropy increase, expanded since the post-shock stage:

ρ γ h p ps = ρ γ ps p H (2.20)
which is strictly equivalent to:

ρ γ-1 h T ps = ρ γ-1 ps T h (2.21)
This directly derives from Hypothesis 6 and from the assumption that the loss of pressure due to the cooling of a fraction of the gas will not signicantly aect the remaining hot fraction.

Hypothesis 11 Turbulence is only carried by the warm phase in the form of relative random motions:

ηρ 1 v 2 1 = ρ w σ 2 = x w ρ w σ 2 = y w ρ 2 σ 2 (2.22)
where σ is the velocity dispersion associated with turbulence, more precisely the nal warm cloud velocity dispersion. Written this way, this term has the dimension of a pressure and precisely corresponds to the Reynolds-averaged stress tensor, when inserted into the Reynolds-averaged Navier-Stokes equation (RANS).

2.4

From our initial phenomenological analysis to a more general model

The model that we have just outlined, with its set of hypotheses, enabled us to successfully derive various relations that are useful for a more general and perhaps more appropriate model for streams passing through an accretion shock. The fact that several quantities are free parameters also allowed us to explore a wide range of parameter space. Some surfaces in the (f, η, φ v,w ) space bound with the subspace where the solutions to the equations exist. This is also one of the reasons why we performed a numerical experiment as described in the next chapter, as a means to constrain parameters and to better understand the initial conditions for the stream before it penetrates the halo.

After a year and a half of exploration, we developed a more complete phenomenological model for streams penetrating halos. The development just described in the previous section was useful for setting the stage for the evolution of the post-shock streaming gas. Now however, to understand the impact of allowing for expansion and fragmentation of the post-shock accreting gas within the context of galaxy evolution, we have to consider various timescales that are relevant for galaxies. The salient time scales that necessary to understand the evolution of the the post-shock gas are: the time necessary for the gas to cool the post-shock cooling time scale; the hot post-shock gas will expand into the ambient halo gas. We estimate this expansion timescale as a relative rate of change in the pressure, assuming the expansion is adiabatic. The leads to the concept of the post-expansion cooling time; to understand how the stream may evolve, we compare these timescales with the dynamical time of the halo. There are other relevant timescales of course, as we describe, but it is the relationships between these characteristic timescales, that are essential for establishing the regimes in which the post-shock gas might become bi-phasic and turbulent. Doing a parameter space search phenomenologically by comparing various timescales for dierent post-shock temperature (halo masses) and stream over-densities, always us to investigate which regimes where the gas will not develop a bi-phasic structure. For example, in what regime the post-shock gas will remain hot or cool quickly reforming a warm stream of gas. To make such an analysis, we rely now on the assumption that through a variety of mechanisms, like obliqueness of the shock or inhomogeneities in the accreting stream, the post-shock gas is inhomogeneous.

With these more general developments and timescale comparisons, we can now investigate: When and how do post-shock gas in streams become multi-phased and turbulent?

2.5 Are cosmological gas accretion streams multiphase and turbulent?

We here reproduce in extenso the article submitted to Astronomy and Astrophysics and whose initial version had been published on ArXiv as Cornuault et al. (2016).

We added two subsections developping and extending the model for a time-evolving distribution which we used in the paper.

Introduction

The realization that we live in a dark matter dominated universe led to the development of the rst comprehensive theory of galaxy formation (e.g. White & Rees, 1978b;Fall & Efstathiou, 1980b). These analytic models embedded simple gas physics into the hierarchical growth of structure whereby smaller halos merged over time forming successively more massive halos (White & Rees, 1978b). Despite the successes of this model in understanding the scale of observed galaxy masses, it was soon realized that there were a number of problems. The most signicant is that modeled galaxies form with a higher fraction of baryons than is observed (e.g., Ferrara et al., 2005;Bouché et al., 2006;Anderson & Bregman, 2010;Werk et al., 2014). This failure was dubbed the over-cooling problem (Benson et al., 2003).

As numerical simulations allowed for galaxy growth to be coupled to the development of large scale structure, they showed that much of the accreting mass may penetrate into the halo as laments of gas and dark matter (Kere² et al., 2005;Ocvirk et al., 2008a). Whether or not these streams pass through a stable accretion shock as they penetrate the halo depends on the mass and redshift of the halo (Birnboim & Dekel, 2003;Dekel & Birnboim, 2006, hereafter BD03 and DB06 respectively). If the shock is not stable, the accretion ow is cold. This cold mode accretion in simulations occurs as streams of warm (10 4 K) gas enter- ing the halo, smooth at kpc-scale and weakly coupled to the infalling dark matter laments (Danovich et al., 2015;Wetzel & Nagai, 2015). Cold mode accretion is ecient in reaching down to a few tenths of a virial radius (Dekel & Birnboim, 2006;Behroozi et al., 2013). The high eciency of gas accretion in some simulations leads to model galaxies with unrealistically high baryon fractions emphasizing the over-cooling problem. To alleviate the problem of excess baryons in simulated galaxies, ecient outows and feedback were introduced (e.g., Hopkins et al., 2012Hopkins et al., , 2016)). Feedback both heats the gas in the halo, preventing it from cooling, and also ejects gas from both the galaxy and halo lowering their total gas content.

The circum-galactic media of galaxies are certainly not devoid of gas, perhaps containing up to approximately half of the total baryon content of the halo (e.g., Werk et al., 2014;Peek et al., 2015). This gas is known empirically to be multiphase. The multiphase nature of halo gas is most evident in local high mass halos, those with masses on the scales of cluster or groups. In clusters, for example, even at constant pressure, a very wide range of gas phases are observed, from hot X-ray emitting gas to cold, dense molecular gas (e.g., Jae et al., 2005;Edge et al., 2010;Salomé et al., 2011;Tremblay et al., 2012;Hamer et al., 2016;Emonts et al., 2016). In galaxy halos, the detection of multiphase gas is mostly through the absorption lines from warm neutral and ionized gas ( 10

4 to ∼10 6
K) and dust via the reddening of background galaxies and quasars (Ménard et al., 2010;Peek et al., 2015, but see Pinto et al. 2014 for the detection of hot gas in X-ray emission lines). Outows from galaxies are also multiphase (e.g., Beirão et al., 2015;Heckman & Thompson, 2017) and are likely crucial for creating and maintaining the multiphase gas in halos (e.g., Gaspari et al., 2012;Sharma et al., 2012a;Borthakur et al., 2013;Voit et al., 2015b;Hayes et al., 2016).

There is only circumstantial evidence for smooth, collimated accretion streams penetrating into galaxy halos (e.g., Martin et al., 2015;Bouché et al., 2016;Vernet et al., 2017a, and references therein). In analogy with analyses of gas in halos and outows, a phenomenological approach may provide additional insights into the nature of ows and halos that galaxy simulations are perhaps not yet achieving (e.g., Sharma et al., 2010Sharma et al., , 2012a,b;,b;Singh & Sharma, 2015;Voit et al., 2015b,a;Thompson et al., 2016). The notion that over-cooling remains a problem in simulations, that there is scant observational evidence for the streams of the type currently simulated, and because high speed collisions of gas can lead to multiphase turbulent media (Guillard et al., 2009(Guillard et al., , 2010;;Ogle et al., 2010;Peterson et al., 2012;Appleton et al., 2013;Alatalo et al., 2015), all motivated us to analyze gas accretion ows phenomenologically.

Just as with the explanation for the lack of cooling ows in clusters (e.g., Peterson et al., 2003;Raerty et al., 2008), our current understanding of accretion ows in galaxy halos may also suer from an overly simplistic view of gas thermodynamics. In clusters, it is now understood that heating and cooling are in approximate global balance, preventing the gas from cooling catastrophically (e.g., Raerty et al., 2008;Sharma et al., 2012a,b;McCourt et al., 2012;Zhuravleva et al., 2014;Voit et al., 2015a). In analogy with gas in cool core clusters, and in contrast to what a number of cosmological simulations currently show, the gas in streams may not cool globally. Instead, if the gas in streams is inhomogeneous and subject to thermal and hydrodynamic instabilities (e.g., Sharma et al., 2010), the dierences in cooling times between the gas phases will lead to fragmentation of the gas. If streams are unstable, their gas will not remain monophasic or laminar.

Thus, our goal in this paper is to investigate the question posed in the title: Are Cosmological Gas Accretion Streams Multiphase and Turbulent?. If yes, the gas energetics may regulate the gas accretion eciency. Dekel et al. ( 2013) estimated the penetration eciency over large halo scales at z∼2 of ∼50% but this estimate only considered macroscopic processes that may inuence the accretion eciency.

Because heating and cooling are controlled by the mass, energy and momentum exchanges between the gas phases, a careful investigation of the gas physics on microscopic scales is required to investigate whether those microphysical processes may further reduce the eciency of gas accretion onto galaxies.

To investigate the question posed in the title, we begin by presenting a qualitative sketch of our scenario, make a quantitative investigation of the impact of expansion in the post-shock gas on an accretion ow, and then discuss the case where the post-shock gas has small uctuations in density and temperature, developing a criterion for when the gas will fragment ( 2.5.2). In 2.5.3, we discuss the consequences of the formation of a multiphase ow on the thermodynamic evolution of the stream after it penetrates the halo. To gauge the astrophysical pertinence of our model, we analyze the evolution of an idealized gas accretion stream into a halo of 10 13 M at z=2 ( 2.5.4). In Section 2.5.5, we discuss why simulations may be missing some ingredients necessary for modeling accretion shocks robustly and outline a few simple consequences of our proposed scenario.

Our framework for multi-phased streams

The idea that gas in halos is multiphase has been suggested for decades (e.g., Binney, 1977;Maller & Bullock, 2004). More recent studies of halo gas attribute the development of multiphase gas to the growth of local thermal instabilities (e.g., Sharma et al., 2010) or galaxy outows (e.g., Thompson et al., 2016;Hayes et al., 2016). Thermal instability is only relevant when the cooling time of the unstable gas is of the same order-of-magnitude or smaller than the dynamical time of the halo (e.g., Sharma et al., 2012a;McCourt et al., 2012). In ambient halo gas and outows from galaxies, the gas must often meet this requirement given that they are observed to be multiphase. However, in an accreting stream of gas, it is dicult to understand how the gas might achieve such a balance in heating and cooling. We may have to consider other processes to determine if it is possible for streams themselves to become multiphased as they ow into the halo. In the following sections, we examine the physics of gas owing into halos from cosmic web laments. The persistent virial shock and the higher pressure of the post-shock gas compared to the ambient halo gas, allows the ow to expand after being shocked (labeled as expansion). The post-shock gas may become unstable, fragmenting to form a biphasic medium. The fragmentation enables a fraction of the initial momentum and energy of the stream to be captured as turbulent clouds of warm gas with a dispersion, σ turb , and a volume-lling factor, φ v,w . The clouds may move beyond the initial radius of the stream, de-collimating the ow. Eventually, the hot post-shock gas mixes with the ambient halo gas which prevents it from cooling further. See text and the Appendix for denitions of the variables.

Qualitative sketch of our specic framework

We briey qualitatively outline our scenario of gas accretion through streams, sketched in Fig. 2.1, introducing the concepts developed later in the paper. We consider a collimated stream of warm gas with a temperature of 10 4 K that pen- etrates into a dark matter halo lled with hot gas at the halo virial temperature.

The smooth density distribution of the hot gas follows the density distribution of the underlying dark matter halo. The ambient halo gas has a long cooling time and has constant density and temperature during the ow. The speed of the ow is set to the virial velocity of the halo and is highly supersonic relative to the sound speed of the gas within the stream. In the following we introduce the three basic physical ingredients of our modeling of streams.

As the stream penetrates at the virial radius, we assume it is shock-heated whenever the hot halo gas provides the necessary pressure support for sustaining the shock ( 2.5.2). The post-shock gas is over-pressurized relative to the ambient halo gas and expands, invalidating the classical one-dimensional analysis of streams (DB06, Mo et al., 2010). The hot post-shock gas mixes with the ambient halo gas, which prevents much of the gas initially in the stream from cooling completely to form a monophasic post-shock stream. We further posit that the post-shock gas will develop inhomogeneities due to, for example, non-planarity or obliqueness of the shock-front or through inhomogeneities in velocity and/or density of the stream before it is shocked. The fragmentation of the gas into hot and warm cloudy phases is central to our scenario. If certain physical conditions are met, the expanding inhomogeneous gas will cool and fragment, forming a two phase medium a hot phase with an embedded warm cloudy phase ( 2.5.2). Density and velocity inhomogeneities in the post-shock gas may be amplied through gas cooling, leading to the formation of a multiphased ow (see 2.5.2).

We expect that part of the kinetic energy of the infalling gas will be converted to turbulence within the warm clouds and random cloud-cloud motions (e.g., Hennebelle & Pérault, 1999;Kritsuk & Norman, 2002a;Heigl et al., 2017). If the level of turbulence is high or if the clouds are formed while the gas is expanding, the warm clouds may spread beyond the initial boundary of the collimated inowing stream (Fig. 2.1 and see 2.5.3). If these conditions are met, the stream will decollimate. The thermodynamic evolution of the post-shock gas in the stream is largely determined by the timescales of several relevant processes gas cooling, expansion of the hot phase, dynamical time of the halo, and the stream disruption due to turbulent motions which we quantify in the remainder of this and the next section.

4.5 5.0 5.5 6.0 6.5 7.0 7.5 log (T ps /K)

2 1 0 1 2 3 log (t/Myr) log (r stream /r vir )=-1 log (r stream /r vir )=-1.5 log (r stream /r vir )=-2 f = 1 0 0 f = 1 0 f = 1 t cool

The Impact of Expansion on the Phenomenology of Accretion Flows

Since the pioneering work of White & Rees (1978b) decades ago, the accretion of gas onto galaxies or into halos has been analyzed in one, radial dimension, assuming homogeneity of the ow (see Mo et al., 2010). The one-dimensional approximation is also used for streams, where it is valid only if one can ignore lateral expansion. To know if this approximation is valid, one has to compare the expansion to cooling time of the gas within the ow. Historically, even the analysis of the stability of the accretion shock neglected the possibility that the post-shock gas may expand into the ambient halo gas gas (DB06; Mo et al., 2010). In our study, we reconsider the sustainability of the accretion shock that occurs as the infalling lament collides with the ambient halo gas. To conduct this analysis, we compare the cooling time of the post-shock gas to various other measures of its dynamical evolution. For simplicity, we only compare the cooling time to the expansion time of the post-shock gas into the surrounding ambient halo gas and the dynamical time of the halo. Intuitively, one can understand that if the cooling time is signicantly shorter than either the expansion or dynamical times, the stream after the shock will quickly cool down and maintain much of its integrity. Such a shock can be unstable and this is essentially the textbook situation that has been considered already (Mo et al., 2010). If the cooling time of the post-shock gas is signicantly longer than either the expansion or dynamical times, then the post-shock gas will mix the ambient halo gas further supplying it with hot gas and entropy. This is another textbook example that has been analyzed one-dimensionally (DB06; Mo et al., 2010). However, if the cooling time is of the same order-of-magnitude of the expansion and halo dynamical times, a wider range of outcomes of the post-shock gas are possible (e.g., Sharma et al., 2010Sharma et al., , 2012b)). This is the point of this paper to describe phenomenologically this more complex regime.

We now consider the impact of expansion on the properties of the post-shock gas. To estimate the cooling, expansion, and dynamical timescales, we introduce two parameters specic to the ow, its over-density relative to that of mean density of gas at the virial radius, f , and the radius of the stream, r stream . All variables used in this and subsequent sections are summarized in Tables 2.1 and 2.2 of the Appendix. As we now show, the gas in the stream immediately after being heated to a high post-shock temperature, T ps , has a pressure, P ps , higher than the ambient halo gas, P H , and thus expands into the surrounding ambient halo gas of density, ρ H . All of the post-shock quantities are given by the normal Rankine-Hugoniot shock relations (see Appendix for appropriate formulae). The halo pressure at the virial radius is,

P H = k B µm p ρ H T H = (γ -1) 2 ρ H v 2 vir = γM 2 1 3f P 0 (2.23)
where T H is the temperature of the ambient halo gas, which we have assumed to the be virial temperature. The pressure ratio just after the shock is,

P ps P H = 2f γ (γ + 1) • g (M 1 ) (2.24)
where g :

M 1 → (2γ/(γ -1) -M -2
1 ) -1 . For γ = 5/3 and within the domain [1,+∞], this function rapidly decreases from 1/4 to 1/5. Hence the pressure ratio, P ps /P H is between 9/5 and 9/4 times the initial stream over-density, f . Since f 1, the pressure of the post-shock gas is higher than the pressure of the ambient halo.

Thus, since the pressure ratio is always greater than one, the ow will expand into the ambient halo gas.

To dene the expansion time of the ow, we approximate this as the inverse of the relative rate of change of pressure in the post-shock gas as it expands.

Quantitatively, this is,

t expand = P ps / Ṗps = -2γr stream / ṙstream ∼ 2γr stream /c ps (2.25)
where r stream is the initial radius of the stream before it expands and c ps is the sound speed of the gas immediately after passing through the shock front. As a rst order approximation, in the case of a homogeneous post-shock medium, one can easily compare the cooling and expansion times. Our denition of the expansion time means it is a dierential measure of the expansion and so the most appropriate comparison is with the instantaneous cooling time of the gas immediately after the shock. The isobaric cooling time is dened as,

t cool (T ) = T dT dt -1 P = 5k B ρT 2µn e n H Λ(T ) ≈ 4.3 × 5k B µm p T 2ρΛ(T ) (2.26)
where k B is the Boltzmann constant, Λ (T ) is the electronic cooling eciency as a function of temperature, T (Sutherland & Dopita, 1993;Gnat & Sternberg, 2007), µ is the mean molecular weight 4 , m p is the mass of the proton, ρ is the mass density of the cooling gas, n e is the electronic density and n H the Hydrogen particle density 5 . Hereafter, the cooling time of the post-shock is denoted by, t cool,ps . We also refer to the cooling time after expansion, noted t cool,pe .

The other timescale with which to compare the cooling time is the dynamical time of the halo. It is simple to estimate the dynamical time. We estimate the 4 µ = 0.6 m p for a fully ionized gas.

5 The factor 4.3 in Eq. 2.26 comes from n e = 1.2n H and n H ≈ ρ/(1.4 m p ). For this analysis, we have assumed log (r stream /r vir )=-1.5. We highlight regions where the cooling times are signicantly longer than the halo dynamical times as hot accretion, regions where the cooling time is signicantly shorter than the dynamical times as cold accretion, and regions where the cooling times are of the same order-of-magnitude as the dynamical times as multiphase accretion (see text for details). These labels are illustrative and are not intended to precisely delineate a clean separation in the timescales.

dynamical time for matter falling from the virial radius with a radial velocity of v 2 (the post-shock velocity) directed at the center of the potential as,

t ff = α r vir v 2 ≈ t dyn,halo (2.27)
For this estimate, we assume a NFW dark matter potential (Navarro et al., 1997) and α, is a factor of-order unity to account for the integrated gravitational acceleration during the fall. Assuming that the characteristic velocity and radius are the virial values, v vir and r vir , the dynamical time is independent of halo mass (Mo et al., 2010). we see that the expansion times are almost constant above M H few10 10 M since the virial shock is suciently strong such that t expand ∝ t dyn,halo . As f increases, the cooling times decrease systematically, which means for low temperatures (low mass halos at z=2), the cooling time is always less than the expansion time. For very high mass halos, even for wide streams with relatively high densities, the cooling time is much longer than the expansion time and can be longer than the halo dynamical time for low to moderately over-dense laments. A wide range of values in f and r stream /r vir lead to cooling times that are less than the halo dynamical time and within an order-of-magnitude of the expansion time. Whenever the expansion time is comparable to the cooling time, the gas ow will globally cool neither purely adiabatically or isobarically.

We can understand the competition between these time scales, t cool,ps , t cool,pe , t dyn,halo , and t expand , more clearly by considering them as a function of the stream over-density, f , and post-shock temperature or halo mass (Fig. 2.3). For the case where t cool,ps >> t expd and t cool,pe > t dyn,halo , the post-shock gas will expand rapidly remaining hot over at least a dynamical time of the halo. In this regime the hot post-shock gas will likely mix with the ambient halo gas preventing signicant cooling. This is akin the hot mode accretion discussed by Dekel and collaborators (e.g., DB06). In Fig. 2.3, we have labeled this regime, "hot accretion". At the other extreme in Fig. 2.3, when t cool,ps << t expand and t cool,ps << t dyn,halo , the gas cools before signicant expansion and before the stream has penetrated deeply into the halo. Accretion in this regime corresponds to their cold mode accretion and have labeled it as such in Fig. 2.3. In between these two regimes, t cool,pe , t cool,ps , t dyn,halo , and t expand are all about the same order-of-magnitude. In this regime, there is not a simple dichotomy between the types of accretion streams they can be both hot and warm if the post-shock gas has a range of temperature and/or densities and the thermodynamic evolution of the post-shock gas will be more complicated. We labeled this regime multiphase accretion in Fig. 2 Density uctuations at constant shock velocity will also lead to inhomogeneities in the post-shock gas in both temperature and density resulting in a range of cooling times (Guillard et al., 2009). As we now discuss, once such dierential cooling sets in, it acts like the thermal instability, leading to phase separation in the ow (Sharma et al., 2012a), but over a nite range of time in the absence of a heating process balancing cooling of the hot phase.

To understand how the inhomogeneous post-shock gas evolves, we investigate how small uctuations in the density and/or temperature are amplied. Neglecting the inuence of any heating process, the radiative cooling is not balanced and there are no xed equilibrium points. Following closely the development in Sharma et al.

(2012a), we begin with a parcel of gas that is over-dense relative to an ambient medium. The magnitude of the over-density is, δ ≡ |δρ/ρ| ∼ |δ T/T| for isobaric conditions. The inverse of the eective cooling time of the over-dense gas parcel relative to the ambient medium is then

1 t cool,eff = - 1 t cool,parcel + 1 t cool,ambient
.

(2.28)

As already shown by Sharma et al. (2012a), for small over-densities, δ ≤1, the inverse of the eective cooling time under isobaric conditions is

1 t cool,eff ∼ -δ ∂ ∂ ln T 1 t cool P = δ t cool 2 - d ln Λ(T ) d ln T .
(2.29) Marginally denser gas cools faster than the gas in which it is embedded and the cooling becomes a runaway process for temperatures below a few 10 5 K. We refer to this process as dierential cooling". As soon as heterogeneities form in approximate pressure equilibrium with their surroundings, denser cooler regions of the gas will cool rapidly, forming clouds, leaving a hotter, rarer, high volume-lling inter-cloud medium. Thus, dierential cooling leads to multiphasic post-shock accretion ows. This is akin to what occurs for thermally unstable gas (Field, 1965) but the phase separation is transient in the absence of a heating process of the hot gas balancing cooling.

Of course, a natural consequence of dierential cooling is that even for halo masses where the post-shock gas has on average a long cooling time, gas with uctuations in temperature or density may locally cool suciently rapidly to form a multiphased stream.

Analytically, inhomogeneities can be accounted for as is done in phenomenological analyses of the formation of stars through the use of probability distribution functions (PDFs) in gas density and/or temperature (e.g., Hennebelle & Chabrier, 2009). We follow the thermodynamical evolution of the temperature and density PDFs using the energy equation, namely,

ė = -ρ Λ(T ) 4.3µm p + P ρ ρ 2 , (2.30)
where e is the specic internal energy. We illustrate the isobaric evolution of inhomogeneities in a ow using such an approach in Fig. 2.4. In this illustration, we consider a log-normal PDF in temperature with a dispersion of σ T = 0.3 (1σ dispersion of log ρ ρ 0 = log T T 0

, where ρ 0 and T 0 are the initial mean values of the density and temperature, with corresponding cooling time t 0 cool ). We have chosen a gas pressure and mean temperature appropriate for the post-shock gas of a ow into a massive, 10 13 M , halo at z=2 (see 2.5.4). For this illustration, the gas pressure is P/k = 5.8 × 10 4 K cm -3 , and the initial mean temperature T 0 =2.7 × 10 6 K. We nd that a substantial fraction of the accreting gas cools to 10 4 K in a fraction of t 0 cool , while the peak of the hot gas temperature PDF does not shift with time (see Fig. 2.4).

We show in Fig. 2.5 both the mass fraction and volume-lling factor of the gas that cools to 10 4 K as a function of time. Note that since the results shown on Figs. 2.4 and 2.3 are plotted as a function of the initial cooling time t 0 cool , to a rst approximation they are not much dependent on the specic values of ρ 0 and T 0 . Those results also apply to the gas cooling after expansion when t expand << t 0 cool . Figure 2.3 shows that the warm gas becomes the dominant mass phase, within about half of the post-shock cooling time, and that the warm gas only lls a small fraction of the volume. For a larger value of the temperature dispersion σ T , Φ m,w increases over a broader range of fractional times distributed around the same mean value.

Figure 2.4: Probability distribution function of the gas temperature at dierent times for isobaric cooling. The initial distribution (black line) is log-normal with a dispersion, σ T = 0.3, and an initial mean gas temperature, T 0 =2.7 × 10 6 K. The distribution at a time equal to the initial isobaric gas cooling time is plotted in blue, that of half and a quarter of this time in red and green, respectively. In the last column of the legend, we indicate the fraction of the gas that has cooled to 10 4 K within the indicated fraction of the cooling time. For example, in this illustration, after one cooling time, approximate 85% of the gas has cooled to 10 4 K.

In Fig. 2.5, the gas is multiphase for only a few t 0 cool . The hot gas eventually cools because we have not considered any heating processes. However, the process of fragmentation of the gas could be sustained if there is heating of the hot gas. Even in absence of heating, after expansion the hot gas will eventually mix with the ambient halo gas thus sustaining the hot phase. Possible local sources of heating for the post-shock gas are the radiative precursor of high velocity shocks, thermal conduction, radiation from the surrounding hot medium, and dissipation of turbulence. The mechanical energy input from active galactic nuclei (AGN), winds generated by intense star formation, and other processes can plausibly balance the cooling of the hot gas globally (e.g., Best et al., 2007;Raerty et al., 2008). Overall, there is more than enough energy, but what is unknown is how and with what eciency this energy is transferred to the streaming gas. Even though simulations do not capture this process, it has been suggested that the energy from the galaxy is transferred eciently through turbulent energy cascade and dissipation (Zhuravleva et al., 2014;Banerjee & Sharma, 2014).

A poignant question to ask is can the owing post-shock gas actually become multiphase while cooling in a halo dynamical time? It is a dicult question to answer in the particular case of accretion streams because it depends on how inhomogeneous the gas is. It will occur if the post-shock conditions are suciently inhomogeneous. The gas will become multiphased around when t cool,pe is the same order-of-magnitude as t dyn,halo . This justies where we placed the multiphase accretion label in Fig. 2.3. This gure shows that this occurs in the important mass range of M halo ∼10 11 to 10 13 M . The values of course depend on the relative over-density of the streams (f ) and through the characteristics of the halos, on redshift. Halos within this mass range is where the bulk of stars have formed in galaxies. Moreover, because the volume lling factor of the warm gas is likely to be always small, even in the cases where the mass fraction is large, multiphase streams are likely to be dicult to identify observationally. We discuss this further in 2.5.5.

Consequences of the formation of multiphased, cloudy accretion ows Turbulence in warm clouds

As the gas fragments, perhaps due to dierential cooling and/or thermal instabilities, we assume that some of the initial kinetic energy of the stream is converted into turbulence within the warm component. We parameterize the turbulence after phase separation as the ratio, η, of the turbulent energy density of the clouds and the initial bulk kinetic energy density of the stream. We dene this ratio as,

η = ρ w v σ 2 turb ρ 1 v 2 1 (2.31)
where ρ w v is the volume-averaged density of the warm clouds ( ρ w v =φ v,w ρ w , where φ v,w is volume-lling factor and ρ w is the density of the warm clouds respectively), σ turb is the cloud-cloud velocity dispersion, ρ 1 and v 1 are the pre-shock gas density and velocity. We assume that v 1 is equal to the virial velocity, v vir . The initial density of the stream is related to the hot halo density, ρ H , as ρ 1 = f ρ H .

The amount of turbulence generated in the post-shock gas is likely determined by as yet poorly understood and undoubtedly complex gas physics. In our model we do not attempt to investigate the complexity involved in this transformation of energy, we simply parameterize the amount of turbulence by η which we allow to be free (see 2.5.3).

As we briey alluded to in 2.5.2, as the turbulent energy dissipates, it reheats the warm clouds, moderating the cooling. The cooling of the clouds will also be moderated by mixing with the ambient halo gas as the stream penetrates deeper into the halo potential (Fragile et al., 2004). Dissipation of turbulent kinetic energy in the hot phase may also contribute to heating the post-shock gas as well as the radiation from the high-speed accretion shock (Allen et al., 2008). The parameter η may be high enough to provide the required heating of the hot gas in the stream through turbulent dissipation and mixing with the halo gas, perhaps instigating thermal instabilities. On the other hand, η must be low enough such that turbulent dissipation and mixing does not prevent instabilities, both thermal and cooling, from growing in the post-shock gas (Banerjee & Sharma, 2014;Zhuravleva et al., 2014). These constraints and considerations may ultimately provide limits on how much or how little turbulence is sustainable in the post-shock gas, but determining these exact values requires further detailed study and will not be considered here.

Mass and momentum budget of the multiphase medium

We assume that as the stream ows into the halo, its mass owate is conserved during the post-shock expansion and does not mix immediately with the ambient halo gas. This leads to the relation,

Sρ 2 v 2 = ρ 1 v 1 (2.32)
where ρ 2 is the density after the hot gas has expanded by a factor, S, and similarly, v 2 is its velocity (Fig. 2.1). The expansion factor, S, is dened as the ratio of the initial over the nal mass uxes (per unit surface perpendicular to the ow).

The post-shock gas will ultimately reach pressure equilibrium with the halo which implies, ρ w T w = ρ h T h = ρ H T H , where ρ w and ρ h are the densities of the warm and hot components and T H is the temperature of the hot components after phase separation in the post-shock gas.

Before there is any momentum exchange with the halo gas, the momentum of the streaming gas is conserved, implying,

S ρ 2 v 2 2 + ηρ 1 v 2 1 + P H = ρ 1 v 2 1 + P 1 (2.33)
where P 1 is the initial pressure of the stream. We assume the fragmenting gas radiates away its heat until reaching a oor temperature, T w =10 4 K. The hot component only cools adiabatically and reversibly (no heat transfer and no entropy increase) expanding after it passed through the shock front, namely, ρ γ h P ps = ρ γ ps P H , where γ is the ratio of specic heats. This approximation holds until the expanding gas reaches pressure equilibrium with the ambient halo gas.

The expansion factor, S, is derived from Eqs. 2.23, 2.32, and 2.33, as,

S = 1 2 η + γ -1 2f -1 1 -1 -4 ρ H ρ 2 ηf + γ -1 2 (2.34)
The expansion of the stream is important in our formulation. It leads to the mixing of the expanding post-shock gas with the ambient halo gas. This mixing couples the hot post-shock gas to the larger energy reservoir of the ambient halo gas which acts as a thermostat preventing the gas from cooling, thereby possibly maintaining the pressure necessary to support a sustained shock. As we discussed in 2.5.2, it is the relative ratios of the thermal cooling time and the expansion time that will inuence how the stream evolves.

Cloudy stream disruption

The relative cloud-cloud motions may lead to the warm clouds spreading beyond the original radius of the stream. So instead of the streams being highly collimated as we assumed they are initially (before the virial shock), the ows may de-collimate. This can be thought of as disruption since the warm clouds expand away from the original trajectory of the stream, thus disrupting the ow. We dene the timescale for disruption as the cloud crossing time of the stream, namely,

t disrupt = r stream σ turb (2.35)
σ turb may be computed from η, f , and the volume-lling factor of the warm clouds, φ v,w . From Eqs. 2.23 and 2.31, assuming pressure equilibrium between gas phases, yields, 53 10 4 10 5 10 6 10 7

10 -27 10 -26 10 -25 10 -24 10 -23 10 -22

T w T h T ps T H ρ H ρ h ρ 1 ρ ps ρ w Σ ps Σ H Σ h
10 -27 10 -26 10 -25 10 -24 10 -23 10 -22 (2) (see text). Σ 1 (red triangle) indicates the density and temperature of the pre-shock gas. The gas is shocked reaching the point Σ ps . Subsequently, the gas cools adiabatically due to the expansion of the stream until the halo pressure is reached at Σ h . Two phases separate. For case (1), phase separation occurs at Σ h , while for case (2), it occurs at Σ ps . In both cases, the hot component mixes with the surrounding halo gas, reaching Σ H . For case (1), the warm component cools radiatively and isobarically to the point Σ w . For case (2), the same point is reached but the gas takes a dierent thermodynamic path. Dashed black lines represent adiabats and isobars labeled A and I respectively. The two blue-dashed curves in the left panel are contours of constant cooling time, t cool ∼ 0.2t dyn,halo (upper curve) and t cool ∼ 0.12t dyn,halo (lower curve). These curves indicate that the cooling time during the expansion remains approximately constant. (right) Analysis of the isobaric dierential cooling instability (Eq. 2.28) of low-, 10 -3 solar (solid line) and solar-metallicity (dashed line) gas as a function of temperature.

T h T ps T H ρ H ρ h ρ 1 ρ w ρ ps Σ ps Σ H Σ h -4 -2 0 2 4 10 4 10 5 10 6 10 7 stable unstable T /K ρ/g.cm -3 Σ w Σ 1 i a ρ/g.cm -3 Σ w Σ 1 i a T /K 2 -d ln Λ/d ln T
When 2d ln Λ/d ln T > 0 the gas can become heterogeneous through accelerated dierential cooling.

σ 2 turb = v 2 1 (ηf /φ v,w )(T w /T H ) = 2 (γ -1) k B µm p (ηf /φ v,w )T w (2.36)
Characterized this way, σ 2 turb /v 2 1 > η. For a wide range of relative amounts of turbulent energy, stream over-density, and volume-lling factor of the warm gas, the cloud-cloud dispersion be up to v vir /2. The dynamical evolution of the stream is determined by the ratio of the disruption and halo dynamical time. If t disrupt t dyn,halo , then the warm clouds within the stream will remain collimated as they ow, otherwise, the streams will de-collimate.

2.5.4 A specic case: 10 13 M halo at z=2

To gauge whether any of the phenomenology we have discussed is pertinent astrophysically, we calculate the stream characteristics for a single dark matter halo of mass, 10 13 M , at redshift 2. We chose this halo mass and redshift because in DB06, halos at this mass and redshift were determined to have substantial accretion rates in the hot mode. Our analysis in 2.5.2 indicated that this halo mass and redshift would be a revealing illustration of how the impact of expansion and accelerated dierential cooling might change the physical characteristics of gas accretion ows (i.e., it would no longer simply be hot mode accretion). Both the halo mass and the redshift set the initial stream density and the characteristics of the halo gas. We provide all the characteristics of the halos and initial gas conditions in the Appendix.

Why are streams cloudy?

There are two limiting cases to specically consider when attempting to understand the development of a biphasic stream. The two cases are: (1) t expand t cool,ps and (2) t expand t cool,ps . In case (1), the warm phase develops only after the expansion has occurred, while in case (2) the clouds form before the expansion. We sketch the thermodynamic evolution (path) of a stream in Fig. 2.6. To make these illustrations, we adopted f = 30 and r stream /r vir =0.01 for case (1), and f = 150 and r stream /r vir =0.1 for case (2).

In case (1), clouds ll the entire expanse of the expanded ow. Adiabatic expansion occurs before radiative cooling becomes important, the stream cools and the density of the hot phase declines without a change in entropy. In case (2), the two phase separate before expansion. In both cases, the hot post-shock gas reaches pressure equilibrium and mixes with the ambient halo gas.

These two conditions are of course for the extreme cases, in reality, the gas will have t expand ∼ t cool,ps (Fig. ). We assume a volume-lling factor of 0.1 (see Fig. 2.5) for the warm clouds and r stream = r vir /10. In regions with values less than 1, the streams are disrupted ( 2.5.4). The shaded region indicates regions that are forbidden because for these values of the parameters, the post-shock pressure is less than the halo pressure. We note that because t disrupt ∝ σ -1 turb , the contours of constant t disrupt /t dyn,halo are shaped like contours of constant σ turb in the same plane. For example, the contour, t disrupt /t dyn,halo =0.2, is close to the contour for σ turb =200 km s -1 , which is almost half the initial velocity of the ow.

evolution is more complex, but the clouds reach the same nal thermodynamical state. The cooling length at the post-shock temperature is the size over which structures can cool isobarically (i.e., cooling length, λ cooling,ps = c ps t cool,ps ). In case (1), the cooling length is much larger than the stream radius, λ cooling,ps >> r stream , clouds may form over all scales within the stream. In case (2), it is much smaller, λ cooling,ps << r stream , and clouds may form over a range of sizes smaller than the stream radius. The expansion of the hot gas does not inhibit the growth of thermal and dierential cooling instabilities because the decrease in the pressure is roughly compensated by the decrease of temperature along an adiabat in the expression of the cooling time. In other words, for the post-shock temperature, i.e., for the halo mass we have adopted, the gas cooling time remains roughly constant as the gas expands (Fig. 2.6; see also Fig. 2.2). Eventually, the warm phase equilibrates at approximately the halo pressure and the hot phase mixes with the halo gas.

We are obviously considering fragmentation on scales much smaller than the scale height of the gravitational potential well and thus we can safely ignore thermal stabilization by convection (Balbus & Soker, 1989;Sharma et al., 2010).

Does the accretion ow disrupt?

The only process we consider in determining whether or not the warm clouds will travel coherently towards the galaxy proper as observed in numerical simulations (e.g. Brooks et al., 2009;Danovich et al., 2015) is the cloud-cloud velocity dispersion. The cloud-cloud dispersion will broaden the stream as it penetrates into the halo. Fig. 2.7 and Fig. 2.8 show contours of t disrupt /t dyn,halo for a constant r stream (Eqs. 2.35 and 2.27). We nd that the disruption time is shorter than or approximately equal to the halo dynamical time. Thus it appears that for a wide range of relative turbulent energy densities, stream over-densities, and volume-lling factors of the warm gas, the ows will not simply fall directly into the potential as a highly collimated, coherent streams. In reality, the clouds are dynamical entities, we expect clouds to keep forming through cooling as other clouds are destroyed by hydrodynamic instabilities and heated by dissipation (Cooper et al., 2009).

Discussion

We now discuss broadly how our ndings relate to aspects of galaxy formation and evolution.

Are virial shocks persistent?

In our scenario, the existence of a hot phase in hydrostatic equilibrium supports a persistent shock (Binney, 1977;Maller & Bullock, 2004). the warm gas and level of turbulence. The contours represent constant ratios of t disrupt /t dyn,halo as labelled (cf. Eqs. 2.35 and 2.27). We assume f =30 and r stream = r vir /10. In regions with values less than 1, the streams are disrupted.

The shaded region has the same meaning as in Fig. 2.7.

appear to show a similar phenomenology as that described in DB06. Depending on the mass of the halo and redshift, streams penetrate at about one to many 100s of km s -1 (van de Voort & Schaye, 2012;Goerdt & Ceverino, 2015b) or much greater than the sound speed of the stream, c s ∼10-20 km s -1 . Simulated accretion shocks are isothermal at high Mach numbers and not stable (e.g., Nelson et al., 2016Nelson et al., , 2015)). Perhaps this uniform isothermality is due to the spatial and temporal resolutions adopted in the cosmological simulations. The scales that simulations must probe are roughly delineated in one-dimensional shock calculations.

Raymond (1979) show that atomic shocks with velocities of ∼100 km s -1 reach their post-shock temperatures within a distance of ≈1-10 ×10 15 cm in less than ∼30 yrs. For higher shock velocities, the spatial and temporal scales will be even shorter (Allen et al., 2008). The gas cools after the shock on timescales that are at most only a couple of orders-of-magnitude longer. In addition, in order to capture the dierential cooling and thermal instabilities in the post-shock gas, resolutions much ner than the Field length are required (Koyama & Inutsuka, 2004;Gressel, 2009). Simulations should be specically designed to capture the multiphase nature of streams penetrating halos to test our scenario by resolving the Field length (Koyama & Inutsuka, 2004). Since the Field length decreases strongly with decreasing temperature, this is most easily done with ad hoc oor temperature higher than 10 4 K but lower than the virial temperature as done for simulating thermal instabilities in cool core clusters (McCourt et al., 2012).

The resolution and temporal scales necessary to resolve high Mach number shocks are not achievable in galaxy-or cosmological-scale simulations. To overcome this limitation, numericists use articial viscosity in the form of a dissipative term either in dynamical equations or dispersion relations, depending on the properties of the gas or the ow (e.g., Kritsuk et al., 2011;Price, 2012;Hu et al., 2014;Beck et al., 2016). Articial viscosity spreads the shock over several resolution elements enabling simulations to resolve heating and cooling across the shock front. The Reynolds number is inversely proportional to the kinetic viscosity of the uid.

If the ow properties are unchanged but the viscosity increased, the Reynolds number of the ow will be articially low. Simulations with articial viscosity have ows with low Reynolds numbers. Simulated low Re ows, Re 1000, tend to be laminar. Those with low spatial and temporal resolutions, due to not resolving the Field length and having unrealistical low Reynolds numbers likely fail to produce biphasic turbulent ows (see, e.g., Kritsuk & Norman, 2002a;Sutherland et al., 2003;Koyama & Inutsuka, 2004;Kritsuk et al., 2011;Nelson et al., 2016, for discussion).

Nature of Flows into Galaxies: Observational tests

As a consequence of our assumption that all energetic quantities scale as v vir , the cloud-cloud dispersion is a simple linear function of η, f , and φ -1 v,w . This relation implies that turbulent velocities of the warm clouds in the post-shock gas are independent of both halo mass and redshift. In principle this means that post-shock streams may be turbulent in any halo at any redshift. The reality is probably much more complex, through both macro-and microscopic gas physics which is not yet well-understood, the 2 parameters, η and φ v,w , likely depend on the accretion velocity and the physical state of the ambient halo gas both of which undoubtedly depend on redshift and halo mass.

Our scenario has observationally identiable consequences. If our scenario is realistic, then observations should reveal: (1) clumpy, turbulent streams; (2) strong signs of the dissipation of turbulent mechanical energy in the warm medium (e.g., Guillard et al., 2009;Ogle et al., 2010;Tumlinson et al., 2011). The situation described in our model where a large fraction of the bulk kinetic energy of the accretion ow is transfered to turbulent motions amongst cold clouds is observed in large-scale galaxy colliding ows, like the situation in the Tay Galaxies or the Stephan's Quintet, where we see evidence for this energy cascade (Peterson et al., 2012;Cluver et al., 2010). In Stephan's Quintet, two atomic gas laments are colliding at ∼1000 km s -1 and yet instead of nding intense X-ray emission from the post-shock gas, most of the bulk kinetic energy is contained in the turbulent energy of the warm molecular gas (Guillard et al., 2009(Guillard et al., , 2010)). Remarkably, roughly 90% of the bulk kinetic energy has not been dissipated in the large-scale shock and is available to drive turbulence. If gas in halos is multiphase and turbulent, then it may be the case as well for accretion streams.

Obviously, a clumpy stream is dicult to identify as such through absorption line spectroscopy and this may explain why streams have not been conspicuously identied so far. This is most obviously seen in Fig. 2.5 where despite a large fraction of the gas is warm, its volume-lling factor is minuscule. Along most lines-of-sight, absorption spectroscopy is expected to sample only the hot, high volume-lling factor halo gas or probe a population of warm ambient halo clouds (Maller & Bullock, 2004;Tumlinson et al., 2011;Werk et al., 2014). The clouds should be looked for in emission. Their emission can be powered by the UV radiation from the galaxy but also through the localized dissipation of turbulent energy and through losses of their gravitational potential energy as they fall into the halo. This may have already been observed in Lyα (e.g., Cantalupo et al., 2014;Martin et al., 2015). In particular, it would be promising to interpret spectralimaging observations such as those provided by MUSE on the ESO/VLT within the context of our scenario (see Borisova et al., 2016;Fumagalli et al., 2016;Vernet et al., 2017a).

Moderating the accretion rate: Biphasic streams and increased coupling between feedback and accretion In our phenomenological model, two mechanisms moderate the accretion eciency on to galaxies: (1) disruption and fragmentation of the ow; (2) interaction between streams and outows of mass, energy, and momentum due to processes occurring within galaxies (e.g., AGN, intense star-formation).

First, streams potentially become multiphase and turbulent leading to short disruption times resulting in de-collimation. Any de-collimation undoubtedly leads to longer accretion times and thus lower overall accretion eciencies compared to smooth isothermal streams (Danovich et al., 2015;Nelson et al., 2016). The post shock gas becomes multiphase over a wide range of halo masses at z=2. A fraction of the initial stream mass ow becomes hot gas and ultimately mixes with the surrounding ambient hot halo gas. Thus even accretion streams potentially feed gas into the hot halo which may have long cooling times compared to the halo free fall time (White & Rees, 1978b;Maller & Bullock, 2004).

Second, simulations indicate that the mass and energy outows from galaxies can interact with streams, regulating or even stopping the ow of gas (e.g., Ferrara et al., 2005;Dubois et al., 2013;Nelson et al., 2015;Lu et al., 2015). Simulated streams are relatively narrow (e.g., Ocvirk et al., 2008a;Nelson et al., 2013Nelson et al., , 2016) ) and generally penetrate the halo perpendicular to the spin axes of disk galaxies and their directions are relatively stable for long periods (e.g., Pichon et al., 2011;Dubois et al., 2014;Welker et al., 2014;Laigle et al., 2015;Codis et al., 2015;Tillson et al., 2015). Feedback due to mechanical and radiative output of intense galactic star formation and active galactic nuclei is observed to be highly collimated in inner regions of disk galaxies (opening angle, Ω ∼π sr, e.g., Heckman et al., 1990;Lehnert & Heckman, 1995, 1996;Beirão et al., 2015). In the case of dwarf galaxies, their outows are generally more weakly collimated (e.g., Marlowe et al., 1995Marlowe et al., , 1997;;Martin, 1998Martin, , 2005)). The geometry of the accretion ows and the signicant collimation of outows from galaxies in simulations, result in only weak direct stream-outow interactions. However, accretion ows in simulations can be moderated or stopped when the halo gas is pumped with mass and energy via feedback to suciently high thermal pressures and low halo-stream density contrasts to induce instabilities in the stream and disrupt it; or when the halo gas develops a suciently high ram pressure in the inner halo due to angular momentum exchange between the gas, dark matter, and galaxy disk to disrupt accretion ows (van de Voort et al., 2011;Dubois et al., 2013;Nelson et al., 2016).

The processes we have described are generic to ows, whether they are inows or outows. It is only the context and timescales that change (Thompson et al., 2016). Just as with the accretion ows modeled here, we also expect the galaxy winds to be highly uncollimated as they ow from the galaxy due to the formation (and destruction) of turbulent clouds. Thus, the generation of turbulent cloudy media in both accretion ows and starburst-driven outows will allow for ecient interaction between these two types of ows. This dynamical interaction likely sustains turbulence in the halo compensating for the dissipation. It is perhaps through this interaction that galaxies become self-regulating on a halo scale (e.g., Fraternali et al., 2013), and not only on a galaxy scale (e.g., Lehnert et al., 2013Lehnert et al., , 2015)).

While the fate of the turbulent clouds is beyond the scope this paper, the qualitative implication is that the gas accretion eciency may be moderated through the generation of turbulence in biphasic ows. The fragmented, turbulent nature of the gas in streams and outows likely makes their dynamical and thermal interaction and coupling ecient. Note that other mechanisms, like the growth of RayleighâTaylor and Kelvin-Helmholtz instabilities, associated with gas cooling, can also trigger the formation of cold clouds in the surrounding halo (e.g. Kere² & Hernquist, 2009). Moderating the overall gas accretion eciency onto galaxies may help to alleviate two signicant challenges in contemporary astrophysics: the distribution of the ratio of the baryonic to total halo mass as a function of halo mass (e.g. Behroozi et al., 2013), where low mass galaxies have especially low baryon fractions; and the requirement for models to drive extremely massive and ecient outows to reduce the baryon content of galaxies (e.g., Hopkins et al., 2012Hopkins et al., , 2016)).

Conclusions

We developed a phenomenological model of lamentary gas accretion, streams, into dark matter halos. We assume both that streams penetrate ambient hot halo gas as homogeneous ows of 10 4 K gas and that they undergo a shock at the virial radius of the halo. The ingredients of the model, those which sets it apart from other phenomenological models of gas accretion, are that we assume the virial shock is sustained, the post-shock gas expands into a ambient hot halo gas, and through several mechanisms or characteristics of the shock front, the post-shock gas is inhomogeneous. To gauge whether this model is astrophysically pertinent, we discuss the thermodynamic evolution of a single stream penetrating a dark matter halo of mass 10 13 M at z=2. From this analysis, we nd that:

• The post-shock gas expands into the halo gas and it can fragment due to dierential cooling and hydrodynamic instabilities. Instabilities lead to the formation of a biphasic ow. It is the formation of a hot post-shock phase which mixes with the ambient hot halo gas, ultimately limiting how much of the gas can cool. As a result of the phase separation and the pressure provided by the hot post-shock gas, we argue that the virial shock may be sustained. However, we have not analyzed the sustainability of the shock in detail in this paper.

• The development of a biphasic medium converts some of the bulk kinetic energy into random turbulent motions in the gas (e.g., Hennebelle & Pérault, 1999;Kritsuk & Norman, 2002a). The turbulent energy cascades from large to small scales and across gas phases. The ows, while retaining signicant bulk momentum as they penetrate into the halo, are turbulent with cloudcloud dispersion velocities that can be up to 1/2 of the initial velocity of the stream.

• For a wide range of turbulent energy densities, our model shows that the stream will lose coherence in less than a halo dynamical time. We emphasize that the turbulent energy density is not in reality a free parameter but is determined by macro-and microscopic multiphase gas physics about which we have only a rudimentary understanding. To understand what processes regulate the amount of turbulence in streams, high resolution simulations of accreting gas need to be made and additional multi-wavelength observations useful for constraining the properties of turbulent astrophysical ows are necessary.

The post-virial shock gas is not isothermal, accretion streams are both hot and cold. The hot-cold dichotomy (see DB06) is no longer a simple function of whether or not the shock is stable, but now relies both on the shock occurring and under what circumstances the post-shock gas becomes multiphase and turbulent.

However, we have discussed may apply if there is no virial shock provided that inowing gas is hot and already inhomogeneous (Kang et al., 2005;Cen & Ostriker, 2006). Thus, even in absence of a virial shock, the gas may become multiphased by compression as it falls deeper into the halo potential.

Moderating the gas accretion eciencies on to galaxies through this and other mechanisms may help to alleviate some signicant challenges in theoretical astrophysics. If gas accretion is actually not highly ecient, then perhaps models will no longer have to rely on highly mass-loaded outows to regulate the gas content of galaxies. It is likely that the underlying physical mechanisms for regulating the mass ow rates and evolution of outows are very similar to those that regulate gas accretion (Thompson et al., 2016). If so, then observing outows in detail can provide additional constraints on the physics of astrophysical ows generally. We do not only have to rely on apparently challenging detections of direct accretion onto galaxies.

Parameters in the model

The quantities that are important in setting the initial conditions of the streamambient halo gas interaction are the mass, virial velocity, and virial radius of the dark matter halo which we denote as M H , v vir , and r vir . The dark matter distribution is given by a NFW prole with a concentration parameter, c, of 10 Navarro et al., 1997). The halo is lled by a hot gas of temperature T H , which we assume to be equal to the virial temperature of the halo, T vir . The density of the hot halo, ρ H , is assumed to follow that of the dark matter density with radius, but is multiplied by the cosmological baryon density relative to the dark matter density, f B = 0.18. This is ≈ 37f B ρ crit , where ρ crit is the critical density of the Universe. The halo pressure, P H , is related to T H and ρ H . The lling factor of this gas is assumed to be one. We are agnostic about how this hot, high volumelling factor halo at the virial temperature formed but note that it likely forms by a combination of accretion of gas from the intergalactic medium and heating through the radiative and mechanical energy output of the galaxy embedded in the halo (e.g., Suresh et al., 2015;Lu et al., 2015). ρ crit /µm p 7.6×10 -5 cm -3 Number density at r vir n 0

(

5.1×10

-4 cm -3

Adiabatic index γ 5/3

Mean particle mass µm p 0.6 × m p

The gas accretes through a stream of infalling gas with radius, r stream , we assume that it passes through a shock and that the properties of the post-shock gas is given by the standard set of shock equations. We simply scale the density of the accreting stream by a factor, f , which is its density contrast of the background dark matter density at the virial radius multiplied by the cosmological baryon fraction (i.e., ρ H ). We further assume that there is a temperature oor in the post-shock gas of 10 4 K. We assumed this temperature mainly because we also assume that the metallicity of the accreting stream is 10 -3 of the solar value. The gas cannot cool much beyond 10 4 K due to it lacking heavy metals (and is likely heated by the meta-galactic ux and the ionizing eld of the galaxy embedded in the halo). This assumption, although naive, is also extremely conservative in

ρ ps = (γ + 1) / (γ -1) M 2 1 / [M 2 1 + 2/(γ -1)] ρ 0
Pressure of the post-shock gas a

P ps = (γ -1) / (γ + 1) [2γ/(γ -1)M 2 1 -1] P 0 Post-
t disrupt = r stream /σ turb Isobaric cooling length Φ ∈ {ps, w, h} λ cooling = c Φ t cool,Φ
Notes.

( †) e grphil representtion of mny of these vriles is shown in pigF PFIF (a) tndrd norml shok eqution from the nkineErugoniot jump onditionsF (b) he eqution ssumes ∆ = 1 -4 [ηf + (γ -1)/2] ρ H /ρ 2 that this implies the post-shock gas will have one of the longest possible radiative cooling time (see Sutherland & Dopita, 1993;Gnat & Sternberg, 2007). We use the cooling curve, Λ(T ), from Gnat & Sternberg (2007) to compute the cooling times in the post-shock gas. We assume that the temperature of the gas in the stream before passing through the shock is also 10 4 K (T 1 ). At those temperatures and very low metallicity, we assume that no molecules form, so that the adiabatic index of the gases is always that of a monatomic gas, namely γ = 5/3.

The parameters we use in the model, given our assumed mass and redshift are given in Table 2.1. We enumerate for completeness and clarity all variables used in our analysis in Table 2.2.

Idealized model of a time-evolving multiphase medium

I implemented a numerical model to follow the dierential cooling of an initially slightly heterogeneous medium. The width of the initial log-normal distribution of the temperatures, δ, is the determinant factor for growing a multiphased medium.

I made simple analytic approximations for the time evolution of mass-and volume fraction of cool gas. Then I correlated the parameters of the analytical function with the initial distribution and draw preliminary physical conclusions. I further show how these correlations may be constrained by the output of our main model and state physical limitations. Afterwards, I implement a turbulent heating source, followed by the implementation of a scheme resembling a typical expansion of the pressurized gas in the hot halo.

Approximation of the time evolution of inhomogeneous gas

I developed a simple model for the evolution of an initially slightly inhomogeneous medium to a multiphase (warm+hot) medium. Assuming

• no cosmological context;

• gas is perfect with adiabatic index γ = 5/3;

• pressure is homogeneous at all times;

• cooling is isobaric (dP /dt = 0);

• temperatures (and thus densities) are initially log-normally distributed, we can numerically compute the thermal evolution of the distribution. This can be done either by a sampling-based or a discretized distribution function-based, nite-dierence method.

The log-normal distribution is usually relevant to isothermal turbulent media (Passot & Vázquez-Semadeni, 1998), but we chose the simplest one knowing that when the eective adiabatic index is more (less) than one, power-law tails should appear at low (high) densities. This is dened by the probability distribution function:

pdf(T ) = 1 T δ √ π exp - log(T /T i ) 2 2δ 2 (2.37)
where T i is the initial peak temperature and δ is the initial dispersion.

The sampling-based method can be described as the following. Given a pressure P , we sample N values Ti (i ∈ N, 1 ≤ i ≤ N ) from a normal probability law. These values represent the temperatures of N particles, through Ti = log(T i ), only coupled by their common pressure. Their densities are derived from the law of perfect gases. Each has a specic energy e i = (γ -1)k B T i /µm p , subject to a (discrete-)time evolution:

∀ i, log 10 e i (t + ∆t) = log 10 e i (t) -∆t/t cool,i (t)

(2.38) and similarly, due to the constant pressure,

∀ i, log 10 ρ i (t + ∆t) = log 10 ρ i (t) + ∆t/t cool,i (t)
(2.39)

where the cooling time of a particle i at time t is dened by:

t cool,i (t) = 4γµm p k B T i (t) (γ -1)ρ i (t)Λ(T i (t), Z) (2.40)
Say that we split the distribution, at any time, in two sets, depending on the value of the temperature T relative to an arbitrary threshold T c = 10 4 K + T , where T T c . Below this value lies the warm component, above it lies the hot one. We are then able to follow the time evolution of the warm mass fraction, as well as its volume-lling factor. The initial dominant cooling time, t cool,0 , is the one corresponding to the peak value of the distribution.

The warm mass fraction is dened by:

φ m,w (t) = #{i|T i (t) < T c }/N (2.41)
and the hot one is simply φ m,h = 1φ m,w . A perfect illustration of this fraction can be found in Fig. 3 of Kritsuk & Norman (2002b). The volume-lling factor of the warm component is:

φ v,w (t) = {i|T i (t)<Tc} 1/ρ i (t) {i} 1/ρ i (t) (2.42)
and the hot one is simply φ v,h = 1φ v,w . Running this code for N = 10 values of δ in [0.01,1], the curves of each fraction is plotted against time. As distribution cools, the process slows down, i.e. the absolute value of time derivatives of mass fractions decrease. Thus we adopt as a time variable the logarithm of t divided by the characteristic time, the initial cooling time of the mean of the PDF in temperature, t cool,0 . Denoting Θ = log 10 (t/t cool,0 ), we found an excellent tting function dened with the two parameters by:

Φ Θ 1/2 ,∆ : Θ → 1 2 1 + erf Θ -Θ 1/2 √ 2∆ (2.43)
where erf is the error function:

erf(x) = 1 √ π x -x e -u 2 du (2.44)
then the following approximation is found:

φ * ,w (Θ) ≈ Φ Θ * 1/2 ,∆ * (Θ) (2.45)
where * ∈ {m, v} denotes a quantity pertaining to the mass or the volume, and Θ * 1/2 and ∆ * are functions of the initial dispersion of the log of temperatures δ.

The former is the log-time at which the fraction reaches 50% and the latter is the lag between ∼31% and ∼69%.

Fitting the two parameters I will apply this calculation to the previously discussed specic halo with a mass of 10 13 M at z = 2. The input parameters must then match the post-shock physical quantities. With the following initial conditions:

• P/k B = 5.8 10 4 K.cm -3

• pdf(log(T )) = N (2.7 10 6 K, δ)

• Z = 0
Results are shown in Fig. 2.5.8 for δ = 0.01, 0.1, 0.3, 0.5. Eq. 2.45 is tted for dierent values of δ ∈ [0.01; 1], and for both φ * ,w , yielding relations Θ * 1/2 (δ) and ∆ * (δ), that are well tted by ane functions: Θ m 1/2 ≈ -0.255

(2.46) ∆ m ≈ 0.854δ (2.47) Θ v 1/2 ≈ -0.264 + 2.19δ
(2.48)

∆ v ≈ 0.438δ

(2.49)

These relations can be summarized as:

• Θ m 1/2 is independent of δ, so that the warm mass fraction reaches 50% at a xed time. That is, half the mass has cooled after ∼0.55 times the initial dominant cooling time, no matter the exact initial dispersion of the PDF.

• ∆ m is proportional to δ, so the higher the initial dispersion, the sooner the medium becomes multiphase and the longer it takes to cool completely.

• Θ v 1/2 increases with δ, so that the warm mass fraction reaches 50% with a delay compared to the increase in the mass fraction. This delay increases linearly with δ. It implies that during the timelag, the warm mass fraction is much higher than the volume-lling factor.

• ∆ v is proportional to δ, implying the same conclusion as for ∆ m . Note that the volume-lling factor reaches 1 at a faster rate than the mass fraction, since ∆ v /∆ m ≈ 1/2. When δ = 0, i.e. the distribution is a Dirac, the mass fraction and the volumelling factor of the warm phase abruptly jumps from 0 to 1 at Θ = Θ m 1/2 = Θ v 1/2 . This latter condition is not reected by the independent linear ts and we shall correct that by adjusting the terms of order 0 in both Θ * 0 expressions with a mean value:

-2 -1.5 -1 -0.5 0 0.5 1 φ m/v,w log 10 (t/t cool,0 ) φm,w φv,w Φ Θ m 1/2 (δ),∆ m (δ) Φ Θ v 1/2 (δ),∆ v (δ)
Θ m 1/2 ≈ -0.260

(2.50)

Θ v 1/2 ≈ -0.260 + 2.19δ
(2.51)

Constraining the two parameters

It is possible to relate those functions to the known densities of our expanded, multiphase, turbulent medium, namely ρ w and ρ h , with the previously derived equation:

φ -1 v,w -1 ρ h = φ -1 m,w -1 ρ w (2.52)
and solving it for (δ, Θ), after having replaced the φ's with their Φ approximations.

φ -1 v,w -1 φ -1 m,w -1 = ρ w ρ h = const.
(2.53) may be translated in:

Φ Θ v 1/2 (δ),∆ v (δ) (Θ) -1 -1 Φ Θ v 1/2 (δ),∆ v (δ) (Θ) -1 -1 = f (Θ, δ) = const.
(2.54) and, as surprising as it may be for such a complicated expression, the locus of f (Θ, δ) =const. is a line in the (Θ, δ)-plane:

δ = α Θ -Θ m 1/2 ≈ α log 10 1.82 t t cool,0 (2.55)
with α a coecient depending on the ratio ρ w /ρ h . This dependence is nonlinear but, again, the function can be numerically interpolated.

What does it mean? If we know the density ratio, there is a one-to-one map between the initial dispersion and the time it took to grow this multiphase medium.

Note that there are important caveats.

Pressure evolution In our model, the gas has expanded so that a perfectly isobaric transformation should not apply. However, this is a realistic way of gaining conservative information on the time the process took and on the initial (postshock) dispersions. Indeed, the loss of pressure by mechanical adjustment to the environment should slow down the cooling so that the times obtained by the method described above are lower bounds to a more physically accurate estimate of the cooling time.

Stationarity Our model explicitly states that the processes are stationary, eventually reaching an asymptotic multiphase state. Here in this sort theoretical study, the whole medium must always end up cool. This can be balanced by a turbulent heating term which stops the cooling at a certain asymptotic mass fraction. Given the order of magnitude of the times considered, an other important ingredient should be added that compensates the coolingnamely the cosmic expansion.

2.5.9 Further: more realistic conditions

Since we wanted to describe the evolution of a cylindrically symmetric ow representing an IGF, we may want to implement additional constraints and processes.

Here are two major propositions on mechanisms that could be implemented to obtain more realistic results, in particular for expanding post-shock streams.

Mimicking the expansion Pressure equilibrium with the ambient halo must be reached. Ideally, without radiative cooling, we expect it to be adiabatic and reversible. To mimic the physical situation, we should write the equations for the expansion so that a vanishing cooling rate implies the adiabatic law for gases (P ∝ ρ γ ). A more physically robust model would include spatial gradients the outer envelope rst expands and the rarefaction wave propagates toward the axis into the medium, at any instant, leading to a pressure eld with a cylindrical geometry. Moreover, the initial conditions should not represent a purely cylindrical discontinuity (there would be unrealistic shocks propagating in the lament and the halo) but perhaps at its most basic level, a layered model that would smooth the radial proles, using knowledge gained from the simulation (see Chap. 3).

Radiative loss and spatial scales Imposing a constant pressure is valid as long as the expansion and the cooling occur one after the other. When they are concurrent, the pressure drops along with the radiative losses. The parcel of gas under this condition either exerts or receives no mechanical work, in which case its average density is conserved, or work is done and it must be accounted for. In a sense, this is scale-dependent, so that the former case corresponds to smallest space scales, the latter to larger ones; their separation is governed by time ratios. For thin cylindrical shells, the sound-crossing time is much shorter than the thermal evolution timescale and the pressure loss is only due to the radiative loss, without work being exchanged between neighboring shells. The outer shells would experience such a work exchange because of higher pressure gradients with their neighbours. One must also account for the fact that the expansion is initially fast and slows down to an asymptotic mechanical equilibrium: the sound-crossing time also evolves.

Concluding remarks

We have developed a phenomenological model for the evolution of the gas stream immediately following a virial shock. However, in several ways, the model left important open questions. These questions are:

• Is there a heating process that prevents the hot post-shock gas from cooling signicantly? In Section 2.5, we mention mixing with the hot halo gas and turbulent heating as such possible processes. These two processes need to be investigated with numerical simulations.

• How strongly do outows and inow couple? Such a coupling is a natural way of regulating or moderating galaxy growth. Through the expansion of the post-shock gas and warm clouds will amplify its cross section outows from the galaxy hosted within the halo. The physics behind such coupling and its eciency should be investigated numerically.

• Are there any observational tests of our scenario? Current observations may already provide observational constraints but we are still missing an interface our scenario and observables. Phenomenological and numerical studies should be post-processed to compute intensity of cooling lines and absorption columns. For example, the gas which is cooling into the warm clouds and its subsequent passage into the halo, may be a relatively prodigious producer of Lyα photons (see Vernet et al., 2017b, for example).

• What are initial physical conditions of gas penetrating halos from cosmic laments? For the stream to become bi-phasic, the temperature and/or density of the post-shock gas must be initially inhomogeneous. It is the degree of this inhomogeneity that sets the timescale over which the medium becomes bi-phasic. Determining this requires us to investigate the thermodynamics of the lament before it penetrates the halo. We discuss this further in the next chapter relying on numerical simulations and analogy with the development presented here.

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First motivations

After having devised a frame in which our phenomenological model applies (see Chap. 2), constraining the free parameters was the new issue. One way to constrain at least the overdensity factor f and the stream radius r stream was to use a numerical simulation. To obtain values as realistic as possible, cosmological conditions had to be set initially. And to be able to extract the relevant quantities, a focused zoom-in was needed. This is why we decided to get closer to Yohan Dubois, who presented us the code Ramses.

An in-depth analysis of an intergalactic lament revealed a great richness of information that could be advantageously related to our problems. The analogy we found between halo accretion and lament accretion has proven to be more profound than initially thought: the very processes involved in our phenomenological modelling, namely thermal fragmentation and turbulence, were as likely to occur in the lamentary context. This convinced us to clarify and organize our ndings.

Rationale

We exposed in the introduction the importance of intergalactic laments (IGFs) as a major channel for the material needed to build galaxies. The study of the socalled cold-mode accretion (eg. Birnboim & Dekel, 2003;Kere² et al., 2005;Dekel & Birnboim, 2006;Ocvirk et al., 2008b;Dekel et al., 2009) was mainly focused on the thermal state of the gas when it has already entered the accreting halo and its possible fate if and when it reaches the hosted galaxy. On the other hand, following the initial theoretical works 1 from Hoyle (1951); Peebles (1969); Doroshkevich (1970); White (1984), further investigations were made about the eect of a tidal torque on laments (may they bind galactic halos or even cluster halos) and the results on the halo spin and galactic rotation by deposition of angular momentum (eg. Pichon et al., 2011;Powell et al., 2011;Stewart et al., 2011;Libeskind et al., 2013a,b;Stewart et al., 2013;Danovich et al., 2015;Tillson et al., 2015;Codis et al., 2015). As well, some of their internal rotational structure in the form of a quadrupole of vorticity has been detailed from numerical simulations (eg. Pichon & Bernardeau, 1999;Laigle et al., 2015;Codis et al., 2015).

As far as lamentary structures are concerned, the most advanced eld so far is that of interstellar laments (ISFs), where gas radiatively cools and is further conned gravitationnally (eg. Heitsch, 2013b,a;Gómez & Vázquez-Semadeni, 2014) or by turbulent motions of the ISM (eg. Padoan et al., 2001;André et al., 2010).

This advance is mainly due to the possibility of observing molecular gas in these 1 Those works contitute the Tidal Torque Theory, or TTT.

structures (Nutter et al., 2008;André et al., 2010;Schmalzl et al., 2010;Arzoumanian et al., 2011;Hill et al., 2011;Juvela et al., 2012). The main dierence between IGFs and ISFs is the role of the cooling: it initiates the condensation of ISFs whereas it is induced by the gravity-driven condensation of IGFs.

This knowledge is however a great benet. It allowed a discrimination and a renement of previous theories owing to cylindrical stationary structures especially in the case of an isothermal gas but also in polytropic cases (eg. Chandrasekhar & Fermi, 1953;Stodólkiewicz, 1963;Simon, 1963;Ostriker, 1964a;Fridman & Polyachenko, 1985). Of course, the isothermal case is well adapted to ISFs, but could fail in a complete description of IGFs where the thermal equilibrium is much longer ot reach. The polytropic case is more general (the isothermal case is recovered when the polytropic index tends to ±∞) and we will test the consequences of the related Lane-Emden equation in our data.

More precisely, the state of the cold core needs to be investigated further, as it is the basis of any cold mode accretion. The conditions under which such a core may collapse due to radiative cooling have been studied by Gray & Scannapieco (2013) by relating the densities and metallicities of the gas with the redshift.

The consequences on the DM-gas decoupling, as well as linear masses of gaseous laments were statistically derived from a cosmological simulation at z ≈ 5 by Harford et al. (2008);Harford & Hamilton (2011).

The gas dynamics in IGFs, its thermal evolution from its outskirts to its axial core and the possible phenomena that would be missed by the simulation but traced through large scale clues, all remain to be inquired into. In order to complete the current knowledge, we will bring some answers to the four questions:

• Is the cylindrical description accurate? Under which conditions?

• What are the typical scales of a quantitative description of an IGF?

• What is the physical state of the gaseous IGF before the halo-entry?

• Do its dynamics aect its thermal state?

Numerical method

We here present the numerical means used to accurately time-integrate a limited set of physical equations, relevant to the evolution of the Universe, and down to the desired scale. The recording of successive states of the integration let us choose an accurate and realistic basis for a numerical study of IGFs.

The Ramses code

Ramses is a code initially designed and programmed by Romain Teyssier (Teyssier, 2002). It is remarkable by its association of an N-body simulation to describe the evolution of a set of dark matter particles with a grid whose local resolution may adapt dynamically and increase in regions of interestnamely an adaptive mesh renementto describe the evolution of elds pertaining to gas physics: density, pressure, velocity components, and metallicity. The numerical scheme used to integrate the hydrodynamic equations employs a high-order Godunov solver with total variation diminishing interpolation of cell-centered quantities at interfaces of cells in order to solve one-dimensional Riemann problems. It is designed to ensure the conservation of physical quantities: mass, momentum, energy, and, if needed, chemical abundances, and so on. It is also very ecient in capturing shocks, which is an ubiquitous phenomenon in astrophysics.

The renement consists in dividing locally the side of a mesh cell by twowhich means splitting each tridimensional cubic cell in eight smaller cubes to reach one higher level of renementfor each cell of gas whose mass crosses a predened ceiling value. When matter condenses, the mass in the cell increases up to the ceiling, then is split in eight until further condensation, so that the mass of a cell of any level is comprised in a relatively narrow interval, all cells having the same mass, in a factor of eight.

Cosmological simulation

We performed rst a dark matter-only cosmological simulation. Cosmological parameters for this simulations include the Hubble constant H 0 = 70.3 km.s -1 .Mpc -1 , a matter-to-total mass-energy ratio Ω m = 0.276, a at Universe (so Ω Λ = 1-Ω m = 0.724), a baryonic fraction f b ≈ 0.1603, an amplitude of the initial matter powerspectrum σ 8 = 0.811. Physical parameters specify that the side of the cubic box is L = 100 Mpc.h -1 in comoving coordinates, i.e. L ≈ 142.2 Mpc with our choice for H 0 . Numerical parameters were set so that the boundary conditions for the simulation box are periodic conditions. The primary, coarse resolution is set to ∆x ≈ 556 kpc and this corresponds to a 256 3 3-d grid, which we will refer from now on as level 8, since 2 8 = 256. Snapshots of the simulation were output from initial redshift of z ini = 100 down to z fin = 0.

The retained output for this study corresponds to an equivalent redshift of z = 2.121. A code used to detect halos and subhalos was then run on the outputs. This program is HaloMaker (Tweed et al., 2009), set up to nd halos with Adaptahop (see the appendix of Aubert et al. (2004)). It provides a halo tree with their masses and equivalent virial parameters. From this catalogue, we selected a halo of 12.6 10 11 M at redshift z ≈ 2.

Zoom-in

Using one of the software tools provided with Ramses, namely geticref, we managed to extract the identity of all dark matter particles contained in a sphere of radius 3 times the virial radius of the desired halo and sharing the same center. The initial positions of all particles being known, it is then possible to determine the envelope of the initial region of space that will condense to form the said halo. Using the software tool MUSIC (implementation and algorithm, see Hahn & Abel (2011)), a new set of initial conditions was produced, so as to reach the renement level 12 in the region of interest. This time, gas is present, cooling follows the interpolated function from Sutherland & Dopita (1993) for a pristine gas (i.e. metal-free) and with no star formation.

A run is then launched with a maximal rening level 17. Let us recall that the nest cells have then a size corresponding to a (2 17 ) 3 simulation, which is 348 pc of side .

From this new simulated data, we extract again a catalogue of halos with

HaloMaker. Inside the region of interest, we noticed that the higher resolution, and especially the ner particles, allowed for fragmentation so that the mass initially contained in the targetted halo has split into several smaller ones. The halo we selected has a mass of 2.78 10 11 M . Its evaluated size is r halo = 190.34 kpc and the virial radius given by the code is 58.5 kpc. This discrepancy already shows that the detected halo is not spherical. The rst value has been used as unitary in all computations.

Study in cylindrical geometry

We here consider the studied lament in cylindrical coordinates. This geometry may seem obvious when tackling objects like laments, but it would also be naive, as we will see, to model laments as purely cylindrically symmetric regions. As matter falls onto the lament and itself and its environment are pulled by a halo gravitational well, all cylindrical coordinates are relevant, because we expect the lament to be at the intersection of sheet-like structures. The possible coordinate separation and the departure from symmetry are then tested.

The radius from the axis is denoted r. The altitude, i.e. the distance to the center of the halo, is ζ. The azimuthal coordinate is θ. The adjective transverse is used whenever ζ is xed, while longitudinal stands for functions depending on ζ only.

The use of cylindrical coordinates (r, θ, ζ) rst requires dening a frame of reference. After having done so (3.4.1), we then set limitations helping dene correct projections of the simulated data (3.4.2). 10 -30 to 10 -22 g.cm -3 , from which is only displayed a thin band around one to a few 10 -28 g.cm -3 .

Choice of a reference frame

Simplest theoretical expectation: axis as a halo-halo linear bond Filaments are by nature denser regions of space connecting even denser nodes, i.e. halos. Dynamical considerations lead to a simple denition: halo centers lie at points where the gravitational acceleration converges from all directions; between two neighbouring halos must exist a saddle-node where convergence is bidimensional and the last dimension is that of a diverging eld; laments are extensions of this saddle-node until they reach the xed points of halos.

Choice of an origin The most natural way of setting the origin of the cylindrical frame is to use the detected center of the nearest halo accreting the studied lament.

Following this choice, the axis of our frame is a ray emerging from the halo and the ζ coordinate may be interpreted as a distance from the halo center. This ts with studies of halos where the relevant coordinate is the spherical radius.

Deviation from linearity Contrary to the simplicity of a cylindrical frame, laments have no reason to be perfect straight lines, since the whole environment is a priori highly heterogeneous and the corresponding vector elds highly anisotropic.

Despite this deviation, the interpretation of our results does not suer from it. A ner approach would consider arcs and an even ner one would track the most probable sites for saddle-nodes with a gradient-based method. For a rst study, such a renement is not needed however. We explain further how to control the amplitude of this deviation.

Studying slices of data

Projecting data is quintessential to multidimensional analysis. The way the data are originally distributed has a crucial impact on optimizing the projection method.

But optimization is a tedious task that can be partly skipped, whenever its benets are in fact negligible. This is the balance we sought for.

Desired accuracy Orders of magnitude and scaling relations One of our main goals being to establish not only narrow ranges of values for physical quantities, but also their gradients and ultimately the possible scaling laws relating them, so as to discriminate between mutually incompatible theoretical predictions Avoiding the trap of a deceptively high precision Firstly, the resolution of the simulation is a frontier. Even excellent methods for interpolating the data will not create additional relevant values. Secondly, even high-precision values varying at the scale of the resolution are neither to be fully trusted as a physically exact variation. Keeping these two considerations in mind, we may design methods of analysis that are both reliable enough and not resource-consuming.

Width of the slice We simply look now for a discretization step in the ζ dimension. Since the axis of the frame is arbitrary, transverse slices are taken along planes that are arbitrarily tilted in comparison with the simulation grid. An excessively thin slice would thus have few to no chance to uniformly capture data points. Moreover, the slices need to contain enough points to enhance the inevitable signal-to-noise ratio resulting from discretization. If too big a cell is cut by the slice but its center falls out of the slab, the space it occupied will be considered empty, damaging the analysis. In a way, we could say that the wider the slice, the smoother the projected elds. And on the other hand, the wider the slice, the more it will capture uctuations, articially blurring gradients.

The best estimate for the width 2δζ took into account the density of data and the fact that maps should noticeably be dierent between ζ and ζ + δζ. Its value has been taken to be δζ ≈ 7.14 kpc.

Assigning a corrected origin to the slice Having cut a slice, and given the fact that the lament itself is not perfectly linear, a local centering is needed. To give a proper origin in the planar frame, we use a simple method called shrinking disk. The principle is to evaluate the 2D coordinates of the barycenter of all points in the 3D slice and then iteratively reduce the size of the disk, evaluate the new barycenter, and so on. Doing so, from a radius of 150 kpc down to the smallest size of a grid cell in that slice (say 350 pc), dividing each radius by 1.1, the loop converges to a density peak inside the slice of lament.

It is possible then to retrieve all detected centers, slice by slice, to draw a polygonal chain that matches more closely the density valley. However, this method does not accurately correct the bias introduced by the linear approximation for the lament axis. This is because the direction of the planes is still taken constant and equal to the initial halo-halo bond direction, rather than parallel to the local line element.

3.5

Anatomy of a transverse slice

Structure and scales

The rst thing to remark about laments is their structure. To highlight it, transverse maps are a good means. After having detailed the distribution of matter, we will present scalar properties of the dynamics.

Density eld

The supporting gure of this subsection is Fig. 3.3, made of four color maps. To exemplify the structure, we chose a transverse slice taken at 343 kpc from the center of the closest halo, which corresponds to about 1.8 times its virial radius.

The colors correspond to rounded values instead of continuously varying ones, so that a color change draws a constant-value contour-line.

Dark matter The lower-left panel shows the dark matter distribution, as well as three circles used as guides to the eye. They are centered on the slice origin, dened in the way described above. Their radii are equally spaced: 30, 90, and 150 kpc.

Structuration voidsheetlament

The contours are far from isotropic outside of the innermost circle and above 90 kpc from the origin, three lobes are apparent: two are roughly aligned in an east-west direction, while the one at south-east seems to host a dense blob. The latter is common to all the transverse slices, meaning that there is a lighter companion lament, that merges with the main one after their plundging in the halo. The former ones indicate the presence of a dense corridor, also found in every slice, thus corresponding to a dense sheetlike structure. The least dense regions on the map are then the ones commonly referred to as voids.

Typical densities The closest to the origin of the gure, the more isotropic are the DM density contours. The region which is apparently relevant to a lamentary description is inside the 90 kpc-radius circle. For the selected halo and at the given altitude, the densities inside this disc range from a few 10 -29 to a few 10 -28 g.cm -3 , with a relatively low gradient. These values are to be compared to the critical density of the Universe at z ∼ 2, which is slightly less than 10 -28 g.cm -3 . The DM overdensity is therefore to be compared to Ω DM ρ crit ≈ 2.4 10 -29 g.cm -3 , thus comprised in a range 1 -10, indicating that the condensation is rather smooth.

Analogy with halos If it was a slice cutting through a halo, we could think of a radially decreasing density with lamentary extensions. We can propose the following analogy; a lament is a dense cylindrical structure connected to other laments by less dense, radially extended sheets in the same way halos are spherical This analogy is to be completed further and this will prove fruitful in understanding the dynamics and growth of laments.

Gas The upper-left panel shows the gas distribution. The three same circles as before are shown.

Sharper structuration

The anisotropies of DM density are recognizable, except that they are much sharper. The east-west sheet, the companion lament as well as the continuation (western) of a dilute sheet aligned with the one crossing the companion (south-western) are visible. As opposed to the DM distribution, the most isotropic contours lie within a radius of 30 kpc and exhibits two small lobes, perpendicular to the sheet direction. We will further explain these.

Typical densities Inside the innermost circle, densities range from a few 10 -29 to up to a few 10 -27 g.cm -3 at the computed origin. This already indicates much stronger radial density gradients than DM. The presence of a much denser, tiny ( 3 kpc radius) region in slices of altitudes down to less than one virial radius follow what we will call the dense axial core, or simply core of the lament.

Compared to Ω b ρ crit ≈ 4.57 10 -30 g.cm -3 , this represents an overdensity ranging from 10 to 1000.

Baryonic fraction

The lower-right panel shows the baryonic fraction, dened as f b = ρ/(ρ + ρ DM ), where ρ and ρ DM are the locally computed densities of gas and dark matter, respectively. Note that the smoothing that resulted from the DM density computation is important to this representation, as sharp DM clumps would have dominated it otherwise.

Interpreting f b The values of the baryonic fraction need to be compared with the cosmic value. This value is a constant all along the cosmological run of the simulation and had been set to f b = 0.1603. On the gure, the black and dark blue regions exhibit a value lower than the cosmic one. The discrepancy between the local and the cosmic values is to be seen as the consequence of a decoupling of gas and dark matter. This decoupling happens whenever the gas is subject to physical processes which do not occur for dark matter, such as shocks or radiative cooling, as we are about to see from temperature maps.

Temperature eld

Three important features are to be remarked in the top-right panel of Fig. 3.3, the one representing the temperature eld in the selected transverse slice.

First, the structure voidssheetslament is as clear as in density maps. Voids appear cold (< 10 4 K), walls are hotter (up to a few 10 5 K). This is consistent with a scheme where the condensation is associated with compression and thus heating.

More, the rightmost sheet, which displays the highest density, appears hotter than its other counterparts, indicating that it must have resulted from a stronger compression through its own gas accretion.

But a strong gradient in temperature, around 90 kpc of radius, makes the lament highly contrasted: a strong shock (density and temperature ratios accross hint at M ∼ 10 -15) propagates outwards into the gas from voids. Closer to the center, the post-shock medium gets cooler and a four-fold structure of a 10 4.5 -10 5 K gas sits there. It will be found later in other elds (see Figs. 3.4,3.14,3.15).

Still consistently with the staged condensation aforementioned, the region where one expects the highest convergence is so compressed that a shock has formed, and further condensation is allowed by the subsequent loss of pressure due to radiative cooling. This is in agreement with the baryonic fraction map panelwhere the post-shock medium extends, gas pressure is so high that it is prevented to collapse at the same rate DM does, but cooling, as a net pressure leakage term, favors a secondary condensation which leads to a higher baryonic fraction close to the axis.

As envisioned by Birnboim et al. (2016) in their cylindrical case, the gas accreting on the IGF can endure a shock. However, we would not try to apply their stability criterion in our case since our ow greatly diers from the one they assumed. Theirs is purely cylindrical while our accreting sheets break the symmetry and the shock fron continuity as well, and theirs is homologous while ours is not even purely radial just beind the shock (see Fig. 3.4).

Nevertheless, we can keep from that work the idea that only the dimension of the accreting object changes from a sheet, to a lament, to a halo. This reinforces the analogyi set out above, according to which a lament is subject to the same kind of accretion-related phenomena as a halo, only with a dierent dimensionality.

Like halos, laments may be surrounded by a shocked corona exhibiting a relative lack of baryons whereas a central region would host an overrepresentative fraction of the like.

Dynamics

The upper-left panel of Fig. 3.4 is a colormap of the radial velocity eld. The bright green contours help identify where it changes sign, that is where the ow is converging on one side and diverging on the other. Outside 90 kpc, the ow is purely falling in towards the lament. Inside that radius, a quadrupolar structure, reminiscent of, and in fact analogous to the one seen in the temperature color map. It displays two facing regions where the ow converges and two other facing ones where it diverges. The four meet at the detected center of the lament.

The constant inow counteracts the central cooling, contributing to support the enclosing shocked gas. The upper-right panel of the same gure is the counterpart of the one described above: it represents the azimuthal velocity eld, the positive (rep. negative) sign corresponding to the counter-clockwise (resp. clockwise) motion. Outside 90 kpc, the eld seems to be divided in two and its vanishing contour (also in bright green) can be considered the locus where gas is equally attracted by two opposing structures, which appear to be the two pairs of accreting sheets. Inside 90 kpc of radius, the quadrupolar structure is visible but even clearer inside 30 kpc.

The lower-right panel is the colormap of a composite eld, that of the ratio of the absolute radial component to the norm of the velocity. Greener colors (i.e. closer to 0) indicate that the azimuthal component dominates, whereas the more purple colors (i.e. closer to 1) stand for a radially dominated eld. The two denser sheets accreting on the lament are noticeable through the fact that their dense inner slab is characterized by a purely radial inow, while their vicinity is marked by a gradient from radial (accretion from void to lament) to orthoradial (accretion from voids to the sheets). This latter gradient is stronger further away from the lament, where its gravitational inuence is lower. The inner quadrupolar structure has a clearer explanation with this panel: gas ows in from two opposite directions, splashes in the two perpendicular directions and the surrounding accretion ow imposes that four recirculation vortices form.

This qualitative description perfectly ts with that of Laigle et al. predominantly aligned with, their laments and that the cross-sections of these laments are typically partitioned into four quadrants with opposite vorticity sign. This is not surprising when one considers IGFs to be lower-scale versions of ICFs. Note however that there is a slight asymmetry in the structure. In fact, the two sheet-like inows that converge in the center are not exactly aligned. This misalignment, or impact parameter, must result in a non-zero overall angular momentum. The whole rotation reinforces the vortices of the same sign whereas it should reduce those of opposite sign. This is shown, in a stylized way, in Fig. 3.6.

From the three panels described above, we can also see that the shock (see Fig. 3.3, upper-right panel) has an eect on the direction of the post-shock velocities. Indeed, as already seen, the two main shock fronts are almost nowhere purely normal to the ow, implying deections of owlines.

The lower-left panel of Fig. 3.4 is a colormap of the axial velocity eld. More precisely, its sign is positive and corresponds to a fall towards the halo marking the end of the studied IGF. The eld seems to be divided in two halves, the division being marked by the two aligned and opposite major sheets: the upper one falls slowlier (∼ 50 km.s -1 ) than the lower one (∼ 100 -15 km.s Let us recall Fig. 3.6. Does it really apply to our lament? To see that, we performed a radial, mass average of the radial and azimuthal components of the velocity eld in the transverse plane. Each is dened as follow. For a collection of N particles, indexed by i and aected with masses, radial positions and velocities m i , R i , v i i≤N , we dene, on a disc of radius r, the mass-averaged radial velocity v r (r) and as:

v r (r) = M (r) -1 R i <r m i v i • u r (3.1)
with M (r) the mass inside the considered disc: M (r) = R i <r m i and u r the radially directed unit vector.

The lefthand panel shows that the total norm of the mass-averaged velocity, being a proxy to the kinetic energy, strongly decreases while approaching the center. Its steeper decline is in the region containing the shock. That energy is dissipated by viscous forces and the resulting heat then radiated away. The radial component is everywhere negative, which is a signature of a ow converging at all scales. This is what allows us to read and interpret this plot from the large to the small scales. Its absolute value follows the same decline as the total norm from 300 kpc down to about 12 kpc. Alike, the azimuthal (or orthoradial) component is everywhere negative. This implies that at any scale, the ow is globally rotating clockwise. This conrms the sketch of Fig. 3.6. The righthand panel explains the drop of radial component at 12 kpc: whatever may be the dissipative eects, the convergent ow is converted into a rotative ow as we approach the origin.

Even if a radial only study erases all anisotropies, this is still an important feature of the gas ow: rotation may stabilize the gas against its collapse.

Accretion rates

A good test of the cylindrical approximation, apart from the qualitative appreciation of transverse colormaps, consists in plotting the accretion rates at dierent radii and as a function of the azimuthal coordinate θ. In a perfect cylindrical geometry, these plots would be vertical lines. To compute such accretion rates, we opted for simplicity. Instead of binning angularly, we simply projected, for each point of the transverse slice, the product of the density by the radial velocity. It is then a mass ux through a rectangular section whose area is the product of the slice's width by the pixel width.

Figure 3.8 reports, in logarithmic scales, those quantities. The points are taken in three rings of radial width 10 kpc and of central radii 150, 90, and 30 kpc. At 150 kpc, the accretion is clearly not cylindrical and we can determine the entry regions of sheets. A black horizontal line serves as a guide to discriminate between accretion from voids and from denser structures. The highest peak corresponds to the companion lament. At 90 kpc, the peaks have dampened and the main accreting sheet dominates. Some blue points are found close to the angle π, meaning that there is a tiny angular sector of outow. At 30 kpc, two main contributions to the inow are found, separated by two outows: this is precisely the vortical quadrupole which is responsible for that. The fact that the angular spans are not all equal hints at the asymmetric picture of Fig. 3.6.

We can now say that the mass ows inwards are neither invariant under rotation nor does the cylindrical accretion (from voids) prevailing.

Unsteady, out-of-equilibrium

Radial binning Only the radial coordinate (i.e. distance to the axis) has been retained to produce radial proles. The loss of angular information then erases a lot of the inner lament structuration (eg. the vorticity quadrupole), but aims at nding a more general, detail-blind model that may subsequently help designing idealized and toy models.

Choice of a slice The selected slice is at ζ ≈ 343 kpc (recall that r vir ≈ 60.8 kpc). In light blue is shown a t of the gas prole by a Plummer prole with parameters ρ 0 = 1.68 10 -27 g.cm -3 and r 0 ≈ 24 kpc (see text for details).

Mass and density prole

Fig. 3.9 present the mass and density radial proles of the selected transverse slice. The radii go below the highest resolution scale of ∼ 110 pc because of the projection eect of a thin slice tilted from the simulation grid axes. We kept these bins because, even if it may appear too ambitious, the proles are rather consistent at these low scales. The histograms are made of 20 bins and have been produced using 8608 cells (total 8.87 10 8 M ) of gas and 1706 dark matter particles (total 4.28 10 9 M ).

As is often the case with polar or cylindrical geometries, the choice of a center is crucial to the radial proles. For the transverse maps, we used the shrinking disk method, computed with all masses, gas and DM. Here, since one of our goals is to compare with self-gravitating models of gaseous laments, we let the code set the origin based on gas masses only.

To produce these proles, interpolation was needed. Like for transverse maps, the discrepancy between data representations is a challenge. Let us recall that gas is computed on a grid whose axes are not align with those of the slice and dark matter is a cloud of mass points. To emulate a lled space, we used the same technique as for maps: both gas and DM are considered mass points and we operated a convolution of the resulting distribution with an adaptive gaussian, bidimensional kernel.

Projecting the distribution of points, indexed by i, on the radial component only requires a transform of the kernel κ i (r):

κ i (r) = 1 2πσ 2 i e - r 2 +r 2 i 2σ 2 i 2π 0 e rr i cos(θ) σ 2 i dθ = 1 σ 2 i e - r 2 +r 2 i 2σ 2 i I 0 ( rr i σ 2 i ) (3.2)
where r i and σ i are the positions and adapted spreadings of the distribution, and I 0 is a modied Bessel function of the rst kind and of order 0.

Such an interpolation allows for an oversampling of the radial bins. But are the results physical? We compared dierent samplings to understand the eects of our choices. First, a linear interpolation of the cumulative mass gives very noisy results, polluted by the dark matter discrete distribution. The fact that DM acts on the ow through its potential (inverse square law) is not accounted for in this case. Then, the choice of the kernel size matters. We scaled it, as would seem natural, with the simulation size of cells: σ i = α∆x i . The applied coecient α spreads the densities. Increasing its value obviously smooths the prole but also changes amplitudes. Above α = 5, the prole does not change anymore. We thus retained this value.

Of course, some features may be smoothed out but the smoothing factor being proportional to the resolution length, which in turn is shorter at small radii, the general behavior of the proles is preserved in logspace representation. The binning was chosen to be regular in logspace, totalling 1000 bins. It even reinforces the continuity of the density eld close to the center, since it would have been highly

uctuating due to the small samples in inner shells.

Typical linear mass range The linear density of our lament at this altitude give us a typical range of values: from r ≈ 30 kpc to r ≈ 150 kpc, the linear mass is comprised in the interval 3 10 7 < λ/M .kpc -1 < 10 9 .

Density proles On the other hand, density seems to fall with a power-law of slope ∼ -1 after a central plateau. Gas density itself declines with an exponent ∼ -2. The gas density prole itself resembles a Plummer-like prole, with a plateau below a characteristic radius (here between 3 kpc and 10 kpc) and a power-law decline. This cylindrical prole is inspired by Ostriker (1964b), where the following law was proposed:

ρ = ρ 0 1 + r r 0 2 p/2 (3.3)
where the plateau is set at a value of ρ 0 up to a radius of r 0 at which it bends down with an asymptotic slope p. This was developed from the Lane-Emden equation 2 for an isothermal gas in cylindrical geometry, with p = 4.

The idea to t our data with an other exponent is consistent with the fact that the gas we study is not the sole source of gravitational potential, that it is certainly not isothermal but maybe evolving from or towards such a case, and that a rotation parameter, function of the radius, may alter the slope by supporting the gas. The gas density prole is well tted by such a prole, with ρ 0 = 1.68 10 -27 g.cm -3 , r 0 = 23.89 kpc and p = 2.070.

Dark matter also seems to follow such a prole, with a much broader plateau.

This is a signature of the impossibility for DM to condense through dissipation.

The lag betwwen the knees of gas and DM explains the decreasing of the baryonic fraction.

DM-gas decoupling The baryonic fraction f b helps understanding the importance of gas physics. When below ∼20 kpc of radius, the baryonic fraction is higher than the cosmic value. At 30 kpc it reaches 50%. Then it decreases toward the cosmic value, reached around 90 kpc, and nally increases again up to 150 kpc. Above 300 kpc, an other dense structure is met. Clearly this indicates a decoupling of gas and dark matter inside the lament. While the gas may condense, dark matter particle y through the space without collision.

Existence of a core The existence of a plateau at low radii helps dening a dense core. But an explanation for this phenomenon still remains to be found. To understand the fact that the gas has neither completely collapsed due to cooling nor has it been prevented from collapsing, we must investigate two families of processesnamely, the connement and the support forces.

3.6

Anatomy of longitudinal structures

Longitudinal slices

In the previously shown gures of transverse slices, the natural coordinate system from the cylindrical geometry would reduce to polar coordinates (r, θ), ζ being a given constant. Setting the center as the origin and the x-axis a the reference, angles θ are completely dened. In each slice, we took a sliver, that is a thin beam along a given direction θ, from each transverse slice. The sliver data array is built by linear interpolation of accordingly rotated transverse slices. We juxtaposed them to build a longitudinal slice corresponding to θ. The width of the sliver was set to match the resolution of transverse slices, that is 365 pc. Each sliver is represented vertically on the longitudinal maps with the width of each transverse slice, that is 7.14 kpc, and is made of 200 data values, i.e. having a (vertical) resolution of 2.9 kpc.

Having done so brings one advantage and one disadvantage. Using the centered transverse slices aligned all those centers in the longitudinal slices, making the lament appear rather straight. This straightness is an artifact. The advantage resides in the setting aside of environmental eects that bended the IGF and thus ensures the visibility of a continuity in longitudinal structures. The drawback is the exact other side of the coin: the hidden bendings could have proven useful to trace either a regular pattern (such as an instability possibly originated from vorticity) or its eects on the internal structure. Such curvature eects are still to be investigated. Ostriker, 1964a). It has been extensively studied, for example by G. Horedt (Horedt, 1970;Viala & Horedt, 1974a,b;Horedt, 1983Horedt, , 1986aHoredt, ,b, 1987a,c,b;,c,b;Oproiu & Horedt, 2008), who published a book specically dedicated to polytropes in astrophysics: Horedt (2004). cylinders.

The simple view of a lament as a tube is found in the longitudinal slice at θ = π/2 (right-hand column of Fig. 3.10). The dense core has an almost constant radius of ∼ 30 kpc and is embedded in a larger tube of shocked gas of radius 60-90 kpc. Two peculiarities are already remarkable: at the halo radius (190 kpc), an accreting subhalo is encountered. It is still interesting to see that its own gaseous halo is in direct continuity with the lament gas. Note that its own shocked gas (over 10 6 K) would be found in transverse cuts between ζ = 240 kpc and ζ = 300 kpc, where the condensation of the subhalo may not be the cause of this shock stronger than at higher altitudes. A strongly shocked envelope may develop at the lament edges with voids and invade the lament: this can explain the presence of a virial-like shock in laments from other numerical works, with an external origin and propagating outside-in.

The second noticeable feature is the presence of small condensates along the axis of the lament. The entropy map is clearer: ve spots of very low entropy are aligned; they have suciently condensed to strongly contrast with their gaseous environment. Their apparent radius is close to 30 kpc. This is reminiscent of, at higher scales, galaxies forming in the cosmic web, or at much lower scales, stars forming in interstellar laments. This common behavior has been studied in Freundlich et al. (2014), where a dispersion relation is derived to describe this pattern emerging from a gravitational instability in a self-gravitating, rotating cylinder.

The other angles of longitudinal slices (left-hand column of Fig. 3.10 and Fig. 3.11) show that what has been learnt from the one transverse slice studied above is almost generic, all along the lament. Indeed, its environment does not vary much in structure with the altitude ζ. The main accreting sheet is almost always found at the same angle θ = π/8, except maybe above ζ = 485 kpc, where it may be slightly tilted. The lament companion is at about θ = 11π/16 all along too, and it merges with our IGF exactly in the accreting subhalo, close to r vir . We can propose the following hypothesis for the formation of that subhalo: a small structure had condensed in the IGF and gravitationally attracted the companion, with the help of the spherical geometry of the potential of the main halo, that naturally tends to make falling objects converge. The entry point where two sheets merge is also constant in position, at θ = 33π/32.

Linear masses

Linear masses are of interest, as they are often assumed constant along IGFs and their value can be considered one of the main input parameters in models (correlated with the central density and exponent of a Plummer prole). linear mass of all matter. Bottom: baryonic fraction at th three radii.

masses of gas only varies smoothly and the positions of localized condensates are rather clear. The ground value for the lament is close to 10 7 M /kpc of gas inside the cylinder of radius 30 kpc, and ∼ 10 8 M /kpc in cylinders broader than 90 kpc. At 150 ± 60 kpc, the accreting subhalo is visible.

Total matter linear masses show that all condensates have a halo-like structure, more extended than their gaseous component. The baryonic fraction shows that above 30 kpc of radius, it is very close to the cosmic value, except in the core of some condensates and in the subhalo. Inside 30 kpc of radius, however, f b oscillates between the cosmic value and 35 -45%, indicating that the gas occupies a dominant role close to the axis. This also means that it experienced a secondary condensation almost everywhere due to radiative cooling. This noisy behavior may be explained by the combination of the partly arbitrary nature of the centering in each slice, and the fact that the density peak may oscillate (for example due to the vicinity of the four vortices).

A notable feature appears at ζ ≈ 336 kpc. Inside the 30 kpc disc, there seems to be an aggregate of dark matter which corresponds to a slight local increase in the gas linear mass. It is also correlated with a relatively low baryonic fraction (more precisely, it is the cosmic value) compared to the neighboring values. Finally, it is seen on the longitudinal maps (Figs. 3.10 and 3.11) that an dense pocket, analogous to those surrounding other condensates, extends around 320 kpc. This may be a new condensate in formation.

Characterizing the condensates

The condensates appear at the following altitudes: 336 (in formation?), 414, 500, 614 and, extrapolating the rightmost linear mass increase, 769 kpc. Their relative spacings is not constant but decreases with ζ. This does not enable us to dene a typical wavelength to compare to the Jeans length or even to models of rotating cylinders, like in Freundlich et al. (2014). If the intervals depend on ζ, and since they must have formed through a gravitational instability, they must be linked to quantities involved in such instabilities and that evolve with ζ. We can suspect the accretion rate on the IGF from the sheets and voids, the imparted torque, and the tidal stretching imposed by the IGF's ending halos. More investigation is needed, especially through outputs at previous redshifts.

They are best seen by use of transverse slices. between the sheets (here the clockwise sense). We hypothesize that either they formed inside the vortices and migrated, leaving a trace behind; or they formed in the more central region (which should also host a weak vortex) and the strongest neighboring vortex sucked its cool gas in. In both cases it must be a region of low shear and high compression. The rst hypothesis is however more consistent with the dynamics described in Laigle et al. (2015), as sketched in their gure 11.

An other interesting fact concerns the shape of the condensates. At 414 and 614 kpc, they look elongated in a north-west to south-east direction, which suggests that they are tridimensional oblate structures whose spin is perpendicular to the lament's axis. The two other slices show, on the contrary, two shapes closer to circles, letting us think of either a disk whose spin is parallel to the main axis, or a sphere cut by the slice. More investigation is needed to see if, like in inter-cluster laments that carry haloes of masses 10 11 < M H < 10 13 M , the spin is correlated with the shear tensor, as suggested by Libeskind et al. (2012). Laigle et al. (2015) studied systematically a number of ICFs and derived a correlation between the spin-lament and the spin-vorticity alignments with the mass of the halo. They derived probability distributions for the cosine of the relevant angles. The spin and the lament are misaligned (almost perpendicular) for high masses and it tends to align descending from 10 13 to 10 12 M . From there, where the alignment was found to be maximal, the angle between the two directions increased again while descending from 10 12 to 10 11 M . Since our condensates barely reach 10 9 M , we can not rely on these correlations to explain our case.

We looked closer at the rst condensate, the one lying at ζ = 414 kpc. Using a principal component analysis to study it in a geometry closer to spherical, we could derive radial information. Cumulative mass proles indicated that the mass, corresponding to a sphere with an average density 200 times higher than the critical one, is 2 10 9 M . The associated radius is 12 kpc in the PCA space, which roughly corresponds to the width of a transverse slice. Gas largely dominates its matter content up to a radius of about 2 kpc and settles in accordance with the cosmic baryonic fraction at about 3 × r 200 . Note however that, even if the sample of particles and grid cells is sucient to derive smooth statistics, the numerical resolution of the central part of the condensate is the highest. This means that no further renement occurred below about 1 kpc of radius.

An other remarkable feature is seen in the temperature-density distribution.

Although it is clear that gas cools while converging, inside a sphere of a few kpc, the distribution follows a compression which is asymptotically adibatic while approaching the center of mass. This is not a physical eect and rather reects the articial polytropic equation of state applied by Ramses to avoid producing singularities when conditions are met to form stars. Since we did not turn the star formation routine on, we only get this signature of a likely star-forming gas in the inner shells. The study of the virialization by comparing pressure, gravitational and centrifugal forces on spherical shells is therefore compromised by this articial bias.

If more appropriate parameters are given (higher resolution and star formation), a fraction of the 10 9 M of gas will end up into stars. This would enrich the IGF gas with metals, critically enhancing its cooling. The scales of masses and sizes point toward a forming dwarf galaxy. It is noticeable, in the sense that IGF could therefore be considered as channels for dense gas, but also dwarf galaxies, directed toward a somehow typical halo.

3.7

Beyond simulation: modelling subgrid features Numerical simulations are of immeasurable help to researchers, as it is the integration of a number of coupled dierential equations that would not be easily solved analytically otherwise. But rst, the cooking depends on the ingredients put in, that is, a nite set of approximate physical laws relying on an other nite set of assumptions. May the numerics get close to the boundaries of the assumed ranges and regimes, it would be highly dangerous to credit the results with full validity.

Second, integrating numerically means using discrete spaces of values. This generates quantization errors which can propagate and self-or cross-amplify during a run. Good codes like Ramses are designed to minimize and keep the errors controlled. However, apart from numerical errors, discretization also annihilates some interesting physical phenomena: scales interactions are damped, and scales outside of the computed scale range are inexistant.

The case of phenomena emerging from small scales are of prime importance here. It is known that instabilities such as cristallizing fronts in solid physics or even the Kelvin-Helmholtz instability in uid physics, as well as a huge number of pattern-generating instabilities, nd their structure in self-amplied modes, because of a local resonance in the phase space close to the instability's onset. When such modes have a time-frequency (resp. space-frequency) higher than allowed by the time-step (resp. resolution), the instability either does not grow or only the boundary modes grow, at an articially slow rate, and notwithstanding the missing nonlinear interactions with the missing modes.

Falling into this category is the thermal fragmentation, where the cooling of a uid with a perturbed pressure, density or temperature eld increases the amplitude of perturbations. In the astrophysical plasmas we are interested in, very localized peaks can appear, for example due to compressive intermittent turbulence, especially in the supersonic case. But if the small scale outliers are smoothed out by resolution, no seed can be numerically found to grow cold clouds in a hotter backgrounds, or at wrong rates and wrong sizes.

Talking about turbulence, all many theories 3 describing it rely on scaling paradigms. Seeing it as a cascade of large structures breaking into smaller ones, like Kolmogorov in 1941, leads to a transfer of energy from a scale to a lower one, ultimately dissipating it at the lowest ones. This seminal idea has been rened over and over but the basic principle remained the same. It stood because it provides a satisfying explanation to the (fractal) structuration of a ow in which energy is injected at some scale, and ultimately dissipated in the form of heat. More, if ow structures resonate or, said otherwise, self-organize, energy is transferred to higher scales, and this sets up a reverse cascade.

In astrophysical ows, the kinetic energy may be injected at the scales characterizing the ow themselves, but also at scales that characterize small-scale phenomena impacting the ow, like hydrodynamical or thermal instabilities. Thus, missing small scales lead rst to the wrong delay between the energy injection and its transformation into heat, and second, to the possible absence of turbulent modes emerging from small-scale instabilities due to reverse cascades.

We adopt here a particular approach. Our aim is to get, from results whose scales are limited in range, clues to underlying processes owing to subscale physics.

It follows the way modellers design so-called subgrid models, implemented as subroutines that are run to phenomenologically mimick missing processes, with the sole knowledge of gridded quantities and their (discrete) derivatives. We will hence try to extrapolate the plausible subgrid features from our simulation data, based on a simple analytical criterion for the growth of a multiphase medium, in Subsection 3.7.1, and with simple scaling arguments for the existence of subresolution turbulence in Subsection 3.7.2.

Multiphase medium?

Out-of-equilibrium dierential cooling

Radiative cooling being more ecient for higher densities and depending on the ionic composition of the gaz, and thus its temperature (and history), a heterogeneously dense, or hot, or pressurized medium may undergo a dierential cooling, in that some of its compounding regions may cool at dierent rates. When these discrepancies tend to self-amplify, small perturbations may grow so that, from slightly heterogeneous, the medium becomes multiphase.

At thermal equilibrium, heating and cooling terms balance each other. Close to this equilibrium, a phase separation may occur and that case is designated thermal instability. Among the many works dealing with it, Sharma et al. (2012a) derived a typical time for the instability to grow under isobaric conditions, depending on the propension to become heterogeneous. Using the panel of Fig. 3.4 relevant to the infalling speeds, we can roughly say that the lament falls at ∼ 100 km.s -1 .

The slice being at ζ = 343 kpc and the virial radius of the accreting halo being r halo = 190 kpc, the approximate time to enter the virial sphere is ∼ 1.5 Gyr.

This indicates that, apart from any other process that may aect the gas, line and recombination emissions may signicantly cool the gas before the entry everywhere exept in the shock-heated regions. Looking at the right-hand panel, almost all of the dense structures such that the lament and its feeding sheets may develop a multiphase medium. Notice that the densest central peak and a ring bounding one of the four vortices are however prevented from it.

This sole slice then suces to indicate that except from the dense core, the four tubular cooler structures are prone to undergo a phase separation before entering the virial sphere. That statement makes the scenarii for an IGF cold core more intricate, but the consequences would greatly modify the envisioned accretion of warm gas through the halo: compression eects, mixing, boundary instabilities, survivability of the dense matter, would all need to be considered as volume and mass fractions of each phase. We refer the reader to Chapter 2 for a broader discussion about the possibly fragmented nature of IGF gas when entering the halo.

Toward a subgrid modelling The smaller the scale, the narrower the density dispersion. A subgrid model could make use of a simple extrapolating scheme to lower scales and virtually grow separate phases with the associated scales for the cooler, denser blobs and the associated timescales for their growth. We did not get further on implementing it, but leave it as a track for future renement of plasma simulations.

There exists a tight connection between the existence of density uctuations and the turbulent nature of the ow. The next subsection may then be fully complementary in understanding the scale-dependence of density uctuations.

Turbulent precursors

Turbulence has often been studied in regimes that are not fully relevant to astrophysical situations. For example, perfectly stationary statistics of the ow describing quantities only apply on some timescales, i.e. when the injection rate can be considered almost constant and when the cascade is fully developed down to the viscous scale. In the class of problems we try to address, even the injection scale itself may vary on a dynamical time.

To track down the possible wrongful absence of turbulent action in simulated data, one must rst establish the relevant scales. Considering the processes that give birth to turbulent motions is the key. The rst one we must think of pertains to the classical view of a multiscale vortical fragmentation, i.e. the so-called solenoidal

∂ω ∂t +(v • ∇) ω +(∇ • v) ω -(ω • ∇) v = ∇ρ × ∇p ρ 2 +∇× ∇ • τ ρ +∇× f ρ (3.5)
where v is the velocity eld, ω = ∇ × v is the vorticity eld, ρ is the density eld, p is the pressure eld, τ is the viscouss stress tensor, f is the resultant eld of all body forces acting on the particle cell, the vector notation is omitted, the parentheses enclosing an operand-free nabla sign ∇ is itself an operator. The equation applies at any point in space and at any time.

Two terms are able to play the role of a source in Eq. 3.5. These are the rst and the third of the right-hand side. The third may represent a forcing term, at which is aected a typical injection scale. The other one is called the baroclinic term. It measures the baroclinity, i.e. the misalignment of the gradients of density and pressure.

On the left-hand side of the same equation, lies a term proportional to the vorticity itself, and to the divergence. Facing the time derivative, and assuming that the divergence evolves more slowly than the vorticity, it acts as a rst-order term. Where the ow diverges, it dampens the vorticity, whereas where the ow converges, it amplies vorticity.

In a Kolmogorov-like development, one would average Eq. 3.5 on a given scale, exhibiting the dependence of each term on the said scale, using a Fourier expansion to eliminate dierential operations. The forcing term would only be relevant at one scale, and the assumption of stationarity of the spectra would be made. The study of turbulence attributes a great importance to the characterization of the forcing.

To estimate its scale, we will make use of the two elds previously proved relevant, the divergence and the rotational of the ow, mapped on the two upper panels of Fig. 3.15. The units of the colorbar is, strictly speaking, a frequency, or the inverse of a time. Here that dimension is of no interest and only the contrasts are to be taken into account. As is not so surprising, regions of strong convergence (i.e. where the divergence is negative) are found along the shock fronts and where the two inows meet. On the other hand, the sites of strong shear (i.e. where the absolute value of the vorticity is high) ar along those two inows, where it slips against the accreting gas previously shocked.

Notice the sign of the vorticity. Its alternation while turning around the center is reminiscent of the quadrupolar vortical structure. It is also in accordance with those rotative tubes where found. The shearing regions seem to span over around 90 kpc each. The compressed regions extend over a total length comparable with the perimeter of the 90 kpc circle, which corresponds to the same order of magnitude as for shear.

Between this estimated injection scale and the maximal resolution scale, there are a number of scales log 10 (90 kpc/348 pc) ≈ 2.4. This is clearly low to consider a cascade which follows scaling laws such as those found in literature, since they should span a wider range of scales to even develop.

The lower-right panel of Fig. 3.15 shows the logarithmic value of the baroclinic term. A rst look at the colorbar makes it remarkable that the term varies over almost nine orders of magnitude. Then we need to add a technical but crucial point. The baroclinic term as computed here is a local contribution to the vorticity increase, but on a tighter mesh than the actual simulation output. This is thus to be considered as a virtual value everywhere the grid is in fact looser. However, interpolation must have smoothed the elds and thus attenuated the gradients; what is shown is thus a lower limit. It appears to be much higher in regions where Note that the impact parameter already described and schematically drawn in Fig. 3.6 will lead to a structuration of vortices of dierent scales. The ow should then evolves towards a tessellation respecting alternating vorticity signs, in the same way one would construct a system of gearings. This tessellation is, on a rst approach, giving rise to a fractal bidimensional ow, i.e. solenoidal turbulence.

The central region of convergence is itself very likely to host a turbulent cascade.

In the assumption of a purely cylindrical geometry, one can trace random motions through the second moment of its radial distribution. In the same way observers infer turbulent motions in molecular clouds, we studied a velocity dispersion as a function of the radial size of the domain over which it is computed.

We mass averaged the squared components of the velocity to obtain dispersion values σ(r). For a collection of N particles, indexed by i and aected with masses, radial positions and velocities m i , R i , v i i≤N , we dene, on a disc of radius r, the mass-averaged velocity v r and the dispersion σ(r) as:

v r = M -1 r R i <r m i v i σ 2 (r) = M -1 r R i <r m i v i -v r 2 (3.6)
with M r the mass inside the considered disc: M r = R i <r m i .

Applied to the exemplied transverse slice, we plot the blue curve of Fig. 3.16.

On the same gure is plotted an equivalent Reynold number, dened as Re = ρσr η(T )

(3.7)

with η(T ) the viscosity taken from Spitzer's formula for highly ionized plamas (Spitzer, 1962):

η(T ) = 2.2 10 -15 T 5/2 log Λ i g.cm -1 .s -1 (3.8)
where Λ i is the ion Coulomb integral. Following Roediger et al. ( 2013), we chose to simply set log Λ i = 40, even if this is slightly wrong when the gas is cooler than a few 10 4 K. Our goal is not to dene an accurate value for the Reynolds number but rather obtain a rough one, whose order of magnitude and radial variations can be interpreted. The corresponding curve in Fig. 3.16 is shown in orange. As far as the RMS velocity is concerned, no clear power-law appears. As expected from its cumulative nature, by denition, it increases with the size of the disc on which it is measured. At very low radii, r < 3 kpc, the dispersion is minimal and, even if the rst bins are not very populated, it appears to be at least 500 km.s -1 . About 300 kpc from the center it reaches almost 10,000 km.s -1 , which gives an average slope of ∼ 0.4.

On the other hand, the Reynolds number displays more interesting features.

First, it varies on over an order of magnitude, from a few 10 5 to almost 10 7 , indicating a regime that should be far from laminar at small scales. It increases up to r ≈ 30 kpc, decreases up to r ≈ 90 kpc and increases again up to a little more than r ≈ 150 kpc. Values above that radius are less reliable since, as seen in transverse maps, the cylindrical geometry is far from adequate. The condensing region 30 kpc < r < 90 kpc is also a mixing region and the Reynolds number increases during convergence. This is in agreement with what we suggested above, based on color maps of turbulent precursors. One must still be careful: it is not an indicator of a fully developed turbulence, as two distant points in a disc of given scale may not interact in a time shorter than other dynamical times.

Regardless of the regime of accretion (supersonic, dilute from voids, or subsonic, dense from sheets), the condensation of gas into a lament must convert a substantial fraction of the incoming ram pressure into small-scale vortices.

The compressive turbulence, on the other hand, is more dicult to trace thanks to such maps. This is unfortunate because this is the component that may, coupled with the radiative cooling, the most strongly aect the homogeneous nature of the ow. But we may consider this mode as wave turbulence, i.e. a structured mixing of acoustic and non-linearly interacting pressure waves. The only clues we can highlight with these data are the irregularity of the shock fronts and the irregular back-propagation of pressure waves from the condensation region (inside 30 kpc). Two processes might dampen them signicantly: viscosity, of course, and radiative cooling, whenever it is sucient to decrease pressure at a higher rate than the waves would increase it.

Main results

Numerical simulations have been used to investigate in detail the gas thermodynamics and kinematics during the formation and evolution of a cosmological lament. The intergalactic lament mainly forms via gas accretion through intersecting sheets. This accretion is anisotropic and the gas dynamics has many characteristics similar to that of turbulent ows. Important to this is the generation of vorticity due converging sheets of gas not being well aligned. The gas in these converging ows has a cooling time shorter than the dynamical time. Its temperature range is such that the dierential cooling discussed in Chapter 2, should also apply during the formation of laments. The thermodynamics of the lament during its formation is, in many ways, analogous to what occurs during the formation of gaseous halos. This nding substantiates our assumption made in Chapter 2 that laments have the initial temperature, velocity, and density structure necessary to seed the growth of inhomogeneities for forming multiphase accretion streams. Thus even if the resolution of our numerical simulations can not account for all physical processes relevant to small-scale perturbations, several clues lead us to think that the gaseous laments may be multiphase and turbulent before entering the halo.

Interestingly, we found that the closer we moved toward the axis of symmetry of the lament, the higher the baryonic fraction, indicating strong condensation of the gas. As part of this investigation, we discovered that dwarf-sized halos with high density gaseous cores formed in a quasi-regular pattern close to the lament axis.

Extending the results

We found during this thesis work, new theoretical properties of lamentary, gaseous ows accreting into galactic halos. Even in the admittedly over-simplied view of laments as homogeneous and laminar ows (Chap. 2), a sustained shock close to the entry into the halo must have several consequences on the post-shock characteristics of the ow. Instead of an isothermal radiative shock which would result in a subsequent laminar ow of cold gas or a completely disrupted ow of hot gas mixing with the ambient halo, depending on characteristic cooling and dynamical timescales, the ow may and likely will be subject to instabilities. Due to its geometry, the post-shock stream will expand in the halo as it cools. This leads to a wider range of phenomena, including perhaps the creation of a turbulent biphasic medium owing into the halo.

When studied with a numerical simulations at redshift 2 (Chap. 3), these laments appear to be very likely to be conned, spinning (vorticity), turbulent and multiphase. Overall, the collimation of the ows, and any spinning or large-scale vorticity they may have, may result in laments like those we assumed in our phenomenological modeling. As we also showed, a virial shock undergone by the lament would potentially amplify the turbulent multiphased nature of the post virial shock ow. This amplication may well occur even if there is not a virial shock but only a compression at around the virial radius of the halo. This is something that needs to be investigated but suggests that a shock at the boundary is not the decisive factor in the development of the multiphased turbulent stream.

The implications of our results for our understanding of the characteristics of gas accretion into galaxies are myriad and important. Having multiphase and turbulent accretion streams under certain conditions must aect the nature of the circumgalactic medium of galaxies. For example, streams of this nature would have a higher potential of interaction with even highly collimated galactic outows. Such interaction may further moderate the accretion eciency of cold gas onto galaxies. Since lamentary accretion is thought to be the major channel of supplying gas to galaxies necessary for sustaining their growth over cosmic time, if the gas accretion rate is moderated through the development of multiphased turbulent streams, then our understanding of the entire balance of inows, star formation and outows must be incomplete. Thus our results are opening new insights into how galaxies get their gas and sustain their growth. This is somehow more complex than what was thought up until recently but moderating accretion rates does help to alleviate some drawbacks of the ndings about cold ows in the context of galactic halos as we outlined in Chapter 2.

More eort should be expended to further include a broader range of physical processes in low resolution, cosmological scale simulations. To reach this goal of understanding the generation of a turbulent and multiphase medium requires, at least initially, a more detailed, higher resolution modelling of the coupling between dynamical and thermal energy conversion (dissipation and motion due to induced local pressure gradients) in idealized simulations. Such modelling will provide us with insights into the hot and cold fractions as well as the typical, randomly distributed, velocities of cold clouds in the post-shock gas and what physical mechanisms determine things like the fraction of hot and cold post-shock gas. Moreover, we can generalize the context of halo entry from a sustained planar shock to any form of compression, especially from the study of the origin of this sudden compression even in the absence of a high Mach shock. Numerical simulations with signicantly higher resolution will also help building statistics and obtaining general relations between the cosmic context and the lamentary ow's nature.

With this general outline in mind, we now present some details of what is to be done next.

Tracks for modelling

We have derived a model based on two methods: a phenomenology derived from timescale ratios, and estimates from a set of conservation equations. We showed that a sustained shock in an IGF would be followed, for suciently massive halos, that the lament's identity may rupture through expansion, thermal fragmentation, turbulent dispersion and (very probably) mixing. Completing this model with other simple phenomenology, we will be able to narrow down less arbitrary and more realistic non-dimensional parameters that will lead to a better understanding of the eect of a lament's disruption on the gas accretion rate onto galaxies and on the properties of the circum-galactic medium.

Relating density and velocity uctuations

As seen in our numerical study (Chap. 3), the incoming gas, before its entry in the halo, is very likely to be already turbulent and multiphase. Relating this result to our phenomenological model, we may rewrite our conservation equations so as to take into account both pre-shock and post-shock fractions (φ m,w and the like) and turbulence (σ). The rst objective is to nd analytical boundaries in the parameter space which delimit regimes where the shock is expected either to amplify or dampen the multiphase and turbulent nature of the ow.

The next step consists of assuming that the uctuations in the velocity eld, accounting for the gas turbulence, and those of the density eld, accounting, among other things, for the density dispersion of the hot component, are tightly connected. This is explained by our stating the following hypotheses that:

1. the condensation of a warm phase induces turbulent motions, 2. the compressible component of turbulence stimulates the phase separation (these rst two items are in fact interrelated), 3. the solenoidal component of turbulence mitigates, on average, the density peaks, 4. the turbulent dissipation heats up the gas, 5. the radiative cooling always acts as the gas evolves with time with the rate depending on temperature and density, 6. the processes 1 to 5 balance each other, resulting in a quasi-static evolution.

We will now focus on the numerical integrator described in 2.5. This relation had been empirically found and theoretically derived for isothermal, supersonic cases, but we will use it as an approximately appropriate guess for M rms = σ ps /c ps . If the compressive modes dampen rapidly, we can take a ducial value of b = 1/3, while a mix of compressive and solenoidal modes yields b ≈ 1/2

(see previous references).

First approach: turbulent decay only

In a rst approach, we can consider that the shock front perturbations are the driver and that the ensuing ow hosts a spontaneously decaying turbulence. Thus, hypothesis 1 would not apply and the turbulent Mach number at each timestep would only result from the dissipation of previous timesteps. Only the initial RMS Mach number is derived from the variance in the initial density.

The turbulent dissipation rate is the inverse of its characteristic time, which is the ratio of the turbulent specic energy (σ 2 ) to the specic dissipation power (nσ 3 /L dissip ) at any given time. This can be expressed as:

t dissip = L dissip σ (4.2)
where L dissip is a typical dissipation length, usually taken as the typical injection length scale. However, doing so is justied in the case of a homogeneously driven turbulence.

The time evolution of σ is given by:

d log(σ) = dt 2t dissip (4.3)
and at each timestep we may calculate the RMS Mach number by dividing σ by the dominant sound speed in the hot medium.

Turbulent motions dissipate into heat at a rate given by the inverse of its characteristic time, dened by the ratio of the thermal specic energy to the specic dissipation power:

t heat = L dissip k B T nσ 3 (4.4)
Since turbulent dissipation is a heating source, a new characteristic time may be constructed to reect the thermal evolution: I already attempted such an implementation and it led, in eect, to long-term saturation of the mass fraction in some cases, where the turbulence is slightly subsonic. No report of this work is currently in progress and so no conclusions can be provided mostly because of the lack of a systematic study of the eect of the two fundamental parameters: b, and L dissip .

t therm = 1 t heat - 1 t cool -1 (4.

New avenues for simulations

While modelling involves a renement and an increase in the number of hypotheses, the numerical simulation used in Chap. 3 relies only on a small set of rst principles (or approximations), which provides a complementary avenue for understanding various phenomena of accretion ows. Tuning parameters in numerical simulations may help alleviate some doubts about what we modelled, while the model will help in deciding what is to be numerically investigated. A lot can be made with this work as a starting point. We now try to orient ourselves towards further developments that may be important for our developing understanding of the formation of laments and accretion ows.

Further numerical processing

Finding scalings An eort must be made to automate the detection and the tracing of intergalactic laments in our data set. This can be done with the help of numerical tools such as DisPerSe (see eg. Sousbie et al. (2011); Sousbie (2011)).

By building a catalogue of laments, characterized by the masses of the halos they ow into, we could systematically extract quantitative properties such as their average Plummer exponents, their average linear density of condensates as well as their average wavelength, their ability to thermally fragment and the level of expected turbulence they have. Such a halo-mass dependent characterization would not only reveal a possibly scale-invariant physics, but would also serve as input into more robust modelling on cosmological scales. Time evolution A logical development would benet from additional outputs from simulations, such as more redshifts, to determine whether our ndings are still valid at higher redshifts, when the lamentary accretion onto galaxies is thought to be most vigorous for all halo masses. The study of earlier snapshots would allow for a deeper insight on dynamics and mass growth of laments, their stability and their ability to condensate warm gas as a function of time, and why, if it is the case, they progressively lose their dominate role of supplying gas relative to mergers after redshift 1.

Further numerical experiments

Increasing resolution The next step may appear obvious. It consists in increasing the resolution of the simulation. The ner the grid, the closer we would get to reveal fragmented elds such as velocity of turbulence and density/temperature for a multiphase medium. This will also allow us to determine more accurate radial proles, leading to a better understanding of the stability of laments and their 123 detailed large scale density distribution. Adaptive mesh renement, the rening condition needs to be changed to understand specic aspects of the characteristics of laments. For example, if boundaries of the laments are to be understood, rening on density gradients, instead of density itself, is of the essence. On the other hand, if rotational eects are to be investigated, such as for understanding the development of turbulence in ows, rening on regions of higher vorticity is the pertinent choice (see Kornreich and Scalo for example).

Running ner physical models Numerous phenomena have been omitted in our zoom-in simulation. An obvious one is the formation of stars in the lament.

Considered time-consuming and possibly without consequence on high-redshift laments, we did not let the gas trigger small scale condensations which may form into star clusters. Apart for articially pressurizing the densest structures, it has the disadvantage of not allowing the pre-enrichment the intergalactic medium with metals. And even if the laments penetrate the halo gas with enough momentum to keep their bulk free from metals, a boundary mixing layer could well form a cooler sheath resulting in an increased warm mass fraction. And if stars may form in some substructures of the laments, they could also self-enriched with metals, which would be a game-changer for galactic models.

The second missing ingredient that could aect laments is the magnetic eld.

With an MHD simulation, plasmas in IGFs would experience weaker shocks and the vorticity structures would interact with the magnetic eld so as to amplify it. The radiative cooling would also be altered, for ion collisions would occur at dierent rates and energies.

Deriving observational constraints

With the use of spatial and thermodynamic conditions, assuming for example a collisional equilibrium, spectra of the gas radiative emission could be computed so as to obtain constraints on observations aiming at nding intergalactic laments.

In particular, emissions would be rather dierent if the medium is homogeneous or heterogeneous. In the latter case, small, dense, cold clouds would represent a high mass fraction but a low volume-lling factor, thus diluting their spectral signature, whereas the highly volume-lling, hot ambient gas would not emit much because of its very low density and low eciency of emission. Not only the spectral lines would be mixed in the case of a heterogeneous medium, but their intensity would be lower.

  framework . . . . . . . . . . . . . . . . . . 7IFIFI gosmi expnsion F F F F F F F F F F F F F F F F F F F F F F U IFIFP hrk nd ryoni mtter F F F F F F F F F F F F F F F F F V IFIFQ snitil density )ututions F F F F F F F F F F F F F F F F F W 1.2 Gravity in action . . . . . . . . . . . . . . . . . . . . . . . 9 IFPFI vrge sle strutures F F F F F F F F F F F F F F F F F F F W IFPFP rlo growth 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 1.3The importance of gas physics . . . . . . . . . . . . . . . 15IFQFI ditive ooling F F F F F F F F F F F F F F F F F F F F F F IT IFQFP hok due to ondenstion F F F F F F F F F F F F F F F F F IU IFQFQ prom wrm gs to stellr light F F F F F F F F F F F F F F IU1.4 Challenges of galaxy evolution . . . . . . . . . . . . . . . 21 IFRFI ixtrinsi explntions for ine0ient glxy formtion F PI IFRFP sntrinsi explntions for ine0ient glxy formtion F F PI IFRFQ eretion!feedk intertionc F F F F F F F F F F F F F F PP 1.1

Figure

  Figure 1.1: SDSS Galaxy map. Galaxies are arranged in large-scale structures. Image Credit: M. Blanton and SDSS. License: Standard SDSS image license.

Figure

  Figure 1.2:

  talked about the migration and the condensation of accreted gas through the halo, but what happens next? The central, oblate region reaches high densities and undergoes further gravitational instabilities that fragments it into much smaller but much denser disks and eventually forming stars. Populated with stars in a gaseous bath, the central regions of halos are now galaxies, within the context of this simple cooling picture.Gaseous lamentary accretion onto galaxiesSince the works ofBirnboim & Dekel (2003);Kere² et al. (2005); Dekel & Birnboim (2006); Dekel et al. (2009), a new paradigm has emerged.Kere² et al. (2005)

Figure 1 . 4 :

 14 Figure 1.4: An important and inuential gure from Dekel & Birnboim (2006).

Figure 1

 1 Figure 1.5: Illustrative gure from Silk & Mamon (2012) A sketch of the observed (blue) and theoretical (red) distributions of galaxies, according to their luminosity, a proxy to their stellar mass. showing the challenges of contemporary galaxy evolution studies. Galaxies are simply not following the development of dark matter and its distribution in halos. Various processes must have made galaxy formation inecient.As can be seen in this gure, the authors concern themselves with the loss term, outows, where active galactic nuclei eject mass from the high mass halos, while supernovae (SN) remove the gas from low mass halos. In this thesis, we concern ourselves with the supply side, how does the gas form into structures and accrete into individual dark matter halos.

Figure 2 . 1 :

 21 Figure 2.1: Sketch of our phenomenological picture of ows of gas passing through a virial shock. The initial inowing gas (in blue) is over-dense relative to the halo gas at the boundary by a factor f (=ρ 0 /ρ H or the stream density divided by the density of the ambient halo gas at the virial radius), shocks at the boundary between the inter-galactic medium (labeled as IGM) and the hot halo gas (labeled as halo). The persistent virial shock and the higher pressure of the post-shock gas

  The cooling time curves for the post-expansion gas are truncated at the minimum temperature spanned by the curves, 10 4 K. The cooling time curves were computed for three values of the initial over-density f , as indicated by green labels on the left side of the panel. Expansion times (solid cyan lines) are shown for three dierent relative lament radii, log 10 r stream /r vir =-1, -1.5, and -2. The dynamical time, t dyn,halo , which is independent of halo mass, is indicated by the horizontal magenta line.

Figure 2

 2 Figure 2.3: Ratio of the post-expansion cooling and halo dynamical timescales as a function of halo mass, M H , and stream over-density, f , at redshift, z=2. The colors represent the log (t cool,pe /t dyn,halo ) (Eqs. 2.26 and 2.27) whose corresponding values are indicated by the color bar. The solid black lines indicate log (t cool,pe /t dyn,halo ) of increasing values of 0 and 1 from the middle to the top of the diagram. Dashed black lines indicate log (t cool,pe /t dyn,halo )=-1. The white contours indicate values of log (t cool,ps /t expand ) (Eqs. 2.25 and 2.26) equal to 0 and 1 (solid), and -1 (dashed).

Fig. 2 .

 2 Fig. 2.2 illustrates how these time scales depend on the post-shock temperature, directly related to the halo mass (top x-axis) and for a range of relative stream over-densities, f , and relative stream radii, r stream /r vir at z=2. The initial stream penetrating the halo gas and the warm post-shock clouds are assumed to have a temperature of 10 4 K, which is maintained through external heating processes, such as the ionization by the meta-galactic ux or ionizing photons from the galaxy in the halo. Since the post-shock temperatures are proportional to velocity, the post-shock temperature translates into dark matter halo mass. From this analysis,

Figure 2

 2 Figure 2.5: The change in the mass fraction, φ m,w , and the volume lling-factor, φ v,w , of the warm, 10 4 K gas as a function of the post-shock cooling time. The time scale is expressed in units of the isobaric cooling time, t 0 cool as dened in the text. The values of the pressure and temperature are the same as those used in Fig. 2.4.

Figure 2

 2 Figure 2.6: (left and middle) Sketch of the thermodynamic path for cases (1) and

Figure 2

 2 Figure 2.8: Disruption of the ow as a function of the volume-lling factor of

  shock speed of sound c ps = γP ps /ρ ps Temperature of the warm phase T w = T 0 Density of the warm phase ρ w = ρ H T H /T w Density of the hot phase (post-expansion) ρ h = ρ ps (P ps /P H ) 1/γ Temperature of the hot phase (post-expansion) T h = ρ H T H /ρ h Volume-averaged density of the warm phase ρ w v = φ v,w ρ w Volume-averaged density ρ 2 = φ v,w ρ w + (1φ v,w ρ h ) Expansion factor of the post-shock stream b S = 1 -√ ∆ / [(γ -1)/f + 2η)] Velocity dispersion of the warm clouds σ turb = 2k B T w ηf /(γ -1)φ v,w µm p Halo dynamical time t dyn,halo = r vir /v vir Cooling time of the phase Φ ∈ {ps, w, h} t cool,Φ = k B µm p T Φ /ρ Φ Λ (T Φ ) Expansion time of the post-shock gas t expand = 2γr stream /c ps Disruption time of the turbulent warm phase

Figure 2

 2 Figure 2.9: Evolution of the warm gas fractions for dierent values of δ.

Figure 2 .

 2 Figure 2.10: Evolution of the warm fractions against the log of normalized time. Fits with Φ functions are shown.

  geometry . . . . . . . . . . . . . . . 80 QFRFI ghoie of referene frme FF F F F F F F F F F F F F F F VI QFRFP tudying slies of dt F F F F F F F F F F F F F F F F F F F VP 3.5Anatomy of a transverse slice . . . . . . . . . . . . . . . 84 QFSFI truture nd sles F

  Beyond simulation: modelling subgrid features . . . . . 105 QFUFI wultiphse mediumc FF F F F F F F F F F F F F F F F F F F IHT QFUFP urulent preursors F F F F F F F F F F F F F F F F F F F F IHV3.8 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 3

 3 Figure 3.1: A view of the halo, with emphasis on the densities characteristic of IGFs. The cube has a side of 381 kpc and densities in the color bar range from

Figure 3 . 2 :

 32 Figure 3.2: The same view as Fig. 3.1, but with a visible slice in the later examplied lament.

  Figure 3.3: Maps of a transverse slice of an IGF, at ζ/r halo = 1.8, i.e. at 343 kpc from the center of the closest end-point halo. Dimensions are 380 × 380 kpc 2 . Marked are three circles as guides to the eye, whose radii are, as written in the bottom-left map, 30, 90, and 150 kpc. Top-left: gas density. Top-right: gas temperature. Bottom-left: dark matter density. Bottom-right: Baryonic fraction. The lament structure is clearly far from purely cylindrical, due to the importance of converging walls.

Figure 3 . 4 :Figure 3

 343 Figure 3.4: Maps of a transverse slice of an IGF, at ζ/r halo = 1.8, i.e. at 343 kpc from the center of the closest end-point halo. Dimensions are 380 × 380 kpc 2 . Marked are three circles as guides to the eye, whose radii are, as written in the bottom-left map, 30, 90, and 150 kpc. Top-left: Radial component of the velocity eld. Bright green contours follow the loci where it changes sign. Top-right: Orthoradial component of the velocity eld. Bottom-left: Infalling speed toward the halo. Bottom-right: Fraction of the absolute radial velocity component over the norm of the total velocity.

Figure 3

 3 Figure 3.6: Stylized view of two converging, unidimensional jets bouncing, giving rise to a quadrupolar vorticity structure, with an impact parameter.

Figure 3 .

 3 Figure 3.5 presents a stylized view of two sheets meeting a producing that kind of quadrupolar structure.

  Figure 3.7: Left : Evolution of the radially mass-averaged velocities. Shown are the averaged radial (orange) and azimuthal (blue) components, as well as the norm (red) of the velocity eld. Right : fraction of the radial (orange) and complementary azimuthal (blue) components in the total norm. The vertical green dashed lines indicate the three guides at 30, 90 and 150 kpc.

Figure 3

 3 Figure 3.8: Mass inow azimuthal proles in a transverse slice of an IGF, at ζ/r halo = 1.8, i.e. at 343 kpc from the center of the closest end-point halo. Represented is product of the density times the radial velocity in M .kpc -2 .yr -1 (log- arithm), at three radii, in rings of width δr ≈ 3 kpc. Red points are for radially converging ows and blue points for radially escaping ows. Top: r = 150 kpc.The horizontal line gives an arbitrary value to discriminate between sheets and voids. Middle: r = 90 kpc. Bottom: r = 30 kpc. This illustrates the fact that the asymmetrical structure due to inowing sheets tends to disappear as the radius is decreased.

Figures 3 .

 3 Figures 3.10 and 3.11 show longitudinal slices constructed the way described above. They count in total four columns, each for a given angle, and three rows, each for a dierent eld: entropy, temperature, and density. The chosen angles are irregularly spaced: θ ∈ {π/8, π/2, 11π/16, π/33, π/32}. They correspond, respectively, to the following directions of the exemplied transverse map (the one at ζ = 343 kpc): that of the most strongly accreting sheet, that crossing the most shocked medium expanding in the voids, that containing the western converging entry point, and that which crosses the interesected companion lament. The trace of the previously marked circles in the transverse maps are here highlighted by green dashed horizontal lines, as if they were generatrices of three coaxial

)Figure 3 )Figure 3

 33 Figure 3.10: Maps of a longitudinal slice of an IGF, at dierent slicing angles. Dimensions of each map are ∆ζ = 704 and ∆r = 570 kpc. The rows show three elds. From top to bottom: the density, the temperature, and the entropy. The columns show two dierent sclicing angles. From left to right: θ = π/8 and θ = π/2, each being illustrated by the green solid lines in the top transverse slices. Horizontal dashed lines correspond to the three circles of transverse slices. The vertical white dashed line corresponds to the exemplied slice.

Figure 3

 3 Figure 3.12: Linear mass λ, in M /kpc as a function of the altitude ζ, taken for each slice at radii r = {30, 90, 150} kpc. Top: linear mass of the gas only. Middle:

Fig. 3 .

 3 Figure 3.13: Maps of four transverse slices of an IGF, at ζ ∈ [414, 500, 614, 764] kpc, from the center of the closest end-point halo. Dimensions are 190 × 190 kpc 2 .Marked are two circles as guides to the eye, whose radii are 30 and 90 kpc. The represented eld is the logarithm of T /ρ 2/3 , used as a proxy to the entropy. The condensates are distinguished by their very low entropy.

A

  look at higher slices reveals that the inside of the vortex tubes are in such a situation far above the virial radius. But a more rigorous approach needs to include the local density contrasts. If well represented by a measure of the dispersion of the density distribution on a given test volume, one could compute a phase separation time at the scale corresponding to the test volume.

turbulence 4 .

 4 It is commonly dened as one component of a turbulent ow, more precisely the divergence-free part. The other component is termed compressive turbulence and is dened, as is mathematically natural when decomposing such a vector eld, as the purely divergent ow, i.e. the rotational-free part.Solenoidal turbulence arises when rotation of particle cells is locally enhanced, that is where shear is strong enough to overcome the dissipative eect of viscosity and where nonlinear terms (namely the advection term of the Navier-Stokes equation for momentum, in hydrodynamics) allow for a stimulation of other spatial frequency modes to grow. The Kelvin-Helmholtz instability is an example among other ones. Let us write the rotational of the Navier-Stokes equation rst divided by the density:

Fig. 3 .Figure 3

 33 Fig. 3.4 involves bending of owlines. It lls almost all of the lament inside a radius of 90 kpc, and, marginally, at the edges of the main accreting sheet. Finally, the lower-left panel of Fig. 3.15 is the projection of the local Mach number, in the barycentric frame computed as in the above sections. Shocks are strong at the lament-void edges (M

Figure 3

 3 Figure 3.16: Velocity dispersion and corresponding Reynolds number averaged on concentric circles as a function of their radii, in a slice at ζ/r halo = 1.8, i.e. at 343 kpc from the center of the closest end-point halo. The dashed tilted line indicates a 3/2 power-law. Green dashed vertical lines correspond to the radii of the concentric circles of transvers slices.

  8 and the modications implied by these new hypotheses. In that new simple case, at each timestep, the density distribution, i.e, the hot post-shock component has a density dispersion, in log, δ, which can be used to derive the turbulent velocity dispersion (σ ∝ M rms √ T ) for the next timestep. How are they be related? Relating δ and M rms If we assume that the initial dispersion of densities is related to the trans-or supersonic turbulence, we can try to apply the density varianceturbulent Mach number relation δ ≈ log(1 + (bM rms ) 2 ) (4.1) with b a constant of order unity and M rms the root-mean-squared Mach number (eg. Passot & Vázquez-Semadeni, 1998; Audit & Hennebelle, 2010; Price et al., 2011; Molina et al., 2012).

  5) relevant for each particle, and varying with time. When it is negative, cooling prevails, whereas when positive, heating dominates. The time evolution of the thermal specic energy is now: 4.6 are then implemented into their discretized versions in the way Eqs. 2.38 and 2.39 were written.

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ompressile )owsD e8eD SIID eUT flus F eFD oker xFD IWVWD heory of lol therml instility in spheril systemsD eptD QRID TII fnerjee xFD hrm FD PHIRD urulene nd ooling in glxy luster oresD wxeD RRQD TVU fek eF wFD et lFD PHITD en improved r sheme for osmologil simultionsD wxeD RSSD PIIH fehroozi F FD ehsler F rFD gonroy gFD PHIQD he everge tr pormtion ristories of qlxies in hrk wtter rlos from z a HEVD eptD UUHD SU 125 feirão FD et lFD PHISD ptilly resolved pitzerEs spetrl mps of the superwind in wVPD wxeD RSID PTRH fenson eF tFD fower F qFD prenk gF FD vey gF qFD fugh gF wFD gole FD PHHQD ht hpes the vuminosity puntion of qlxiescD eptD SWWD QV fest F xFD von der vinden eFD uu'mnn qFD rekmn F wFD uiser gF FD PHHUD yn the prevlene of rdioEloud tive glti nulei in rightest luster glxiesX implitions for eqx heting of ooling )owsD wxeD QUWD VWR finney tFD IWUUD he physis of dissiptionl glxy formtionD eptD PISD RVQ firnoim FD hekel eFD PHHQD iril shoks in glti hloescD wxeD QRSD QRWfirnoim FD dnos hFD inger iFD PHITD he rydrodynmi tility of qseous gosmi pilmentsD eptD VQPD vR flumenthl qF FD per F wFD rimk tF FD ees wF tFD IWVRD pormtion of glxies nd lrgeEsle struture with old drk mtterD xtureD QIID SIU forisov iFD et lFD PHITD iquitous qint vyα xeule round the frightest usrs t z∼QFS eveled with wiD eptD VQID QW forthkur FD rekmn FD triklnd hFD ild FD himinovih hFD PHIQD he smpt of trursts on the girumglti wediumD eptD UTVD IV fouhé xFD vehnert wF hFD éroux gFD PHHTD he missing metls prolem E ssF row mny metls re in z£a PFP glxiescD wxeD QTUD vIT fouhé xFD et lFD PHITD ossile igntures of goldE)ow hisk from wi sing z∼I qlxyEusr ir towrd h tIRPPEHHHID eptD VPHD IPI frooks eF wFD qovernto pFD uinn FD frook gF fFD dsley tFD PHHWD he ole of gold plows in the essemly of qlxy hisksD eptD TWRD QWT fryn qF vFD xormn wF vFD IWWVD ttistil roperties of Ey glustersX enlyti nd xumeril gomprisonsD eptD RWSD VH gntlupo FD errigoniEftti pFD rohsk tF FD rennwi tF pFD wdu FD PHIRD e osmi we (lment reveled in vymnEα emission round luminous highEredshift qusrD xtureD SHTD TQ gen FD ystriker tF FD PHHTD here ere the fryonsc ssF peedk i'etsD eptD TSHD STH ghndrsekhr FD permi iFD IWSQD rolems of qrvittionl tility in the resene of wgneti pieldFD eptD IIVD IIT gluver wF iFD et lFD PHIHD owerful r 2 vine gooling in tephn9s uintetF sF wpping the igni(nt gooling thwys in qroupEwide hoksD eptD UIHD PRV godis FD ihon gFD ogosyn hFD PHISD pin lignments within the osmi weX theory of onstrined tidl torques ner (lmentsD wxeD RSPD QQTW gooper tF vFD fiknell qF FD utherlnd F FD flndErwthorn tFD PHHWD trurstE hriven qlti indsX pilment pormtion nd imission roessesD eptD UHQD QQH gornuult xFD vehnert wFD foulnger pFD quillrd FD PHITD ere gosmologil qs eE retion trems wultiphse nd urulentcD preprintD @arXiv:1609.04405A hnovih wFD hekel eFD rhn yFD geverino hFD rimk tFD PHISD pour phses of ngulrE momentum uildup in highEz glxiesX from osmiEwe strems through n extended ring to dis nd ulgeD wxeD 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  .3.

	Inhomogeneous infalling streams and post-shock gas: dierential cool-
	ing
	We now discuss the physical mechanisms behind the development of multiphasic
	accretion streams. Although our previous discussion of the relevant timescales
	considered homogeneous streams, density and velocity inhomogeneities may arise
	through the dynamics of the shock itself (Kornreich & Scalo, 2000; Sutherland

et al., 2003)

. Simulations show that stream are not accreted homogeneously but have substructure in both density and velocity

(Nelson et al., 2016)

. Inhomogeneities may arise due to a range of curvature in accretion shock-fronts, translating into a range of Mach numbers and post-shock temperatures and densities.

Table 2

 2

		.1: Halo and gas parameters for example in 2.5.4
	Parameter Name	Symbol	Value
	Halo mass	M H	10 13 M
	Baryonic fraction	f B	0.18
	Redshift	z	2
	Virial radius	r vir	220 kpc
	Virial velocity	v vir	440 km s -1
	Critical density

Table 2

 2 Density of the halo gas at r vir ρ H = ρ NFW (r vir ) ≈ 37.0 × f B ρ crit (z) Pressure of the halo gas at r virP H = k B T H ρ H /µm p

	.2: Model variables and their relationships	† .

The simplest case is that of an isothermal sphere at pressure equilibrium. An alternative would make use of an NFW density prole.

The eciency of such an interaction depends on several factors, fro example the geometry of the ows: if an AGN produces a jet perpendicular to the galactic plane while a warm lament reaches the ISM parallel to that plane, they would not directly interact.

We can either consider that turbulence is produced or that the turbulent velocity dispersion in the pre-shock medium was negligible compared to the total pressure but strongly amplied by the shock.

We refer the reader toLandau & Lifshitz (1987) for the early, albeit deeply insightful, developments of the theory of turbulence.

The vast majority of literature about turbulence aiming at understanding human-or Earth- scale ow motions, that is that of liquids and a stratied atmosphere, it concerns incompressible ows. Hence, the term solenoidal is never used therein,

Chapter 3

Structure and dynamics of intergalactic laments at z ≈ 2 Abstract By means of an adaptive mesh-renement code, we performed a zoom-in simulation of a specic halo and extracted the data pertaining to one of its accreting intergalactic laments. After having presented the need for a simulation, we shortly describe the code Ramses. We address the common view of an IGF as a pure cylinder and detail the radial and angular properties of its gaseous component.

Some key quantities are studied as they evolve along the lament and we derive a rened paradigm to study IGFs, as well as consequences regarding their fate after entering the target halo. We nally made use of the accessible data to extrapolate some subgrid physical processes which the simulation must have missed.

-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 3. Propension to become heterogeneous Marked are three circles as guides to the eye, whose radii are, as written in the ooling times in millions of years. Right: Propensity to become heterogeneous by use of the index dened in the text. When the value is positive, the accelerated dierential cooling is active. The higher the value, the stronger the eect.

the inverse of the density contrast. When this time gets to innity, i.e. the growth rate vanishes, the instability does not occur. This situation is met when

where Λ is the eective cooling power per unit density. The equation itself carries less information than the typical rate, as we lose the quantitative value, but changing the equal sign into a less-than sign constitutes a qualitative criterion for the propension to increase density gradients.

Even if the heating provided by metagalactic uxes are present into our numerical computation, it is not sucient to balance the cooling at densities that characterize the interior of laments. It is therefore inaccurate to look for a possible thermal instability there. Nevertheless, a self-amplied dierential cooling may act on relevant timescales. The use of the aforementionned criterion (Eq. 3.4) is still useful to determine whether a medium is likely to become multiphase. The quantitative information will then simply come from the cooling time of the gas.

We projected in Fig. 3.14 two elds for the already exemplied transverse slice.

The left-hand panel shows the cooling time of each pixel, while the right-hand one shows the value of the spread to Eq. 3.

RFPFI elting density nd veloity

RFQFI purther numeril proessing

Even further

The shock transition itself could be more accurately modelled. Breaking the assumption that the shock is planar and stationary should inform us about the transition parameter η and about the shock stability itself. First, we would study oblique-symmetric, concave, convex and perturbed shocks and derive turbulent amplitudes by staying close to the derivation of the Richtmyer-Meshkov instability (Richtmyer, 1960;Meshkov, 1969). The origin of the shock itself should be investigated, either by means of an analytical model of an inward-propagating strong compression and with a numerical simulation of an idealized lament.

We also discussed exploring the parameter space of galaxy mass, redshift, halo environment, etc on the stability of the shock. Doing so, it is possible to nd regions wherein our phenomenological modeling is invalid. This constitutes an interesting and somehow general way of constraining the relevance of our combination of hypotheses. Indeed, crossing some bounding surfaces of the parameter space questions the assumption of stationarity. If the ow is not stationary, would it result in losing or reinforcing the necessary lower supporting forces for sustaining the virial shock? In the rst case, the disruption of the lamentary warm ow would vanish, the same way as with the unstable shock from Dekel & Birnboim (2006), but for more complex reasons. In the second case, that we could call overstability, it would result in the disappearance of the collimated ow at the edge of the halo. An even more subtle but perhaps more realistic possibility is that the shock oscillates, alternating between under-and over-stable phases, due to the halo pressure evolving as the ow penetrates deeper. Determining a stability criterion for a fully 3-dimensional shock is another avenue we must consider.

There are also interesting avenues through which the accretion eciency onto the galaxy proper may also be estimated. Bullock, 2004) would strongly improve may predictions for the mass acquisition rate of warm gas by the galaxy and has the gas balance in galaxies.