A Tension between the Early and Late Universe: Could Our Underdense Cosmic Neighbourhood Provide an Explanation?

A Tension between the Early and Late Universe:

Could Our Underdense Cosmic Neighbourhood Provide an Explanation?

Dissertation in partial fulfillment of the requirements for the degree of Master in Science with a Major in Physics

Student: Supervisors:

Sveva Castello Edvard M¨ortsell,

Marcus H¨og˚as, Martin Sahl´en Subject reader:

Erik Zackrisson

COSMOLOGIA Sono giunta

Negli abissi della mente E ho fatto un paragone Con il cosmo.

E ho realizzato, Incredula,

Che la trama di astri Si ripresenta identica Negli intrecci dei pensieri E nelle forme dell’essere.

E specchiandomi

Nei filamenti di galassie, Ho trovato me stessa E l’armonia celeste Ha fatto breccia Nel mio animo,

Ricordandomi di essere Una luce

Tra miliardi di luci, Nate per donare Un bagliore d’umanit`a All’Universo.

COSMOLOGY I reached

The abysses of my mind And I made a comparison With the cosmos.

Then I realized, Surprisingly,

That the starry arabesques Look identical

To the twines of my thoughts And the shape of my being.

And while contemplating The filaments of galaxies, I found my true self And the celestial harmony Conquered my soul, Reminding me That I am just A shining sparkle

Among billions of others, Striving to provide

A glimpse of hope

Throughout the Universe.

Sveva Castello

Abstract

In recent years, the increasingly precise constraints on the value of the Hubble constant, H₀, have highlighted a discrepancy between the results arising from early-time and late-time measurements. A potential solution to this so-called Hubble tension is the hypothesis that we reside in a cosmic void, i.e. an under- dense cosmic neighbourhood characterized by a faster local expansion rate. In this thesis, we model this scenario through the Lemaˆıtre-Tolman-Bondi formal- ism for an isotropic but inhomogeneous universe containing matter, curvature and a cosmological constant, which we denote by ΛLTB. We numerically im- plement this framework with two different formulations for the local matter density profile, respectively based upon a more realistic Gaussian ansatz and the idealized scenario of the so-called Oppenheimer-Snyder model. We then constrain the background cosmology and the void parameters involved in each case through a Markov Chain Monte Carlo analysis with a combination of recent data sets: the Pantheon Sample of type Ia supernovae, a collection of baryon acoustic oscillations data points from different galaxy surveys and the distance priors extracted from the latest Planck data release. For both models, the resulting bounds on the investigated parameter space suggest a preference for a −13% density drop with a size of approximately 300 Mpc, interestingly matching the prediction for the so-called KBC void already identified on the basis of independent analyses using galaxy distributions. We quantify the level of improvement on the Hubble tension by analyzing the ΛLTB constraints on the B-band absolute magnitude of the supernovae, which provides the cali- bration for the local measurements of H0. Since no significant difference is observed with respect to an analogous fit performed with the standard ΛCDM model, we conclude that the potential presence of a local void does not resolve the tension.

Popul¨arvetenskaplig sammanfattning

Hubblekonstanten, vanligtvis betecknad H₀, ¨ar en grundl¨aggande kosmologisk pa- rameter som beskriver universums nuvarande expansionshastighet och ¨ar relaterad till b˚ade dess ˚alder och observerbara storlek. Baserat p˚a v˚ar vanliga kosmologiska modell kan v¨ardet p˚a H₀ m¨atas med m˚anga olika tekniker, som involverar olika typer av teoretiska antaganden och data. Under de senaste ˚aren har precisionen i s˚adana m¨atningar ¨okat radikalt, vilket har resulterat i en avvikelse mellan resultaten som h¨arr¨or fr˚an observationer vid tidig och sen kosmologisk tid, som motsvarar stora och sm˚a avst˚and fr˚an jorden. Trots den anstr¨angning som kosmologisamfundet har

¨

agnat sig ˚at f¨or att hitta en tillfredsst¨allande f¨orklaring ˚aterst˚ar orsaken till denna s˚a kallade Hubblekontrovers att uppt¨ackas.

Bland m˚anga potentiella l¨osningar bygger ett s¨arskilt intressant f¨orslag p˚a hypotesen att Vintergatan ¨ar bel¨agen i en region i Universum som har en l¨agre t¨athet ¨an genom- snittet, det vill s¨aga ett “kosmiskt tomrum”. Regionen skulle k¨annetecknas av en snabbare kosmisk expansionshastighet, vilket potentiellt f¨orklarar den observerade lokala ¨okningen av Hubblekonstantens v¨arde. Denna “tomrumsl¨osning” har redan diskuterats i tidigare studier, vilket lett till mots¨agelsefulla resultat. Syftet med denna avhandling ¨ar att unders¨oka detta komplicerade scenario genom att anv¨anda mer generella och rigor¨osa antaganden, samt en mer komplett upps¨attning observa- tioner.

F¨or att uppn˚a detta har vi utvecklat en Pythonkod d¨ar vi numeriskt har imple- menterat det s˚a kallade Lemaˆıtre-Tolman-Bondi-ramverket, vilket g¨or det m¨ojligt att modellera ett lokalt kosmiskt tomrum centrerat kring Vintergatan. Vi har j¨amf¨ort denna teoretiska beskrivning med data med m˚alet att begr¨ansa egenskaperna f¨or ma- terians densitetsprofil i det lokala Universumet. Denna analys pekar p˚a en preferens f¨or ett tomrum som ungef¨ar motsvarar en t¨athetsminskning p˚a −13% med en storlek p˚a cirka 300 Mpc, vilket intressant nog matchar tidigare f¨oruts¨agelser som erh˚allits fr˚an galaxunders¨okningar. V˚ara resultat indikerar ocks˚a att inf¨orandet av s˚adana lokala tomrum ger observationerna en b¨attre beskrivning ¨an den vanliga kosmolo- giska modellen, vilket skulle betyda att vi verkligen bor i en kosmisk region med en ovanligt l˚ag t¨athet. Emellertid, genom att noggrant analysera dess potentiella inverkan p˚a de kosmologiska parametrarna i v˚ar beskrivning, drar vi slutsatsen att detta scenario inte l¨oser Hubblekontroversen.

Popular Science Summary

The Hubble constant, usually denoted H₀, is a fundamental cosmological parameter that encodes the current expansion rate of the Universe and is related to both its age and observable size. Based upon our standard cosmological model, the value of H0

can be measured with a variety of techniques, involving different types of theoretical set-ups and data. In recent years, the precision of such measurements has radically increased, highlighting a discrepancy between the results arising from observations at early and late cosmological times, respectively corresponding to large and small distances from Earth. Despite the effort that the cosmology community has devoted to find a satisfactory explanation, the root of this so-called Hubble tension remains as of today poorly understood.

Among many potential solutions, a particularly interesting proposal relies on the hypothesis that the Milky Way is located in a region of the Universe that has a lower matter density than the average, i.e. a “cosmic void”. Because of this density drop, this region would be characterized by a faster cosmic expansion rate, potentially explaining the observed local increase in the value of the Hubble constant. This “void solution” has already been discussed in previous studies, leading to contradictory results. The objective of this thesis is to shed further light on this complex scenario by approaching it under more general and more rigorous assumptions and by employing a more complete set of observations.

In order to achieve this, we have developed a Python code where we have numerically implemented the so-called Lemaˆıtre-Tolman-Bondi framework, allowing to model a local cosmic void centered on the Milky Way. We have compared this theoretical description with data, with the objective of constraining the properties of the mat- ter density profile of the local Universe. This analysis suggests a preference for a void approximately corresponding to a −13% density drop with a size of around 300 Mpc, interestingly matching earlier predictions obtained on the basis of galaxy sur- veys. Our results also indicate that the inclusion of such a local void yields a better description to the observations than the standard cosmological model, suggesting that we could indeed reside in an underdense cosmic neighbourhood. However, by closely analyzing its potential impact on the cosmological parameters involved in our description, we conclude that this scenario does not ease the Hubble tension.

List of abbreviations

BAO Baryon Acoustic Oscillations CDM Cold Dark Matter

CMB Cosmic Microwave Background ESS Effective Sample Size

FLRW Friedmann-Lemaˆıtre-Robertson-Walker GG Generalized Gaussian

KBC Keenan-Barger-Cowie LTB Lemaˆıtre-Tolman-Bondi MCMC Markov Chain Monte Carlo OS Oppenheimer-Snyder

SN Supernova

TRGB Tip of the Red Giant Branch

Contents

Abstract i

Popul¨arvetenskaplig sammanfattning ii

Popular Science Summary iii

List of abbreviations iv

1 Introduction 1

2 Lemaˆıtre-Tolman-Bondi models 5

2.1 The metric . . . . 5

2.2 The generalized Friedmann equation . . . . 7

2.3 Numerical solution . . . . 9

2.4 The distance relations . . . . 12

3 Numerical investigation 14 3.1 Implementation of the ΛLTB formalism . . . . 14

3.1.1 The generalized Gaussian ansatz . . . . 15

3.1.2 The Oppenheimer-Snyder ansatz . . . . 18

3.1.3 Sample numerical results . . . . 19

3.2 Bayesian parameter inference and MCMC analysis . . . . 21

3.2.1 Data sets and likelihood formulation . . . . 23

3.2.2 Parameter space and priors . . . . 28

4 Results and discussion 31 4.1 Parameter constraints . . . . 31

4.2 Model comparison . . . . 36

4.3 Is the tension reconciled? . . . . 37

5 Outlook 39

Post Scriptum 41

Appendices 43

A Solving the ΛLTB equations of motion 43

B Derivation of δ(r) for the generalized Gaussian ansatz on (r) 47

C The Oppenheimer-Snyder model 49

D A simplified set-up for the ΛLTB formalism at high redshifts 51

E Technical details about the MCMC analysis 53

F ΛCDM triangle plot 55

References 56

1 Introduction

At the time of writing, cosmology has reached an interesting turning point that has sparked much enthusiasm and speculation over the past few years: the so-called Hubble tension (Freedman, 2017; Verde et al., 2019). The backstory that is required to understand the current impasse dates back to around a century ago, when Edwin Hubble and Georges Lemaˆıtre independently derived the well-known linear relation between the distances and radial velocities of nearby galaxies (Lemaˆıtre, 1927; Hub- ble, 1929). The proportionality constant in the law, H₀, now carries Hubble’s name and has become a key parameter in the standard paradigm for Big Bang cosmology, the ΛCDM¹ model. Within this framework, H0 encodes the expansion rate of the Universe at present time and is directly related to both the age of the Universe and its observable size. Therefore, much effort has been devoted to measure the value of this parameter with different techniques, involving data from several cosmic epochs.

In the past years, however, the increasingly precise constraints on H₀ have revealed a significant discrepancy between the results arising from observations at “early” and

“late” cosmological times.

Early-time observations provide an indirect way to determine H₀ by calibrating the six free parameters in the ΛCDM model and consequently characterizing the evolu- tion of the Universe until today. A standard procedure to achieve this consists in comparing the ΛCDM prediction with measurements of the temperature and polar- ization fluctuations in the spectrum of the cosmic microwave background (CMB), which is believed to arise during the epoch of cosmic recombination. Through this technique, the recent high-precision CMB observations performed by the Planck mis- sion yielded a value of H₀ = 67.4 ± 0.5 km s⁻¹ Mpc⁻¹ (Planck Collaboration, 2020).

This result is consistent with the observationally independent early-time constraint arising from estimates of the primordial deuterium-to-hydrogen ratio, in combination with the baryon acoustic oscillation (BAO) imprint in the clustering of galaxies (The Dark Energy Survey and the South Pole Telescope Collaborations, 2018).

On the other hand, late-time, “local” determinations of H₀can be performed directly by constructing a “distance ladder”, whose rungs consist of successive geometric cal- ibrations for the luminosities and distances of specific objects, typically Cepheid variable stars and type Ia supernovae (SNe Ia). Among others, the SH0ES Team (Riess et al., 2021) employed this technique to obtain a value of H0 = 73.2 ± 1.3 km

1The name of this model refers to the two components that are believed to provide the dominant contributions to the total cosmic density: Λ is a label for dark energy in the form of a cosmological constant and CDM stands for “cold dark matter”.

s⁻¹ Mpc⁻¹, implying a 4.2σ difference with respect to early-time constraints. This result is compatible with the measurements performed by calibrating the distance ladder with alternative sources, such as the surface brightness fluctuations of distant galaxies (Blakeslee et al., 2021) and bright red-giant-branch stars, through the “tip of the red giant branch” method (TRGB, see e.g. Soltis et al. 2021). Similar local findings were also reported by employing the time delays between the strongly lensed images of background quasars (Wong et al., 2020), observations of water megamasers orbiting around supermassive black holes (Pesce et al., 2020) and the Tully-Fisher relation between the rotational velocity and the intrinsic luminosity of spiral galaxies (Kourkchi et al., 2020; Schombert et al., 2020). Overall, when combined with the aforementioned late-time measurements, the SH0ES result implies a 4σ to 6σ dis- crepancy with respect to early-time constraints (Verde et al., 2019), suggesting the presence of a schism in our interpretation of data from different cosmic epochs.

This tension is clearly portrayed in Figure 1, which is reproduced from Di Valentino et al. (2021) and contains a compilation of the most recent high-precision measure- ments obtained with different techniques. The cosmology community has responded to this issue with much interest and excitement, and a large variety of potential explanations has been proposed. Di Valentino et al. (2021) have catalogued all the solutions that have been investigated to date, invoking systematic effects in the measurements, modifications in the description of the early-time physical processes or alternative models to ΛCDM. While some late-time proposals have been clearly proven to rely on incorrect assumptions (Efstathiou, 2021), several explanations are still debated. In particular, an interesting solution that does not require the intro- duction of new physics consists in postulating the presence of a matter underdensity in the local Universe, i.e. a cosmic void. This would result in a faster local expan- sion, leading to a larger local value of H0 and therefore alleviating the tension at least partially.

Previous studies have already searched for observational signatures of such underden- sity and estimated the potential effects on the calibration of distance ladders. Based upon the reconstructed luminosity density from a combination of galaxy surveys, Keenan et al. (2013) provided evidence for the existence of the so-called Keenan- Barger-Cowie (KBC) void, corresponding to a −30% underdensity with a radius of around 300 Mpc roughly centered on the Milky Way. Whitbourn and Shanks (2014) and B¨ohringer et al. (2020) found similar indications on the scale of a few hundreds Mpc, with slight variations in size and depth with respect to the KBC void. These results, however, were recently questioned by Kenworthy et al. (2019), who employed the Pantheon sample of SNe Ia (Scolnic et al., 2018) to show that there is no evidence

for the presence of abrupt density variations at low redshifts. Moreover, N-body sim- ulations (Odderskov et al., 2014; Wu and Huterer, 2017) suggested that, even if the presence of a local void is not excluded, the extreme depth required to reconcile the Hubble tension is not likely to be attained in the standard ΛCDM framework. On the other hand, Lombriser (2020) focused on a much smaller density fluctuation on a 40 Mpc scale and concluded that this could guarantee a borderline consistency between the local SNe Ia measurement of H₀ and the Planck one.

This interestingly complex scenario is the starting point for this thesis, which aims at approaching this “void solution” to the Hubble tension under more general as- sumptions than in the aforementioned studies. Most of the existing literature, in fact, focuses on specific void sizes and depths and evaluates the impact on the local value of H₀ by solely incorporating the constraints arising from SNe Ia. In this study, instead, we choose to model a generic local underdensity in the standard framework of the Lemaˆıtre-Tolman-Bondi metric (LTB, Lemaˆıtre 1997; Tolman 1934; Bondi 1947) and we numerically implement this formalism with a more rigorous and physi- cally meaningful set of boundary conditions than in most examples in literature. We then employ the tools of Bayesian parameter inference (Trotta, 2017) and standard Markov Chain Monte Carlo (MCMC) techniques (Foreman-Mackey et al., 2013) to perform a fit to a combination of three data sets covering a broad redshift range:

the Pantheon Sample of SNe Ia by Scolnic et al. (2018), a collection of BAO data points from different galaxy surveys and the distance priors extracted from the latest Planck data release (Chen et al., 2019). This approach generalizes the analysis by Cai et al. (2020) and allows to constrain the properties of the local density profile with increased precision. Moreover, it provides an immediate way to quantify the level of improvement in the Hubble tension by performing a comparison with an analogous fit to the same data sets for the ΛCDM model.

This report contains a detailed description of the analysis outlined above and is struc- tured as follows. The next section is devoted to a review of the Lemaˆıtre-Tolman- Bondi metric and its use to model distance measures in the presence of a local cosmic void. Section 3 then contains a description of the numerical investigation carried out to constrain the shape of the underdensity through the aforementioned data sets. In Section 4, we discuss the resulting bounds on the investigated parameter space and assess the degree to which the Hubble tension can be alleviated. A summary of the analysis and some suggestions for future improvements are provided in Section 5, while the Appendices contain some additional mathematical derivations, plots and technical details that were not included in the main text.

Figure 1: Compilation of early-time (indirect) and late-time (direct) measurements of the Hubble constant H₀ performed with different techniques. The image is repro- duced from Di Valentino et al. (2021), where the indicated references can be found.

The red and the blue band respectively refer to the 68% confidence levels in the measurements by Planck (Planck Collaboration, 2020) and the SH0ES Team (Riess et al., 2021), while the dotted vertical line corresponds to H₀ = 69.3 km s⁻¹ Mpc⁻¹ and is included to provide an easier visualization of the division between early- and

2 Lemaˆıtre-Tolman-Bondi models

The first step in understanding the impact of a local matter underdensity on cosmo- logical observations is to identify a suitable theoretical framework to characterize the space-time geometry of such system. LTB solutions to Einstein’s field equations of general relativity (Lemaˆıtre, 1997; Tolman, 1934; Bondi, 1947) describe an isotropic but inhomogeneous universe, i.e. involving a matter density that is the same in all directions but depends on the spatial point considered. Therefore, these solutions provide a simplified but realistic formalism to model a spherically symmetric cos- mic void with a matter distribution that varies along the radial coordinate. In this section, we briefly review the key equations and concepts in the LTB framework following Enqvist (2008), while leaving a more detailed mathematical derivation to Appendix A.

2.1 The metric

The standard LTB formalism (Bondi, 1947) is constructed by considering a space- time manifold characterized by the following assumptions:

1. the geodesics, i.e. the trajectories of hypothetical freely-falling point-like par- ticles, are non-intersecting;

2. the metric is spherically symmetric.

The first point corresponds to Weyl’s postulate (Robertson, 1933), which is also an underlying assumption of the standard Friedmann-Lemaˆıtre-Robertson-Walker met- ric (FLRW) involved in the ΛCDM model (see e.g. Carroll 2019). Instead, the second point implies that the LTB universe is isotropic but not necessarily homo- geneous, thus violating the so-called cosmological principle that provides the second fundamental pillar for the FLRW metric.

The set-up given by assumptions 1 and 2 allows to identify a “comoving” reference frame x^µ= (t, r, θ, φ), where t is the proper time (and we have set the speed of light c to 1) and the spatial coordinates “co-move” (dxⁱ/ dt = 0) with the aforementioned point-like particles along the geodesics². Additionally, a generic non-rotating matter content is included, with the assumption that

2Here and in the following, Greek indices range between 0 and 3 and are employed to denote the components of four-vectors. On the other hand, Latin indices range between 1 and 3 and only refer to the spatial coordinates.

3. its mass density ρ_m(t, r) (where the subscript m refers to “matter”) is finite at any point.

By choosing the spatial origin xⁱ = 0 as the symmetry center, it can be shown that the most general ansatz for a cosmological metric satisfying the requirements 1 - 3 can be written in the form

ds² = − dt²+ X²(t, r) dr²+ R²(t, r) (dθ² + sin θ²dφ²), (2.1) where X(t, r) and R(t, r) are generic functions containing both a temporal and a radial dependence. In order to obtain an explicit expression for these functions, the metric can be applied to Einstein’s field equations of general relativity, yielding the LTB equations of motion. This allows to characterize the evolution of the LTB universe by solving the resulting system of differential equations. The details of these computations can be found in Appendix A and we simply quote the key results below.

First, one equation of motion allows to rewrite the LTB metric in its usual form (see equation (A.9))

ds² = − dt²+ R⁰²(t, r)

1 − k(r)dr²+ R²(t, r) (dθ²+ sin θ²dφ²), (2.2) where R⁰(t, r) ≡ ∂R(t, r)/∂r and k(r) < 1 is a function related to the curvature of the hypersurfaces t ≡ const. We note that the FLRW metric can be recovered as a special case of equation (2.2) with the substitutions

R(t, r) → a(t)r and k(r) → kr², (2.3) where a(t) is the standard cosmological scale factor and k is a constant describing the intrinsic spatial curvature of the metric. Therefore, in the context of a generic LTB model, R(t, r) can be interpreted as a generalized scale factor with an additional radial dependence and k(r) can be seen as a generalized curvature term.

In order to solve the remaining equations of motion, it is necessary to specify the matter and energy content of the system. In the original LTB formulation, the universe is assumed to only contain matter in the form of pressureless dust, i.e.

a perfect fluid that only interacts gravitationally and therefore moves along the geodesics. With this set-up, the equations of motion can be solved analytically and a unique solution for R(t, r) can be singled out by specifying a set of initial conditions.

In the context of this study, instead, we choose to consider the more general class of

“ΛLTB” models, which also incorporate the contribution of a cosmological constant Λ with density ρ_Λ= const. In this case, we can obtain a numerical solution for R(t, r) and the cosmological parameters involved in the description through the procedure outlined below.

2.2 The generalized Friedmann equation

The ΛLTB equations of motion yield the following relation (see equation (A.23)):

R˙²(t, r)

R²(t, r) = κρ_Λ

3 − k(r)

R²(t, r) + M (r)˜

R³(t, r), (2.4)

with ˙R(t, r) ≡ ∂R(t, r)/∂t and the Einstein gravitational constant κ = 8πG/c⁴, where G is Newton’s constant of gravitation and c is the speed of light (set to 1 throughout). The LTB mass function ˜M (r) is defined as

M (r) = κ˜ Z r

0

d˜r R⁰(t, ˜r)R²(t, ˜r)ρ_m(t, ˜r) (2.5) and we note that its value at a generic radius r only depends on the integrated matter content within that radius³. This implies that the LTB formalism describes a “bubble” that evolves independently of the surrounding universe and it is thus ideal to model a matter density variation, as in the presence of a cosmic void.

In order to achieve a clearer physical understanding of this scenario, we can perform a comparison with the more familiar case of a FLRW universe containing matter, curvature and a cosmological constant. We begin by assuming that these cosmic components have density parameters Ω_m(t), Ω_k(t) and Ω_Λ respectively, measured in units of the critical density ρ_c = 3H²(t)/κ, where H(t) is the Hubble parameter.

With this set-up, we can write the standard Friedmann equation (see e.g. Carroll 2019), which describes the time evolution of the Hubble parameter starting from an initial time t = t₀ (usually corresponding to today):

H²(t)

H₀² = Ω_m,0

 a₀ a(t)

3

+ Ω_k,0

 a₀ a(t)

2

+ Ω_Λ in FLRW, (2.6) where

H(t) ≡ ˙a(t)

a(t) and Ω_m,0+ Ω_k,0+ Ω_Λ = 1. (2.7)

3The value of ˜M is also time independent, as can be seen from equation (A.18)

Here, the subscript 0 indicates quantities evaluated at t₀ and we note that we have defined the Hubble constant H₀ = H(t₀). The comparison with equation (2.4) sug- gests to interpret the latter as a generalized Friedmann equation and to rewrite it as

H²(t, r)

H₀²(r) = Ω_m,0(r) R₀(r) R(t, r)

3

+ Ω_k,0(r) R₀(r) R(t, r)

2

+ Ω_Λ(r) in ΛLTB, (2.8) where the initial time t = t₀(r) now contains a radial dependence, such that R₀(r) = R(t₀(r), r). We have thus defined a generalized version of the Hubble parameter⁴, the Hubble constant and the density parameters evaluated at t0(r):

H(t, r) ≡

R(t, r)˙

R(t, r) =⇒ H0(t) ≡ R˙₀(r)

R₀(r), (2.9)

Ω_m,0(r) ≡

M (r)˜

H₀²(r)R₀³(r), Ω_k,0(r) ≡ − k(r)

H₀²(r)R²₀(r) and Ω_Λ(r) ≡ κρΛ

3H₀²(r), (2.10) again with

Ωm,0(r) + Ωk,0(r) + ΩΛ(r) = 1. (2.11) The key difference with respect to the FLRW case is that all quantities now contain an additional radial dependence, incorporating inhomogeneities both in the expansion rate (through H₀(r)) and in the matter distribution (through the present-day matter density ρ_m,0(r)). Even though their evolution is related via Einstein’s equation, these two functions consist of two physically distinct degrees of freedom, since they are completely independent of each other as boundary conditions. Thus, a (Λ)LTB universe could be characterized by both types of inhomogeneities or also by only one of them. For example, a homogeneous matter distribution at present time could arise from an inhomogeneous big bang occurring at different times in different spatial points.

In order to solve the generalized Friedmann equation in (2.8), it is necessary to fix the gauge freedom of the generalized scale factor R(t, r). Once more, we can achieve a better intuition about this by drawing an analogy with the FLRW case, where a₀ from equation (2.6) is a normalization constant and can be set to any positive number. The corresponding generalized scale factor at t₀ in (Λ)LTB, R₀(r), can

4We note that, because of the radial dependence of the generalized scale factor, it is also possible to introduce a “radial” Hubble parameter HR(t, r) ≡ ˙R⁰(t, r)/R⁰(t, r). In an inhomogeneous uni- verse, this is in principle different from the “transverse” Hubble parameter entering the generalized Friedmann equation in (2.8) - (2.9), where we have dropped the subscript T for simplicity.

instead be assumed to be any smooth and invertible positive function and, in the following, we adopt the standard choice

R₀(r) ≡ R(t₀(r), r) = r. (2.12) This allows to rewrite the generalized Friedmann equation (2.8) as follows:

H²(t, r)

H₀²(r) = Ω_m,0(r)

 r

R(t, r)

3

+ Ω_k,0(r)

 r

R(t, r)

2

+ Ω_Λ(r), (2.13) with

H(t, r) ≡

R(t, r)˙

R(t, r) =⇒ H₀(t) = R˙0(r)

r , (2.14)

Ω_m,0(r) ≡ M (r)˜

H₀²(r)r³, Ω_k,0(r) ≡ − k(r)

H₀²(r)r² and Ω_Λ(r) ≡ κρ_Λ

3H₀²(r). (2.15)

2.3 Numerical solution

We will now present a numerical approach to solve the generalized Friedmann equa- tion (2.13) for R(t, r), which, as we will discuss below, enters the distance relations for a ΛLTB universe and thus plays a fundamental role in the comparison with obser- vations. We begin by considering equation (2.15), where we notice that we need to determine the four quantities ˜M (r), k(r), ρ_Λ(or Ω_m,0(r), Ω_k,0(r), Ω_Λ(r) respectively) and H₀(r). However, we will now show that it is sufficient to only specify two initial conditions, mirroring the aforementioned two types of inhomogeneities in a ΛLTB universe.

We focus on ˜M (r) first, which only depends on r and is the only function where the physical matter density ρ_m(t, r) is involved. Therefore, the time dependence of ρ_m is unimportant, implying that we can specify an expression for ρ_m at a given t₀(r) as an initial condition:

ρ_m,0(r) ≡ ρ(t₀(r), r). (2.16) We then choose to set t0 to the elapsed time since the Big Bang, tBB, which we assume to be the same at all radii: t₀ ≡ t_BB = const. This allows to rewrite the

definition of ˜M (r) from equation (2.5) as M (r) = κ˜

Z r 0

d˜r R⁰(t, ˜r)R²(t, ˜r)ρ_m(t, ˜r)    t=t0

= κ Z r

0

d˜r R⁰₀(˜r)R²₀(˜r)ρm,0(˜r)

= κ Z r

0

d˜r ˜r²ρm,0(˜r),

(2.17)

where we have employed the gauge choice R₀(r) = r from (2.12) and the fact that R⁰(t, r)

t0= const. = dR₀(r)

dr = 1. (2.18)

Instead of referring to ρm,0(r), however, it is generally more convenient to consider the dimensionless density contrast δ(r), which is defined as

δ(r) ≡ ρ_m,0(r) − ρ^asy_m,0

ρ^asy_m,0 , with ρ^asy_m,0 = 3Ω^asy_m,0(H₀^asy)²

κ , (2.19)

where the label “asy” denotes asymptotic quantities evaluated at r → ∞. The parametrization with δ(r) is particularly suitable when modelling a cosmic void, since it provides an intuitive description where the formulation for δ(r) can be interpreted as the void density profile. With this set-up, the relation for ˜M (r) in (2.17) can be rewritten by inserting

ρm,0(r) = 3Ω^asy_m,0(H₀^asy)²(1 + δ(r))

κ , (2.20)

and, from the definition of Ω_m,0(r) in (2.15), we obtain Ωm,0(r) ≡

M (r)˜

H₀²(r)r³ = 3Ω^asy_m,0(H₀^asy)² H₀²(r) r³

Z r 0

d˜r ˜r²(1 + δ(˜r)). (2.21) We have now reduced the set of quantities to specify to k(r), ρ_Λ and H₀(r), and we notice that equation (2.11) allows to rewrite Ω_k,0(r) in terms of the other two density parameters, such that no additional assumption concerning k(r) is required:

Ω_k,0(r) = 1 − Ω_m,0(r) − Ω_Λ(r). (2.22) Moreover, if we adopt the standard assumption that the space-time is asymptoti- cally flat, i.e. that the spatial curvature vanishes (Ω_k,0 → 0) for r → ∞ , we can

employ equation (2.11) to obtain a relation for the asymptotic values of the density parameters:

Ω^asy_Λ = 1 − Ω^asy_m,0, (2.23) Since

Ω_Λ(r)H₀²(r) = Ω^asy_Λ (H₀^asy)² = κρ_Λ

3 = const., (2.24)

we obtain

Ω_Λ(r) = (1 − Ω^asy_m,0) H₀^asy H₀(r)

2

. (2.25)

To summarize, we have found the set of relations















Ω_m,0(r) = 3Ω^asy_m,0 r³

 H₀^asy H₀(r)

2 Z r 0

d˜r ˜r²(1 + δ(˜r)).

Ω_Λ(r) = (1 − Ω^asy_m,0) H₀^asy H₀(r)

2

Ωk,0(r) = 1 − Ωm,0(r) − ΩΛ(r),

(2.26)

where, for a specified δ(r) and fixed values of Ω^asy_m,0 and H₀^asy, the density parameters are given as functions of H₀(r) only. Therefore, we can now determine the latter by inserting these expressions into the generalized Friedmann equation (2.13) and by numerically solving it for H₀(r). This can be achieved by rewriting (2.13) as

Z R0(r)≡r 0

dR H₀(r) R

q

Ω_m,0(r) _R^r
3

+ Ω_k,0(r) _R^r
2

+ Ω_Λ(r)

=

Z t0≡tBB

0

dt Z r

0

dR

H₀(r)pΩ_m,0(r) r³R⁻¹+ Ω_k,0(r) r²+ Ω_Λ(r) R² = tBB = const.,

(2.27)

where we need to specify the value of t_BB. Since we have assumed that the space- time is asymptotically flat and that t_BB does not vary with r, we can calculate t_BB in the limit r → ∞, where the equations reduce to the FLRW case with zero curvature.

Thus, by employing the standard Friedmann equation (2.6) with Ω_k,0= 0 and a₀ = 1, such that and 0 ≤ a ≤ 1, we obtain

tBB = Z 1

0

da H₀^asyq

Ω^asy_m,0a⁻¹+ (1 − Ω^asy_m,0) a²

. (2.28)

We can now employ the result for H₀(r) from equation (2.27) and the relations for the density parameters at t₀ in (2.26) to numerically solve the generalized Friedmann

equation (2.13) for R(t, r) over a predefined (t, r) grid. This completes the derivation and we note that, as mentioned in the beginning of this section, we have characterized the radial dependence of all ΛLTB quantities by setting initial conditions on two functions only: ρ_m,0(r) (or δ(r)) via (2.16) and t₀(r) via (2.28), corresponding to inhomogeneities in the matter distribution and the expansion rate respectively.

2.4 The distance relations

The procedure described up to this point allows to determine the behaviour of the key cosmological parameters in the ΛLTB formalism and, as we will discuss in Sec- tion 3, we are now interested in comparing these theoretical predictions with ob- servations. This requires to specify the cosmological distance relations for a ΛLTB universe, which can be obtained by relating the propagation of light rays to the inho- mogeneities involved in H₀(r) and ρ_m,0(r). A full derivation for an observer located in the symmetry center⁵ is presented in Section 3 in Enqvist (2008) and we simply summarize the main results here with a slight change of notation.

In the framework of general relativity, light travels along the null geodesics associated with a given line element, i.e. trajectories with ds² = 0. Here, we focus on the ΛLTB space-time described by the line element in equation (2.2) and consider an observer at the origin r = 0, which, as mentioned in Section 2.1, coincides with the symmetry center. Because of spherical symmetry, there must exist null geodesics with dθ = dφ = 0, such that the light rays propagate radially. By imposing these requirements, we obtain a pair of differential equations relating the coordinates t and r to the cosmological redshift z, which measures the increase in wavelength in an expanding space-time:









 dt

dz = − R⁰(t, r) (1 + z) ˙R⁰(t, r) dr

dz = − p1 − k(r) (1 + z) ˙R⁰(t, r).

(2.29)

Here, the minus signs arise from the choice to focus on incoming light rays and we note that these expressions contain a manifest dependence on the inhomogeneities via k(r), which, according to equation (2.15), can be written as

k(r) ≡ −H₀²(r) r²Ω_k,0(r) = −H₀²(r) r²(1 − Ω_m,0(r) − Ω_Λ(r)) . (2.30)

5A derivation for an off-center observer in a LTB universe involving matter and curvature can instead be found in Blomqvist and M¨ortsell (2010).

The null geodesic equations can be numerically solved over a given redshift interval by employing the results for the ΛLTB cosmological parameters and R(t, r) obtained through the procedure discussed in Section 2.3. The output for t(z) and r(z) can then be used to calculate the generalized scale factor as a function of redshift only, corresponding to the angular diameter distance d_A(z) for a ΛLTB universe:

R(z) = R(t(z), r(z)) ≡ d_A(z). (2.31) Moreover, we can obtain the comoving angular diameter distance

DA(z) ≡ (1 + z) dA(z) = (1 + z) R(z), (2.32) and the luminosity distance

d_L(z) ≡ (1 + z)²d_A(z) = (1 + z)²R(z), (2.33) to be compared with, for example, observations of SNe Ia.

3 Numerical investigation

The ΛLTB formalism presented in the previous section provides a suitable framework to model a hypothetical matter underdensity centered on the Milky Way, which could explain the observed increase in the local value of H0. In order to assess whether this is a viable solution to the Hubble tension, we can simultaneously constrain the properties of the local matter distribution and of the background cosmology by nu- merically implementing this set-up and performing a comparison with observations.

This allows to verify whether such a local underdensity indeed exists and, if so, whether it provides a better description to the data than the standard cosmological model, ΛCDM.

In this study, we have approached these questions through the widely used tools of Bayesian parameter inference (Trotta, 2017), allowing to employ a set of observations to “update” the knowledge about a collection of parameters. The computations were performed with a private Python code specifically developed for this thesis and based upon the emcee package (Foreman-Mackey et al., 2013), which implements the affine invariant MCMC ensemble sampling from Goodman and Weare (2010). In the following, we will describe the key phases in the analysis and the set-up for the inference.

3.1 Implementation of the ΛLTB formalism

The initial module of the aforementioned Python script contains a numerical im- plementation of the ΛLTB formalism. First, the generalized Friedmann equation (2.13) is numerically solved over a predefined (t, r) grid according to the procedure outlined in Section 2.3. This yields a solution for the generalized scale factor R(t, r) and the cosmological parameters Ω_m,0(r), Ω_k,0(r), Ω_Λ(r) and H₀(r), which can be numerically interpolated over the grid. The reliability of the results was checked by setting the contribution of the cosmological constant to zero (ρΛ = 0) and by com- paring the numerical output with the analytical expressions that can be obtained in matter-curvature LTB models (see e.g. Section 2 in Blomqvist and M¨ortsell 2010).

The numerical solution for R(t, r) is then used to obtain the time and radial deriva- tives ˙R(t, r) and R⁰(t, r). Together with the interpolated expressions for the cosmo- logical parameters, these allow to solve the pair of differential equations (2.29) for incoming radial null geodesics, yielding the relations t(z) and r(z) over a specified redshift interval. By inserting these into the interpolated solution for R(t, r), we

determine the cosmological distance measures for a ΛLTB universe from equations (2.31) - (2.33), which can be compared with observations. A consistency check of the numerical implementation was performed by verifying that the code output in the absence of an underdensity matched the expected results for a FLRW metric.

The computations described above require to set an ansatz on the void density profile δ(r) entering the set of relations (2.26). A possible choice consists in considering the standard formulations that are employed to fit galaxy data, such as the “universal profile” proposed by Nadathur et al. (2015). However, the shape of these functions is generally controlled by a large set of free parameters and it is difficult to exclude the combinations that lead to nonphysical results. Instead, we perform our main analysis with a generalized Gaussian (GG) ansatz, which provides a simplified but realistic description. Moreover, we also consider the idealized scenario of the so-called Oppenheimer-Snyder (OS) model (Oppenheimer and Snyder, 1939), whose density profile consists in a simple step function. In both cases, based upon the set-up for the null geodesics discussed in Section 2.4, we assume that the observer is located at the center of the void, i.e. at r = 0. We will now briefly discuss the key features and the implementation of both profiles and present some sample plots of the results obtained with a combination of parameter values roughly corresponding to the KBC void.

3.1.1 The generalized Gaussian ansatz

We begin by focusing on the generalized Gaussian set-up, which requires a short preamble. When specifying an ansatz on the physical matter density ρ_m,0(r) or, equivalently, on the density profile δ(r), it is possible to characterize the radial de- pendence of all ΛLTB quantities by calculating the integral in the expression for Ω_m,0(r) from (2.21). In practical applications, however, this integral often becomes computationally expensive and unstable when inserted into the relation (2.27) to numerically obtain H₀(r). Therefore, unless very simple geometries are considered (as in the OS model discussed below), it is generally more convenient to set an ansatz on Ω_m,0(r) directly, as in Hoscheit and Barger (2018), or on the combination Ω_m,0(r) H₀²(r), as in Kenworthy et al. (2019). Such set-ups are widely used in litera- ture and are only valid as long as they are consistently implemented and the physical implications for the void shape are discussed in terms of ρ_m,0(r) or δ(r). As we will discuss below, neither Hoscheit and Barger (2018) nor Kenworthy et al. (2019) are entirely rigorous from this point of view, and we seek to improve on their approach.

Following Kenworthy et al. (2019), we choose to set an ansatz on the variation of