Nicolas Queyriaux

Eugen S Andreiadis

Stéphane Torelli

Jacques Pécaut

Brad S Veldkamp

Eric A Margulies

Michael R Wasielewski

Murielle Chavarot-Kerlidou
email: murielle.chavarot-kerlidou@cea.fr

Vincent Artero
email: vincent.artero@cea.fr.

Jacques Pecaut

CuAAC-based assembly and characterization of a ruthenium-copper dyad containing a diimine-dioxime ligand framework

Keywords: Copper, ruthenium, CuAAC coupling, light-activated dyad, EPR, photolysis

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Introduction

Artificial photosynthetic systems efficiently achieving light-harvesting and solar energy conversion into a chemical potential have been extensively investigated. [1][2][3][4][5][6][7] In the field of solar fuels, their capability to ultimately drive multielectronic redox catalytic processes such as water oxidation, hydrogen evolution or CO 2 reduction is a strong motivation for their development and integration into functional devices. [8][9][10][11][12][13] The design and study of novel molecular dyads combining a light-harvesting unit with an electroactive moiety is thus a major target in the field of artificial photosynthesis. In these multicomponent structures, the establishment of a long-lived charge-separated state through photoinduced intramolecular electron transfer is a crucial requirement for producing functional systems.14 This is achieved by a judicious choice of the two active units, but also strongly relies on the chemical nature of the covalent bridge. The latter plays a key role in both the charge transport and the charge-separated state lifetime; however, this role is very complex to understand both from a structural and an electronic point of view. [15][16][17][18][19] There is therefore a great interest in having in hand modular and straightforward synthetic methodologies to synthesize and study structurally-diverse dyad assemblies. In that context, the Copper (I)catalysed Azide-Alkyne Cycloaddition (CuAAC), one of the best-known examples of "click chemistry", appears as a unique synthetic tool20: this cycloaddition is efficient under mild conditions, high-yielding and compatible with most organic and metallo-organic functions. For these reasons, CuAAC has been applied to the construction of sophisticated molecular architectures in order to study photoinduced processes.21-23 This strategy has been particularly utilized for the functionalization of ruthenium polypyridine photoactive units24 by various organic moieties,25-27 nickel28, 29 or iron30 complexes and for their attachment onto surfaces.31-33 However, to the best of our knowledge, CuAAC has never been employed to assemble a Ru tris-diimine photosensitizer with a copper complex, although copper also displays interesting redox properties. Photoinduced electron transfers in heterodinuclear ruthenium copper complexes have for instance been exploited to develop luminescent probes,34-36 molecular switches,37 dye-sensitized solar cells38 and also for the photochemical reduction of nitrite.39 Recently, a RuCu dyad for the photocatalytic oxidation of organic substrates mediated by dioxygen activation has also been reported. 40 In this contribution, we describe the synthesis and full characterization of the first rutheniumcopper dinuclear complex assembled by CuAAC click chemistry. This dyad consists of a ruthenium polypyridine photoactive unit covalently linked to a copper (II) diimine dioxime complex. EPR-monitored photolysis experiments have been undertaken in combination with detailed photophysical investigations to shed light on the photoinduced electron transfer processes. its subsequent metallation with Ru(bpy) 2 Cl 2 .45 Copper (II)-diimine dioxime complexes have been initially reported as biomimetic models for copper-containing metalloproteins. 46 We previously reported the synthesis of the azido-substituted tetradentate diimine dioxime ligand.47, 48 Reaction with copper perchlorate hexahydrate was successfully achieved in 82% yield, as reported for the parent complex [Cu(DO)(DOH)pn(OH 2 )](ClO 4 ).49 Deprotonation of one oxime function upon coordination to the divalent metallic salt, characteristic for the diimine dioxime complexes,50 yields the monocationic H-bridged Cu II complex 2. Its molecular structure solved by single crystal X-ray diffraction reveals a square pyramidal geometry for the pentacoordinated copper centre. A displacement of the axiallycoordinated water molecule by one oxime oxygen from a neighboring complex is observed and leads to the formation of a polymeric network within the lattice (Fig. S1), as previously observed for [Cu(DO)(DOH)pn(OH 2 )](ClO 4 ).50

Results and discussion

Synthesis

Classical CuAAC procedures, recently applied to related ruthenium tris-diimine complexes, 26, 51 rely on the combination of copper (II) sulfate with sodium ascorbate to generate insitu the catalytically-active copper (I) species. The reactivity of 1 under these conditions was first tested with a commercially-available benzyl azide as model partner. In our hands, the procedure was optimized by employing a homogeneous ternary solvent mixture (CH 2 Cl 2 / MeOH/H 2 O) and benzyltriazole-functionalized complex 3 was isolated in 65 % yield after purification (see S.I.). These new conditions were then applied for the coupling of 1 with azide-functionalized complex 2. After purification by chromatography on silica gel, dyad 4 was isolated as a dark-orange powder in 55 % yield.

H NMR spectroscopy

1 H NMR spectra have been recorded in CD 3 CN for the three Ru complexes. Spectra of compounds 3 (Fig. S2) and 4 (Fig. S3) are characterized by the complete disappearance of the alkyne proton signal, observed at 3.56 ppm for 1. In addition, spectrum of 3 displays two new signals, a singlet at 5.63 ppm for the benzylic CH 2 protons (Fig. S2) and a singlet at 8.24 ppm attributed to the aromatic proton on the newly-formed 1,2,3-triazole unit (Fig. 1, red trace, arrow).

The presence of the paramagnetic Cu II centre in 4 however prevents the detection of the CH 3 , CH 2 and CH groups from the diimine dioxime ligand in the aliphatic region of the spectrum. This is also true for the triazolyl proton, expected to appear as a singlet in the aromatic region, by comparison with 3, and for the phenylene protons ortho to the triazole ring (7.67 ppm in 1 and 8.06 ppm in 3). In addition, the signal of the two other phenylene protons appears as the large signal in the baseline around 8.4 ppm (Fig. 1, green trace, dashed arrow) and a light broadening is observed at 9.04 and 7.80 ppm for the H 4/7 and H 3/8 protons of the phenanthroline ring respectively (Fig. 1, green trace, dashed arrows), more distant from the Cu II centre.

Electrochemistry

The electrochemical properties of dyad 4 and of the two precursors 1 and 2 were investigated by cyclic voltammetry. Data are summarized in Table 1 and the voltammograms are displayed in Fig. S5. Complexes 1 and 4 exhibit redox features characteristic for ruthenium tris-diimine complexes, with a reversible Ru III /Ru II process at +0.88 V and +0.93 V vs Fc +/0 , respectively, thus only slightly affected by the coupling. Three reduction processes take place between -1.7 and -2.4 V vs Fc +/0 , corresponding to the reduction of the three diimine ligands in the Ru II coordination sphere (the first one more likely occurring on the π-acceptor imidazolophenanthroline ligand). An additional cathodic wave is observed at -0.95 V (E pa ) for 4, attributed to the Cu II /Cu I process by comparison with the azide precursor 2 (Fig. S5). Loss of reversibility is observed for this process in 4, indicative for a fast chemical reaction or structural rearrangement that can be partially outrun at higher scan rates (≥ 10 V.s -1 ). Overall, redox properties of dyad 4 compare well with those from the independent Ru and Cu subunits 1 and 2, thus underlining that the two metallic centres are not coupled in the electronic ground state.

UV-vis absorption spectroscopy

Absorption spectra were recorded for 1, 2 and 4 at a concentration of 2 x 10 -5 M in acetonitrile. Compounds 1 and 4 display a broad absorption band in the visible region, centred at 455 nm with a shoulder at 430 nm, characteristic of the Metal-to-Ligand-Charge-Transfer ( 1 MLCT) transition of the ruthenium tris-diimine moiety. A slight contribution from the absorption of the copper diimine-dioxime moiety can be observed in the UV region for 4, by comparison with 2. The spin-forbidden d-d transition of the copper complex can be observed at 480 nm for this family of complexes,52 with a very low molar absorptivity (≈ 200 L.Mol -1 .cm -1 , Fig. S6). Overall, the absorption spectrum of dyad 4 can be simply described as the superimposition of its individual components 1 and 2 (dotted line in Fig. 2) with a slight contribution at 326 nm tentatively attributed to the newly formed triazole ring; this observation is again in good agreement with the absence of significant ground state electronic interaction between the two electroactive moieties.

The Cu I spectroscopic features have been determined by spectroelectrochemical reduction on the parent complex [Cu(DO)(DOH)pn(OH 2 )](ClO 4 ). Disappearance of the weak Cu II signal around 480 nm concomitantly with the appearance of a new signal at 665 nm attributed to the reduced Cu I complex49, 53 were observed during the course of an electrolysis experiment at an applied potential of -0,70 V vs Ag/AgCl (Figure S6).

Steady-state fluorescence spectroscopy

Emission spectra were recorded for isoabsorptive solutions of 1 and 4 in acetonitrile. Typical emission signals for ruthenium tris-diimine photosensitizers are obtained, without any shift of the maximum emission wavelength (λ em = 604 nm). Dyad 4 is, on the other hand, characterized by a 50 % quenching of the emission of the ruthenium centre, compared to 1. Two different photoinduced processes could account for this quenching: either energy transfer or intramolecular electron transfer from the ruthenium excited state (*Ru) to the copper (II) moiety. Although the d-d band of 2 partially overlaps the emission signal of the ruthenium centre, energy transfer is very unlikely to occur: the very low absorption coefficient of this metal-centered transition indeed strongly disfavors the Förster mechanism and the Dexter mechanism is spin forbidden. According to (i) the Cu II/I reduction potential (Table 1), (ii) the Ru III/II oxidation potential (Table 1) and (iii) the *Ru excited state energy E 00 (2.12 eV),54, 55 the oxidative quenching of *Ru by photoinduced electron transfer to the copper (II) moiety is estimated to be feasible with a relatively low thermodynamic driving force (ΔG ET = -0.24 eV). [START_REF]CuI species by electron transfer from Ru* to CuII was calculated using the equation: DG1° = E[END_REF] The ability of 1,2,3-triazole bridges to mediate charge transport has been particularly evaluated with porphyrin-based dyads and triads, albeit with some differences in their efficiencies.57-60 When considering Ru tris-diimine-based assemblies, however, the few reported examples by Aukauloo and co-workers gave evidence for the efficiency of such triazole-based bridging units.25, 26, 28

EPR-monitored photolysis experiment

EPR spectroscopy is the technique of choice to bring any metal redox state modification to light. It has been previously employed in the literature to characterize light-induced redox processes such as oxidation of a Ni II centre29 or reduction of a Cu II centre40 in rutheniumbased dyads, under photolysis conditions. Reductive photolysis experiments are typically performed in the presence of a large excess of a sacrificial electron donor, such as triethylamine (TEA); the latter rapidly reacts with the oxidized photosensitizer to regenerate Ru II , thus avoiding charge recombination and leading to accumulation of the targeted reduced species. To probe the occurrence of a photoinduced electron transfer with 4, a photolysis experiment was monitored using EPR. The spectrum of 4 (Fig. S7) is characteristic for a mononuclear copper centre in axial symmetry (g// > g⊥) with a hyperfine coupling of 20.3 mT, similar to the azide-precursor 2 (Fig. S7) and to the literature.53 Thus, no significant modification of the copper diiimine dioxime geometry occurs, either upon coupling with 1 (Fig. S7), or in presence of 200 equivalents of TEA (Fig. 3; t = 0 min). However, the EPR signal decreases upon visible light irradiation and is almost completely silent after 30 minutes (Fig. 3).

Additional experiments have showed that light and TEA are both required for the disappearance of signal (Fig. S8 andS9). The complete disappearance of the EPR signal associated with the Cu II centre in 4 thus clearly establishes that photoinduced electron transfer occurs upon light-activation of dyad 4 in presence of a sacrificial electron donor, leading to the formation of the EPR-silent reduced Cu I species. Two mechanisms can operate under these photolysis conditions: (i) the oxidative quenching pathway, previously discussed in the "steady-state fluorescence spectroscopy" section, which forms a Ru III Cu I intermediate before regeneration of the Ru II centre by the sacrificial electron donor TEA; (ii) the reductive quenching pathway, where *Ru is first quenched by TEA, yielding a Ru I Cu II species, that ultimately produced Ru II Cu I . The oxidative quenching process was unambiguously established, on the basis of time-resolved spectroscopic studies, for two dyads we previously studied: a ruthenium-cobalt one, also relying on an imidazo[4,5-f] [1,10]phenanthroline bridging ligand,41 and more recently a ruthenium-copper assembly.40 Moreover, previous reports have established that quenching of *[Ru(bpy) 3 ] 2+ and related Ru-based photosensitizers by aliphatic amines (TEA, TEOA) was not efficient, even at high molar concentrations of the quencher,61-63 supporting the fact that a reductive quenching process is quite unlikely to occur in our system.

Transient absorption spectroscopy

Femtosecond and nanosecond transient absorption studies have been carried out in order to obtain the kinetics for the photoinduced intramolecular processes. The femtosecond transient absorption spectra of complexes 1, 3 and 4 are similar (Fig. S10-S12). Photoexcitation of the ruthenium centre with 414 nm light generates *Ru, which exhibits broad absorption at wavelengths longer than 500 nm together with ground state bleach near 450 nm. After the initial rise of these features, the spectra exhibit little change as the pump to probe delay is increased step-wise to 5.4 ns. Specifically, no new signal arises around 665 nm for 4, which would be characteristic for Cu I .

In the nanosecond transient absorption experiments (Fig. S13-S14), stimulated emission from the *Ru excited state is observed between 565 nm and 725 nm, in addition to ground state bleaching and excited state absorption. For 3, the excited state lifetime is 1.07 ± 0.05 μs, based on the kinetics at 460 nm, 540 nm, 620 nm and 800 nm. For dyad 4, a faster excited state decay is observed, with a lifetime of 0.38 ± 0.01 μs. However, the formation of an intermediate state with a longer lifetime is very unlikely as the ground state bleach, the excited state absorbance and the emission all decay at nearly equal rates. The origin of this shortened lifetime is not clear. The presence of the paramagnetic Cu II species may enhance the rate of intersystem crossing from the Ru-based 3 *MLCT state back to the ground state. Alternatively, electron transfer from the *Ru excited state to the Cu(II) centre followed by faster charge recombination can also explain the increased decay rate without observation of an intermediate species. Similar behavior has been previously reported for another dyad featuring a triazole bridge, where charge separation was slower than charge recombination, thus preventing accumulation of the charge separated state to any appreciable extent.19, 59

Conclusion

There is an important need for developing modular and efficient methodologies to assemble molecular photosensitizers and catalysts into covalent photocatalytic systems. Diiminedioxime cobalt complexes are well-known as hydrogen-forming electrocatalysts.64 As a first step towards the formation of a cobalt-based photocatalytic dyad, we exploited the versatility of the CuAAC methodology to covalently couple a ruthenium tris-diimine photosensitizer with a copper diimine dioxime redox-active unit. Dyad 4 is the first example of a dinuclear Ru II Cu II complex assembled through click chemistry. It has been characterized extensively by 1 H NMR, cyclic voltammetry, UV-vis absorption, fluorescence and EPR spectroscopies. A 50 % quenching of the ruthenium-centred fluorescence is observed in dyad 4, although specific signatures and kinetics related to the intramolecular photo-induced electron transfer could not be obtained from ultrafast spectroscopy experiments, possibly due to a fast recombination process. Under continuous photolysis conditions, the sacrificial electron donor, however, regenerates the ruthenium ground state faster than charge recombination and allows for photoaccumulation of the Ru II Cu I compound. This result opens interesting perspectives for the application of dyad 4 and derivatives in the field of photocatalysis.

Typical procedure for the CuAAC coupling 4. In a round bottom flask, 1 (80 mg, 0,08 mmol) and 3 (70 mg, 0.16 mmol) were dissolved in a homogeneous solvent mixture of MeOH, CH 2 Cl 2 and H 2 O (3:3:1; 15 mL). The solution was degassed with argon before the addition of sodium ascorbate (80 mg, 0.39 mmol) and copper sulfate (8 mg, 0.03 mmol). The reaction mixture was stirred at 50°C under argon in the dark over the weekend. The organic solvents were evaporated and the desired compound was precipitated by dropwise addition of a saturated KPF 6 aqueous solution. The orange precipitate was filtered, washed with water and dried under vacuum. The crude product was purified by column chromatography on silica gel (MeCN/0.4M aqueous KNO 3 , 80:20). The fractions were concentrated and the product precipitated by dropwise addition of a saturated KPF 6 aqueous solution. The precipitate was filtered, washed with water and dried under vacuum to give 65 mg (55 % yield) of 4 as a dark orange powder. Absorption spectra of 1 (blue), 2 (orange) and 4 (green) recorded at 2.10 -5 M in acetonitrile and room temperature emission spectra (λ exc = 450 nm) of isoabsorptive solutions of 1 (dashed blue) and 4 (dashed green) in acetonitrile. CuAAC coupling between alkyne-substituted ruthenium tris-diimine complex 1 and azidesubstituted copper (II) diimine dioxime complex 2, yielding RuCu dyad 4. Redox properties a of compounds 1, 2 and 4 determined by cyclic voltammetry at 100 mV.s -1 in a CH 3 CN solution of n-Bu 4 NBF 4 (0.1 mol.L -1 ). Faraday Discuss. Author manuscript; available in PMC 2018 June 02.

Dyad 4 (

 4 Scheme 1) was synthesized by coupling an alkyne-functionalized Ru-trisdiimine light-harvesting moiety 1 with an azide-substituted copper (II) diimine dioxime redox-active unit 2. The structure of 1, bearing a functionalized imidazo[4,5-f][1,10]phenanthroline ligand, directly stems from supramolecular ruthenium-based assemblies previously studied in our group;41, 42 additionally, properties of closely related Ru photosensitizers, displaying the same ligand pattern, are well-studied.43, 44 The alkyne function was readily introduced via the preparation of the (4-ethynylphenyl)imidazolo[4,5-f][1,10]phenanthroline ligand and

1 H

 1 NMR (400 MHz, CD 3 CN) δ 9.04 (sbr, 2H), 8.53 (dd, J = 16.1, 8.1 Hz, 4H), 8.44 (sbr, 4H), 8.11 (dd, J = 7.8 Hz, 2H), 8.01 (dd, J = 17.2, 6.3 Hz, 4H), 7.87 (d, J = 5.2 Hz, 2H), 7.80 (sbr, 2H), 7.61 (d, J = 5.3 Hz, 2H), 7.46 (dd, J = 6.4 Hz, 2H), 7.23 (dd, J = 6.3 Hz, 2H). MS (ESI+) m/z 1368 [M -H 2 O -PF 6 ] + , m/z 611 [M -H 2 O -2PF 6 ] 2+ , m/z 538 [M -H 2 O -3PF 6 -H] 2+ , m/z 359 [M -H 2 O -3PF 6 ] 3+ . HRMS (ESI+) calculated for C 52 H 46 CuN 15 O 2 Ru 359.0764; found 359.0767. Elemental analysis (%), calculated for C 52 H 48 CuF 18 N 15 O 3 P 3 Ru + 0.8 KPF 6 : C 37.93, H 2.94, N 12.76; found: C 38.27, H 3.20, N 12.36.

Figure 1 .

 1 Figure 1. Aromatic region of the 1 H NMR spectra recorded for 1, 3 and 4 in CD 3 CN.

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3. X band EPR spectra of complex 4 (1 mM) recorded at 42 K before (a) and after 15 min (b) and 30 min (c) irradiation time in CH 3 CN in presence of TEA (200 molar eq.). Microwave frequency: 9.4 GHz; modulation amplitude: 12 G; modulation frequency: 100 kHz; microwave power: 0.5 mW).

  Scheme 1.

  wave potentials E 1/2 determined as (E pa + E pc )/2 and reported versus Fc +/0 b Measured on a Pt working electrode.c Measured on a glassy carbon working electrode.

Table 1

 1 Queyriaux et al. Page 13 Faraday Discuss. Author manuscript; available in PMC 2018 June 02.

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Faraday Discuss. Author manuscript; available in PMC 2018 June 02.

Acknowledgements

This work was supported by the Life Science Division of CEA (2011 DSV-Energy program), the COST Action CM1202 PERSPECT-H 2 O, the French National Research Agency (Labex program, ARCANE, ANR-11-LABX-0003-01) and the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013)/ERC Grant Agreement n.306398. Work at Northwestern University was supported by the Argonne-Northwestern Solar Energy Research (ANSER) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, under award number DE-SC0001059. Colette Lebrun and Daniel Imbert (CEA, INAC-SCIB) are acknowledged for the ESI-MS and the fluorescence measurements, respectively.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.


Galliano Valent
email: galliano.valent@orange.fr

André Tilquin
email: tilquin@cppm.in2p3.fr

Thomas Schücker
email: thomas.schucker@gmail.com

The Lemaître-Hubble diagram in axial Bianchi IX universes with comoving dust

Keywords: 98.80.Es, 98.80.Cq Key-Words: cosmological parameters, supernovae, anisotropies

Axial Bianchi IX universes have compact, simply connected, homogeneous three-spaces and feature minimal symmetry breaking of the cosmological principle. We consider a positive cosmological constant, comoving dust and a mildly anisotropic expansion in the recent past, z < 6, which might be observable in the near future with the James Webb Space Telescope and the Vera Rubin Observatory. To this end we compute the direction dependent redshift and apparent luminosity of type 1a supernovae.

Introduction

Bianchi I universes have a long history in cosmology. As early as 1933, Georges Lemaître (1933), following a recommendation by Albert Einstein, considers Bianchi I universes without using this name. The entire Bianchi family [START_REF] Bianchi | Engineers and Scientists[END_REF] of universes is introduced to cosmology in the late sixties (Ellis & MacCallum 1969;Hawking 1969;Collins & Hawking 1973). These universes are characterized by homogeneous 3-spaces of simultaneity with three Killing vectors. In Bianchi I, the Abelian model in the Bianchi family, the Killing vectors commute. The maximal number of Killing vectors of a 3-space is six. A theorem by Guido Fubini (1903) is central in our work.

Theorem: Consider a Riemannian manifold of dimension d ≥ 3. Its isometry group cannot be of dimension n = d (d + 1)/2 -1.

It tells us that the sub-maximal number of Killing vectors in 3-space is four. Nevertheless the corresponding universes are contained in Bianchi's classification as special cases, let us call them axial Bianchi universes. Some authors, e.g. King & Ellis (1973), use the adjective 'locally rotationally symmetric (L.R.S.)' instead of axial.

For example, the Bianchi I universes have three independent scale factors, a(t), b(t), c(t),

dτ 2 = dt 2 -a(t) 2 dx 2 -b(t) dy 2 -c(t) 2 dz 2 , (1)
with the three Killing vectors, ∂ x , ∂ y , ∂ z , generating the three translations. Axial Bianchi I universes, a = b, have sub-maximally symmetric 3-spaces, and their fourth Killing vector, x ∂ y -y ∂ x , generates rotations around the z-axis. Note that in the limit of small anisotropies (c -a)/a → 0 the axial Bianchi I universes converge to the maximally symmetric flat Robertson-Walker universes.

There are only two more families of universes with the above two properties: (i) being submaximally symmetric (Fubini 1903) and (ii) admitting convergent limits of small anisotropies to universes complying with the cosmological principle. Besides the axial Bianchi I universes these two families are the axial Bianchi V universes (negative curvature) and the axial Bianchi IX universes (positive curvature). The 3-spaces of the latter are also known as Berger spheres (Berger 1961).

The Cosmic Microwave Background data shows small, but significant anisotropies of the order of 10 -5 motivating property (ii). All Bianchi universes close to Robertson-Walker universes, have been used to try and decipher a 3-dimensional subgroup of the 6-dimensional isometry groups in these data. We recommend the reader-friendly overview on this topic by Pereira & Pitrou (2019).

Besides via observation of the Cosmic Microwave Background, deviations from the cosmological principle may also be observed via drifts of extra-galactic sources (Darling 2014;Marcori et al. 2018), via privileged directions in the Lemaître-Hubble diagram of type 1a supernovae (Kolatt & Lahav 2001) and of quasars and gamma-ray bursts (Secrest et al. 2021;Luongo et al. 2022), via weak lensing data (Pitrou, Pereira & Uzan 2015), via galaxy cluster scaling relations (Migkas et al. 2021), via radio galaxies (Siewert, Schmidt-Rubart & Schwarz 2021) and perhaps via black hole or neutron star mergers in the future (Pitrou, Cusin & Uzan 2020).

We mention four recent model-independent analyses of type 1a supernovae, z < 2.26, with respect to anisotropies: Rahman et al. (2002), Deng & Wei (2018), [START_REF] Zhao | [END_REF] and Perivolaropoulos (2023). All four find no significant deviations from isotropy. This is mainly due to the inhomogeneous distribution of the observed supernovae in the sky.

Our interest in the Lemaître-Hubble diagram for supernovae is motivated by an expected experimental break-through:

In 2024 the Vera Rubin Observatory (previously referred to as the Large Synoptic Survey Telescope, LSST) should start operating and observe 50 000 supernovae per year up to redshift z = 1, improving the precision of the Lemaître-Hubble diagram by one to two orders of magnitude (Schücker, Tilquin & Valent 2014). Complementarily, the James Webb Space Telescope is expected to observe some 200 type 1a supernovae up to redshift z = 6 in the next six years (Lu et al. 2022). Also the Chinese Space Station Telescope should start operating in 2024 and observe some 1800 type 1a supernovae below a redshift of z = 1.3 in a time span of two years (Li et al. 2023).

So far only Bianchi I universes have been used to fit the Lemaître-Hubble diagram (Saunders 1969;Koivisto & Mota 2008a,b;[START_REF] Campanelli | [END_REF]Schücker et al. 2014;Fleury, Pitrou & Uzan 2015) and the expected break-through encourages us to go beyond the Abelian model.

Computing the Lemaître-Hubble diagram one commonly admits Einstein's equations with a cosmological constant. One further assumes that radiation can be neglected and that galaxies, supernovae and dark matter are modeled by comoving dust. 'Comoving' means that there is a system of coordinates with respect to which the energy-momentum tensor has vanishing time-space components and 'dust' means that the space-space components vanish. With these assumptions Farnsworth (1967) has shown that axial Bianchi V universes are reduced to Friedman universes with negative curvature. In our more detailed companion paper Valent, Tilquin & Schücker (2023) the interested reader may find an alternative proof (Valent, 2009).

Farnsworth's result explains our interest in axial Bianchi IX universes on the theoretical side. On the observational side we note that a spherical Friedman universe is still compatible with data (Uzan, Kirchner & Ellis 2003), (Di Valentino, Melchiorri & Silk 2019), (Liu, Hu, Tang & Wu 2023) and we think it is worthwhile to consider a model featuring both small positive curvature and small anisotropies.

In section 2 we compute the Lemaître-Hubble diagram and Einstein's equations in axial Bianchi IX universes. Our approach is pedestrian. The considered universe has four Killing vectors and therefore four conserved Noether quantities. With those we compute the geodesics of a photon coming from a supernova of 'known' absolute luminosity. Varying infinitesimally the emission time, we derive the direction-dependent redshift and varying infinitesimally the initial direction of the photon we derive the apparent luminosity.

In section 3 we linearize our results in small anisotropies (c -a)/a.

In section 4 we recall the exact solutions of Friedman's equations in presence of a positive cosmological constant and comoving dust as given by Edwards (1972) in terms of Jacobi elliptic functions. They will be useful to speed up the fitting of the observed Lemaître-Hubble diagram to the one calculated in axial Bianchi IX universes with small anisotropies. Section 5 outlines our blueprint of the fitting procedure.

Axial Bianchi IX

Bianchi IX universes have compact, simply connected, homogeneous 3-spaces. The isometry group, SO(3), is nonAbelian and compact, on the diametrical opposite of the Abelian isometry group of Bianchi I. Much is known about the dynamics of Bianchi IX universes. Wald (1983): The far future of all initially expanding Bianchi I -VIII universes satisfying Einstein's equations with positive cosmological constant approaches the de Sitter universe exponentially fast. This only remains true for Bianchi IX universes if the cosmological constant is sufficiently large with respect to spacial curvature. "A positive cosmological constant provides an effective means of isotropizing homogeneous universes."

In the far past, Bianchi IX universes have complicated dynamics, oscillatory singularities, and chaos, (Misner 1969) (Belinsky, Khalatnikov &Lifshitz 1971 and1972). Intriguingly the axial Bianchi IX universes are better behaved than the tri-axial ones. Dechant, Lasenby & Hobson (2009) and ( 2010) add a scalar field to axial Bianchi IX universes in order to soften the initial singularity. Let us stress that we will modestly remain in our recent past, z < 6. Giani, Piattella & Kamenshchik (2022) have used Bianchi IX universes to model the gravitational collapse of matter inhomogeneities. They map the Bianchi IX field equations into those for a spherical Friedman universe filled with two scalar fields, only one scalar field in the axial case. We will meet the Klein-Gordon equation of this scalar field when we linearize the anisotropy, equation (52).

Metric

The axial Bianchi IX universe can be defined [START_REF] Bianchi | Engineers and Scientists[END_REF]Terzis 2013) by:

dτ 2 = dt 2 -1 4 a 2 dx 2 -1 4 (a 2 cos 2 x + c 2 sin 2 x) dy 2 -1 4 c 2 dz 2 -1 2 c 2 sin x dy dz , (2)
with the Euler angles x ∈ ( 3 2 π, 3 2 π + 2π), y ∈ (0, 2π) and z ∈ (0, 2π). It has four Killing vectors:

sin y ∂ x -tan x cos y ∂ y + cos y cos x ∂ z , cos y ∂ x + tan x sin y - sin y cos x ∂ z , ∂ y ; ∂ z . (3)
(For vanishing anisotropies, c -a = 0, we recover the spherical Robertson-Walker universe.)

The Euler angles x, y, z are related to the familiar polar coordinates (King 1991) by:

χ = arccos cos x -3 2 π 2 sin y -z 2 , x = 2 arccos 1 -sin 2 χ sin 2 θ + 3 2 π, (4)
θ = arctan tan x -3 2 π 2 / cos y -z 2 , y = ϕ + arccos sin χ cos θ 1 -sin 2 χ sin 2 θ , (5)
ϕ = y + z 2 , z = ϕ -arccos sin χ cos θ 1 -sin 2 χ sin 2 θ . (6)

Geodesics

Now we are ready to prove that our coordinates are comoving with respect to test masses. With the four Killing vectors, Emmy Noether's theorem yields four conserved quantities on any geodesic x µ (q) with affine parameter q and ˙:= d/dq:

A := a 2 sin y ẋ + (c 2 -a 2 ) sin x cos x cos y ẏ + c 2 cos x cos y ż , (7) Ā := a 2 cos y ẋ -(c 2 -a 2 ) sin x cos x sin y ẏ -c 2 cos x sin y ż , (8) B := (a 2 cos 2 x + c 2 sin 2 x) ẏ + c 2 sin x ż, (9)

C := c 2 sin x ẏ + c 2 ż , (10)
or equivalently

ẋ = ± 1 a 2 A 2 + Ā2 - B sin x -C cos x 2 1/2 , ẏ = B -C sin x a 2 cos 2 x , (11)
ż = C c 2 -sin x B -C sin x a 2 cos 2 x , A cos y -Ā sin y = - B sin x -C cos x . (12)
Any dust test-particle with vanishing initial 3-velocity, A = B = C = 0, will remain at rest with respect to the coordinates (x, y, z). For a massless test particle, we have ẋµ g µν ẋν = 0 or equivalently:

ṫ2 = A 2 + Ā2 + B 2 -C 2 4 a 2 + C 2 4 c 2 . ( 13
)

Redshift

We are now ready to compute the redshift in axial Bianchi IX universes. Since every Bianchi space is homogeneous (in the sense that it forms a single orbit under the group action), we may place the emitter at the north pole, χ e = 0. This choice considerably simplifies the computations, but we must pay due attention to the coordinate singularity at the poles. Equations ( 6) fix x e = 3 2 π and z e = y e ± π. At cosmic time t e our emitter at the north pole radiates a photon. To have a unique solution of the geodesic equation in the interval [q e , q 0 ] we choose initial conditions at q e : t(q e ) = t e ,

x(q e ) = x e = 3 2 π, y(q e ) = y e , z(q e ) = z e = y e + π, ṫ(q e ) = ṫe > 0, ẋ(q e ) = R > 0, ẏ(q e ) = S, ż(q e ) = że . ( 14)

They determine the conserved quantities by equations ( 7)-( 10):

A sin y e = Ā cos y e = a 2 e R, A 2 + Ā2 = a 4 e R 2 , B = -C = c 2 e (S -że ), (15)
and evaluating the first of equations ( 12) at emission we obtain the initial condition

że = 1 -2 a 2 e c 2 e S, (16)
where we have used the short-hand a e := a(t e ), c e := c(t e ). Other abbreviations will be useful:

s := 2S R , W (t) := 2 1 a 2 + s 2 c 2 -1/2 , f (x) := 1 + sin x cos x , V (x) := 1 -s 2 f (x) 2 -1/2 . (17)
In appendix A.1 we show how to eliminate the Noether constants from the geodesic equations (11,12,13) in favor of the initial conditions. Let us trade the equation for ż in favor of the equation for φ = ( ẏ + ż)/2 and eliminate the affine parameter q in favor of cosmic time t.

Then we remain with only three equations for the trajectory of the photon.

dx dt = 1 a 2 W V , dϕ dt = 1 2 s 1 a 2 - 1 c 2 W, sin(y -y e ) = s f (x). ( 18
)
We suppose that the photon arrives today, t 0 = t(q 0 ), in our telescope situated at x 0 := x(q 0 ) and integrate the first two of these three equations by separation of variables,

t 0 te W a 2 = x 0 xe V = 2 √ 1 + s 2 arctan( √ 1 + s 2 f 0 V 0 ), ( 19
)
s 2 t 0 te 1 a 2 - 1 c 2 W = ϕ 0 -ϕ e = ϕ 0 -y e -1 2 π. ( 20
)
Consider now a second photon emitted at the north pole an atomic period T later in cosmic time than the first photon and arriving in our comoving telescope still at x 0 at cosmic time t 0 + T D . (• D stands for Doppler.) We suppose that the atomic periods, T and T D , are much smaller than the time of flight t 0 -t e and the geodesics of the two photons, x µ (q) and xµ (q), q ∈ [q e , q 0 ] are infinitesimally close. For the second geodesic, the initial conditions at q e are: t(q e ) = t e + T , x(q e ) = x e = 3 2 π, ỹ(q e ) = ỹe , z(q e ) = ỹe + π, ṫ(q e ) > 0, ẋ(q e ) = R > 0, ẏ(q e ) = S, ż(q e ) = 1 -2

ã2 e c2 e S. ( 21
)
We want to adjust the initial 'velocity' (or direction) of the second photon, ỹe =:

y e [1 + δ], s := 2 S R =: s[1 + ], (22)
such that its final comoving position coincides with that of the first photon, ˜ x 0 = x 0 . The adjustment being infinitesimal, we may keep only linear terms in T, T D , δ and . In this approximation, that we also indicate by ∼, we obtain 4

T D W 0 - T W e ∼ 0. ( 23
)
You find the details in appendix A.2. For non-vanishing T , we have the redshift

z := T D -T T ∼ W 0 W e -1. ( 24
)
This formula agrees with the general definition of redshift, that you find in Pierre Fleury's beautiful thesis (Fleury 2015) as equation (1.71).

To avoid confusion with the third Euler angle, we underline the redshift z temporarily.

Apparent luminosity

The trajectory of our photon is described by the lightlike geodesic x(t) with initial conditions ( 14) and given by equations (18-20). We obtained the redshift from the infinitesimal variation of the initial conditions in (14):

t e → t e + T, y e → y e (1 + δ), s → s(1 + ). ( 25
)
Requiring that the two geodesics end at the same point in space x 0 = ˜ x 0 we computed the proper time difference at reception T D = t0 -t 0 . For the apparent luminosity we need two more infinitesimally close lightlike geodesics

x δ (t) = x(t) + δ(t) and x (t) = x(t) + (t) (26)
with varied spatial initial conditions y e → y e (1 + δ) and s → s(1 + ) respectively. All three geodesics are emitted simultaneously at t e from the point x e . The two 3-vectors δ(t) and (t), which should not be confused with the infinitesimal numbers δ and , are interpreteted as tangent vectors at x(t) just as the velocity x (t). The prime denotes the derivative with respect to cosmic time t. As a function of time, δ(t) and (t) define a family of infinitesimal triangles. They are perpendicular to the trajectory x(t). The triangles represent the beam of photons emitted by the supernova at x(t e ). The photons illuminate a triangular photo-plate now and here x(t 0 ), see figure 1. The plate is placed perpendicular to the beam and its area is dS(t 0 ) =: dS 0 . Our task is to compute the area dS(t) of the infinitesimal triangles and their solid angle dΩ e at emission. Then the apparent luminosity is:

= L dΩ e dS 0 (z + 1) -2 , ( 27
)
with the absolute luminosity L of the supernova. As detailed in appendix B, the infinitesimal triangles defined by δ(t) and (t) are perpendicular to the ray x (t) and their areas as a function of time dS(t), see figure 1, are given by

dS = 1 2 ( δ • δ)( • ) -( δ • ) 2 1/2 = 1 2 δ y e s 1 + s 2 a 2 c W (1 + sin x) 1 -1 2 I 3 f V , (28)
where we denote by δ • the negative of the scalar product of 3-vectors defined by the spatial part of the metric tensor g µν in equation ( 2), and with

I 3 (t) := 1 4 t te W 3 a 2 1 a 2 - 1 c 2 . ( 29
)
Of course at emission time t e the area vanishes and to compute the solid angle it sustains at the north pole x(t e ) we evaluate dS(t e +dt) to leading order in dt. From the first of equations ( 18) we have dx ∼ W e /a 2 e dt and:

1 + sin x(t e + dt) ∼ 1 2 W 2 e a 4 e dt 2 , f (x(t e + dt)) ∼ 1 2 W e a 2 e dt. ( 30
)
We linearize the integral I 3 defined in equation ( 29) with respect to dt,

I 3 (t e + dt) ∼ 1 4 W 3 e a 2 e 1 a 2 e - 1 c 2 e dt. ( 31
) Therefore 1 -1 2 I 3 f V = 1 4 1 + s 2 c 2 e W
and the apparent luminosity:

= L 32π W 5 e (1 + s 2 ) a 2 e c e a 2 0 c 0 W 0 (1 + sin x 0 ) 1 -1 2 I 3 0 f 0 V 0 -1 . ( 34
)
Note that the initial condition y e drops out from the redshift and the apparent luminosity. Therefore only the initial condition s remains to be interpreted. It is related to the angle between the direction here and now -x (t 0 ) pointing to the supernova at the north pole and the privileged direction of the axial Bianchi IX universe, the fourth Killing vector ∂ z . Let us denote this angle by γ 0 . We have:

cos γ = -x • ∂ z ∂ z • ∂ z = s c 2 a 2 + s 2 or s = c a cot γ = c 0 a 0 cot γ 0 . (35)

Einstein's equations

To write down the non-vanishing components of the Einstein tensor we will use the notations H := a /a and H c := c /c,

G tt = 3H 2 + 2H(H c -H) + 1 a 2 4 - c 2 a 2 , (36)
G xx = - a 2 4 H + H c + H 2 + H 2 c + HH c + c 2 a 4 , (37)
G zz = - c 2 4 2 H + 3 H 2 + 1 a 2 4 -3 c 2 a 2 , (38)
G yy = cos 2 x G xx + sin 2 x G zz , G yz = sin x G zz . ( 39
)
Therefore we may assume that the energy-momentum tensor has only one non-vanishing component, the mass density T tt = ρ(t), 'matter is comoving dust'. The covariant conservation of energy, ρ + (2H + H c )ρ = 0, yields mass conservation:

ρ(t) = a 2 0 c 0 ρ 0 a(t) 2 c(t) , a 0 := a(t 0 ), c 0 := c(t 0 ), ρ 0 := ρ(t 0 ). ( 40
)
The Einstein equations read:

3H 2 + 2H(H c -H) + 1 a 2 4 - c 2 a 2 = Λ + 8π Gρ, (41)
H + H c + H 2 + H 2 c + HH c + c 2 a 4 = Λ, ( 42
) 2 H + 3 H 2 + 1 a 2 4 -3 c 2 a 2 = Λ. ( 43
)
Let us use the mass conservation to reduce the three Einstein equations to the tt component and to the xx component minus the zz component. We obtain the equivalent system:

3H 2 + 2H(H c -H) + 1 a 2 4 - c 2 a 2 = Λ + 8π G ρ 0 a 2 0 c 0 a 2 c , ( 44
) (H c -H) + (2H + H c )(H c -H) - 4 a 2 1 - c 2 a 2 = 0. ( 45
)
3 Linearizing axial Bianchi IX

Linearizing Einstein's equations

Consider the Robertson-Walker universe with one scale factor as function of cosmic time a F (t), • F for Friedman, and its Hubble parameter H F := a F /a F . The first Friedman equation is the tt component of the Einstein equation and can be written

3 H 2 F + 3 κ a 2 F = Λ + 8π G ρ F 0 a 3 F 0 a 3 F , (46)
with four parameters:

• the 'initial' condition today a F (t 0 ) := a F 0 , (Of course we mean 'final', because we want to compute the past.)

• the cosmological constant Λ,

• 8π G ρ F 0 with G being Newton's constant and ρ F 0 the mass density of dust today,

• κ = 0, 1, -1 for the 3-spaces with zero, positive and negative curvature.

A fifth parameter is important: today's Hubble parameter also more appropriately called Lemaître-Hubble constant, H F (t 0 ) =: H F 0 . Its absolute value is determined by the first Friedman equation evaluated today. At the same time the choice of the other four parameters must obey the inequality,

Λ + 8π G ρ F 0 -3 κ a 2 F 0 ≥ 0. ( 47
)
Note also that in the flat case κ = 0, the initial condition a F 0 is not essential. Being a first order ordinary differential equation, the Friedman equation has a unique local solution once we have chosen the four parameters in compliance with the inequality.

If the universe is not static, the first Friedman equation ( 46) implies the second Friedman equation (the xx component of the Einstein equation),

2 H F + 3 H 2 F + κ a 2 F = Λ, (48)
and the mass conservation, ρ F + 3 H F ρ F = 0, is satisfied if we set

ρ F := ρ F 0 a 3 F 0 a 3 F . ( 49
)
For a static universe (Einstein) we must have κ = 1 and a -2

F 0 = Λ = 4π G ρ F 0 .
It has seven Killing vectors, but it is unstable. From now on we will ignore the static universe and use the standard dimensionless parameters:

Ω m0 := 8π G ρ F 0 3H 2 F 0 , Ω Λ0 := Λ 3H 2 F 0 , Ω κ0 := -κ a 2 F 0 H 2 F 0 = 1 -Ω m0 -Ω Λ0 . ( 50
)
Then a big bang or a bounce occurs and we take it as origin of cosmic time a F (0) = 0 or H F (0) = 0. Now consider positive curvature, κ = 1. The following linearization in η(t) is mandatory:

a = a F [1 -1 2 η], c = a F [1 + η]. (51)
Indeed in Bianchi I universes, κ = 0, we used the simpler linearization with a = a F , implying however ρ 0 = ρ F 0 (Schücker et al. 2014). For Bianchi V and IX this simpler linearization together with Einstein's equations implies isotropy, η = 0. In our long version (Valent et al. 2022) we justify our linearization (51) and show that it implies ρ 0 = ρ F 0 . Let us linearize now the two Einstein equations in Bianchi IX universes under the form (44) and ( 45) using (51). To first order in η we obtain the first Friedman equation ( 46) and the Klein-Gordon equation for η.

η + 3 H F η + 8 η a 2 F ∼ 0. ( 52
)
Note that the function η is the Klein-Gordon field in Giani, Piattella & Kamenshchik (2022) and our equation ( 52) agrees with their equation (3.6).

Our four essential initial conditions, a 0 , c 0 , H 0 , H c0 , fix ρ 0 by the tt component of the Einstein equation ( 44). They translate into the four essential initial conditions:

a F 0 = 2 3 a 0 + 1 3 c 0 , H F 0 ∼ 2 3 H 0 + 1 3 H c0 , η 0 = c 0 -a 0 a 0 + 1 2 c 0 , η 0 ∼ 3 2 (H c0 -H 0 ). ( 53
)
We call the last two 'ellipticity' and 'Hubble stretch'.

Linearizing redshift and apparent luminosity

We start by linearizing W (t) defined by the second of equations ( 17),

W ∼ 2 √ 1 + s 2 a F 1 -1 2 1 -2s 2 1 + s 2 η , (54)
from which we obtain the linearized redshift,

z + 1 ∼ a F 0 a F e 1 -1 2 1 -2s 2 1 + s 2 (η 0 -η e ) . ( 55
)
Next we linearize the integral I 3 (t), equation ( 29),

I 3 ∼ 6 (1 + s 2 ) 3/2 t te η a F , (56)
and linearize the apparent luminosity:

∼ L 2π a 2 F e a 4 F 0 1 (1 + s 2 )(1 + sin x 0 ) 1 + 1 2 1 -2s 2 1 + s 2 (η 0 -5η e ) + 3 (1 + s 2 ) 3/2 χ e0 ηe0 f 0 V 0 , (57)
with the definitions:

χ e (t) := t te 1 a F , χ e0 := χ e (t 0 ), ηe (t) := 1 χ e (t) t te η a F , ηe0 := ηe (t 0 ). (58)
Remember that we used χ for the radial coordinate in equation ( 6). This χ is the geodesic distance from the north pole on the 3-sphere. We have set the speed of light to unit. Therefore the conformal time χ e (t) coincides with the comoving geodesic distance of our photon emitted at the north pole χ = 0 at cosmic time t e in the underlying Friedman universe, η = 0.

To compute the apparent luminosity of our photon in the axial Bianchi IX universe in linear approximation, we still need f (x) V (x) to zeroth order in x -x F and 1 + sin x to first order. To this end we linearize equation ( 19) and obtain:

f (x F ) V (x F ) = tan χ e √ 1 + s 2 and 1 + sin x ∼ 2 1 + s 2 sin 2 χ e 1 + 1 + 4s 2 1 + s 2 χ e ηe tan χ e . (59)
Now, the apparent luminosity takes the form:

∼ F 1 + 1 2 1 -2s 2 1 + s 2 η 0 -5η e + 4 χ e0 tan χ e0 ηe0 (60) with F = L 4π a 2 F 0 sin 2 χ e0 a F e a F 0 2 . ( 61
)
Note that the initial condition s appears in the linearized redshift z (dropping the underline) and in the linearized apparent luminosity always multiplied by η. Therefore to first order, we may replace s here simply by s ∼ cot γ 0 . Note that in this approximation the angle γ also coincides with the coordinate θ, which is constant for geodesics coming from the north pole. Therefore:

z + 1 ∼ a F 0 a F e 1 - 1 -3 cos 2 θ 2 (η 0 -η e ) , (62)
∼ F 1 + 1 -3 cos 2 θ 2 η 0 -5η e + 4 χ e0 tan χ e0 ηe0 . ( 63
)
In the limit of zero curvature, a 0 and c 0 → ∞, we have

lim sin χ e0 = χ e0 , lim χ e0 tan χ e0 = 1, ( 64
)
and the differential equation for η in Bianchi IX ( 52) reduces to the one in Bianchi I. Taking due account of our change of definitions, η I = 3 2 η IX and χ I = 1/a while here χ IX = 1/a F , we reproduce our perturbative results (Schücker et al. 2014) of axial Bianchi I.

Note the cos 2 θ in equations ( 62) and ( 63) ensuring that the Lemaître-Hubble diagram in Bianchi I and IX universes has forward-backward symmetric anisotropies (quadrupole). They can be distinguished easily from the anisotropies induced by a peculiar velocity of the observer (the kinetic dipole), that do not share this symmetry.

Exact solutions of Friedman's equations with comoving dust

To speed up the fitting of axial Bianchi IX universes with the Lemaître-Hubble diagram, it will be useful to have exact solutions of the Friedman equations ( 46). Exact solutions of the Friedman equations in presence of a cosmological constant, curvature, dust and radiation are well-known, for example in terms of Weierstraß elliptic functions (Lemaître 1933;Coquereaux & Grossmann 1982;Coquereaux 2015). We will not have to include radiation, which considerably simplifies the solutions and we will use Jacobi elliptic functions, following Edwards (1972).

The scale factor a F as function of conformal time χ

Let us change the independent variable in the scale factor a F from cosmic time t to the dimensionless conformal time = comoving geodesic distance,

χ(t) := t 0 1 a F , 0 ≤ t. ( 65
)
We will set the big bang at t = 0, a F (0) = 0, but the corresponding singularity will be integrable. Note the link with our former definition (58):

χ e (t) = χ(t) -χ(t e ). ( 66
)
By abuse of notations we write the inverse function of χ(t) as t(χ).

Let us also change dimensionless parameters assuming that the cosmological constant is positive:

Λ > 0, γ := 4 Ω m0 Ω Λ0 , ν := - 4 3 Ω κ0 Ω Λ0 . ( 67
)
Note that ν and κ carry the same signs or vanish both. Let us assume that

γ 2 -ν 3 = 16 Ω 2 m0 Ω 2 Λ0 1 + 4 27 Ω 3 κ0 Ω Λ0 Ω 2 m0 > 0, ( 68
)
is positive. This implies that there is a big bang. If γ 2 -ν 3 vanishes we have Einstein's static universe with seven Killing vectors. Universes with negative γ 2 -ν 3 are spherical and undergo one bounce. They contain the maximally symmetric de Sitter spacetimes (positive curvature, ten Killing vectors). Nevertheless we will ignore this last class, because it is frankly incompatible with today's observations, which we would like to try and fit in section 5 with universes breaking minimally the isometry group in the spaces of simultaneity (positive curvature, four Killing vectors). Since γ 2 -ν 3 is positive, a further parameter,

s := -1 2 γ + γ 2 -ν 3 1/3 -1 2 γ -γ 2 -ν 3 1/3 , (69)
is real and s2 -ν > 0. This inequality allows us to define three positive constants

A := s2 -3 4 ν < B := 3s 2 -3 4 ν, k 2 := 1 2 + 3 4 s2 -ν/2 AB , (70)
and to check that k 2 < 1 since we have

k 2 (1 -k 2 ) = 4A 2 B 2 -9(s 2 -ν/2) 2 16A 2 B 2 = 3 s2 (s 2 -ν) 16A 2 B 2 > 0. ( 71
)
(Of course the constants A and B have nothing to do with the conserved quantities defined in equations ( 7) and ( 9).)

We need one last dimensionless positive constant,

σ := a F 0 Λ AB 3 . ( 72
)
Now the solution of Friedman's equation ( 46) can be written explicitly as function of conformal time using the Jacobi and related functions defined in appendix C (suppressing the argument k 2 ),

a F (t(χ)) = a F 0 |s| A A + B 1 -cn (σχ) cn (σχ) -cn (σ χ) , cn (σ χ) := A -B A + B . (73)
The big bang takes place at χ = 0, t(0) = 0, because cn 0 = 1. The age of the universe in conformal time is:

χ 0 = 1 σ arc cn A (1 + |s|) -B A (1 + |s|) + B . (74)
Note that at finite conformal time χ the scale factor tends to infinity. The corresponding cosmic time t( χ) is infinite, indeed the integral

∞ 0 dt a F (t) = χ (75)
is convergent.

The relation ( 73) was first given by Edwards (1972).

Cosmic time t as function of conformal time χ

We use the first of the four Theta functions θ 1 (v, q) defined in appendix C and Jacobi's notations:

q := e -πK /K , H(u) := θ 1 πu 2K , q , ζ(u) := d du ln H(u). (76)
(The function H(u) has nothing to do with the Hubble parameter.) Then we can write the cosmic time as function of conformal time

t(χ) = 3 Λ 1 2 ln H(σ(χ + χ)) H(σ(-χ + χ)) s dn (σχ) + d sn (σχ) s dn (σχ) -d sn (σχ) -ζ(σ χ) + d s σχ , (77)
with the abbreviations:

s := sn u = 2 √ AB A + B , c := cn u = A -B A + B , d = dn u = |s| A + B , u = σ χ. (78)
An ab initio proof of the expressions a F (t(χ)) and t(χ) can be found in our long version (Valent et al. 2022).

Unfortunately we do not have the explicit inverse χ(t), which we compute numerically by interpolation.

The Lemaître-Hubble diagram in axial Bianchi IX universes with small anisotropies

The Lemaître-Hubble diagram is the parametric plot ( (t e ), z(t e )) whose parameter is the cosmic emission time t e . As the light arriving in our telescope cannot tell us its emission time it is convenient to use as parameter the conformal emission time χ(t e ) instead of the cosmic emission time t e . Then the fit with the supernova data follows the following steps:

1. Choose a spherical or flat Friedman universe with comoving dust characterized by three positive parameters:

H F 0 , Ω Λ0 , Ω m0 , such that Ω Λ0 + Ω m0 ≥ 1 (positive or zero curvature, Ω κ0 ≤ 0), (79) 1
+ 4 27 Ω 3 κ0 Ω Λ0 Ω 2 m0 > 0 (initial big bang). (80)
Then the age of the universe in conformal time χ 0 is given by equation ( 74) and in cosmic time t 0 = t(χ 0 ) by equation ( 77). The scale factor today is

a F 0 = 1 H F 0 √ -Ω κ0 . (81)
For any conformal time χ the scale factor a χ (χ) := a F (t(χ)) is given by equation ( 73).

2. Fix the z-axis in the sky by two parameters, e.g. (right ascension, declination).

3. Choose two more small parameters: the final conditions of the infinitesimal anisotropy η(t) today. Since we will work with the conformal time, let us define η χ (χ) := η(t(χ)) and the final conditions today by:

η χ (χ 0 ) = η(t 0 ) = η 0 'ellipticity', (82)
d dχ η χ (χ 0 ) = a F 0 d dt η(t 0 ) 'Hubble stretch'. (83)
Both should be smaller in absolute value than 10 %. Note that in the flat case the ellipticity vanishes by the choice a 0 = c 0 and Bianchi I only has the Hubble stretch.

4. Rewriting equation (52) in terms of conformal time, we have:

d 2 dχ 2 η χ + 2 H χ d dχ η χ + 8 κ η χ ∼ 0, κ = 1, 0, (84)
with the conformal Hubble parameter

H χ := d dχ a χ /a χ = σ(1 -c) sn u dn u (1 -cn u)(cn u -c) . (85)
Solve this linear differential equation numerically for times prior to χ 0 with the final conditions: ellipticity and Hubble stretch.

5. Choose a supernova with its redshift z, luminosity obs and its direction. Let θ be the angle between this direction and the chosen z-axis.

Invert numerically

z + 1 ∼ a F 0 a χ (χ e ) 1 - 1 -3 cos 2 θ 2 (η 0 -η χ (χ e )) (86)
to obtain the emission time χ e := χ(t e ) for a given redshift and angle θ. Attention, the conformal emission time is a number and should not be confused with the function χ e (t) := t te a -1 F , equation ( 58). We have χ e (t 0 ) =: χ e0 = χ 0 -χ e and η χ (χ e ) = η(t e ) = η e .

For consistency of the linearization, η e must remain smaller in absolute value than 10 %.

∼ L 4π a 2 F 0 sin 2 χ e0 (z + 1) 2 1 + 1 -3 cos 2 θ 2 η 0 -5η e + 4 χ e0 tan χ e0 ηe0 , (88)
and compare it to the observed apparent luminosity obs .

8. Let ∆ i be obs -for the ith supernova and let V be the covariance matrix encoding the experimental error bars. Define

χ 2 := ij ∆ i V -1 ij ∆ j .
(Of course this χ 2 has nothing to do with conformal time.) 9. Minimize χ 2 over the set of the seven parameters:

H F 0 , Ω Λ0 , Ω m0 , (right ascension, declination), ellipticity and Hubble stretch in their domains of definition. Axial Bianchi I is the five-parameter sub-model with vanishing curvature, Ω Λ0 + Ω m0 = 1, and vanishing ellipticity, η 0 = 0.

Note that the direction dependence is captured by the quadrupole coefficient (1-3 cos 2 θ)/2 in both redshift and apparent luminosity as function of emission time. This coefficient varies from -1 for θ = 0 to 1/2 for θ = 90 • and vanishes around θ = 54.7 • . In this direction the Lemaître-Hubble diagrams of the axial Bianchi IX and Bianchi I universes coincide with the direction-independent Lemaître-Hubble diagrams of their underlying spherical and flat Friedman universes.

As an example we display in figure 2 the Lemaître-Hubble diagrams (z) for θ = 0 simultaneously for the Bianchi IX universe with Ω Λ0 = 0.35, Ω m0 = 0.75, η 0 = 0, d/dχ η χ (χ 0 ) = 0.04 and the Bianchi I universe with Ω Λ0 = 0.3, Ω m0 = 0.7, η 0 = 0, d/dχ η χ (χ 0 ) = -0.01 and their underlying Friedman universes.

The example suggests that small positive curvature and small anisotropies do co-exist peacefully in axial Bianchi IX universes during our recent past.

In 2014 we confronted the Bianchi I model with the 740 type 1a supernovae below a redshift of z = 1.3 in the Joint Light curve Analysis by Betoule et al. (2014) and found only a 1-σ signal (Schücker et al. 2014). Again in 2014 Darling (2014) used the tri-axial Bianchi I model (seven parameters) to fit the apparent motion of 429 extra-galactic radio sources measured by Titov & Lambert (2013) using Very Long Baseline Interferometry. His main Hubble stretch has the same sign as ours but is ten times larger and the results are again compatible statistically.

Waiting and preparing for the mentioned supernovae to be observed, a combined analysis of axial Bianchi IX universes with the Lemaître-Hubble diagram, the Cosmic Microwave Background, the drift of extra-gallactic sources and Baryonic Acoustic Oscillations (and maybe with weak lensing or black-hole mergers) is called for.

Conclusions

Our motivation for this work is to use Bianchi universes to search for anisotropies in the Lemaître-Hubble diagram of type 1a supernovae. These universes are necessarily close to the maximally symmetric Friedman universes and, if we want to add only the minimal number of degrees of freedom to the fit of the standard model, the Bianchi universes must have four isometries, i.e. are axial. Therefore we remain only with axial Bianchi I, V and IX universes, see the cartoon in figure 3.

Axial Bianchi V universes are incompatible with comoving dust (Farnsworth 1967), an asymmetry between negative and positive curvature that we find most remarkable.

Therefore we computed the Lemaître-Hubble diagram for axial Bianchi IX universes. In the limit of zero curvature, it reproduces the diagram for axial Bianchi I universes. Equations ( 13), ( 11)-( 12

) become ṫ = a 2 e R W -1 , ẋ = a 2 e a 2 R V -1 , (93) ẏ
= a 2 e a 2 R s f x , ż = -ẏ + a 2 e R s 1 a 2 - 1 c 2 , (94)
sin(y -y e ) = s f (x). ( 95)

Let us compute the initial polar angles (6). For θ we use equation ( 95) and write

tan θ = f cos y-z 2 = 1 s sin(y -y e ) cos y-z 2 = 1 s cos(y -y e -1 2 π) cos y-z 2 . ( 96
)
In this last form we can evaluate tan θ safely at the north pole and find tan θ e = 1/s = 1 2 ẋe / ẏe . For ϕ we obtain ϕ e = (y e + z e )/2 = y e + π/2.

Note that deriving equation ( 95) with respect to the affine parameter q and using the equation for ẋ in (93) reproduces the equation for ẏ in (94), which we may therefore drop.

A.2 Second photon

Equation ( 95) yields

δy e ∼ - sf 0 1 -s 2 f 2 0 = -sf 0 V 0 . (97)
With

W := 2 1 a 2 + s2 c 2 -1/2 ∼ W 1 - s 2 4c 2 W 2 , V := 1 -s2 f 2 -1/2 ∼ V 1 + s 2 f 2 V 2 , ( 98
)
the two equations ( 19),( 20

1 c 2 W 3 a 2 + 2 f 0 V 0 . ( 100
) T D W 0 - T W e ∼ s 2 1 4 t 0 te 1 a 2 + s 2 c 2 W 3 a 2 + (1 + s 2 ) x 0 xe f 2 V 3 -2f 0 V 0 = s 2 t 0 te W a 2 + (1 + s 2 ) x 0 xe f 2 V 3 -2f 0 V 0 = s 2 x 0 xe V + (1 + s 2 ) x 0 xe f 2 V 3 -2f 0 V 0 = s 2 x 0 xe (1 + f 2 )V 3 -2f 0 V 0 = s 2 2f V | x 0 xe -2f 0 V 0 = 0, ( 101
)
where we have used successively the definition of W (the second of the four definitions ( 17)), equation ( 19), the definition of V (the fourth of the definitions ( 17)), the second of the two integrals (91) and the first of the limits (92).

B Apparent luminosity

Explicitely the three 3-vectors are:

x = sW a 2    (sV ) -1 f x -f x + 1 -a 2 c 2    , δ = δ y e   0 1 1   , (102)
= s 2 1 + s 2    f   -2 (sV ) -1 -(sV ) -1   + sI 3   (sV ) -1 f x -f x + (1+s 2 ) s 2      , (103)
with

I 3 (t) := 1 4 t te W 3 a 2 1 a 2 - 1 c 2 .
The derivation of equations ( 102) is straight-forward. For equation ( 103) it is convenient to start with equation ( 19) in the form

t te W a 2 = x xe V = 2 √ 1 + s 2 arctan( √ 1 + s 2 f V ) =: G(x, s), (104)
and to replace s → s(1 + ) and x → x + x ,

t te W a 2 = G(x + x , s + s ), (105)
with W given by the first of equations ( 98):

W ∼ W 1 - s 2 4c 2 W 2 . ( 106
)
For the linearization we also need,

∂ x G = V, and ∂ s G = s 1 + s 2 (2f V -G), (107)
and with the algebraic identity,

1 4 W 2 c 2 - 1 1 + s 2 = - 1 1 + s 2 1 4 W 2 1 a 2 - 1 c 2 , ( 108
)
we obtain the x-component of in equation ( 103). The y-and z-components then are computed by linearizing the third of equations ( 18), sin(y -y e ) = sf (x), and equation ( 20).

Let us denote by δ • the negative of the scalar product of 3-vectors defined by the spatial part of the metric tensor g µν in equation ( 2). In particular x • x = 1, since we have set the speed of light to one. For the other five scalar products we find:

δ • x = 0, • x = 0, δ • δ = 1 4 δ 2 y 2 e cos 2 x [a 2 + c 2 f 2 ], • = 2 s 2 (1 + s 2 ) 2 s 2 a 2 f 2 + 1 4 a 2 f 2 + c 2 cos 2 x V 2 -2I 3 a 2 c 2 W 2 cos x V + I 2
C Jacobi elliptic and related functions

The Jacobi elliptic functions cn (u, k 2 ), sn (u, k 2 ), dn (u, k 2 ) with k 2 ∈ [0, 1] interpolate between trigonometric functions for k 2 = 0 and hyperbolic ones for k 2 = 1 according to cn (u, 0) = cos u, sn (u, 0) = sin u, dn (u, 0) = 1, (110) cn (u, 1) = tanh u, sn (u, 1) = 1/cosh u, dn (u, 1) = 1/cosh u.

(111)

These functions may be defined from the differential system (dropping the second argument, which will always be the fixed number k (113)

Figure 1 :

 1 Figure 1: Three infinitesimally close geodesics and two perpendicular triangles

  the theoretical apparent luminosity,

Figure 3 :

 3 Figure 3: Robertson-Walker;

  -sn u dn u, d du sn u = cn u dn u, d du dn u = -k 2 cn u sn u,(112)with the initial conditions: sn 0 = 0, cn 0 = 1, dn 0 = 1. These definitions imply the algebraic relations cn 2 u + sn 2 u = 1, dn 2 u + k 2 sn 2 u = 1.

  2 e + O(dt) and dS(t e + dt) ∼ 1 16 δ y e s

							W 3 e a 2 e c e	dt 2 .	(32)
	Finally the solid angle is
	dΩ e =	dS(t e + dt) 4π dt 2	=	δ 64π	y e s	W 3 e a 2 e c e	,

  Therefore we think that a fit of the seven-parameter axial Bianchi IX model to the Lemaître-Hubble diagram of supernovae becomes reasonable only once we can include the data expected from the Vera Rubin Observatory, the James Webb Space Telescope and the Chinese Space Station Telescope.Also in 2014Cea (2014) fitted the axial Bianchi I model to the WMAP and Planck data at redshift 1090. He also found a 1-σ signal. Although his Hubble stretch has opposite sign and is smaller than ours by eight orders of magnitude, our results are compatible with his.

	2	DB9&1short.nb
							blue : Bianchi 9, orange : Friedman,
							green : Bianchi 1,
							red : Friedman flat
							2.8
							2.7
							2.6
							2.5
						Out[1102]=	2.4
							2.3	Bianchi IX
							Friedman spherical
							2.2	Friedman flat
							1.12	1.14	1.16	1.18	1.20	Bianchi I
			Figure 2: apparent luminosity in arbitrary units versus redshift z
			blue, continuous: Bianchi IX,	orange, dashed: Friedman spherical,
			0.025	green, dotted: Bianchi I,	red, dashed dotted: Friedman flat
			0.024
							Printed by Wolfram Mathematica Student Edition
	Out[1097]=	0.023
			0.022
			1.16	1.17	1.18	1.19	1.20
					Printed by Wolfram Mathematica Student Edition

Acknowledgements: We thank our two referees for their constructive feedback.

The project leading to this publication has received funding from Excellence Initiative of Aix Marseille University -A*MIDEX, a French "Investissements d'Avenir" programme (AMX-19-IET-008 -IPhU).

Data availability: The data underlying this article will be shared on reasonable request to the corresponding author.

A Redshift

A.1 First photon

We have the following derivatives,

and limits at the north pole,

We will indicate the inverse functions by arc. Of paramount importance are the elliptic integrals K and K (not a derivative) defined by

In fact Jacobi even defined the functions cn (u), sn (u) and dn (u) for complex values of u, revealing that their squares are meromorphic and doubly periodic functions with periods (2K, 2iK ). What is special with elliptic functions is that besides their double periodicity they must have either a double pole or two simple poles in their period parallelogram, otherwise they are reduced to a constant. We also need the first of the four Theta functions and follow the conventions of [START_REF] Whittaker | A Course of Modern Analysis[END_REF],

For the user's convenience table 1 summarizes a few commands accessing Jacobi elliptic functions and others in Mathematica and Maple.

Mathematica

Table 1: A few access commands in Mathematica and Maple. Note the absence of the square on k in Maple. Note also that EllipticTheta [i, v, q] are not elliptic functions.