11institutetext: 1 PPGCosmo, Universidade Federal do Espírito Santo,
29075-910,Vitória, ES, Brazil
2 Núcleo Cosmo-ufes & Departamento de Física, Universidade Federal do Espírito
Santo, 29075-910, Vitória, ES, Brazil
3 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, I-34143
Trieste, Italy
4 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151
Trieste, Italy
5 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de
Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, F-31400 Toulouse, France
6 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg
16, 69120 Heidelberg, Germany
7 Université St Joseph; Faculty of Sciences, Beirut, Lebanon
8 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, E-28049 Madrid,
Spain
9 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa,
Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal
10 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, PT-1749-016 Lisboa, Portugal
11 Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay F-91191
Gif-sur-Yvette Cedex, France
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Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
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Catone, Italy
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Porto, Portugal
15 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP,
Rua das Estrelas, PT4150-762 Porto, Portugal
16 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008,
Santiago, Chile
17 School of Physics and Astronomy, Queen Mary University of London, Mile End
Road, London E1 4NS, UK
18 Dipartimento di Fisica, Universitá degli Studi di Torino, Via P. Giuria 1,
I-10125 Torino, Italy
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Torinese (TO), Italy
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University, D-52056 Aachen, Germany
23 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
24 Centre National d’Etudes Spatiales, Toulouse, France
25 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, Astrophysique,
Instrumentation et Modélisation Paris-Saclay, 91191 Gif-sur-Yvette, France
26 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
27 Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capitá 2-4,
08034 Barcelona, Spain
28 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405,
Orsay, France
29 ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo,
28692 Villanueva de la Cañada, Madrid, Spain
30 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, UK
31 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via
Piero Gobetti 93/3, I-40129 Bologna, Italy
32 Dipartimento di Fisica e Astronomia ”Augusto Righi” - Alma Mater Studiorum
Università di Bologna, via Piero Gobetti 93/2, I-40129 Bologna, Italy
33 INFN-Sezione di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy
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di Genova, via Dodecaneso 33, I-16146, Genova, Italy
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Napoli, Italy
37 Mullard Space Science Laboratory, University College London, Holmbury St
Mary, Dorking, Surrey RH5 6NT, UK
38 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
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(Barcelona), Spain
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I-80126, Napoli, Italy
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Universitá di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy
43 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125,
Firenze, Italy
44 Institut national de physique nucléaire et de physique des particules, 3
rue Michel-Ange, 75794 Paris Cédex 16, France
45 Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
46 European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma,
Italy
47 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822,
F-69622, Villeurbanne, France
48 Observatoire de Sauverny, Ecole Polytechnique Fédérale de Lau- sanne,
CH-1290 Versoix, Switzerland
49 Department of Astronomy, University of Geneva, ch. dÉcogia 16, CH-1290
Versoix, Switzerland
50 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
51 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK
52 INFN-Padova, Via Marzolo 8, I-35131 Padova, Italy
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Scienza dello Spazio (OAS), Via Gobetti 93/3, I-40127 Bologna, Italy
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I-40126 Bologna, Italy
55 INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, I-35122
Padova, Italy
56 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1,
D-85748 Garching, Germany
57 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-
Universität München, Scheinerstrasse 1, 81679 München, Germany
58 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, N-0315 Oslo, Norway
59 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Drive, Pasadena, CA, 91109, USA
60 von Hoerner & Sulger GmbH, SchloßPlatz 8, D-68723 Schwetzingen, Germany
61 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
62 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg,
Germany
63 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin
katu 2, 00014 University of Helsinki, Finland
64 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk
4, 7991PD, Dwingeloo, The Netherlands
65 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71,
53121 Bonn, Germany
66 Department of Physics, Institute for Computational Cosmology, Durham
University, South Road, DH1 3LE, UK
67 INFN-Bologna, Via Irnerio 46, I-40126 Bologna, Italy
68 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix,
Switzerland
69 European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The
Netherlands
70 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade
120, DK-8000 Aarhus C, Denmark
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00133 Roma, Italy
72 Institute of Space Science, Bucharest, Ro-077125, Romania
73 Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, E-38204, San
Cristóbal de La Laguna, Tenerife, Spain
74 Departamento de Astrofísica, Universidad de La Laguna, E-38206, La Laguna,
Tenerife, Spain
75 Aix-Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
76 Dipartimento di Fisica e Astronomia ”G.Galilei”, Universitá di Padova, Via
Marzolo 8, I-35131 Padova, Italy
77 Aix-Marseille Univ, CNRS, CNES, LAM, Marseille, France
78 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
79 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Tapada da Ajuda, PT-1349-018 Lisboa, Portugal
80 Universidad Politécnica de Cartagena, Departamento de Electrónica y
Tecnología de Computadoras, 30202 Cartagena, Spain
81 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700
AV Groningen, The Netherlands
82 Infrared Processing and Analysis Center, California Institute of
Technology, Pasadena, CA 91125, USA
83 INAF-Osservatorio Astronomico di Brera, Via Brera 28, I-20122 Milano, Italy
84 Institut d’Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université,
98 bis boulevard Arago, 75014 Paris, France
85 Junia, EPA department, F 59000 Lille, France

#  Euclid: Testing the Copernican principle with next-generation
surveys††thanks: This paper is published on behalf of the Euclid Consortium.

D. Camarena1<EMAIL_ADDRESS>V. Marra2,3,4 Z. Sakr5,6,7 S. Nesseris8 A. Da
Silva9,10 J. Garcia-Bellido8 P. Fleury11 L. Lombriser12 M. Martinelli13 C. J.
A. P. Martins14,15 J. Mimoso9,10 D. Sapone16 C. Clarkson17 S. Camera18,19,20
C. Carbone21 S. Casas22 S. Ilić23,24,5 V. Pettorino25 I. Tutusaus12,26,27,5 N.
Aghanim28 B. Altieri29 A. Amara30 N. Auricchio31 M. Baldi32,31,33 D. Bonino20
E. Branchini34,35 M. Brescia36 J. Brinchmann15 G.P. Candini37 V. Capobianco20
J. Carretero38,39 M. Castellano13 S. Cavuoti36,40,41 A. Cimatti42,43 R.
Cledassou24,44 G. Congedo45 L. Conversi46,29 Y. Copin47 L. Corcione20 F.
Courbin48 M. Cropper37 H. Degaudenzi49 F. Dubath49 C.A.J. Duncan50,51 X.
Dupac29 S. Dusini52 A. Ealet47 S. Farrens25 P. Fosalba26,27 M. Frailis3 E.
Franceschi31 M. Fumana21 B. Garilli21 B. Gillis45 C. Giocoli53,54 A. Grazian55
F. Grupp56,57 S.V.H. Haugan58 W. Holmes59 F. Hormuth60 A. Hornstrup61 K.
Jahnke62 A. Kiessling59 R. Kohley29 M. Kunz12 H. Kurki-Suonio63 P. B. Lilje58
I. Lloro64 O. Mansutti3 O. Marggraf65 F. Marulli32,31,33 R. Massey66 M.
Meneghetti67,31 E. Merlin13 G. Meylan68 M. Moresco32,31 L. Moscardini32,31,33
E. Munari3 S.M. Niemi69 C. Padilla38 S. Paltani49 F. Pasian3 K. Pedersen70 G.
Polenta71 M. Poncet24 L. Popa72 L. Pozzetti31 F. Raison56 R. Rebolo73,74 J.
Rhodes59 G. Riccio36 Hans-Walter Rix62 E. Rossetti32 R. Saglia56,57 B.
Sartoris57,3 A. Secroun75 G. Seidel62 C. Sirignano76,52 G. Sirri33 L. Stanco52
C. Surace77 P. Tallada-Crespí78,39 A.N. Taylor45 I. Tereno9,79 R. Toledo-
Moreo80 F. Torradeflot78,39 E.A. Valentijn81 L. Valenziano31,33 Y. Wang82 G.
Zamorani31 J. Zoubian75 S. Andreon83 D. Di Ferdinando33 V. Scottez84,85 M.
Tenti67

###### Abstract

Context. The Copernican principle, the notion that we are not at a special
location in the Universe, is one of the cornerstones of modern cosmology. Its
violation would invalidate the Friedmann-Lemaître-Robertson-Walker metric,
causing a major change in our understanding of the Universe. Thus, it is of
fundamental importance to perform observational tests of this principle.

Aims. We determine the precision with which future surveys will be able to
test the Copernican principle and their ability to detect any possible
violations.

Methods. We forecast constraints on the inhomogeneous Lemaître-Tolman-Bondi
(LTB) model with a cosmological constant $\Lambda$, basically a cosmological
constant $\Lambda$ and cold dark matter (CDM) model but endowed with a
spherical inhomogeneity. We consider combinations of currently available data
and simulated Euclid data, together with external data products, based on both
$\Lambda$CDM and $\Lambda$LTB fiducial models. These constraints are compared
to the expectations from the Copernican principle.

Results. When considering the $\Lambda$CDM fiducial model, we find that Euclid
data, in combination with other current and forthcoming surveys, will improve
the constraints on the Copernican principle by about $30\%$, with $\pm 10\%$
variations depending on the observables and scales considered. On the other
hand, when considering a $\Lambda$LTB fiducial model, we find that future
Euclid data, combined with other current and forthcoming datasets, will be
able to detect gigaparsec-scale inhomogeneities of contrast $-0.1$.

Conclusions. Next-generation surveys, such as Euclid, will thoroughly test
homogeneity at large scales, tightening the constraints on possible violations
of the Copernican principle.

###### Key Words.:

Cosmology: observations – (Cosmology:) cosmological parameters – Space
vehicles: instruments – Surveys – Methods: statistical – Methods: data
analysis

##  1 Introduction

Modern cosmology relies on several fundamental assumptions, such as the
hypothesis that we do not occupy a special location in the Universe. Thanks to
this assumption, called the Copernican principle, cosmologists have made
tremendous advances in the understanding of the Universe and laid the
foundation of the standard cosmological model. The Copernican principle, along
with the fact that the Universe appears to be statistically isotropic, implies
that our Universe is homogeneous and isotropic on sufficiently large scales,
eliminating any possible spatial dependence in the cosmological parameters.
Equivalently, the space-time is accurately described by the Friedmann-
Lemaître-Robertson-Walker (FLRW) metric. Clearly, any violation of the
Copernican principle indicates a breakdown of the FLRW paradigm and,
therefore, of the standard cosmological model. Thus, testing the Copernican
principle is an essential task in cosmology.

One of the most fundamental tests of the Copernican principle comes from
observations of our motion with respect to the cosmic microwave background
(CMB) rest frame, which induces a kinematic dipole that has already been
observed in the CMB (Aghanim et al., 2014; Akrami et al., 2020; Ferreira &
Quartin, 2021; Saha et al., 2021), the local bulk flow (Colin et al., 2011;
Feindt et al., 2013; Hudson et al., 2004; Carrick et al., 2015), X-ray
clusters (Migkas et al., 2020, 2021), type Ia supernovae (SNe; Mohayaee et
al., 2021; Rahman et al., 2022), high redshift radio sources (Colin et al.,
2017; Bengaly et al., 2018; Siewert et al., 2021), and distant quasars
(Secrest et al., 2021).111See also (Aluri et al., 2022), and references
therein, for a recent review of observational tests of the FLRW paradigm. Many
of these observations have intrinsic systematic errors that have to be taken
into account (Dalang & Bonvin, 2022) in order to avoid theoretical biases.
Another route is to perform null tests of the FLRW metric (see Nesseris et
al., 2022, for recent forecasts) or to estimate the homogeneity scale (Yadav
et al., 2010; Kim et al., 2022).

Additionally, it is also possible to test the Copernican principle by assuming
an inhomogeneous metric, such as that of the Lemaître-Tolman-Bondi (LTB) model
(Garcia-Bellido & Haugboelle, 2008a; February et al., 2010; Valkenburg et al.,
2014; Redlich et al., 2014). In fact, current observations can meaningfully
test the Copernican principle, leading to constraints on deviations from the
FLRW metric at almost the cosmic variance level (Camarena et al., 2021).

In this paper we explore the precision with which next-generation surveys will
probe for violations of the Copernican principle. Specifically, we focus on
Euclid (Laureijs et al., 2011), which is an M-class space mission of the
European Space Agency planned to be launched in 2023. The satellite will carry
two instruments on board – the visible imager (VIS; Cropper et al., 2018) and
the near-infrared spectrophotometric instrument (Prieto et al., 2012;
Maciaszek et al., 2016) – which will carry out a photometric and spectroscopic
galaxy survey covering over $15\,000\,\mathrm{deg}^{2}$ of the sky, with the
aim of measuring the growth of the large-scale structure (LSS) up to a
redshift of $z\sim 2$ (Euclid Collaboration et al., 2022).

Euclid will have two main, complementary cosmological probes, namely galaxy
clustering and weak lensing from the photometric survey and galaxy clustering
from the spectroscopic survey. While photometric surveys image a larger number
of galaxies than spectroscopic ones, they also have larger redshift
uncertainties. On the other hand, spectroscopic galaxy surveys have much
higher radial precision, but target many fewer objects. Euclid has very high
spectroscopic accuracy; it will be able to make very precise measurements of
galaxy clustering that also include the radial dimension, that is to say, it
will be able to probe clustering along the line of sight.In this work we
create mock baryon acoustic oscillation (BAO) data, in accordance with
Euclid’s spectroscopic survey specifications, based on the Fisher matrix
approach of Euclid Collaboration: Blanchard et al. (2020, hereafter EC20)

Furthermore, we also stress some of the possible synergies between Euclid and
other contemporary surveys. The latter include the Legacy Survey of Space and
Time (LSST) performed at the Vera C. Rubin Observatory (Abell et al., 2009)
and that of the Dark Energy Spectroscopic Instrument (DESI; Aghamousa et al.,
2016) since they will be complementary to Euclid in terms of redshift, thus
significantly extending the possible redshift range of our analysis.

Finally, forecast constraints on deviations from the Copernican principle were
presented in Amendola et al. (2018), where a joint analysis between Euclid
(Laureijs et al., 2011) and a stage IV SN mission (Albrecht et al., 2006,
assuming SNAP as a concrete example) was performed. Here we update the
constraints of this analysis by using more recent Euclid specifications (see
EC20, ) while also considering synergies with other surveys.

This paper is organised as follows: In Sect. 2 we briefly review the dynamics
of a spherically inhomogeneous space-time based on the LTB metric but with the
addition of a cosmological constant, $\Lambda$ (i.e. the $\Lambda$LTB model)
and discuss our particular choices for its arbitrary functions. In Sect. 3 we
present the data used in our analysis and explain how mock catalogues are
produced considering particular fiducial cosmologies, while in Sect. 4 we
define and discuss the Copernican prior. Our results are presented and
discussed in Sect. 5 and Sect. 6. We conclude in Sect. 7.

##  2 Spherically symmetric inhomogeneous models with a cosmological constant

A spherically inhomogeneous space-time can be modelled using the $\Lambda$LTB
model, which practically is a standard cosmological constant $\Lambda$ and
cold dark matter (CDM) model endowed with a spherical inhomogeneity. Here, we
aim to test the homogeneity of the Universe, and thus, we neglect anisotropic
degrees of freedom by placing the observer at the centre of the spherical
inhomogeneity. In this section we briefly review the $\Lambda$LTB model
presented in Camarena et al. (2021);222The notation adopted here differs from
the notation used in Camarena et al. (2021). a comprehensive review is given
in Marra et al. (2022).

Hereafter, we use a prime to denote a partial derivative with respect to the
radial coordinate, $r$, while we use a dot to denote a partial derivative with
respect to the time coordinate, $t$.

###  2.1 Dynamics

The LTB metric can be written as

$\mathrm{d}s^{2}=-c^{2}\mathrm{d}t^{2}+\frac{R^{\prime
2}(t,r)}{1-K(r)\,r^{2}}\mathrm{d}r^{2}+R^{2}(r,t)\,\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\phi^{2}\right)\,,$
(1)

where the curvature $K(r)$ is an arbitrary function of the radial coordinate.
The FLRW metric can be recovered by imposing $K\ =\textrm{constant}$ and
$R=a(t)\,r$, with $a(t)$ as the FLRW scale factor. From the line element Eq.
1, we can define the transverse and longitudinal scale factors,
$a_{\perp}=R(r,t)/r$ and $a_{\parallel}=R^{\prime}(r,t)$, respectively. The
two scale factors define two different expansion rates given by

$\displaystyle H_{\perp}(t,r)\equiv\frac{\dot{a}_{\perp}}{a_{\perp}}\,,\qquad
H_{\parallel}(t,r)\equiv\frac{\dot{a}_{\parallel}}{a_{\parallel}}\,.$ (2)

Solving Einstein’s equations with a cosmological constant, we obtain an
equation analogue to the first Friedmann equation,

$\frac{H_{\perp}^{2}}{H_{\perp 0}^{2}}=\Omega_{\rm m,0}\,\left(\frac{a_{\perp
0}}{a_{\perp}}\right)^{3}+\Omega_{k,0}\,\left(\frac{a_{\perp
0}}{a_{\perp}}\right)^{2}+\Omega_{\Lambda,0}\,,$ (3)

where the present-day density parameters are now functions of $r$,

$\displaystyle\Omega_{\Lambda,0}(r)$ $\displaystyle=\frac{\Lambda
c^{2}}{3H_{\perp 0}^{2}}\,,$ (4) $\displaystyle\Omega_{k,0}(r)$
$\displaystyle=-\frac{K(r)\>c^{2}}{H_{\perp 0}^{2}\,a_{\perp 0}^{2}}\,,$ (5)
$\displaystyle\Omega_{\rm m,0}(r)$ $\displaystyle=\frac{2\,m(r)}{H_{\perp
0}^{2}\,a_{\perp 0}^{3}\,r^{3}}\,,$ (6)

which satisfy $\Omega_{\rm m,0}(r)+\Omega_{k,0}(r)+\Omega_{\Lambda,0}(r)=1$.
It should be noted that we have defined $H_{\perp 0}\equiv H_{\perp}(t_{0},r)$
and $a_{\perp 0}\equiv a_{\perp}(t_{0},r)$.

From Eq. 6 we can see that Einstein’s equations introduce another arbitrary
function: the Euclidean mass $m(r)$. This function arises as a constant of
integration and is defined via

$m(r)=\int_{0}^{r}\mathrm{d}r^{\prime}\>4\pi G\,\rho_{\rm
m}(t,r^{\prime})\,a_{\parallel}\,a_{\perp}^{2}r^{\prime 2}\,,$ (7)

where $\rho_{\rm m}(t,r)$ is the local matter density.

Another arbitrary function of the $\Lambda$LTB model is the Big Bang function,
$t_{\rm BB}(r)$. This can be interpreted as the time corresponding to the Big
Bang singularity surface, and it emerges from integration of Eq. 3:

$t-t_{\rm BB}(r)=\frac{1}{H_{\perp
0}(r)}\int_{0}^{\chi}\frac{\mathrm{d}x}{\sqrt{\Omega_{\rm
m,0}(r)x^{-1}+\Omega_{k,0}(r)+\Omega_{\Lambda,0}(r)x^{2}}}\,,$ (8)

where $\chi=a_{\perp}(t,r)/a_{\perp 0}$. We note that the last integral
defines the age of the Universe, $t_{0}$, when integrated from zero to one.

Finally, from the line element Eq. 1 it follows that the geodesic equations
are

$\displaystyle\frac{\mathrm{d}t}{\mathrm{d}z}=-\frac{1}{(1+z)\,H_{\parallel}(t,r)}\,,\qquad\frac{\mathrm{d}r}{\mathrm{d}z}=-\frac{c\sqrt{1-K(r)r^{2}}}{(1+z)\,a_{\parallel}(t,r)\,H_{\parallel}(t,r)}\,.$
(9)

###  2.2 Free functions

As shown above, the $\Lambda$LTB model has three arbitrary functions: $K(r)$,
$m(r)$, and $t_{\rm BB}(r)$. One of these functions is just a gauge freedom
that can be fixed by re-scaling the radial coordinate. Here, we fix the radial
coordinate $r$ such that the mass function satisfies $m(r)\propto r^{3}$; we
will later present the missing normalisation of the Euclidean mass. On the
other hand, a Big Bang function different from zero introduces decaying modes
into the matter density (Silk, 1977). The presence of these modes leads to a
disagreement with the standard scenario of inflation (Zibin, 2008). Thus, to
ensure the absence of decaying modes on the matter density we set $t_{\rm
BB}(r)=0$, which also implies that the Big Bang singularity happens everywhere
simultaneously.

Hence, we end up with just one arbitrary function, the curvature profile
$K(r)$. Here, we adopt the monotonic compensated profile given by

$K(r)=K_{\rm B}+(K_{\rm C}-K_{\rm B})\,P_{3}(r/r_{\rm B})\,,$ (10)

where $r_{\rm B}$ is the comoving radius of the spherical inhomogeneity,
$K_{\rm B}$ is the background curvature, $K_{\rm C}$ is the central curvature,
and the function $P_{3}$ follows,

$\displaystyle
P_{3}(x)=\left\\{\begin{array}[]{ll}1-\exp\big{[}-(1-x)^{3}/x]&\mbox{ for
}0\leq x<1\\\ 0&\mbox{ for }1\leq x\end{array}\right.\,,$ (13)

with $x=r/r_{\rm B}$. Thus, Eq. 10 ensures a smooth transition between the LTB
and FLRW metrics; the $\Lambda$LTB model asymptotes to the $\Lambda$CDM model
at scales $r\geq r_{\rm B}$.

We note that while we have fixed $m(r)$ and $t_{\rm BB}(r)$ using physical
arguments, our choice of $K(r)$ remains arbitrary and could have a significant
impact on our analysis. In Appendix A we discuss our choice and compare it
with another $\Lambda$LTB model. We also present an extra analysis performed
using an extension of Eq. 10. This further analysis shows that a more
generalised curvature profile could weaken the constraints by up to a factor
of $\sim 2.$

Once the three arbitrary functions are fixed, we can compute $\rho_{\rm
m}(r,t)$ in order to determine the matter density contrast using

$\delta(r,t):=\frac{\rho_{\rm m}(r,t)}{\rho_{\rm m}(r_{\rm B},t)}-1\,,$ (14)

and we can also compute the mass (integrated) density contrast via

$\displaystyle\Delta(r,t_{0})$
$\displaystyle=\frac{4\pi\int_{0}^{r}\mathrm{d}\bar{r}\,\delta(\bar{r},t_{0})\,a_{\perp}^{2}(\bar{r},t_{0})a_{\parallel}(\bar{r},t_{0})r^{2}}{4\pi\,a_{\perp}^{3}(r,t_{0})\,r^{3}/3}$
(15) $\displaystyle=\frac{m(r)}{4\pi G\,R^{3}(r,t_{0})/3\,\rho_{\rm
m}^{\rm{out}}(t_{0})}-1=\frac{\Omega_{\rm m,0}(r)}{\Omega_{\rm
m,0}^{\rm{out}}}\left[\frac{H_{\perp 0}(r)}{H_{\perp
0}^{\rm{out}}}\right]^{2}-1\,.$

Here we use the superscript ‘out’ to denote the FLRW background quantities
outside the inhomogeneity, for example $\rho_{\rm m}(r_{\rm
B},t_{0})=\rho_{\rm m}^{\rm{out}}(t_{0})$. We additionally make use of the
FLRW comoving coordinate at the present time, which is defined as

$r^{\rm out}:=r\>a_{\perp 0}(r)/a^{\rm out}(t_{0})\,.$ (16)

The top panel in Fig. 1 shows the matter and integrated mass density contrast
as a function of $r^{\rm out}$ at $t=t_{0}$ for a deep and gigaparsec-scale
void. Also displayed in the figure are $\Omega_{\rm
m,0}(r^{\mathrm{out}})-\Omega^{\mathrm{out}}_{\rm m,0}$ and
$\Omega_{k,0}(r^{\mathrm{out}})-\Omega^{\mathrm{out}}_{k,0}$, giving the
deviations of the matter and curvature densities with respect to their
$\Lambda$CDM counterparts (middle panel), along with the deviations of the
transverse and longitudinal expansion rates with respect to $H_{0}^{\rm out}$
(bottom panel).

It is important to highlight that the assumption of the profile Eq. 10
implicitly introduces a compensating scale, here denoted by $r^{\rm out}_{\rm
L}[\delta(r^{\rm out}_{\rm L},t)=0]$, at which the central overdense or
underdense region makes a transition to the surrounding mass-compensating
underdense or overdense region. The $r^{\rm out}_{\rm L}$ and the $r^{\rm
out}_{\rm B}$ scales are represented in Fig. 1 by the dotted vertical lines.
Furthermore, one can note that at the centre of the inhomogeneity
$\Delta(0,t)=\delta(0,t)$, while at the boundary shell we have $\Delta(r_{\rm
B},t)=\delta(r_{\rm B},t)=0$. Since the LTB and FLRW metrics perfectly match
at the boundary shell, then we have that $r^{\rm out}_{\rm B}=r_{\rm B}$.
Finally, from Eq. 6 is possible to determine the missing normalisation of the
Euclidean mass; specifically we find $m(r)=\Omega_{\rm
m,0}^{\rm{out}}\,(H_{0}^{\rm{out}})^{2}\,r^{3}/2$, while Eq. 5 leads to
$K_{\rm B}=-\Omega_{k,0}^{\rm{out}}\,(H_{0}^{\rm{out}})^{2}$.

Figure 1: Several $\Lambda$LTB quantities as a function of the FLRW comoving
coordinate $r^{\rm{out}}$. Top: Density contrast of matter,
$\delta(r)=\rho_{\rm m}(r,t_{0})/\rho_{\rm m}(r_{\rm B},t_{0})-1$, and
integrated mass, $\Delta(r)$, as functions of the FLRW comoving coordinate. At
radius $r_{\rm L}^{\rm out}$ the central structure ends and the compensating
shell begins, while $r_{\rm B}^{\rm out}$ is the boundary of the spherical
inhomogeneity. Middle: Deviations from the background density of matter,
$\smash{\Omega^{\mathrm{out}}_{\rm m,0}}$, and curvature,
$\smash{\Omega^{\mathrm{out}}_{k,0}}$, as a function of the FLRW comoving
coordinate due to the radial dependence on $\Omega_{\rm m,0}(r)$ and
$\Omega_{k,0}(r)$. We note that the FLRW background quantities are recovered
for $r\geq r^{\mathrm{out}}_{\rm B}$. Bottom: Transverse and longitudinal
fluctuations in the expansion rates as a function of the FLRW comoving
coordinate.

###  2.3 Parameter space

We used the monteLLTB code to solve the dynamical equations and then to sample
the parameter space of the $\Lambda$LTB
model.333https://github.com/davidcato/monteLLTB The monteLLTB code combines
montepython (Audren et al., 2013; Brinckmann & Lesgourgues, 2019) for the
Markov chain Monte Carlo parameter space exploration and likelihoods, class
(Blas et al., 2011) for the CMB computation and voiddistances2020 (Valkenburg,
2012) for the $\Lambda$LTB metric functions via a wrapper that translates the
montepython trial vector into an effective FLRW vector that is suitable for
class (see Camarena et al. 2021 for details).

Since the LTB metric asymptotes to FLRW at $r\geq r_{\rm B}$, the background
expansion of our model is specified by the standard six $\Lambda$CDM
parameters: the normalised Hubble constant $h:=H_{0}/100$, the baryon density
$\Omega_{\mathrm{b,0}}$, the cold dark matter density
$\Omega_{\mathrm{cdm,0}}$, the optical depth $\tau$, the amplitude of the
power spectrum $A_{\rm s}$ and its tilt $n_{\rm s}$.

On the other hand, the spherically inhomogeneous region is fixed by the
boundary redshift, $z_{\rm B}$, and the mass density contrast at the centre of
the inhomogeneity, $\Delta_{\rm C}\equiv\Delta(0,t)$. The parameters $z_{\rm
B}$ and $\Delta_{\rm C}$ are related to the parameters $r_{\rm B}$ and $K_{\rm
C}$, which explicitly appear in the definition of the curvature profile. As
discussed in Sect. 4, we conveniently present our results using the
compensating scale $r_{\rm L}^{\rm out}$ and the mass contrast at such a
scale, $\Delta_{\rm L}\equiv\Delta(r_{\rm L},t_{0})$.

We note that for $\Delta_{\rm C}\,\approx\,0$, the parameter space is highly
degenerate so that the model could feature arbitrarily large values of $z_{\rm
B}$ and be still allowed by the data. In order to overcome this issue, we
adopt, as in Camarena et al. (2021), the flat prior $z_{\rm B}\in[0,0.5]$.
This prior allows us to map non-Copernican structures of up to $r_{\rm L}^{\rm
out}\sim 1$ Gpc.

##  3 Data

Here we present both the forecast and current data used to constrain the
$\Lambda$LTB model. Although our goal is to forecast constraints on the
Copernican principle given the forthcoming surveys, the inclusion of current
data is needed to tightly constrain the $\Lambda$LTB parameter space at small
scales.

Forecasted datasets are generated considering four fiducial models, which are
based on the $\Lambda$CDM and $\Lambda$LTB models (see Table 1). By
considering the $\Lambda$CDM forecast data we aim to determine the precision
with which next-generation surveys will be able to probe for deviations from
the FLRW metric. Meanwhile, by considering the inhomogeneous $\Lambda$LTB
fiducial model, we aim to investigate the ability of future surveys to detect
a violation of the Copernican principle. In order to assume a consistent
fiducial model, current data have been re-scaled to agree with the
aforementioned fiducial models. Such a re-scaling is performed following the
procedure described in Appendix B. Furthermore, we also assume that there are
no tensions among the different datasets; this includes tensions between early
and late determinations. The theoretical predictions for the observables
implemented here follow the equations discussed in Sect. 3 of Camarena et al.
(2021). We note that, in order to fully calibrate the SN distances, we also
assume a fiducial value for the absolute magnitude of SN data, that is,
$M_{0}=-19.3$.

###  3.1 Forecast data

In order to forecast how Euclid and other forthcoming surveys will constrain
deviations from the Copernican principle, we created mock SN and BAO data; the
former provide information on the luminosity distance, while the latter
concern the Hubble parameter and the angular diameter distance. In particular,
the recipes described here were used for the $\Lambda$CDM catalogues, while
the $\Lambda$LTB catalogues are obtained by suitably re-scaling the former
(see Appendix B).

Table 1: Parameter values for the fiducial models that are used for the mocks. The values used for the $\Lambda$CDM follow the fiducial of EC20; in particular, spatial flatness is assumed. The Hubble constant $H_{0}$ is shown in units of km s-1 Mpc-1, while the absolute magnitude of the SN $M_{0}$ is shown in units of mag.  model | $M_{0}$ | $\Omega_{\rm m,0}$ | $\Omega_{\rm b,0}h^{2}$ | $H_{0}$ | $\Delta_{\rm C}$ | $z_{\rm B}$
---|---|---|---|---|---|---
$\Lambda$CDM | $-19.3$ | $0.32$ | $0.02225$ | $67$ | - | -
$\Lambda$LTB $1$ | $-19.3$ | $0.32$ | $0.02225$ | $67$ | $-0.5$ | $0.05$
$\Lambda$LTB $2$ | $-19.3$ | $0.32$ | $0.02225$ | $67$ | $-0.1$ | $0.4$
$\Lambda$LTB $3$ | $-19.3$ | $0.32$ | $0.02225$ | $67$ | $-0.1$ | $0.8$

The four fiducial cosmologies based on the $\Lambda$CDM and the $\Lambda$LTB
model that we consider here are shown in Table 1; the former was also used in
EC20. To make the mocks for the $\Lambda$CDM model, we calculated the redshift
evolution of the Hubble parameter, along with the luminosity and angular
diameter distances, using the recipe described in the next section, which is
based on the specifications of Euclid and other LSS surveys. On the other
hand, as mentioned before, the $\Lambda$LTB mock catalogues are obtained
following the process described in Appendix B. Since computing correlation
matrices for models far from the $\Lambda$CDM model is currently not possible
(Harnois-Deraps et al., 2019; Friedrich et al., 2021; Ferreira & Marra, 2022),
we first compute the correlation matrix assuming the $\Lambda$CDM model. Then,
we apply the method described in Appendix B to obtain the corresponding
$\Lambda$LTB matrices.

Since future surveys like Euclid are expected to provide observations with
high precision, it is important to be convinced that our analysis methods will
be robust. Hence, in order to understand and take possible observational
systematic uncertainties that can affect the measurements into account,
several analyses, such as that of Euclid Collaboration: Paykari et al. (2020),
have been performed. In the latter, the observational systematic effects of
the Euclid VIS instrument were studied, taking the modelling of the point
spread function and the charge transfer inefficiency into account. Since these
systematic effects are expected to be better understood by the time the data
arrive, in this analysis we assume that they will be under control in the
final data products. In any case, in what follows we in fact include several
astrophysical systematic effects, such as the galaxy bias, as discussed in
what follows.

####  3.1.1 SN surveys

In our analysis we focus on two forthcoming SN surveys, the first of which is
based on the proposed Euclid DESIRE survey (Laureijs et al., 2011; Astier et
al., 2014), while the second one is based on the specifications of the LSST.
In particular, we assume that the Euclid DESIRE survey will observe $1700$ SNe
in the redshift range $z\in[0.7,1.6]$, while the one from the LSST survey will
observe $8800$ SNe in the redshift range $z\in[0.1,1.0]$, thus resulting in a
total of $10\,500$ points.

In either case we consider the redshift distributions of the SN events as
described in Astier et al. (2014), assuming the points are not correlated with
each other. Even though the Euclid SN survey is not currently guaranteed to
take place, we decided to include it in order to extend the redshift range of
LSST at high $z$. For the SN mocks we include an observational error of the
form
$\sigma_{\textrm{tot},i}^{2}=\delta\mu^{2}_{i}+\sigma^{2}_{\textrm{flux}}+\sigma^{2}_{\textrm{scat}}+\sigma^{2}_{\textrm{intr}}$,
where the terms corresponding to the intrinsic contributions, the scatter and
the flux are the same for all events: $\sigma_{\textrm{intr}}=0.12$,
$\sigma_{\textrm{scat}}=0.025$, and $\sigma_{\textrm{flux}}=0.01,$
respectively. Finally, we also include an error on the distance modulus
$\mu=m-M_{0}$ that scales linearly with $z$ as $\delta\mu=e_{\rm
M}\leavevmode\nobreak\ z$, where $e_{\rm M}$ follows a Gaussian distribution
with zero mean and standard deviation $\sigma(e_{\rm M})=0.01$ (see Gong et
al., 2010; Astier et al., 2014), which includes the possible redshift
evolution of SNe not taken into account by the distance estimator (see Astier
et al. 2014). However, while a value of $e_{\rm M}=0.01$ is required to take a
possible systematic evolution into account, this would be added quadratically
to an effective term of $e_{\rm M}=0.055$ arising from SN lensing. The latter
has been theoretically calculated by several authors to be of the order of
$\sigma_{\rm lens}\simeq 0.055\,z$, for example $\sigma_{\rm lens}=0.052\,z$
(Marra et al., 2013; Quartin et al., 2014), and $\sigma_{\rm lens}=0.056\,z$
(Ben-Dayan et al., 2013), while observationally it was determined, via the
Supernova Legacy Survey to be $\sigma_{\rm lens}=(0.055\pm 0.04)\,z$ (Jonsson
et al., 2010) and $\sigma_{\rm lens}=(0.054\pm 0.024)\,z$ (Kronborg et al.,
2010).

####  3.1.2 Local prior on the Hubble constant

We also forecast a $1\%$ measurement of the Hubble constant, which is the
grand goal of the SH0ES collaboration,

$\displaystyle H_{0}$ $\displaystyle=\begin{cases}67.00\pm
0.67\,$\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{Mpc}}^{-1}$&\textrm{for}\
\Lambda\textrm{CDM}\\\\[2.84526pt] 67.62\pm
0.68\,$\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{Mpc}}^{-1}$&\textrm{for}\
\Lambda\textrm{LTB}\leavevmode\nobreak\ 1\\\\[2.84526pt] 68.22\pm
0.68\,$\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{Mpc}}^{-1}$&\textrm{for}\
\Lambda\textrm{LTB}\leavevmode\nobreak\ 2\\\\[2.84526pt] 68.45\pm
0.68\,$\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{Mpc}}^{-1}$&\textrm{for}\
\Lambda\textrm{LTB}\leavevmode\nobreak\ 3\end{cases},$ (17)

where the central value is given by the fiducial $H_{0}$ value for the
$\Lambda$CDM fiducial model, meanwhile for the $\Lambda$LTB models this
central value is the expected value given the methodology described in
Appendix C. Here, as mentioned earlier, we consider a scenario in which there
is no tension between early and late determinations of the Hubble constant. By
assuming a single consistent fiducial model, we focus on the constraining
potential of future surveys to test the Copernican principle, leaving the
issue of the Hubble tension to other studies. This is in part justified since
Camarena et al. (2021, 2022) shown that a large inhomogeneity cannot explain
away the Hubble tension. Finally, we impose the Gaussian prior of Eq. 17 on
$H_{0}^{\rm L}$; the corresponding Hubble constant value for an inhomogeneous
model (see Appendix C for a detailed discussion).

####  3.1.3 Large-scale structure surveys

Here, we now briefly describe our procedure for creating mock BAO data based
on the specifications of Euclid via a Fisher matrix approach, following the
methodology of EC20 for the spectroscopic survey, on which we focus since we
are interested in obtaining precise measurements of the angular diameter
distance $D_{\mathrm{A}}(z)$ and the Hubble parameter $H(z)$. We do not
consider weak lensing by Euclid, nor other perturbation level observables such
as redshift space distortions, because there is not yet a fully developed
linear perturbation theory on inhomogeneous backgrounds such as the LTB.
444See, however, Moss et al. (2011); Ishak et al. (2013) for a comparison with
observations. A discussion and the numerical simulation of the LSS on an LTB
background is provided in Marra et al. (2022) and references therein.

As was extensively discussed in EC20, the main targets of the Euclid survey
will be emission line galaxies (ELGs), which are bright emitters in specific
lines, such as $H_{\alpha}$ and [O III], that can be seen in the redshift
range $z\in[0.9,1.8]$, and can be used to measure the galaxy power spectrum.
In particular, Euclid will determine approximately $30$ million spectroscopic
redshifts with an uncertainty of $\sigma_{z}=0.001(1+z)$ (Pozzetti et al.,
2016), which will provide the galaxy power spectrum with information on the
distortions due to the redshift uncertainty, the residual shot noise, the
Alcock-Paczynski effect, the redshift space distortions and the galaxy bias.
Furthermore, non-linear effects, such as a non-linear smearing of the BAO
feature or a non-linear scale-dependent galaxy bias that distorts the shape of
the power spectrum, have also been taken into account (see Wang et al. 2013
and de la Torre & Guzzo 2012, respectively).

In this work we again make use of the same binning scheme as in Martinelli et
al. (2020, 2021), which differs from that of EC20. In particular, instead of
four equally spaced redshift bins, we now consider nine bins of width $\Delta
z=0.1$. After re-binning the data provided in EC20, we obtain the following
specifications for the galaxy number density $n(z)$, given in units of
$\mathrm{Mpc}^{-3}$, and that of the galaxy bias $b(z)$:

$\displaystyle n(z)$
$\displaystyle\\!\\!=\\!\\!\\{2.04,2.08,1.78,1.58,1.39,1.15,0.97,0.7,0.6\\}\\!\times\\!10^{-4}\\!\\!,$
(18) $\displaystyle b(z)$
$\displaystyle\\!=\\!\\{1.42,1.5,1.57,1.64,1.71,1.78,1.84,1.90,1.96\\}\,.$
(19)

In Martinelli et al. (2020) we tested our choice for the binning scheme
against that of EC20, and we found the results were in agreement.

In the case of the $\Lambda$CDM mocks, the Fisher matrix for the cosmological
parameters, along with the associated covariance matrix, can be derived by
following the methodology described in EC20. The cosmological parameters we
consider for the $\Lambda$CDM mocks include the background quantities
$\\{\omega_{\rm m}=\Omega_{\rm m,0}h^{2}$, $h$, $\omega_{\rm b}=\Omega_{\rm
b,0}h^{2}$, $n_{\rm s}\\}$, two non-linear parameters $\\{\sigma_{\rm
p},\,\sigma_{\rm v}\\}$ (see EC20) and the five redshift-dependent parameters
$\\{\ln D_{\rm A},\,\ln H,\,\ln f\sigma_{8},\,\ln b\sigma_{8},\,P_{\rm s}\\}$,
which are estimated in every redshift bin. Here we have defined
$f\sigma_{8}\equiv f(z)\sigma_{8}(z)$ as the linear growth rate multiplied by
$\sigma_{8}$, which corresponds to the RMS fluctuations in the matter mass
density in a comoving sphere of $8\,h^{-1}\,\mathrm{Mpc}$, while
$b\sigma_{8}\equiv b(z)\sigma_{8}(z)$ and $P_{\rm s}$ are the galaxy bias and
the shot noise, respectively (see EC20). From this we can then estimate the
expected uncertainty of the measurements of the Euclid survey for both the
angular diameter distance $D_{\rm A}(z)$ and the Hubble parameter $H(z)$, in
every redshift bin, while all other parameters are marginalised over.
Furthermore, we apply the approach presented in Appendix B to obtain the
corresponding $\Lambda$LTB mock data.

Since the spectroscopic survey of Euclid will only cover the redshift range
$z\in[0.9,1.8]$, this limits the range where SN and BAO data will be obtained.
Hence, in order to cover smaller redshifts we complement our analysis by using
fiducial data products from the DESI survey as well. DESI has already
initiated survey operations in 2021 and will eventually obtain spectra for
tens of millions of galaxies and quasars up to $z\sim 4$, thus making
redshift-space distortion and BAO analyses possible. To create DESI mocks,
assuming the $\Lambda$CDM model, we follow the methodology for both the
angular diameter distance $D_{\rm A}(z)$ and the Hubble parameter $H(z)$, as
described in Aghamousa et al. (2016). These Fisher matrix forecasts were also
derived using the full anisotropic galaxy power spectrum (i.e. measurements of
the matter power spectrum as a function of the angle with respect to the line
of sight), as described in Font-Ribera et al. (2014). This approach is similar
to that of the Euclid forecasts and it also includes all information from the
two-point correlation function. In particular, the baseline DESI survey will
cover approximately $14\,000\,\mathrm{deg}^{2}$ and will target ELGs, luminous
red galaxies, bright galaxies, and quasars, all in the redshift range
$z\in[0.05,3.55]$, although the precision of the measurements will depend on
the target population. Regarding the specific populations, the bright galaxies
will be in the range $z\in[0.05,0.45]$ in five equally spaced redshift bins,
while the ELGs and the luminous red galaxies will be in the range
$z\in[0.65,1.85]$ in $13$ equally spaced bins. Finally, the Ly-$\alpha$ forest
quasars will be in the range $z\in[1.96,3.55]$ in $11$ equally spaced bins and
we assume that the points are uncorrelated with each other.

In the case when we used the combination of Euclid and DESI data together, in
order to avoid overlap between the two surveys at late times, we only
considered the DESI points that do not overlap with those of Euclid, because
an overlap will lead to undesired correlations between the surveys. Moreover,
since the DESIRE + LSST SN points will only reach at most $z=1.6$, we included
the DESI data up to $z=0.9$, thus omitting the Ly-$\alpha$ forest
observations. However, when used separately we considered the full redshift
range of the datasets.

###  3.2 Current data

As shown in Camarena et al. (2021), CMB and SN data are necessary in order to
obtain tight constraints on the $\Lambda$LTB model. For our particular case,
this means that the presence of real data (i.e. Planck 2018 and Pantheon SNe)
is needed even though our analysis aims to forecast the contribution of
forthcoming surveys. The inclusion of CMB data is crucial to constrain the
background parameters, while the usage of low-$z$ SNe allows us to break the
degeneracy of the $\Lambda$LTB parameters model at small scales. As discussed
at the beginning of the present Sect., we rescale current data according to
the predictions of the fiducial models shown in Table 1 and following the
procedure described in Appendix B.

####  3.2.1 Cosmic microwave background

When the $\Lambda$CDM forecast data are considered, we perform our analysis
including the latest Planck CMB data555http://www.esa.int/Planck (Aghanim et
al., 2020b). We use the high-$\ell$ TT+TE+EE, low-$\ell$ TT, and low-$\ell$ EE
likelihoods. Particularly, we use the compressed version of high-$\ell$ data,
that is, the likelihood normalised over all nuisance parameters except
$A_{\mathrm{planck}}$. We note that typical constraints obtained for
$\Lambda$CDM using these likelihoods include the fiducial values adopted for
the forecast data (Table 1) within $68\%$ uncertainties, allowing the
combination of CMB and forecast data without the necessity of applying the re-
scaling technique.

On the other hand, the $\Lambda$LTB cosmologies presented on Table 1 could
significantly change the CMB power spectra and lead to disagreements of these
with the constraints of the aforementioned likelihoods. Thus, one should
change the Planck CMB data according to the $\Lambda$LTB fiducial cosmologies.
This is not a trivial task given the complex structure of the CMB likelihoods
and our limited understanding of perturbations on the inhomogeneous models.
Thus, for our analyses of the $\Lambda$LTB mock data we use the CMB distance
priors on the shift parameter $R$, the acoustic scale $l_{A}$, the amount of
baryons $\Omega_{\rm b}h^{2}$, and the tilt of the power spectrum $n_{\rm s}$.
We build the mock CMB priors considering the current measurements given by
Chen et al. (2019).

####  3.2.2 SN surveys

The lack of SN data at very low redshifts $z\sim 0.01$ – the lowest LSST point
lies at $z=0.1$ – increases the degeneracy between $\Delta_{\rm C}$ and
$z_{\rm B}$, loosening the constraints on the $\Lambda$LTB model. To overcome
this issue, we include the Pantheon SN compilation (Scolnic et al., 2018).

####  3.2.3 Large-scale structure surveys

We also include BAO data from 6dFGS (Beutler et al., 2011), SDSS-MGS (Ross et
al., 2015) and BOSS-DR12 (Alam et al., 2017) surveys. The isotropic
measurements from 6dFGS and SDSS-MGS allow us to access redshifts 0.1 and
0.15, respectively, while BOSS provides anisotropic measurements at redshifts
0.38, 0.51, and 0.61. We note that these current data overlap with our
forecast DESI catalogues, but we assume no correlations between these
datasets. Hereafter we collectively refer to this set of data as BAOs. We note
that our analysis does not include the latest eBOSS data (Ross et al., 2020;
Raichoor et al., 2020; Lyke et al., 2020; du Mas des Bourboux et al., 2020)
chiefly because the eBOSS dataset spans over all the redshift range of our
forecast Euclid data.

####  3.2.4 $y$-Compton distortion and the kSZ effect

Finally, when forecast data from $\Lambda$CDM were analysed, we introduced
priors on the $y$-Compton distortion and the kinetic Sunyaev-Zeldovich (kSZ)
effect. For the $y$-Compton distortion, we adopted the upper limit prior at
$95.4\%$ uncertainty provided by COBE-FIRAS $y<1.5\times 10^{-5}$ (Fixsen et
al., 1996). Meanwhile, for the kSZ effect we adopted the $\sim 47\%$
constraint from SPT-SZ and SPTpol surveys (Reichardt et al., 2021).
Considering the $\Lambda$CDM fiducial, we implemented the Gaussian prior on
the kSZ amplitude as $D_{3000}=$3.49\pm
1.63\text{\,}\mathrm{\SIUnitSymbolMicro K}$$.

Priors on the $y$-Compton distortion and the kSZ effect were not used for our
analysis of the $\Lambda$LTB forecast data since they do not improve upon
constraints given by the combinations of the other datasets.

##  4 Copernican prior

Figure 2: $95\%$ and $99\%$ confidence level constraints on the integrated
mass contrast, $\Delta_{\rm L}$, and the comoving size, $r^{\textrm{out}}_{\rm
L}$, for three different data combinations as compared to the constraints from
the Copernican prior convolved with the CMB likelihoods.

In the absence of the Copernican principle, the LSS of the Universe may
feature arbitrary radial inhomogeneities. In an FLRW model, instead, those
structures are constrained by the Copernican principle. Such constraints can
be obtained through linear perturbation theory. Assuming that the density
contrast $\Delta(r)$ is a Gaussian field, we can compute its RMS by

$\sigma^{2}(r)=\int_{0}^{\infty}\frac{\mathrm{d}k}{k}\>\left[\frac{k^{3}P_{\rm
m,0}(k)}{2\pi}\frac{3j_{1}(rk)}{rk}\right]^{2}\,,$ (20)

where $P_{\rm m,0}(k)$ is the standard power spectrum today and $j_{1}$ is the
spherical Bessel function of the first kind. The aforementioned quantities can
be used to define a prior that establishes the probability of finding an
inhomogeneous deviation from the FLRW at a given scale. Such a prior is the
so-called Copernican prior and can be used to constrain $\Delta_{\rm C}$ and
$z_{\rm B}$ through (Camarena et al., 2021)

$\mathcal{P}(\Delta_{\rm C},z_{\rm
B})\propto\exp{\left[-\frac{1}{2}\frac{\Delta^{2}(r_{\rm
L},t_{0})}{\sigma^{2}(r_{\rm L}^{\rm out})}\right]}\,,$ (21)

where $\Delta(r,t_{0})$ is given by Eq. 15, $r_{\rm L}(\Delta_{\rm C},z_{\rm
B})$ is the radius of the central under/overdensity, and $r_{\rm
L}^{\mathrm{out}}(\Delta_{\rm C},z_{\rm B})$ is the latter radius in the FLRW
comoving coordinates of Eq. 16. We note that $r_{\rm
L}^{\mathrm{out}}(\Delta_{\rm C},z_{\rm B})$ is the scale of interest since it
defines the size of the central under/overdensity. Additionally, by definition
the Copernican prior vanishes at the matching shell, $r_{\rm
B}^{\mathrm{out}}$, since the matter and mass fluctuations disappear. We
present our results using the radius $r_{\rm L}^{\mathrm{out}}$ and the mass
contrast $\Delta_{\rm L}\equiv\Delta(r_{\rm L},t_{0})$.

Despite the fact that Eq. 21 can constrain the deviations from the FLRW model
by constraining $\Delta_{\rm C}$ and $z_{\rm B}$, this prior does not
constrain the cosmological parameters needed to assess the information
contained in perturbations, for instance $P_{\rm m,0}$. On the other hand, CMB
observations should describe the early Universe at any point and, in
particular, also at our observing position if the Copernican principle is
valid. That is, under the assumption of the Copernican prior, CMB information
such as the power spectrum should constrain $\Delta_{\rm C}$ and $z_{\rm B}$
(and the background cosmological parameters).

We then compared the cosmological constraints on $\Lambda$LTB with the ones
from the Copernican prior convolved with the CMB likelihood to obtain $P$, the
probability distribution of $\Delta_{\rm C}$ and $z_{\rm B}$, given the
initial conditions obtained from the CMB and their uncertainty, which, under
the Copernican principle, describe matter perturbations around us:

$P(\Delta_{\rm C},z_{\rm B})=\int\mathrm{d}p_{i}\,\mathcal{P}(\Delta_{\rm
C},z_{\rm B})\,\mathcal{L}_{\rm CMB}(p_{i},\Delta_{\rm C},z_{\rm B})\,,$ (22)

where $p_{i}$ denotes the standard $\Lambda$CDM parameters and
$\mathcal{L}_{\rm CMB}$ is the CMB likelihood of Sect. 3.2.

##  5 Results

As mentioned in Sect. 2.3, we explore the parameter space using the monteLLTB
code: a cosmological solver and sampler for the $\Lambda$LTB model. Most of
the plots shown in this Sect. have been produced using getdist (Lewis, 2019).

Specifically, we constrained the $\Lambda$LTB model using several combinations
of current and forecast data. We defined as a baseline analysis (hereafter
‘Base’) the combination of CMB, Pantheon SN, LSST, and $H_{0}$ data; however,
we neglected possible correlations between LSST and Pantheon. We also defined
the baseline analysis relative to current data (hereafter ‘Base C’) as the
combination of CMB, Pantheon, and $M_{\rm B}$ data, with the last being the
B-band absolute magnitude of SNe as inferred by the Cepheid distances (see
Camarena et al., 2022). We neglected any possible correlation between the
future DESI and Euclid dataset with the current BAOs. When DESI and Euclid
data are combined, we replaced DESI measurements between $z\in[0.95,1.75]$
with the Euclid data points.

Table 2: Ratios of the areas of the $95\%$ contours from observations and the Copernican principle (see Fig. 2). We also include (last row) the case with background curvature, $K_{\rm B}\neq 0$ in Eq. 10. Observables |  $A_{\rm obs}$/$A_{\rm CP}$
---|---
$0\\!\leq\\!r^{\textrm{out}}_{\rm L}$ | $190\,\mathrm{Mpc}\\!\leq\\!r^{\textrm{out}}_{\rm L}$
Flat background FLRW metric
Base (CMB+Pantheon+LSST+$H_{0}$) | $0.82$ | $2.1$
Base + BAO + Euclid | $0.80$ | $2.0$
Base + BAO + DESI | $0.78$ | $1.9$
Base + BAO + Euclid \+ DESI | $0.75$ | $1.9$
Data above + $y$-dist. + kSZ | $0.75$ | $1.7$
Curved background FLRW metric
Data above | $0.82$ | $1.9$

We now present separately our results for the cases of the $\Lambda$CDM and
$\Lambda$LTB fiducial models of Table 1. As said earlier, we use the
$\Lambda$CDM fiducial model to test how well future data can constrain
deviations from the FLRW metric, while we use the $\Lambda$LTB fiducial models
to see if future data can detect a violation of the Copernican principle.

###  5.1 $\Lambda$CDM mocks

####  5.1.1 The Copernican principle in light of the forthcoming surveys

Figure 3:  $95\%$ and $99\%$ confidence level constraints on the contrast at the centre, $\Delta_{\rm C}$, and the redshift of the boundary, $z_{\rm B}$, for the $\Lambda$LTB mock catalogues of Table 1 as compared to the constraints from the Copernican prior convolved with the CMB likelihood. The black star is placed at the fiducial values for the LTB parameters, i.e. $\Delta_{\rm C}=-0.5$ and $z_{\rm B}=0.05$ (top row, $\Lambda$LTB 1), $\Delta_{\rm C}=-0.1$ and $z_{\rm B}=0.4$ (middle row, $\Lambda$LTB 2), and $\Delta_{\rm C}=-0.1$ and $z_{\rm B}=0.8$ (bottom row, $\Lambda$LTB 3). We note that the $z_{\rm B}$ axis is not same for all figures. Table 3: Percent improvement on constraints on radial inhomogeneity from next-generation surveys as compared to present-day constraints.  Observables considered in this analysis |  Present-day observables considered in Camarena et al. (2021) | Percent improvement
---|---|---
$0\\!\leq\\!r^{\textrm{out}}_{\rm L}$ | $190\,\mathrm{Mpc}\\!\leq\\!r^{\textrm{out}}_{\rm L}$
Base |  Base C (CMB + Pantheon + $M_{0}$) |  29$\%$ |  28$\%$
Base C + BAO + HZ |  26$\%$ |  28$\%$
Base C + BAO + HZ + $y$-dist. + kSZ |  20$\%$ |  0$\%$
Base + BAO + Euclid \+ DESI | Base C |  35$\%$ |  34$\%$
Base C + BAO + HZ |  32$\%$ |  34$\%$
Base C + BAO + HZ + $y$-dist. + kSZ |  26$\%$ |  10$\%$
Base + BAO + Euclid \+ DESI + $y$-dist. + kSZ | Base C |  35$\%$ |  41$\%$
Base C + BAO + HZ |  32$\%$ |  41$\%$
Base C + BAO + HZ + $y$-dist. + kSZ |  26$\%$ |  19$\%$

In Fig. 2 we show the marginalised constraints at the $95\%$ and $99\%$
confidence levels on the integrated mass contrast, $\Delta_{\rm L}$, and the
comoving size, $r^{\textrm{out}}_{\rm L}$, for three different data
combinations as compared to the constraints coming from the Copernican prior
convolved with the CMB likelihoods.

The constraining power of future surveys on the radial inhomogeneity can be
quantitatively compared to the expectation from the Copernican prior and CMB
by comparing the ratio of the 95% confidence regions in the parameter space
(see Table 2). Considering all scales, the ratio is always less than one,
showing the capability of future surveys to rule out non-Copernican
structures. However, at large scales, constraints provided by data still allow
for non-Copernican mass density fluctuations since for $r^{\textrm{out}}_{\rm
L}\geq 190\,{\rm Mpc}$ the ratio is approximately equal to two. We note that,
for both cases, the combination Base + DESI + Euclid provides constraints
comparable to those obtained from the combination of all data, pointing out
the important role that forthcoming LSS surveys will have to test the
Copernican principle.

We also consider the case of non-zero background curvature, that is, $K_{\rm
B}\neq 0$ in Eq. 10. The result is shown in the last row of Table 2. The
inclusion of background curvature degrades the constraints by approximately
$10\%$ compared to the flat case, still providing a competitive constraint on
the non-Copernican parameters.

####  5.1.2 Comparison with present-day constraints

In order to quantify the role of future surveys in constraining inhomogeneity
around us, we compare our constraints with the ones from current data only, as
obtained in Camarena et al. (2021). Specifically, we compute the improvement
on the observed area $A_{\textrm{obs}}$ considering the data combinations
presented in Table 3. Our present analyses do not include a cosmic chronometer
dataset as contributions of this kind of data are expected to be secondary as
compared with SNe and BAOs (Camarena et al., 2021). We note that our previous
implementation of such data did not include the full covariance matrix
presented in Moresco et al. (2020), revised and discussed in Moresco et al.
(2022).

Our Base analysis shows an improvement upon the current constraints by more
than $20\%$, when all scales are considered, and provides an improvement of
$28\%$ when compared to the constraints from Base C and Base C + BAO + HZ at
scales $r^{\textrm{out}}_{\rm L}\geq 190\,\mathrm{Mpc}$, where HZ denotes the
cosmic chronometers dataset used in Camarena et al. (2021). It is interesting
to note that our forecast Base analysis provides constraints comparable to
those obtained with all the latest cosmological data available, Base C + BAO +
HZ + $y$-dist + kSZ case, showing the importance of forthcoming SN surveys and
$1\%$ prior on the Hubble constant.

On the other hand, LSS surveys will play an important role in testing the
Copernican principle. As shown in Table 3, future measurements from Euclid and
DESI will sharpen the current constraints of Base C by approximately $35\%$,
both at $0\\!\leq\\!r^{\textrm{out}}_{\rm L}$ and
$190\,\mathrm{Mpc}\\!\leq\\!r^{\textrm{out}}_{\rm L}$. The inclusion of Euclid
and DESI will also tighten the parameter space by more than $30\%$ compared to
the combination Base C + BAO + HZ. When compared to the combination Base C +
BAO + HZ + $y$-dist. + kSZ, our analysis with the forthcoming Euclid and DESI
data shows an improvement of $26\%$ for $0\\!\leq\\!r^{\textrm{out}}_{\rm L}$
and $10\%$ for $190\,\mathrm{Mpc}\\!\leq\\!r^{\textrm{out}}_{\rm L}$.

Finally, the combination of all data considered here will tighten our current
constraints, leading to improvements up to $41\%$ for scales at
$190\,\mathrm{Mpc}\\!\leq\\!r^{\textrm{out}}_{\rm L}$ and $35\%$ for
$0\\!\leq\\!r^{\textrm{out}}_{\rm L}$ (see Table 3).

###  5.2 $\Lambda$LTB mocks

In Fig. 3 we show the marginalised constraints at the $95\%$ and $99\%$
confidence levels on $\Delta_{\rm C}$ and $z_{\rm B}$, for the three
$\Lambda$LTB fiducial cosmologies, as compared to the constraints coming from
the Copernican prior and CMB observations.

From the analysis relative to $\Lambda$LTB 1 (top row), we can see that future
data will be able to probe the local structure. This means that the effect of
the cosmic variance on the position of the observer will be reduced thanks to
the forthcoming surveys.

On the other hand, from the analysis relative to $\Lambda$LTB 2 (middle row)
and 3 (bottom row), we see that inhomogeneities that are large, but relatively
shallow, can be detected with high significance thanks to future data. More
precisely, one can note that our analyses exclude the FLRW case ($\Delta_{\rm
C}=0$ and $z_{\rm B}=0$) by $\gtrsim 3\sigma$ (pink contours). This stresses
the important roles of the next-generation surveys in testing the Copernican
principle.

##  6 Discussion

###  6.1 The role of large-scale structure data

We have seen from the results of Sect. 5.1 on the $\Lambda$CDM mocks that
future surveys, such as Euclid, will grant a $\approx\,$30% improvement on
inhomogeneity around the observer. In particular, for scales greater than 190
Mpc, the combination of all data will constrain inhomogeneity to only 1.7
times the area of the region allowed by standard cosmology. Given the fact
that Euclid probes the redshift range $0.9<z<1.8$, one may wonder if the
improvement due to Euclid comes directly from better constraints on the shape
of the angular diameter distance and Hubble rate or indirectly from better
constraints on the cosmological parameters.

In order to answer the previous question we show in Fig. 4 the fluctuations in
the apparent magnitude, Hubble rate and angular diameter distance for the
$\Lambda$LTB model as compared to the fiducial $\Lambda$CDM one. The 68% and
95% bands are obtained by evaluating the relevant functions at every point of
the chains. We compare three analyses: the Base one, Base with present BAO and
Euclid, and Base with present BAOs and DESI. From this plot, it appears that
the shape of the various functions does not change when adding Euclid or DESI.
In other words, these two surveys do not improve the constraints in specific
redshift ranges but rather they help at tightening the overall uncertainties.
From this we conclude that the improvement due to Euclid comes mostly from
better constraints on the cosmological parameters, although this works in
synergy with DESI and the other observables.

Figure 4: Fluctuations in the apparent magnitude (top row), Hubble rate
(middle row), and angular diameter distance (bottom row) for the $\Lambda$LTB
model as compared to the fiducial $\Lambda$CDM one. The 68% and 95% bands are
obtained by evaluating the relevant functions at every point of the chains.
The red points show LSST and Euclid DESIRE SN data, the black ones Euclid
data, and the purple ones DESI data. See Sect. 6.1.

###  6.2 Beyond the central observer

As mentioned earlier, our aim is to test radial homogeneity around us,
neglecting anisotropies. We then placed the observer at the centre of the
spherical over/underdensity. However, in an inhomogeneous universe beyond
FLRW, neglecting anisotropies could not be justified because anisotropies may
affect observables as much as radial inhomogeneities. In other words, the
modelling adopted in this work implies a spherically symmetric inhomogeneity
and a fine-tuning of the observer’s position.

From the results of Sect. 5.1 on the $\Lambda$CDM mocks we see, a posteriori,
that large structures with shallow contrasts are allowed by future data. If,
for example, we consider a contrast of $\delta=-0.1$, the corresponding change
in the Hubble rate is approximately $\delta
H_{0}/H_{0}=-f(\Omega_{m})\delta/3\simeq 0.017$, where $f\simeq 0.5$ is the
present-day growth rate for the concordance $\Lambda$CDM model. The CMB
dipole, if the observer were at, for example, a distance $d_{\rm obs}=300$ Mpc
from the centre, using $v=\Delta H\,d_{\rm obs}$, is then

$\displaystyle\beta={v\over c}\simeq 1.2\times 10^{-3}\,,$ (23)

which is basically the observed CMB dipole (Aghanim et al., 2020a). As the
structures that we consider in this work extend to, at most, 1000 Mpc (see
Fig. 2), the required fine-tuning has a chance of less than 1 in 40. In other
words, the fine-tuning required to satisfy the CMB dipole is rather mild and
therefore the motivation for considering an off-centre observer is to provide
a better description of possibly anisotropic data, rather than to relieve the
fine-tuning of the observer’s position.

It is worth mentioning that the fine-tuning is instead very severe when
considering void models as alternatives to dark energy, a possibility that was
not explored here and not favoured by data (see Marra et al. 2022). Indeed, in
this case the underdensity has a radius of $\approx\,$3 Gpc and $\delta
H_{0}/H_{0}\approx\,0.2$ so that the observer has to be within $\approx\,$30
Mpc from the centre, giving rise to a fine-tuning of one in a million (Marra &
Notari, 2011). We note, however, as pointed out in Garcia-Bellido & Haugboelle
(2008b), that it is possible to alleviate this improbability by displacing the
observer and then making them move towards the centre. For distances of a few
hundred Mpc and velocities of a few thousand km s-1, the effect is
indistinguishable from the observed CMB dipole. In a way, one exchanges an
improbability in location for an improbability in the direction of motion. The
overall effect is to reduce the coincidence to a few parts in a thousand.

##  7 Conclusions

Testing fundamental assumptions of cosmology is a crucial step towards
improving our understanding of the Universe and firmly establishing the
foundations of the standard cosmological paradigm. In this work we have tested
the Copernican principle by placing constraints on the $\Lambda$LTB model
using current and forecast data products. Specifically, we focused on the
capability of Euclid to test the Copernican principle in conjunction with data
from current and forthcoming surveys, such as SH0ES, DESI, and LSST.

In particular, we compared constraints on the $\Lambda$LTB model coming from
the forecast and current data against constraints drawn from the Copernican
prior—the statistical counterpart of the Copernican principle. This comparison
allowed us to quantify how well we can constrain deviations from the
Copernican principle.

We have considered two types of fiducial models: the standard $\Lambda$CDM
model and the inhomogeneous $\Lambda$LTB model. By analysing the latter we
aimed to determine if next-generation surveys will be able to detect
deviations from the Copernican principle, while our analysis of $\Lambda$CDM
data aimed to investigate if forthcoming data can successfully test the
Copernican principle.

We have found that the inclusion of data from Euclid, and other future
surveys, will improve the current constraints on the Copernican principle by
up to $40\%$. This improvement will be especially important at scales $r_{\rm
B}\geq 190$ Mpc, where the inclusion of Euclid, and other forthcoming surveys,
will reduce the constrained area of the space parameters by a factor of $<2$
as compared with the area allowed by the Copernican prior. Furthermore, we
find that using the forthcoming Euclid data, and data from other future
surveys, we will be able to detect inhomogeneous deviations of the FLRW
metric, including gigaparsec-scale inhomogeneities of contrast $-0.1$. Our
analyses show that, given the precision of Euclid and other forthcoming
surveys, a detection of this kind would allow us to rule out the FLRW space-
time ($\Delta_{\rm C}=0$ and $z_{\rm B}=0$) by $\gtrsim 3\sigma$.

Our results rely on the assumption of a particular curvature profile, and, as
shown in Appendix A, constraints could be weakened by up to a factor of $\sim
2$ under the assumption of a more general profile. This drawback in our
analysis, produced by the choice of a particular curvature profile, could be
overcome by introducing data-driven methods that allow us to reconstruct the
local distribution of matter in a more robust way. We will implement
approaches of this sort in future research.

In summary, this work highlights the importance of synergies between Euclid
and external probes in testing the Copernican principle, which is one of the
fundamental assumptions of the standard cosmological paradigm.

###### Acknowledgements.

DC thanks CAPES for financial support. VM thanks CNPq and FAPES for partial
financial support. This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 888258. JGB, MM and SN acknowledge support from the
research project PGC2018-094773-B-C32, and the Spanish Research Agency
(Agencia Estatal de Investigación) through the Grant IFT Centro de Excelencia
Severo Ochoa No CEX2020-001007-S, funded by MCIN/AEI/10.13039/501100011033. LL
was supported by a Swiss National Science Foundation Professorship grant (Nos.
170547 & 202671). The work of CJAPM work was financed by FEDER—Fundo Europeu
de Desenvolvimento Regional funds through the COMPETE 2020—Operational
Programme for Competitiveness and Internationalisation (POCI), and by
Portuguese funds through FCT - Fundação para a Ciência e a Tecnologia in the
framework of the project POCI-01-0145-FEDER-028987 and PTDC/FIS-
AST/28987/2017. MM also received support from “la Caixa” Foundation (ID
100010434), with fellowship code LCF/BQ/PI19/11690015. DS acknowledges
financial support from the Fondecyt Regular project number 1200171. AdS
acknowledges the support from the Fundação para a Ciência e a Tecnologia (FCT)
through the Investigador FCT Contract No. IF/01135/2015 and POCH/FSE (EC) and
in the form of an exploratory project with the same reference. JPM and AdS
acknowledge the support from FCT Projects with references EXPL/FIS-
AST/1368/2021, PTDC/FIS-AST/0054/2021, UIDB/04434/2020, UIDP/04434/2020,
CERN/FIS-PAR/0037/2019, PTDC/FIS-OUT/29048/2017. This work made use of the CHE
cluster, managed and funded by COSMO/CBPF/MCTI, with financial support from
FINEP and FAPERJ, and operating at the Javier Magnin Computing Center/CBPF.
This work also made use of the Virgo Cluster at Cosmo-ufes/UFES, which is
funded by FAPES and administrated by Renan Alves de Oliveira. The Euclid
Consortium acknowledges the European Space Agency and a number of agencies and
institutes that have supported the development of Euclid, in particular the
Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy,
the Canadian Euclid Consortium, the French Centre National d’Etudes Spatiales,
the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research
Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de
Ciencia e Innovación, the National Aeronautics and Space Administration, the
National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool
Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the
State Secretariat for Education, Research and Innovation (SERI) at the Swiss
Space Office (SSO), and the United Kingdom Space Agency. A complete and
detailed list is available on the Euclid web site (http://www.euclid-ec.org).

## References

  * Abell et al. (2009) Abell, P. A. et al. 2009, arXiv e-prints, arXiv:0912.0201
  * Aghamousa et al. (2016) Aghamousa, A. et al. 2016, arXiv e-prints, arXiv:1611.00036
  * Aghanim et al. (2014) Aghanim, N. et al. 2014, A&A, 571, A27
  * Aghanim et al. (2020a) Aghanim, N. et al. 2020a, A&A, 641, A1
  * Aghanim et al. (2020b) Aghanim, N. et al. 2020b, A&A, 641, A6, [Erratum: A&A 652, C4 (2021)]
  * Akrami et al. (2020) Akrami, Y. et al. 2020, A&A, 644, A100
  * Alam et al. (2017) Alam, S. et al. 2017, MNRAS, 470, 2617
  * Albrecht et al. (2006) Albrecht, A., Bernstein, G., Cahn, R., et al. 2006, arXiv e-prints, arXiv:astro-ph/0609591
  * Aluri et al. (2022) Aluri, P. K. et al. 2022 [arXiv:2207.05765]
  * Amendola et al. (2018) Amendola, L. et al. 2018, LRR, 21, 2
  * Astier et al. (2014) Astier, P. et al. 2014, A&A, 572, A80
  * Audren et al. (2013) Audren, B., Lesgourgues, J., Benabed, K., & Prunet, S. 2013, JCAP, 1302, 001
  * Ben-Dayan et al. (2013) Ben-Dayan, I., Gasperini, M., Marozzi, G., Nugier, F., & Veneziano, G. 2013, JCAP, 06, 002
  * Bengaly et al. (2018) Bengaly, C. A. P., Maartens, R., & Santos, M. G. 2018, JCAP, 04, 031
  * Beutler et al. (2011) Beutler, F., Blake, C., Colless, M., et al. 2011, MNRAS, 416, 3017
  * Blas et al. (2011) Blas, D., Lesgourgues, J., & Tram, T. 2011, JCAP, 07, 034
  * Brinckmann & Lesgourgues (2019) Brinckmann, T. & Lesgourgues, J. 2019, PDU, 24, 100260
  * Camarena et al. (2021) Camarena, D., Marra, V., Sakr, Z., & Clarkson, C. 2021, MNRAS, 509, 1291
  * Camarena et al. (2022) Camarena, D., Marra, V., Sakr, Z., & Clarkson, C. 2022, Class. Quant. Grav., 39, 184001
  * Carrick et al. (2015) Carrick, J., Turnbull, S. J., Lavaux, G., & Hudson, M. J. 2015, MNRAS, 450, 317
  * Chen et al. (2019) Chen, L., Huang, Q.-G., & Wang, K. 2019, JCAP, 02, 028
  * Colin et al. (2017) Colin, J., Mohayaee, R., Rameez, M., & Sarkar, S. 2017, MNRAS, 471, 1045
  * Colin et al. (2011) Colin, J., Mohayaee, R., Sarkar, S., & Shafieloo, A. 2011, MNRAS, 414, 264
  * Cropper et al. (2018) Cropper, M., Pottinger, S., Azzollini, R., et al. 2018, Proc. SPIE Int. Soc. Opt. Eng., 10698, 1069828
  * Dalang & Bonvin (2022) Dalang, C. & Bonvin, C. 2022, Mon. Not. Roy. Astron. Soc., 512, 3895
  * de la Torre & Guzzo (2012) de la Torre, S. & Guzzo, L. 2012, MNRAS, 427, 327
  * du Mas des Bourboux et al. (2020) du Mas des Bourboux, H. et al. 2020, ApJ, 901, 153
  * Efstathiou (2021) Efstathiou, G. 2021, MNRAS, 505, 3866
  * Euclid Collaboration et al. (2022) Euclid Collaboration, Scaramella, R., Amiaux, J., et al. 2022, A&A, 662, A112
  * Euclid Collaboration: Blanchard et al. (2020) Euclid Collaboration: Blanchard, A., Camera, S., Carbone, C., et al. 2020, A&A, 642, A191
  * Euclid Collaboration: Paykari et al. (2020) Euclid Collaboration: Paykari, P., Kitching, T., Hoekstra, H., et al. 2020, A&A, 635, A139
  * February et al. (2010) February, S., Larena, J., Smith, M., & Clarkson, C. 2010, MNRAS, 405, 2231
  * Feindt et al. (2013) Feindt, U. et al. 2013, A&A, 560, A90
  * Ferreira & Quartin (2021) Ferreira, P. d. S. & Quartin, M. 2021, PRL, 127, 101301
  * Ferreira & Marra (2022) Ferreira, T. & Marra, V. 2022, Mon. Not. Roy. Astron. Soc., 513, 5438
  * Fixsen et al. (1996) Fixsen, D., Cheng, E., Gales, J., et al. 1996, ApJ, 473, 576
  * Font-Ribera et al. (2014) Font-Ribera, A., McDonald, P., Mostek, N., et al. 2014, JCAP, 05, 023
  * Friedrich et al. (2021) Friedrich, O. et al. 2021, MNRAS, 508, 3125
  * Garcia-Bellido & Haugboelle (2008a) Garcia-Bellido, J. & Haugboelle, T. 2008a, JCAP, 04, 003
  * Garcia-Bellido & Haugboelle (2008b) Garcia-Bellido, J. & Haugboelle, T. 2008b, JCAP, 09, 016
  * Gong et al. (2010) Gong, Y., Cooray, A., & Chen, X. 2010, ApJ, 709, 1420
  * Harnois-Deraps et al. (2019) Harnois-Deraps, J., Giblin, B., & Joachimi, B. 2019, A&A, 631, A160
  * Hudson et al. (2004) Hudson, M. J., Smith, R. J., Lucey, J. R., & Branchini, E. 2004, MNRAS, 352, 61
  * Ishak et al. (2013) Ishak, M., Peel, A., & Troxel, M. A. 2013, PRL, 111, 251302
  * Jonsson et al. (2010) Jonsson, J., Sullivan, M., Hook, I., et al. 2010, MNRAS, 405, 535
  * Kim et al. (2022) Kim, Y., Park, C.-G., Noh, H., & Hwang, J.-c. 2022, Astron. Astrophys., 660, A139
  * Kronborg et al. (2010) Kronborg, T. et al. 2010, A&A, 514, A44
  * Laureijs et al. (2011) Laureijs, R. et al. 2011, arXiv e-prints, arXiv:1110.3193
  * Lewis (2019) Lewis, A. 2019, arXiv e-prints, arXiv:1910.13970
  * Lyke et al. (2020) Lyke, B. W. et al. 2020, ApJS, 250, 8
  * Maciaszek et al. (2016) Maciaszek, T., Ealet, A., Jahnke, K., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9904, Space Telescopes and Instrumentation 2016: Optical, Infrared, and Millimeter Wave, ed. H. A. MacEwen, G. G. Fazio, M. Lystrup, N. Batalha, N. Siegler, & E. C. Tong, 99040T
  * Marra et al. (2022) Marra, V., Castro, T., Camarena, D., Borgani, S., & Ragagnin, A. 2022, Astron. Astrophys., 664, A179
  * Marra & Notari (2011) Marra, V. & Notari, A. 2011, CQG, 28, 164004
  * Marra et al. (2013) Marra, V., Quartin, M., & Amendola, L. 2013, PRD, 88, 063004
  * Martinelli et al. (2020) Martinelli, M. et al. 2020, A&A, 644, A80
  * Martinelli et al. (2021) Martinelli, M. et al. 2021, A&A, 654, A148
  * Migkas et al. (2021) Migkas, K., Pacaud, F., Schellenberger, G., et al. 2021, A&A, 649, A151
  * Migkas et al. (2020) Migkas, K., Schellenberger, G., Reiprich, T. H., et al. 2020, A&A, 636, A15
  * Mohayaee et al. (2021) Mohayaee, R., Rameez, M., & Sarkar, S. 2021, EPJ ST, 230, 2067
  * Moresco et al. (2022) Moresco, M., Amati, L., Amendola, L., et al. 2022, arXiv e-prints, arXiv:2201.07241
  * Moresco et al. (2020) Moresco, M., Jimenez, R., Verde, L., Cimatti, A., & Pozzetti, L. 2020, ApJ, 898, 82
  * Moss et al. (2011) Moss, A., Zibin, J. P., & Scott, D. 2011, PRD, 83, 103515
  * Nesseris et al. (2022) Nesseris, S. et al. 2022, Astron. Astrophys., 660, A67
  * Pozzetti et al. (2016) Pozzetti, L., Hirata, C. M., Geach, J. E., et al. 2016, A&A, 590, A3
  * Prieto et al. (2012) Prieto, E., Amiaux, J., Auguères, J.-L., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, ed. M. C. Clampin, G. G. Fazio, H. A. MacEwen, & J. Oschmann, Jacobus M., 84420W
  * Quartin et al. (2014) Quartin, M., Marra, V., & Amendola, L. 2014, PRD, 89, 023009
  * Rahman et al. (2022) Rahman, W., Trotta, R., Boruah, S. S., Hudson, M. J., & van Dyk, D. A. 2022, Mon. Not. Roy. Astron. Soc., 514, 139
  * Raichoor et al. (2020) Raichoor, A. et al. 2020, MNRAS, 500, 3254
  * Redlich et al. (2014) Redlich, M., Bolejko, K., Meyer, S., Lewis, G. F., & Bartelmann, M. 2014, A&A, 570, A63
  * Reichardt et al. (2021) Reichardt, C. L. et al. 2021, ApJ, 908, 199
  * Ross et al. (2015) Ross, A. J., Samushia, L., Howlett, C., et al. 2015, MNRAS, 449, 835
  * Ross et al. (2020) Ross, A. J. et al. 2020, MNRAS, 498, 2354
  * Saha et al. (2021) Saha, S., Shaikh, S., Mukherjee, S., Souradeep, T., & Wandelt, B. D. 2021, JCAP, 10, 072
  * Scolnic et al. (2018) Scolnic, D. et al. 2018, ApJ, 859, 101
  * Secrest et al. (2021) Secrest, N. J., von Hausegger, S., Rameez, M., et al. 2021, ApJL, 908, L51
  * Siewert et al. (2021) Siewert, T. M., Schmidt-Rubart, M., & Schwarz, D. J. 2021, A&A, 653, A9
  * Silk (1977) Silk, J. 1977, A&A, 59, 53
  * Valkenburg (2012) Valkenburg, W. 2012, GRG, 44, 2449
  * Valkenburg et al. (2014) Valkenburg, W., Marra, V., & Clarkson, C. 2014, MNRAS, 438, L6
  * Wang et al. (2013) Wang, Y., Chuang, C.-H., & Hirata, C. M. 2013, MNRAS, 430, 2446
  * Yadav et al. (2010) Yadav, J. K., Bagla, J. S., & Khandai, N. 2010, Mon. Not. Roy. Astron. Soc., 405, 2009
  * Zibin (2008) Zibin, J. P. 2008, PRD, 78, 043504

##  Appendix A The curvature profile

The analyses presented in this paper rely on the assumption of the compensated
profile introduced in Sect. 2.2, which was chosen in order to ensure that the
$\Lambda$CDM background is recovered at $r\geq r_{\rm B}$, a crucial feature
in order to confront CMB data consistently using an effective FLRW model.

Figure 5: Comparison between our model and the GBH parametrization. Top:
Density contrast of matter today, $\delta(r,t_{0})$, as a function of the FLRW
comoving coordinate. Our compensated model (solid blue line) satisfies
$\delta(r_{\rm L}^{\mathrm{out}},t_{0})=\delta(r_{\rm
B}^{\mathrm{out}},t_{0})=0$, while the $\Lambda$CDM background is not exactly
recovered for the GBH models (dashed red and dotted green lines). Middle:
Deviations from the background density of matter, $\Omega^{\mathrm{out}}_{\rm
m,0}$, as a function of the FLRW comoving coordinate. The GBH model with
$r_{0}=r_{\rm B}$ (red dashed line) largely deviates from
$\Omega^{\mathrm{out}}_{\rm m,0}$ at $r^{\mathrm{out}}=r_{\rm
B}^{\mathrm{out}}$. On the other hand, the choice of $r_{0}=r_{\rm L}$ for the
GBH leads to deviations of approximately $-0.01$ from the background at
$r^{\mathrm{out}}=r_{\rm B}^{\mathrm{out}}$. The model assumed in this work
fully recovers $\Omega^{\mathrm{out}}_{\rm m,0}$ at any $r^{\mathrm{out}}\geq
r_{\rm B}^{\mathrm{out}}$. Bottom: Deviations from the background Hubble
expansion as a function of the FLRW comoving coordinate. GBH models (dashed
red and dotted green lines) do not perfectly match the $\Lambda$CDM background
expansion history at $r^{\mathrm{out}}=r_{\mathrm{B}}^{\mathrm{out}}$.

Here, we compare our model to the Garcia-Bellido and Haugboelle (GBH) model
(Garcia-Bellido & Haugboelle 2008a), which parametrises the LTB metric by
imposing

$\displaystyle\Omega_{\rm m,0}(r)$ $\displaystyle=\Omega^{\mathrm{out}}_{\rm
m,0}+\left(\Omega^{\mathrm{in}}_{\rm m,0}-\Omega^{\mathrm{out}}_{\rm
m,0}\right)\left\\{\frac{1-\tanh\left[(r-r_{0})/2\Delta
r\right]}{1+\tanh\left[r_{0}/2\Delta r\right]}\right\\}\,,$ $\displaystyle
H_{0}(r)$ $\displaystyle=H_{\perp 0}^{\mathrm{out}}+\left(H_{\perp
0}^{\mathrm{in}}-H_{\perp
0}^{\mathrm{out}}\right)\left\\{\frac{1-\tanh\left[(r-r_{0})/2\Delta
r\right]}{1+\tanh\left[r_{0}/2\Delta r\right]}\right\\}\,,$

where $r_{0}$ is the size of the void, $\Delta r$ the transition scale,
$\Omega^{\mathrm{in}}_{\rm m,0}\equiv\Omega_{\rm m,0}(r=0)$, and $H_{\perp
0}^{\mathrm{in}}\equiv H_{\perp 0}(r=0)$. Fig. 5 shows the differences between
our model (solid blue line) and the GBH model, where we have adopted
$r_{0}=r_{\rm L}$ (dotted green line) and $r_{0}=r_{\rm B}$ (dashed red line).
When the size of the GBH inhomogeneity is fixed to $r_{\rm B}$, a scale
greater that $r_{\rm B}$ is needed to recover the $\Lambda$CDM background. On
the other hand, if one assumes $r_{0}=r_{\rm L}$, the GBH model tends to the
$\Lambda$CDM background at $r=r_{\rm B}$. In contrast, our model perfectly
matches the $\Lambda$CDM background at any scale $r\geq r_{\rm B}$. The
compensating behaviour of our model is particularly notable in the top panel
of the Fig. 5, where we note that $\delta(r_{\rm
L}^{\mathrm{out}},t_{0})=\delta(r_{\rm B}^{\mathrm{out}},t_{0})=0$ for our
model while the GBH models does not satisfy $\delta(r^{\mathrm{out}},t_{0})=0$
for all $r^{\mathrm{out}}\geq r_{\rm B}^{\mathrm{out}}$.

Furthermore, to investigate the dependence of our results on the chosen
profile, we performed an additional analysis using the following
generalisation of Eq. 10:

$\displaystyle P_{3}(x,\alpha)=\begin{cases}1&\mbox{ for }0\leq x<\alpha\\\
1-\exp\big{[}-\frac{1-\alpha}{x-\alpha}(1-\frac{x-\alpha}{1-\alpha})^{3}]&\mbox{
for }\alpha\leq x<1\\\ 0&\mbox{ for }1\leq x\end{cases}\,,$ (24)

with $0<\alpha<1$. This new parameter will modify the smoothness of the
transition between the inner and compensating region, leading to sharpened
profiles when $\alpha$ approximates to $1$. Results from this extra analysis,
which is performed using the combination Base + BAO + Euclid \+ DESI +
$y$-dist. + kSZ from the $\Lambda$CDM fiducial, shows that the inclusion of
the $\alpha$ parameter weakens the constraints on $\Delta_{\rm C}$ and $r_{\rm
L}^{\mathrm{out}}$ by a factor of two, compared to the results from Eq. 10.

##  Appendix B Re-scaling datasets

Covariance matrices are fundamental pieces of forecast analyses. However,
their production for forthcoming surveys is an open issue when non-standard
cosmologies are considered (Harnois-Deraps et al. 2019; Friedrich et al. 2021;
Ferreira & Marra 2022). This complicates the construction of forecast data for
$\Lambda$LTB cosmologies. Nesseris et al. (2022) has overcome this issue by
neglecting the error due to the non-standard cosmology. Here, we apply a re-
scaling method to convert the $\Lambda$CDM forecast data and its covariance
matrices into $\Lambda$LTB catalogues.

Consider a given dataset, with $x_{i}$ being the observed quantity, $z_{i}$
the corresponding redshift, and $C_{ij}$ the covariance matrix. This dataset
can be re-scaled to agree with a particular model via the following steps.

First, we define $R_{ij}=C_{ij}/x_{i}x_{j}$, a new matrix that contains the
relative uncertainties and correlations from the original covariance matrix.
Second, we compute with the theoretical prediction of the new model the
fiducial values at the relevant redshifts, such that $x^{\mathrm{f}}_{i}\equiv
x^{\mathrm{fid}}(z_{i})$. Third, using the above defined quantities, we
compute the new correlation matrix as
$\tilde{C}_{ij}=x^{\mathrm{f}}_{i}x^{\mathrm{f}}_{j}\>R_{ij}$. Finally, we
then draw a random realisation, $\tilde{x}_{i}$, of the multivariate-normal
distribution $\mathcal{N}(x^{\mathrm{f}}_{i},\tilde{C}_{ij})$.

We note that this method assumes that relative error and correlations are not
changed by a non-standard model. As discussed through this paper, the
procedure above is also applied to re-scale real data according the fiducial
models presented on Table 1; this ensures that all data are consistently
described by a particular fiducial model.

##  Appendix C The inhomogeneous Hubble constant

The $\Lambda$LTB model features a profile function $H_{0}(r)$ that depends on
the radial distance from the centre of the void, instead of a constant value
like $H_{0}$ in the $\Lambda$CDM model. Since there is not a preferable scale
to set the rate of expansion of the Universe, the definition of $H_{0}$
remains arbitrary. To overcome this issue, we extend the FLRW definitions and
mimic the observational procedure to locally constrain the Hubble constant.
Explicitly, we adopt the definition $H_{0}^{\rm L}$ for inhomogeneous
cosmological models that was introduced in Camarena et al. (2022). This
method, which is applied for every sample point of the parameter space,
follows the following steps.

First, we create a mock catalogue using the redshifts of Pantheon SNe at
$0.023<z<0.15$ and the $\Lambda$LTB luminosity distances at the corresponding
redshifts. Second, the mock data are fitted using an extension of the
cosmographic expansion given by

$\displaystyle D_{L}(z)$ $\displaystyle=\frac{cz}{H^{\rm
L}_{0}}\left[1+\frac{(1-q^{r}_{0})z}{2}\right]\,,$ (25) $\displaystyle
q^{r}_{0}(r)$ $\displaystyle=\left[\frac{\Omega_{\rm
m}(r)}{2}-\Omega_{\Lambda}(r)\right]\left[\frac{H_{0}(r)}{H^{\rm
L}_{0}}\right]^{2}\,,$ (26)

where $q^{r}_{0}$ is the radial-dependent deceleration parameter. Finally, the
best-fit value of $H_{0}^{\rm L}$ is adopted as the measured Hubble constant.

It is interesting to point out that this procedure mimics the standard cosmic
distance ladder analysis of SNe that follow the Hubble flow, while taking into
account the effect of the inhomogeneity on the measurement of the Hubble
constant. We note that other authors have previously proposed similar
approximations (Redlich et al. 2014; Efstathiou 2021).