# Euclid: Cosmological forecasts from the void size function

S. Contarini, G. Verza, A. Pisani, N. Hamaus, M. Sahlén, C. Carbone, S. Dusini, F. Marulli, L. Moscardini, A. Renzi, et al.

► To cite this version:

S. Contarini, G. Verza, A. Pisani, N. Hamaus, M. Sahlén, et al. Euclid: Cosmological forecasts from the void size function. Astron.Astrophys., 2022, 667, pp.A162. 10.1051/0004-6361/202244095. hal-03691899

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# Euclid: Cosmological forecasts from the void size function*

S. Contarini<sup>1,2,3,***</sup> D, G. Verza<sup>4,5</sup> D, A. Pisani<sup>6,7,8</sup> D, N. Hamaus<sup>9</sup> D, M. Sahlén<sup>10,11</sup> D, C. Carbone<sup>12</sup>, S. Dusini<sup>4</sup>, F. Marulli<sup>1,2,3</sup>, L. Moscardini<sup>1,2,3</sup>, A. Renzi<sup>4,5</sup>, C. Sirignano<sup>4,5</sup>, L. Stanco<sup>4</sup>, M. Aubert<sup>13</sup>, M. Bonici<sup>11</sup>, G. Castignani<sup>1,3</sup>, H. M. Courtois<sup>13</sup>, S. Escoffier<sup>14</sup>, D. Guinet<sup>13</sup>, A. Kovacs<sup>15,16</sup>, G. Lavaux<sup>17</sup>, E. Massara<sup>18,19</sup>, S. Nadathur<sup>20</sup>, G. Pollina<sup>8</sup>, T. Ronconi<sup>21,22</sup>, F. Ruppin<sup>23</sup>, Z. Sakr<sup>24,25</sup>, A. Veropalumbo<sup>26</sup>, B. D. Wandelt<sup>6,27</sup>, A. Amara<sup>20</sup>, N. Auricchio<sup>3</sup>, M. Baldi<sup>2,3,28</sup>, D. Bonino<sup>29</sup>, E. Branchini<sup>30,31</sup>, M. Brescia<sup>32</sup>, J. Brinchmann<sup>33</sup>, S. Camera<sup>29,34,35</sup>, V. Capobianco<sup>29</sup>, J. Carretero<sup>36,37</sup>, M. Castellano<sup>38</sup>, S. Cavuoti<sup>32,39,40</sup>, R. Cledassou<sup>41</sup>, G. Congedo<sup>42</sup>, C. J. Conselice<sup>43</sup>, L. Conversi<sup>44,45</sup>, Y. Copin<sup>23</sup>, L. Corcione<sup>29</sup>, F. Courbin<sup>46</sup>, M. Cropper<sup>47</sup>, A. Da Silva<sup>48,49</sup>, H. Degaudenzi<sup>50</sup>, F. Dubath<sup>50</sup>, C. A. J. Duncan<sup>51</sup>, X. Dupac<sup>44</sup>, A. Ealet<sup>23</sup>, S. Farrens<sup>52</sup>, S. Ferriol<sup>23</sup>, P. Fosalba<sup>53,54</sup>, M. Frailis<sup>55</sup>, E. Franceschi<sup>3</sup>, B. Garilli<sup>11</sup>, W. Gillard<sup>14</sup>, B. Gillis<sup>42</sup>, C. Giocoli<sup>56,57</sup>, A. Grazian<sup>58</sup>, F. Grupp<sup>59,8</sup>, L. Guzzo<sup>60,61,62</sup>, S. Haugen<sup>63</sup>, W. Holmes<sup>64</sup>, F. Hormuth<sup>65</sup>, K. Jahnke<sup>66</sup>, M. Kümmel<sup>8</sup>, S. Kermiche<sup>14</sup>, A. Kiessling<sup>64</sup>, M. Kilbinger<sup>67</sup>, M. Kunz<sup>68</sup>, H. Kurki-Suonio<sup>69</sup>, R. Laureijs<sup>70</sup>, S. Ligori<sup>29</sup>, P. B. Lilje<sup>63</sup>, I. Lloro<sup>71</sup>, E. Maiorano<sup>3</sup>, O. Mansutti<sup>55</sup>, O. Marggraf<sup>72</sup>, K. Markovic<sup>64</sup>, R. Massey<sup>73</sup>, M. Melchior<sup>74</sup>, M. Meneghetti<sup>2,3</sup>, G. Meylan<sup>46</sup>, M. Moresco<sup>1,3</sup>, E. Munari<sup>55</sup>, S. M. Niemi<sup>70</sup>, C. Padilla<sup>37</sup>, S. Paltani<sup>50</sup>, F. Pasian<sup>55</sup>, K. Pedersen<sup>75</sup>, W. J. Percival<sup>18,19,76</sup>, V. Pettorino<sup>52</sup>, S. Pires<sup>52</sup>, G. Polenta<sup>77</sup>, M. Poncet<sup>78</sup>, L. Popa<sup>79</sup>, L. Pozzetti<sup>3</sup>, F. Raison<sup>59</sup>, J. Rhodes<sup>64</sup>, E. Rossetti<sup>1</sup>, R. Saglia<sup>8,59</sup>, B. Sartoris<sup>55,22</sup>, P. Schneider<sup>72</sup>, A. Secroun<sup>14</sup>, G. Seidel<sup>66</sup>, G. Sirri<sup>2</sup>, C. Surace<sup>80</sup>, P. Tallada-Crespi<sup>36,81</sup>, A. N. Taylor<sup>42</sup>, I. Tereno<sup>48,82</sup>, R. Toledo-Moreo<sup>83</sup>, F. Torradeflot<sup>36,81</sup>, E. A. Valentijn<sup>84</sup>, L. Valenziano<sup>3,2</sup>, Y. Wang<sup>85</sup>, J. Weller<sup>8,59</sup>, G. Zamorani<sup>3</sup>, J. Zoubian<sup>14</sup>, S. Andreon<sup>61</sup>, D. Maino<sup>11,60,62</sup>, and S. Mei<sup>86</sup>

*(Affiliations can be found after the references)*

Received 23 May 2022 / Accepted 10 August 2022

## ABSTRACT

The Euclid mission – with its spectroscopic galaxy survey covering a sky area over 15 000 deg² in the redshift range 0.9 < z < 1.8 – will provide a sample of tens of thousands of cosmic voids. This paper thoroughly explores for the first time the constraining power of the void size function on the properties of dark energy (DE) from a survey mock catalogue, the official Euclid Flagship simulation. We identified voids in the Flagship light-cone, which closely matches the features of the upcoming Euclid spectroscopic data set. We modelled the void size function considering a state-of-the art methodology: we relied on the volume-conserving (Vdn) model, a modification of the popular Sheth & van de Weygaert model for void number counts, extended by means of a linear function of the large-scale galaxy bias. We found an excellent agreement between model predictions and measured mock void number counts. We computed updated forecasts for the Euclid mission on DE from the void size function and provided reliable void number estimates to serve as a basis for further forecasts of cosmological applications using voids. We analysed two different cosmological models for DE: the first described by a constant DE equation of state parameter, w, and the second by a dynamic equation of state with coefficients w₀ and wₐ. We forecast 1σ errors on w lower than 10% and we estimated an expected figure of merit (FoM) for the dynamical DE scenario FoM<sub>w₀wₐ</sub> = 17 when considering only the neutrino mass as additional free parameter of the model. The analysis is based on conservative assumptions to ensure full robustness, and is a pathfinder for future enhancements of the technique. Our results showcase the impressive constraining power of the void size function from the Euclid spectroscopic sample, both as a stand-alone probe, and to be combined with other Euclid cosmological probes.

**Key words.** dark energy – cosmology: theory – galaxies: statistics – catalogs – surveys – methods: data analysis

## 1. Introduction

Cosmic voids are vast under-dense regions filling most of the volume of the present-day Universe. With sizes up to hundreds of megaparsec (Gregory & Thompson 1978; Tikhonov & Karachentsev 2006; Thompson & Gregory 2011; Szapudi et al. 2015) they are the largest observable structures in the cosmic web (Zeldovich et al. 1982; Bond et al. 1996) – the pattern arising in the galaxy distribution. Voids constitute a unique cosmological probe: their interiors, spanning a large range of scales and fea-

turing low matter density, make them particularly suited to study dark energy (DE) and modified gravity (Lee & Park 2009; Biswas et al. 2010; Li & Efstathiou 2012; Clampitt et al. 2013; Spolyar et al. 2013; Cai et al. 2015; Pisani et al. 2015a; Pollina et al. 2016; Zivick et al. 2015; Achitouv 2016; Sahlén et al. 2016; Falck et al. 2018; Sahlén & Silk 2018; Paillas et al. 2019; Perico et al. 2019; Verza et al. 2019; Contarini et al. 2021), as well as massive neutrinos (Massara et al. 2015; Banerjee & Dalal 2016; Kreisch et al. 2019, 2022; Sahlén 2019; Schuster et al. 2019), primordial non-Gaussianity (Chan et al. 2019), and physics beyond the standard model (Peebles 2001; Reed et al. 2015; Yang et al. 2015; Baldi & Villaescusa-Navarro 2018). Cosmic voids are becoming an effective and competitive new probe of cosmology thanks to the advent of current and upcoming sky surveys such as 6dF

<footnote>* This paper is published on behalf of the Euclid Consortium.</footnote>

<footnote>** Corresponding author: S. Contarini, e-mail: sofia.contarini3@unibo.it</footnote>
Galaxy Survey (6dFGS, Jones et al. 2009), VIMOS Public Extra-
galactic Redshift Survey (VIPERS, Guzzo et al. 2014), SDSS-
III's Baryon Oscillation Spectroscopic Survey (BOSS, Alam et al.
2017) and Extended Baryon Oscillation Spectroscopic Survey
(eBOSS, Dawson et al. 2016) from the Sloan Digital Sky Sur-
vey (SDSS, Blanton et al. 2017), Dark Energy Survey (DES,
Dark Energy Survey Collaboration 2016), Dark Energy Spec-
troscopic Instrument (DESI, DESI Collaboration 2016), Prime
Focus Spectrograph (PFS, Tamura et al. 2016), the Roman Space
Telescope (Spergel et al. 2015), Spectro-Photometer for the His-
tory of the Universe and Ices Explorer (SPHEREx, Doré et al.
2018), and Large Synoptic Survey Telescope (LSST, Ivezić et al.
2019). Studying voids requires redshift surveys of very large vol-
ume, deep enough in the red band to measure a huge number of
redshifts also for low-mass galaxies, and to map in detail signifi-
cant contiguous fractions of the observable Universe. The Euclid
survey, expected to sample the sky over 15 000 deg², will pro-
vide a unique opportunity to capitalise on cosmic voids, to lever-
age on measurements of the galaxy distribution at large scales
and to improve our knowledge on cosmology and fundamental
physics. Voids hold the keys to shed light on some of today's
open problems in cosmology (Pisani et al. 2019, and references
therein).

Cosmic voids from recent galaxy surveys have been used
in a wide range of cosmological applications. They are sensi-
tive to geometric effects, such as the Alcock–Paczynski effect
(Alcock & Paczynski 1979; Lavaux & Wandelt 2012; Sutter
et al. 2012, 2014a; Hamaus et al. 2016; Mao et al. 2017a)
and baryonic acoustic oscillations (Kitaura et al. 2016; Liang
et al. 2016; Chan & Hamaus 2021; Forero-Sánchez et al. 2022;
Khoraminezhad et al. 2022), as well as redshift-space distor-
tions (RSD, Paz et al. 2013; Hamaus et al. 2014a, 2015, 2017,
2020; Cai et al. 2016; Chuang et al. 2017; Achitouv et al. 2017;
Achitouv 2019; Hawken et al. 2017, 2020; Correa et al. 2019,
2022; Nadathur & Percival 2019; Nadathur et al. 2019, 2020;
Aubert et al. 2022), weak lensing (Melchior et al. 2014; Clampitt
& Jain 2015; Chantavat et al. 2016, 2017; Gruen & Dark Energy
Survey Collaboration 2016; Cai et al. 2017; Sánchez et al.
2017b; Baker et al. 2018; Brouwer et al. 2018; Fang et al. 2019;
Vielzeuf et al. 2021; Davies et al. 2021; Bonici et al. 2022), the
integrated Sachs-Wolfe effect and cross-correlation with the cos-
mic microwave background (CMB, Baccigalupi 1999; Granett
et al. 2008; Pápai & Szapudi 2010; Cai et al. 2010, 2014a,b,
2017; Nadathur et al. 2012; Flender et al. 2013; Ilić et al. 2013;
Nadathur & Crittenden 2016; Kovács et al. 2017, 2019, 2022a,b;
Kovács 2018; Dong et al. 2021; Hang et al. 2021). See e.g. Pisani
et al. (2019) and Moresco et al. (2022) for a review on cosmic
void applications for cosmology.

In this paper we consider the void size function, which
describes the number density of voids as a function of their size.
Over the last two decades, studies of the hierarchical evolution of
the void population in the excursion-set framework have allowed
the construction of a theoretical void size function model built
from first principles, the so-called Sheth & van de Weygaert
model (Sheth & van de Weygaert 2004), later extended by
Jennings et al. (2013). The void size function and its link to voids
detected in galaxy surveys have been explored in depth with cos-
mological simulations (Furlanetto & Piran 2006; Platen et al.
2007; Paranjape et al. 2012; Jennings et al. 2013; Pisani et al.
2015a; Ronconi et al. 2019; Contarini et al. 2019; Verza et al.
2019; Contarini et al. 2021) and recently this statistic has proved
to be a promising tool to constrain cosmology (Pisani et al.
2015a; Sahlén 2019; Contarini et al. 2019; Verza et al. 2019;

Kreisch et al. 2022). The void size function has already been
measured in surveys (see e.g. Nadathur 2016; Mao et al. 2017b;
Aubert et al. 2022; Hamaus et al. 2020), and used for extreme-
value statistics cosmology constraints (Sahlén et al. 2016), but
the void size function as a stand-alone probe has not yet been
used to derive cosmological constraints.

In this work we focus on the power of the void size func-
tion from the *Euclid* survey to constrain cosmological parame-
ters. This study relies on the largest *Euclid*-like light-cone, the
*Flagship* simulation (Potter et al. 2017). The paper belongs to
a series of companion papers investigating the scientific return
that can be expected from voids observed by the *Euclid* mis-
sion. It aims at measuring and theoretically modelling the void
size function from the Flagship simulation, providing a state-of-
the-art forecast for void numbers to be expected from the *Euclid*
survey. Our model allows us to estimate the constraining power
of the void size function on the DE equation of state while also
varying the total matter density of the Universe and the total
mass of neutrinos. This analysis is focused on voids found in the
spectroscopic galaxy distribution, for which the identification of
voids is particularly accurate and reliable. We note that we leave
for future work the measurement of the void size function in
the photometric galaxy distribution from *Euclid*, for which the
data treatment greatly differs from the spectroscopic one (see e.g.
Pollina et al. 2019; Bonici et al. 2022).

The paper is organised as follows: in Sect. 2 we introduce the
Flagship simulation, and describe the void finder and the clean-
ing algorithm used to obtain the void catalogue; in Sect. 3 we
present the theoretical model of the void size function (Sect. 3.1),
describe how to self-consistently align the measured void cat-
alogue with the theoretical description (Sect. 3.2), discuss the
Bayesian statistical analysis used to perform the cosmologi-
cal forecasts (Sect. 3.3), and finally introduce the cosmologi-
cal models considered in this work (Sect. 3.4). In Sect. 4 we
fit the theoretical model to the measured void size function in
the Flagship simulation (Sect. 4.1) to obtain constraints on the
DE equation of state and the remaining considered cosmolog-
ical parameters, for different adopted approaches (Sect. 4.2);
we conclude giving a discussion and a summary of our results
in Sect. 5.

## 2. Galaxy and void catalogues

We now introduce the main tools for our work: the simulation and
the void catalogues. This section also includes a brief description
of the void finder and of the void catalogue preparation.

### 2.1. *Flagship simulation*

In this work we employed the *Euclid* Flagship mock galaxy cata-
logue<fnref n="1" /> (Castander et al., in prep.). This catalogue was created
running a simulation of two trillion dark matter particles in a
periodic box of $L = 3780 h^{-1}$ Mpc per side (Potter et al. 2017),
with a flat $\Lambda$-cold dark matter ($\Lambda$CDM) cosmology characterised
by the parameters $\Omega_m = 0.319$, $\Omega_b = 0.049$, $\Omega_{de} = 0.681$,
$\sigma_8 = 0.83$, $n_s = 0.96$ and $h = 0.67$, as obtained by *Planck* in
2018 (Planck Collaboration VI 2020). The simulation box was
converted into a light-cone and the dark matter haloes have been
identified using the *Rockstar* halo finder (Behroozi et al. 2013).
These haloes were populated with central and satellite galaxies

<footnote n="1">¹ Version 1.8.4.</footnote>
<table><caption>**Table 1.** Void counts measured in the redshift-space mock galaxy catalogue considering the redshift bins and selections used for this analysis.</caption><thead><tr><th>z range</th><th>Shell volume [(h<sup>-1</sup> Gpc)<sup>3</sup>]</th><th>MGS [h<sup>-1</sup> Mpc]</th><th>f<sub>cut</sub>(z)</th><th>All voids</th><th>Voids after cleaning</th></tr></thead><tbody><tr><td>0.950–1.035</td><td>1.157</td><td>10.28</td><td>2.30</td><td>4989</td><td>343</td></tr><tr><td>1.035–1.126</td><td>1.329</td><td>11.02</td><td>2.24</td><td>4935</td><td>343</td></tr><tr><td>1.126–1.208</td><td>1.269</td><td>11.74</td><td>2.18</td><td>4232</td><td>342</td></tr><tr><td>1.208–1.318</td><td>1.796</td><td>12.63</td><td>2.12</td><td>5302</td><td>341</td></tr><tr><td>1.318–1.455</td><td>2.363</td><td>13.51</td><td>2.06</td><td>5935</td><td>342</td></tr><tr><td>1.455–1.700</td><td>4.490</td><td>14.45</td><td>2.00</td><td>8435</td><td>343</td></tr><tr><td>0.950–1.700</td><td>12.40</td><td>13.69</td><td>2.15</td><td>33 828</td><td>2054</td></tr></tbody></table>

**Notes.** The first column represents the minimum and the maximum redshift values for each bin, while the second and the third columns provide the volume in units of $(h^{-1} \text{ Gpc})^3$ corresponding to each shell of the sky octant, and the mean separation between galaxies (MGS), respectively. The fourth column reports the factor, $f_{\text{cut}}(z)$, used to select voids unaffected by the incompleteness of counts. The last two columns show the number counts of voids identified by the VIDE void finder with $R > f_{\text{cut}}(z)$ MGS and of voids obtained after the cleaning procedure with $R_{\text{eff}} > f_{\text{cut}}(z)$ MGS, respectively. In the last row we show the total volume of all redshift shells, the mean MGS and $f_{\text{cut}}(z)$ values and the total void counts corresponding to the entire range of redshifts. A table with equi-spaced redshift bins is provided in Appendix A to serve as a reference for future forecast analyses needing void numbers.

voids whose centre is closer than $30 h^{-1} \text{ Mpc}$ to the edge and corrected the model accordingly for the selected volume. We then pruned voids at low and high redshifts to further avoid selection effects given by redshift boundaries of the light-cone, and we divided the sample in six redshift bins. This number is found as the optimal compromise between maximising the number of redshift shells and keeping void numbers in bins high enough to avoid falling in the shot-noise dominated regime. In order to have shells with roughly the same number of cleaned voids identified in redshift space and to avoid border effects at the light-cone redshift boundaries, we selected the following redshift bin edges: $z_i = [0.950, 1.035, 1.126, 1.208, 1.318, 1.455, 1.700]$. Each shell contains at least 340 voids, within the range of effective radii considered in the analysis of the measured void size function described below.

Tracer sparsity leads to a drop of counts for small voids in the measured void size function (Sutter et al. 2014b; Verza et al. 2019). The incompleteness depends on the mean galaxy separation and therefore on the redshift of the sample (Jennings et al. 2013; Ronconi et al. 2019; Contarini et al. 2019; Verza et al. 2019). Modelling the drop of counts for small voids is not trivial. To avoid falling in this regime, we conservatively excluded from the analysis voids with radii falling in the range of scales affected by incompleteness. We removed voids with radii smaller than MGS $f_{\text{cut}}(z)$, where MGS is the mean galaxy separation and $f_{\text{cut}}(z)$ is a factor dependent on the redshift of the sample. We computed the value of mean galaxy separation as $\text{MGS} = (V_{\text{shell}}/N_{\text{gal}})^{1/3}$, where $V_{\text{shell}}$ is the volume of the redshift shell analysed and $N_{\text{gal}}$ is the number of galaxies present in it. The factor $f_{\text{cut}}(z)$ is chosen empirically based on the drop of void counts and on the steep departure from the theoretical model. We found that values spanning from 2.3 (lowest redshift bin) to 2 (highest redshift bin) for $f_{\text{cut}}(z)$ ensure the exclusion of spatially unresolved voids in redshift space.

Since we expect the resulting void size function in redshift space to be shifted towards greater effective radii due to the effects of RSD (Pisani et al. 2015b; Zhao et al. 2016; Nadathur 2016; Correa et al. 2021), we extended the minimum radius for the real-space case, adding an extra bin at small radii while keeping the same binning of the redshift-space case for higher bins. We verified that these choices allow us to be outside of the incompleteness regime, for both the void size function in real and redshift space.

In Table 1 we show the number counts of voids selected from the redshift-space mock galaxy catalogue. For each of the

redshift bins with edges $z_i$, we report the volume occupied by the shell and the MGS of the tracers, together with the factor $f_{\text{cut}}(z)$ used to compute the minimum void radius considered in this analysis. For completeness, we show the void number counts both before and after the cleaning procedure aimed to line up observed voids to theoretical voids, according to the void size function model. The sharp decrease of the void number is an expected outcome of the cleaning procedure, that selects the largest in volume and deepest underdensities identified by VIDE and rescales their sizes towards smaller values, causing a more severe rejection of voids during the removal of the spatial scales affected by the incompleteness of counts. Although this conservative approach leads to a loss of the void size function constraining power, it ensures the selection of an high-purity void sample and a robust treatment of void number counts. In future works, different approaches will be explored to improve the void selection also at small radii: among these, the application of machine learning techniques (Cousinou et al. 2019) is promising to carefully remove only spurious voids and consequently enhance the performance of the void size function as a cosmological tool.

## 3. Theory and methods

In this section we introduce the theoretical background necessary for this work. We first discuss the model of the void size function, then we present the prescriptions applied to extend this model to voids identified in the distribution of biased tracers. We describe the Bayesian statistical analysis used to provide forecasts on the DE equation of state and on the sum of neutrino masses. Finally, we present the cosmological scenarios considered in our analysis.

### 3.1. Theoretical void size function

To estimate the constraining power of the void size function, i.e. the distribution function of void radii, we first need a theoretical model. The void size function model most widely used in the literature relies on the excursion-set formalism, developed within the framework of the halo mass function (Peacock & Heavens 1990; Cole 1991; Bond et al. 1991; Mo & White 1996). This model was first proposed by Sheth & van de Weygaert (2004) and extended by Jennings et al. (2013). The distribution of fluctuations that become voids, i.e. the multiplicity function, is obtained as the conditional first crossing distribution
of the matter density contrast filtered at decreasing Lagrangian radius in a double barrier problem: a fluctuation becomes a void at a radius $R_v$ if the filtered density contrast first crosses the void formation threshold $\delta_v^L$ at $R_v$, without having crossed the threshold for collapse $\delta_c^L$ at any larger scale<fnref n="4" /> (Sheth & van de Weygaert 2004). The multiplicity function of Sheth & van de Weygaert (2004) is derived for spherical fluctuations in Lagrangian space, i.e. the initial density field linearly evolved to the epoch of interest, while the observed voids live in the fully nonlinear evolved density field in comoving coordinates, i.e. the Eulerian space. The spherical approximation allows us to easily go back and forth from Lagrangian to Eulerian space in all the computations.

The void size function probes the inner region of cosmic voids and in contrast to the collapsing case, i.e. halo formation (Monaco 1995; Sheth & Tormen 2002), the spherical approximation is accurate enough for this purpose, at least for voids of scales detectable by Euclid (Icke 1984; Verza et al. 2019).

The linear threshold for collapse is fixed at $\delta_c^L = 1.686$, according to the collapse of a spherical fluctuation. This value corresponds in an Einstein-de Sitter model to the full collapse in linear theory, when the halo virialises. The void case is different: if the initial underdensity identifying a void is deep enough, its evolution is not marked by any specific event, and it continues its outward-directed expansion forever. It is common to consider the shell-crossing<fnref n="5" /> condition as the event that identifies the void formation (Blumenthal et al. 1992; Sheth & van de Weygaert 2004; Jennings et al. 2013), but this condition strictly depends on the initial density profile of the underdensity. For an initial density profile represented as a step function, shell-crossing happens at a nonlinear matter density contrast of $\delta_{sc}^{NL} \approx -0.8$, corresponding to a linear threshold of $\delta_{sc}^L \approx -2.7$. Considering more physical density profiles (e.g. Hamaus et al. 2014b; Massara & Sheth 2018), shell-crossing in voids does not necessarily happen and, if it does, it may occur at even lower threshold values. Given the considered thresholds, our voids remain far from the shell-crossing regime, therefore, in principle, it is always possible to map the measured Eulerian density profile to the corresponding Lagrangian one. As we do not reach shell-crossing, we have the freedom to choose any threshold value to define void formation (Ronconi et al. 2019; Contarini et al. 2019, 2021; Verza et al. 2019).

The multiplicity function, as given by Sheth & van de Weygaert (2004), is:

$$f_{\ln \sigma}(\sigma) = 2 \sum_{j=1}^{\infty} \exp\left(-\frac{(j\pi x)^2}{2}\right) j\pi x^2 \sin(j\pi D), \quad (1)$$

with

$$D = \frac{|\delta_v^L|}{\delta_c^L + |\delta_v^L|}, \quad x = \frac{D}{|\delta_v^L|} \sigma, \quad (2)$$

where $\sigma$ is the square root of the variance of linear matter perturbations on the Lagrangian scale $r_L$. All these quantities are computed in the linear regime, on which the excursion-set formalism relies. The void size function in Lagrangian space is then readily derived as (Sheth & van de Weygaert 2004; Jennings et al. 2013):

$$\frac{dn_L}{d \ln r_L} = \frac{f_{\ln \sigma}(\sigma)}{V(r_L)} \frac{v \ln \sigma^{-1}}{d \ln r_L}, \quad (3)$$

where $V(r_L) = 4\pi r_L^3/3$ is the volume of the spherical fluctuation of radius $r_L$. Conversely to the case of the halo mass function, the void size function in Eulerian space is different from the one in Lagrangian space. Firstly, the expansion of voids from linear to nonlinear theory has to be taken into account. The evolution of perturbations in the nonlinear regime provides the conversion from the linear to the nonlinear shell radius:

$$\frac{r}{r_L} = \left(\frac{\bar{\rho}}{\rho_v}\right)^{1/3}, \quad (4)$$

where $\bar{\rho}$ is the mean density of the Universe and $\rho_v$ is the average density within the void. Secondly, to prevent the fraction of the volume occupied by voids from exceeding unity in the transition from linearity to nonlinearity, we fix the void volume fraction of the Universe to be equal in the linear and in the nonlinear regimes (Jennings et al. 2013):

$$V(r) dn = V(r_L) dn_L|_{r_L=r_L(r)}. \quad (5)$$

With this requirement the model ensures void volume conservation – hereafter Vdn model, following Jennings et al. (2013) – and from Eq. (3) we can derive the final definition of the theoretical void size function adopted in this paper:

$$\frac{dn}{d \ln r} = \frac{f_{\ln \sigma}(\sigma)}{V(r)} \left. \frac{d \ln \sigma^{-1}}{d \ln r_L} \right|_{r_L=r_L(r)}. \quad (6)$$

## 3.2. Methodology

To compare the measured and the theoretical void size functions, we need to link objects found by the void finder in the tracer distribution with the ideal spherical and isolated voids described by the void size function theoretical model (Jennings et al. 2013). Any watershed void finder defines a region spanning from its density minimum to its overdense ridge (Roerdink & Meijster 2000; Neyrinck 2008; Platen et al. 2007; Sutter et al. 2015). On the contrary, the theoretical voids are matter density fluctuations for which the mean density contrast in a sphere reaches a specific threshold value at a radius $R_{eff}$. Previous papers attempted to mitigate this difference by modifying the threshold of the model (Pisani et al. 2015a; Sahlén et al. 2016; Sahlén 2019), in particular considering marginalisation over the threshold, for cosmological uses of the model.

It is useful to recall that the Vdn model describes voids evolving in the total matter density field, but that in our case (and when dealing with data) we can only identify voids in the galaxy density field. Therefore, to align these objects to those modelled by the theory, we need to relate the characteristic density threshold used in the theoretical model, $\delta_v^L$, to the corresponding one in the galaxy density field. To accomplish this purpose, we relied on the following two steps for data preparation: first, we measured the mean density profiles of cosmic voids to find the radius of the sphere at which the mean density contrast reaches the desired value $\delta_{v,tr}$ in the galaxy distribution

<footnote n="4">⁴ In this paper the density contrasts derived in linear and nonlinear theory are indicated with the superscripts L and NL, respectively. In absence of any superscript, we take for granted the reference to the nonlinear counterpart.</footnote>

<footnote n="5">⁵ In a purely theoretical framework, cosmic voids can be represented by negative perturbations modelled as a set of concentric shells. During their evolution, the inner shells will expand faster than the outer ones and will eventually overtake the more external ones, giving rise to the so-called 'shell-crossing' phenomenon. During this event the trajectories of two fluid elements in Lagrangian coordinates cross each other, breaking the one-to-one map between Lagrangian and Eulerian space. See Massara & Sheth (2018) for a detailed study on the mapping between Lagrangian and Eulerian voids.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Summary of the underdensity thresholds and corresponding relationships introduced in this work. In the first row we report the values relative to the shell-crossing phenomenon (subscript sc) together with their corresponding numerical values, shown in the white tags. In the second row we list the density contrasts values used in our analysis. The quantities circled in red refer to density contrasts defined in nonlinear theory (superscript NL), while those circled in yellow are defined in linear theory (superscript L). The latter are used to compute the Vdn model and are obtained from the former by means of Eq. (8). All the quantities listed in the central and in the right columns are defined in the density field traced by unbiased objects, i.e. DM particles (subscript DM). On the far left we report instead the underdensity threshold used to define void in this paper. It has a value of -0.7 and is computed in the density field traced by galaxies (subscript tr). To find its corresponding value in the underlying DM field we make use of Eq. (7).</figcaption></figure>

(Jennings et al. 2013; Ronconi & Marulli 2017; Ronconi et al. 2019; Contarini et al. 2019; Verza et al. 2019), i.e. the resized radius, $R_{eff}$. Second, it is necessary to find the corresponding density contrast in the underlying matter density distribution, within the resized radius. Recently the properties of voids in the galaxy distribution, as well as of galaxies and tracer bias within cosmic voids, have been explored extensively (Furlanetto & Piran 2006; Sutter et al. 2014b; Neyrinck et al. 2014; Pollina et al. 2017, 2019; Contarini et al. 2019, 2021). To recover the matter density contrast corresponding to the threshold value in the galaxy density field, we need to model the galaxy distribution inside cosmic voids taking into account tracer bias. To describe tracer bias, i.e the bias of the cosmological objects chosen to trace voids, various possibilities have been considered, including a full theoretical description (see discussion in Verza et al. 2019 and Desjacques et al. 2018 for an extensive review), or a robust modelling of bias inside voids based on simulations (Pollina et al. 2017, 2019; Contarini et al. 2019, 2021). In this analysis we chose to rely on the latter, following Contarini et al. (2019, 2021). These works showed that it is possible to extend the Vdn model by considering a linear relationship between tracer and matter density contrast in cosmic voids, $\delta_{v,tr}^{NL}$ and $\delta_{v,DM}^{NL}$, with a dependence only on the large-scale effective bias $b_{eff}$:

$$ \delta_{v,DM}^{NL} = \frac{\delta_{v,tr}^{NL}}{\mathcal{F}(b_{eff})}, \qquad (7) $$

where $\delta_{v,DM}^{NL}$ is the value of the threshold in the dark matter field to be used in the Vdn model, after its conversion in linear theory (Jennings et al. 2013). For $\Lambda$CDM and the DE equations of state considered in this work, the conversion from nonlinear to linear density contrast in the matter field is cosmology and redshift independent with very high accuracy (Jennings et al. 2013; Pace et al. 2017), allowing us to exploit the fast and precise Bernardeau (1994) fitting formula:

$$ \delta_v^L = C [1 - (1 + \delta_v^{NL})^{-1/C}], \quad \text{with } C = 1.594. \qquad (8) $$

Comparing the void density profiles computed both in the Friend-of-Friends (FoF) halos and dark matter particle field, Contarini et al. (2019) found that the function $\mathcal{F}$, reported in Eq. (7), is well modelled as a linear relation of the large-scale effective bias $b_{eff}$:

$$ \mathcal{F}(b_{eff}) = B_{\text{slope}} b_{eff} + B_{\text{offset}}, \qquad (9) $$

where $B_{slope}$ and $B_{offset}$ are the values of the first and second coefficients of the linear function, respectively. This relation will be calibrated in this work using the samples of galaxies and voids extracted from the Flagship light-cone and the resulting values of $B_{slope}$ and $B_{offset}$, together with the associated uncertainties, will be presented in Sect. 4.1. The parametrisation introduced in equations Eq. (7), (9) was tested also in Contarini et al. (2021), using different selection criteria for the halo identification and verifying its negligible dependence on the cosmological model. In particular, this relation is tested varying the neutrino mass and the parameters of the $f(R)$ class of modified gravity models, in the form introduced by Hu & Sawicki (2007). The quantity represented by the function $\mathcal{F}(b_{eff})$ parametrises the value of the tracer effective bias measured inside cosmic voids and it has been denoted $b_{punct}^6$ in Contarini et al. (2019, 2021). To facilitate the reader’s comprehension of the adopted methodology, we summarise in Fig. 1 all the negative density contrasts mentioned in this paper and their relative relations.

To convert the underdensity threshold of the Vdn model according to the function $\mathcal{F}(b_{eff})$, we first need to compute accurately the large-scale effective linear bias of our galaxy sample. For this estimate we followed the same prescriptions described in Marulli et al. (2013, 2018). In particular, we exploited the galaxy two-point correlation function (2PCF), performing a Bayesian statistical analysis to infer the effective bias, $b_{eff}$. We computed the angle-averaged galaxy 2PCF $\hat{\xi}(r)$ in real space creating a random catalogue 10 times larger than the original one and using the Landy & Szalay estimator (Landy & Szalay 1993).

We then estimated the covariance matrix, which measures the variance and correlation between the different bins of the 2PCF. For this purpose we applied the Bootstrap method, dividing the original catalogues in 125 sub-catalogues and constructing 100 realisations by resampling from the sub-catalogues,

<footnote n="6">6 We note that $b_{punct}$ and $\mathcal{F}(b_{eff})$ refer to the same quantity, i.e. the value of the tracer bias computed inside cosmic voids. This quantity represents the relation between the void density profiles computed using biased (e.g. galaxies) and unbiased (e.g. dark matter particles) mass tracers (see Pollina et al. 2017, 2019; Contarini et al. 2019, 2021). Nevertheless, we keep two different notations since their computation is different: $b_{punct}$ is measured for each bin of redshift, while $\mathcal{F}(b_{eff})$ is given as a function of $b_{eff}$ and varies linearly with it by construction (see Sect. 4.1).</footnote>
with replacement. In the end we performed a full Markov chain Monte Carlo (MCMC) analysis of the 2PCF, using a Gaussian likelihood function. The 2PCF model, $\xi_{\text{mod}}(r)$, is computed as follows:

$$ \xi_{\text{mod}}(r) = b_{\text{eff}}^2 \xi_m(r), \qquad (10) $$

where $\xi_m(r)$ is the matter 2PCF, which is estimated by Fourier transforming the matter power spectrum, $P_m(k)$, computed with the Code for Anisotropies in the Microwave Background (CAMB<fnref n="7" />, Lewis et al. 2000). Then we accurately estimated the effective bias parameter $b_{\text{eff}}$ by sampling its posterior distribution with the MCMC modelling in the range of scales of [20–40] h⁻¹ Mpc.

We underline that the relative error associated to $b_{\text{eff}}$ is expected to be relatively small because of the strategy used to compute this quantity relying on the galaxy catalogue in real space and assuming the true cosmological parameters of the simulation. A more complete and realistic treatment will be performed in the future, including in the analysis the modelling of the multipoles of the 2PCF, which will allow us to take into account the effects of redshift-space and geometrical distortions (see e.g. Scoccimarro 2004; Taruya et al. 2010; Beutler et al. 2017; Pezzotta et al. 2017).

We finally recall that another approach to compute effective bias, analogous to that applied in this work, is to measure the 2PCF in Fourier space and to model it via the theoretical matter power spectrum $P(k)$ (see e.g. Beutler et al. 2017). Additionally, an alternative methodology to extract Flagship galaxy bias is to follow e.g. Tutusaus et al. (2020), who parametrised the Flagship galaxy bias as a function of $z$, albeit for the photometric redshift selection.

### 3.3. Bayesian statistical analysis

In this work we used a reliable forecast method for the sensitivity of the void size function in the *Euclid* survey to constrain the cosmological model, based on a parameter extraction from Bayesian analysis with MCMC (Perotto et al. 2006; Wang et al. 2009; Lahav et al. 2010; Martinelli et al. 2011; de Bernardis et al. 2011; Wolz et al. 2012; Hamann et al. 2012; Khedekar & Majumdar 2013; Audren et al. 2013).

In order to forecast the sensitivity of void counts with an MCMC analysis in *Euclid*, we have to consider that the Flagship simulation covers about one third of the *Euclid* survey. We obtained the *Euclid* predicted void number counts relying on the theoretical void size function model validated on the Flagship simulation (see Sect. 4.1), that is assuming a fiducial ΛCDM cosmology with the cosmological parameters of the Flagship and the calibration in redshift space of the Vdn model described in Sect. 4.1. We assumed the same binning of void radii employed in our Flagship analysis but consider a survey area matching the one expected for *Euclid* (roughly three times the Flagship area), rescaling the Poissonian errors of the void number counts consistently by a factor $\sqrt{3}$.

This allows us to use MCMC analysis to explore the likelihood distribution in the parameter space without any assumption on the Gaussianity of parameters and local approximations around the fiducial value, as in Fisher forecasts. Moreover, according to the Cramér–Rao inequality, the Fisher matrix gives a lower bound on the error on a parameter (Kendall et al. 1987), while the MCMC is proven to be more realistic, in particular in the presence of degeneracies (Perotto et al. 2006; Wolz et al. 2012; Audren et al. 2013; Sellentin et al. 2014). Finally, this kind

of approach allows us to compute unbiased constraints, with confidence contours centred on the Flagship simulation cosmological parameters and on the calibrated nuisance parameters $B_{\text{slope}}$ and $B_{\text{offset}}$.

According to Bayes’s theorem, given a set of data $\mathcal{D}$, the distribution of a set of parameters $\Theta$ in the cosmological model considered is given by the posterior probability:

$$ \mathcal{P}(\Theta|\mathcal{D}) \propto \mathcal{L}(\mathcal{D}|\Theta) p(\Theta), \qquad (11) $$

where $\mathcal{L}(\mathcal{D}|\Theta)$ is the likelihood and $p(\Theta)$ the prior distribution. Since in this work we consider the number counts of cosmic voids, the likelihood can be assumed to follow Poisson statistics (Sahlén et al. 2016):

$$ \mathcal{L}(\mathcal{D}|\Theta) = \prod_{i,j} \frac{N(r_i, z_j|\Theta)^{N(r_i, z_j|\mathcal{D})} \exp[-N(r_i, z_j|\Theta)]}{N(r_i, z_j|\mathcal{D})!}, \qquad (12) $$

where the product is over the radius and redshift bins, labelled as $i$ and $j$ respectively. The $N(r_i, z_j|\mathcal{D})$ quantity corresponds to the number of voids in the $i$th radius bin and $j$th redshift bin, while $N(r_i, z_j|\Theta)$ corresponds to the expected value in the cosmological model considered, given a set of parameters $\Theta$. In our work, the former is obtained from the Flagship analysis (with the void size function model validated on the Flagship simulation, but considering that the *Euclid* area will be three times larger), while the latter is given by the predictions of the void size function model varying the considered cosmological parameters $\Theta$.

In performing the MCMC analysis, the mapping between redshift and comoving distance changes with the cosmological parameters assumed at each step of the chain. This introduces geometrical distortions for all the considered sets of cosmological parameters (different from the true one). We used a fiducial cosmology to build up the void catalogue, and, in computing the likelihood, we theoretically accounted for the distortion effects on the quantities we measured. In particular, geometrical distortions can be modelled with two effects: they vary the inferred survey comoving volume and introduce the Alcock–Paczyński (Alcock & Paczynski 1979) distortion. The effect on the survey volume impacts the number of voids expected in the survey. The theoretical void size function model predicts the number density of voids in each radius and redshift bin. Therefore, to obtain the total number of voids, the number density has to be multiplied by the volume, which is impacted by the cosmology. On the other hand, the Alcock–Paczyński (Alcock & Paczynski 1979) distortion affects the size of voids and introduces an anisotropy between the orthogonal and the parallel direction with respect to the line-of-sight. These quantities change according to (Sánchez et al. 2017a):

$$ r_{\parallel}^{\prime} = \frac{H(z)}{H'(z)} r_{\parallel} = q_{\parallel}^{-1} r_{\parallel}, \qquad r_{\perp}^{\prime} = \frac{D'_{A}(z)}{D_{A}(z)} r_{\perp} = q_{\perp}^{-1} r_{\perp}; \qquad (13) $$

where $r_{\parallel}$ and $r_{\perp}$ are the comoving distances between two objects at redshift $z$ projected along the parallel and perpendicular direction with respect to the line-of-sight, $H(z)$ is the Hubble parameter and $D_A(z)$ the comoving angular-diameter distance. The primed quantities refer to the calculation at the fiducial cosmology, the non-primed to the true cosmology, assumed in a MCMC step. It follows that the volume of a sphere with radius $R$ appears modified according to $R = q_{\parallel}^{1/3} q_{\perp}^{2/3} R'$ (Ballinger et al. 1996; Eisenstein et al. 2005; Xu et al. 2013; Sánchez et al. 2017a; Hamaus et al. 2020; Correa et al. 2021), so the void size function expected in the survey is shifted accordingly. We checked

<footnote n="7">⁷ http://camb.info</footnote>
the validity of this relationship varying the cosmology used to get the comoving distances from redshifts and consequently correcting the radius $R_{\text{eff}}$ at which voids reach the underdensity threshold $\delta_{\text{v,tr}}^{\text{NL}}$.

We assumed the void centres to remain at the same locations at different cosmologies. While void shapes can suffer from symmetric geometrical distortions, this marginally affects the identification of void centres, and the effect is even smaller since the void size function is an averaged quantity. Furthermore, the variation caused by the change of the cosmological parameters on void radii is taken into account by the modelling of the Alcock-Paczyński effect, therefore the cleaning procedure (see Sect. 2.2) is applied only once to the void sample, considering a fiducial $\Lambda$CDM cosmology. We note that the combination of the two effects – volume effect acting on the expected number density, and the Alcock-Paczyński effect acting on the void sizes – enhances the constraining power of the void size function.

### 3.4. Cosmological models

The aim of this work is to investigate the constraining power of the void number count statistic on cosmological parameters, focusing in particular the DE equation-of-state parameters. We considered two cosmological models, extending the standard $\Lambda$CDM with different DE equation of states. The first model, wCDM, implements a constant DE equation of state w; the second one, $w_0w_a$CDM, parametrises dynamical DE models with the popular Chevallier-Polarski-Linder (CPL) equation of state (Chevallier & Polarski 2001; Linder 2003):

$$w_{\text{CPL}}(z) = w_0 + w_a \frac{z}{z+1}. \qquad (14)$$

Both cosmological models consider a flat universe and do not include spatial fluctuations of the DE, which are negligible given the scales considered in this work (see e.g. Khoraminezhad et al. 2020). We performed the MCMC analysis of each cosmological model focusing on different sets of free cosmological parameters: together with the DE equation of state parameters (i.e. w or $w_0$ and $w_a$, depending on the cosmological model) the density parameter $\Omega_m$ or the sum of neutrino masses $M_v$ are allowed to vary. Moreover, we analysed both the cases with two different approaches: firstly, fixing the parameters of the extended Vdn model, $B_{\text{slope}}$ and $B_{\text{offset}}$, to the median values obtained from the calibration performed with Flagship data (label: fixed calibration); secondly, allowing $B_{\text{slope}}$ and $B_{\text{offset}}$ to vary in the parameter space described by a 2D Gaussian distribution centred on their median values and given by the calibration with the Flagship simulation (label: relaxed calibration).

The two adopted approaches are meant to demonstrate the impact of the calibration that will be performed in Sect. 4.1 on the cosmological forecast. In this work the constraints on the parameters $B_{\text{slope}}$ and $B_{\text{offset}}$ are indeed limited to the statistical relevance of the number counts of voids identified by means of the Flagship galaxies. The case in which the cosmological forecasts are computed fixing $B_{\text{slope}}$ and $B_{\text{offset}}$ to their exact calibrated values represents therefore an optimistic evaluation of the results that we may obtain in the future thanks to the usage of larger mock catalogues, or by means of a fully theoretical modelling of the tracer bias inside cosmic voids (see Sect. 3.2).

The cosmological model considered for the analysis is characterised by a primordial comoving curvature power spectrum amplitude fixed to the Flagship simulation value, $A_s = 2.11 \times 10^{-9}$. We followed the strategy to fix this parameter in order to mimic the future application to real data, which will be supported

by the impressive constraints obtained from the study of CMB anisotropies by Planck Collaboration VI (2020). Thanks to this approach, for each MCMC step we derived $\sigma_8$, i.e. the root mean square mass fluctuation in spheres with radius $8 h^{-1}$ Mpc. We relied on CAMB to compute this quantity as a derived parameter, which depends on all the cosmological parameters involved in the evolution of the matter power spectrum $P_m(k)$.

The density parameter $\Omega_m$ is computed as the sum of cold dark matter, baryon and neutrino energy densities, $\Omega_m = \Omega_{\text{cdm}} + \Omega_b + \Omega_v$, and its variation in the Bayesian statistical analysis is balanced by the changing of the DE density parameter, $\Omega_{\text{de}}$, to keep flat the universe geometry, $\Omega_{\text{de}} = 1 - \Omega_m$.

The implementation of massive neutrinos in the MCMC analysis was performed considering the sum of the mass of neutrinos as a free parameter in the cosmological model. Neutrinos were modelled with one massive eigenstate and two massless ones, assuming an effective number of neutrino species $N_{\text{eff}} = 3.04$ (Froustey et al. 2020; Bennett et al. 2021) and relating the neutrino mass to the neutrino density parameter as (Mangano et al. 2005):

$$\Omega_v = \frac{M_v}{93.14 h^2 \text{ eV}}, \qquad (15)$$

where we denote $M_v = \sum m_v$ as the sum of the neutrino mass eigenstates.

Since the thermal free-streaming of massive neutrinos suppresses density fluctuations, the abundance of voids changes with massive neutrinos, with respect to the massless neutrinos case (see e.g. Kreisch et al. 2019; Schuster et al. 2019; Contarini et al. 2021, for a discussion). We included the variation of the neutrino density parameter, $\Omega_v$, in the MCMC analysis, by keeping the value of the total matter density $\Omega_m$ fixed (see Sect. 2.1), thus rescaling consistently the cold dark matter density parameter $\Omega_{\text{cdm}}$. We tested the effect of considering both $\Omega_m$ and $\Omega_v$ as free parameters of the model, finding a strong degeneracy between the two. We chose therefore to separate into different scenarios the models having either $\Omega_m$ or $\Omega_v$ unconstrained: our goal is to investigate the sensitivity of the void size function to these two parameters independently, aiming at using this void statistic in combination with other cosmological probes.

We relied on CAMB for the computation of the total matter power spectrum used to predict the theoretical model of the void size function. The region of the parameter space characterised by a DE equation of state with $w_0 + w_a > 0$ is not covered by CAMB.

## 4. Results

The aim of this section is to compare our theoretical predictions with the void size function measured from the Flagship simulation. We then provide forecasts for the Euclid survey, using a Bayesian statistical analysis to predict constraints on the parameters of the DE equation of state, modelling the void size function according to the theoretical prescriptions reported in Sect. 3.

### 4.1. Void size function analysis

To compare the theoretical void size function with the number counts of voids measured in the galaxy distribution, we need to convert the threshold $\delta_{\text{v,tr}}^{\text{NL}}$ fixed in measurements to the corresponding one in the matter distribution, as described in Sect. 3.2. First of all, we verified the calibration of the relation $\mathcal{F}(b_{\text{eff}})$ reported in Eq. (9) using the Flagship simulation. To this end,
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Calibration of the relation $F(b_{eff})$ from Eq. (9), required for the conversion of the threshold $\delta_{v,\text{tr}}$ in Eq. (7). Left: the 68% and 95% confidence levels in the $B_{\text{slope}}-B_{\text{offset}}$ plane for the void catalogues built both in real (blue) and in redshift space (orange). Right: the solid lines represent the resulting linear relations $F(b_{eff})$ obtained with the calibrated coefficients $B_{\text{slope}}$ and $B_{\text{offset}}$ for real (blue) and redshift space (orange), while the shaded regions indicate an uncertainty of $2\sigma$ on the relationships. The markers represent the calibration obtained for each bin of redshift, leaving $b_{\text{punct}}$ as the only free parameter of the void size function model when fitting the measured void number counts. This alternative calibration provides a value of $b_{\text{punct}}$ for each redshift of the sample and is associated with the value of the effective bias $b_{\text{eff}}$ of the Flagship galaxies at that specific redshift. As a comparison we also show the linear function calibrated using FoF dark matter haloes in real space by Contarini et al. (2019), displayed with a dashed grey line.</figcaption></figure>

we extracted the value of $B_{\text{slope}}$ and $B_{\text{offset}}$ by leaving them as free parameters with uniform priors of the extended Vdn model and fitting the measured void number counts in the selected redshift bins, considering also a Gaussian prior for $b_{\text{eff}}$ at each redshift. We notice that, since the error on the effective bias only corresponds to a few percent of its value, the variation allowed for this parameter during the fit is small. All the remaining cosmological parameters were kept fixed to the Flagship simulation values during this calibration.

With this prescription we obtained the confidence levels reported on the left panel of Fig. 2, for the void size function measured in both real and redshift space in light blue and orange, respectively. The resulting coefficients for the calibrated relations are:

$$F(b_{\text{eff}}) = (0.96 \pm 0.04) b_{\text{eff}} + (0.44 \pm 0.07) \quad \text{and} \quad (16)$$

$$F(b_{\text{eff}}) = (0.96 \pm 0.03) b_{\text{eff}} + (0.26 \pm 0.06), \quad (17)$$

for the redshift-space and the real-space void abundance, respectively.

We show on the right panel of Fig. 2 the corresponding linear relations obtained with these calibrations, with a shaded area representing an uncertainty of $2\sigma$. As a comparison, we present in the same plot the values computed for $b_{\text{punct}}$, leaving it as the only free parameter of the model and fitting separately the measures at different redshifts. This analysis is aimed at testing the precision of the calibrated relations for each redshift: in the right plot of Fig. 2 the markers with best match to the linear relations correspond in Fig. 3 to the redshift bins for which the calibrated model more accurately reproduces the measured void number counts, while points that depart from the linear relationship in Fig. 2 (right plot) will lead to a slightly worse agreement between theory and model in Fig. 3.

Finally, we report also the calibration performed in Contarini et al. (2019) using the dark matter haloes of the COu-

pled Dark Energy Cosmological Simulations (CODECS, Baldi 2012), and represented in grey in the right panel of Fig. 2. At lower redshifts the calibration we measure in this paper is in good agreement with the calibration from the CODECS simulation, characterised by a cosmology consistent with the results of the seven-year Wilkinson Microwave Anisotropy Probe (WMAP7, Komatsu et al. 2011), but it slightly deviates from the latter at higher redshift values. The reason for this minor deviation is twofold. Firstly it is linked to the kind of cosmic tracers (i.e. dark matter haloes or galaxies) and the selection criteria (i.e. minimum mass or magnitude) used to identify voids (Contarini et al. 2021). Secondly it is related to the fact that in Contarini et al. (2019) the calibration was performed for redshift from 0 to 1, while here we are testing this relationship beyond this range. The physics underlying the function $F(b_{\text{eff}})$ and its relation with the mass tracers used to identify voids will be investigated in future papers.

More importantly, since the void size function will be measured on real data from the Euclid survey, we have to deal with voids detected in redshift space. The overall effect of RSD on voids, relevant for the void size function, is an apparent enlargement of the voids’ volume, due to the elongation along the line of sight. This is reflected in a mean shift of the measured void size function towards greater radii. Even if this effect can in principle be theoretically modelled (Pisani et al. 2015b; Correa et al. 2021), we decide to parametrise it empirically as described below. Indeed, the theoretical approach requires knowledge of the void matter density profile for the entire void population, which has to be characterised in simulations and may introduce some model dependencies. We found that the parametrisation of $F(b_{\text{eff}})$ can be exploited to encapsulate also the modifications on the void sizes caused by the enlargement of cosmic voids in redshift space. This approach has the advantage of being both simple to model and robust, allowing us to take into account,
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Comparison between the measured void number counts as a function of $R_{\text{eff}}$ (the void radii rescaled by the cleaning algorithm), and the theoretical predictions given by the extended Vdn model, in six different redshift bins. The dark green circles and the dark red diamonds represent the measured void size functions in real and redshift space, respectively, while the corresponding model predictions are depicted in light blue and orange. The shaded regions indicate the uncertainty of $2\sigma$ assigned to the model through the calibration of the extended Vdn parameters. *Bottom panels:* report the residuals computed as the difference of data points from the relative theoretical model, divided by the Poissonian error associated with each data point. The hatched regions represent a band with amplitude 2 useful to check if the data points, considered with a $2\sigma$ error, are compatible with the main theoretical curve.</figcaption></figure>

with the same parameter, both the impact of tracer bias in voids and of the RSD. Moreover, this approach is fully agnostic and does not require any assumption about the void density profile, nor any other modelling, making it particularly suited to survey analyses.

It is worth noting that the relation obtained for voids in redshift space shows a greater offset but almost the same slope with respect to its analog in real space. This difference reflects the

increase of void sizes in redshift space. It also opens the way to test theoretical implementations in future work, indicating that a simple modelling of those effects should suffice to extract robust constraints.

Equipped with these calibrated relations, we now have all the elements necessary to compare the measured void size function with the theoretical predictions given by the extended Vdn model, in which the underdensity threshold is converted as
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Cosmological forecasts for the *Euclid* mission from the void size function for the wCDM model, characterised by a DE component described by a constant *w*. The contours represent the 68% and 95% confidence levels obtained by means of the Bayesian statistical analysis described in Sect. 3.3. **Left:** forecasts for a cosmological model with *w* and Ω_m as free cosmological parameters. We report the constraints obtained by fixing the calibration parameters with blue contours marked by a solid line and the results obtained by relaxing the calibration constraints with light-blue contours marked by a dashed line (see Sect. 3.4). **Right:** forecasts for a cosmological model with *w* and M_ν as free cosmological parameters. We represent the results of the fixed calibration case as red confidence contours having solid borders and those of the relaxed calibration case as orange contours having dashed borders. For each plot we show also the constraints on σ_8, computed as a derived parameter. The true values of the parameters are shown by a black dashed line.</figcaption></figure>

described in Sect. 3.2. Figure 3 provides the main results of our Flagship analysis. We show the comparison between the measured void number counts and the corresponding theoretical void size functions, both in real and redshift space, for the six equi-populated bins in redshift. The Poissonian errors related to the data are represented by the error bars, while the uncertainty related to the theoretical model is shown as a shaded region. The latter is computed associating an error to $F(b_{\text{eff}})$ given by the interval delimited by the coloured bands in Fig. 2. The residuals are reported at the bottom of each sub-plot and are calculated as the difference from the theoretical model, in units of the data errors. The latter show an excellent agreement between simulated data and theoretical models, even when considering voids identified in the Flagship galaxy catalogue in redshift space. The measured void number counts are indeed within an uncertainty of $2\sigma$, shown by the hatched coloured bands in the bottom panels, represented in units of the data errors.

To test the goodness of the fits shown in Fig. 3 we computed the reduced $\chi^2$ using the weighted sum of squared deviations of the two data sets from their corresponding models and dividing the results by the degrees of freedom of the two systems, $\nu = N_{\text{bin}} - N_{\text{par}}$, where $N_{\text{bin}}$ is the number of bins in radius used to compute the void counts and $N_{\text{bin}}$ is the number of free parameters of the model. In our case $N_{\text{bin}} = 50$ and $N_{\text{par}} = 2$, since we are fitting the void counts simultaneously for all the redshift shells and we are considering $B_{\text{slope}}$ and $B_{\text{offset}}$ only as free parameters of the model. The results are $\chi_v^2 = 1.60$ and $\chi_v^2 = 1.02$ for real and redshift space, respectively.

## 4.2. Cosmological forecasts

In this section we provide the cosmological forecasts obtained using the void size function in redshift space in the perspective of

the *Euclid* mission. We applied the statistical analysis described in Sect. 3.3 to derive constraints on the parameters of the two cosmological models analysed, labelled as wCDM and $w_0w_a$CDM, following the two approaches described in Sect. 3.4. For the model wCDM we assumed a flat prior for all the remaining free cosmological parameters of the model, and for the model $w_0w_a$CDM we assumed a Gaussian prior distribution with standard deviation $\sigma = 5$ for $w_0$ and $\sigma = 15$ for $w_a$, both centred on the true values of these parameters, given by the Flagship simulation cosmology ($w_0 = -1, w_a = 0$). We preferred to use very wide Gaussian priors instead of uniform ones to improve the numerical stability of the whole pipeline, but we tested that uniform priors yield consistent results. The remaining cosmological parameters analysed in this work ($\Omega_m$ and $M_\nu$) were included in the void size function modelling with uniform prior distributions.

In Fig. 4 we present the 68% and 95% confidence levels of the constraints on the model wCDM. In the left plot we show the *Euclid* forecasts from a void size function model characterised by *w* and Ω_m as free cosmological parameters. We represent with different colours and borders the results obtained with the two approaches described in Sect. 3.4: in blue with solid contours the forecasts obtained by fixing the extended Vdn parameters $B_{\text{slope}}$ and $B_{\text{offset}}$, in light-blue with dashed contours those obtained by relaxing the calibration constraints by means of a 2D Gaussian prior on $B_{\text{slope}}$ and $B_{\text{offset}}$, which distribution is represented in the left panel of Fig. 2. In the right plot we represent the same forecasts but considering a void size function model with the neutrino total mass $M_\nu$ as free parameter instead of the matter density Ω_m. In this case we show the fixed and the relaxed calibration approach results in red and orange, respectively. In both the presented cases $\sigma_8$ is computed as derived parameter. As expected, the effect of relaxing the calibration constraints is to broaden the confidence contours.
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Same as Fig. 4 but for the cosmological model labelled as $w_0w_a$CDM, having a dynamical DE component described by the CPL parametrisation (see Sect. 3.4).</figcaption></figure>

<table><caption>**Table 2.** Cosmological forecasts computed for the Euclid mission from the void size function for the cosmological model wCDM.</caption><thead><tr><th>Model</th><th>w</th><th>σ<sub>8</sub></th><th>Ω<sub>m</sub></th><th>M<sub>ν</sub> [eV]</th><th>B<sub>slope</sub></th><th>B<sub>offset</sub></th></tr></thead><tbody><tr><td rowspan="2">Fixed calib.</td><td>-1.01<sup>+0.09</sup><sub>-0.11</sub></td><td>0.83 ± 0.03</td><td>0.319<sup>+0.005</sup><sub>-0.004</sub></td><td>0</td><td>0.96</td><td>0.44</td></tr><tr><td>-0.99<sup>+0.06</sup><sub>-0.04</sub></td><td>0.83<sup>+0.1</sup><sub>-0.2</sub></td><td>0.319</td><td>&lt;0.03</td><td>0.96</td><td>0.44</td></tr><tr><td rowspan="2">Relaxed calib.</td><td>-1.0 ± 0.1</td><td>0.84 ± 0.04</td><td>0.318<sup>+0.008</sup><sub>-0.005</sub></td><td>0</td><td>0.96 ± 0.02</td><td>0.44 ± 0.04</td></tr><tr><td>-0.98<sup>+0.10</sup><sub>-0.07</sub></td><td>0.83<sup>+0.02</sup><sub>-0.03</sub></td><td>0.319</td><td>&lt;0.06</td><td>0.95 ± 0.02</td><td>0.46 ± 0.04</td></tr></tbody></table>

**Notes.** In this table we report the results of the two analysis strategies adopted in this work: considering the parameters $B_{\text{slope}}$ and $B_{\text{offset}}$ fixed to the respective median calibrated values (label: fixed calib.) or with a multivariate Gaussian with the same median value but a constraining power given by the calibration procedure with Flagship (label: relaxed calib.). For each of the two cases we present, in the upper and lower line, the forecasts obtained fixing $M_ν$ or $\Omega_m$ to the Flagship simulation true values, respectively. All the constraints are reported with errors with a $1\sigma$ confidence level.

<table><caption>**Table 3.** Same as Table 2 but for the $w_0w_a$CDM scenario.</caption><thead><tr><th>Model</th><th>w<sub>0</sub></th><th>w<sub>a</sub></th><th>σ<sub>8</sub></th><th>Ω<sub>m</sub></th><th>M<sub>ν</sub> [eV]</th><th>B<sub>slope</sub></th><th>B<sub>offset</sub></th><th>FoM<sub>w<sub>0</sub>,w<sub>a</sub></sub></th></tr></thead><tbody><tr><td rowspan="2">Fixed calib.</td><td>-1.0 ± 0.2</td><td>-0.1<sup>+0.7</sup><sub>-0.9</sub></td><td>0.84<sup>+0.04</sup><sub>-0.03</sub></td><td>0.32 ± 0.01</td><td>0</td><td>0.96</td><td>0.44</td><td>4.9</td></tr><tr><td>-1.0<sup>+0.2</sup><sub>-0.6</sub></td><td>-0.1<sup>+0.3</sup><sub>-0.8</sub></td><td>0.83<sup>+0.02</sup><sub>-0.03</sub></td><td>0.319</td><td>&lt; 0.08</td><td>0.96</td><td>0.44</td><td>17</td></tr><tr><td rowspan="2">Relaxed calib.</td><td>-0.8<sup>+1.6</sup><sub>-0.6</sub></td><td>-0.9<sup>+3.6</sup><sub>-9.6</sub></td><td>0.86 ± 0.04</td><td>0.32 ± 0.01</td><td>0</td><td>1.01<sup>+0.03</sup><sub>-0.04</sub></td><td>0.35<sup>+0.08</sup><sub>-0.05</sub></td><td>0.78</td></tr><tr><td>-0.9<sup>+0.3</sup><sub>-0.2</sub></td><td>-0.5<sup>+0.9</sup><sub>-1.3</sub></td><td>0.86<sup>+0.02</sup><sub>-0.05</sub></td><td>0.319</td><td>&lt; 0.08</td><td>0.99<sup>+0.01</sup><sub>-0.04</sub></td><td>0.38<sup>+0.07</sup><sub>-0.01</sub></td><td>2.3</td></tr></tbody></table>

**Notes.** In this case we present in the last column also the values computed with Eq. (18) to estimate the FoM for the DE equation of state.

In Fig. 5 we show the same contours represented in Fig. 4 but considering the $w_0w_a$CDM scenario. The free cosmological parameters of the void size function model are the coefficients of the DE equation of state, $w_0$ and $w_a$, together with $\Omega_m$ (left plot) or $M_ν$ (right plot). Also in this case the relaxation of the constraining condition of the calibration parameters causes an enlargement of the confidence contours. In this scenario however, the strongest impact of the calibration constraints is on the $w_0-w_a$ parameter plane, in particular along the diagonal where these parameters become degenerate. The effect of the calibration constraints on $\Omega_m$ and $M_ν$ has a lower impact.

In Tables 2 and 3 we report the values, with relative $1\sigma$ errors, of the cosmological constraints derived for the wCDM and $w_0w_a$CDM scenario, respectively. The constraints on the sum of neutrino masses $M_ν$ are expressed as a $1\sigma$ upper limit. For each table we show the results for the two approaches followed in this paper: fixing and relaxing the calibration constraints on the void size function model. The calibration parameter are reported in the columns $B_{\text{slope}}$ and $B_{\text{offset}}$ for completeness. Notice that each quantity reported without any uncertainty is considered fixed in the specific scenario presented in that table row.
For the $w_0w_a$ CDM scenario, in order to evaluate the constraining power of the void size function on the DE equation of state, we derived the figure of merit (FoM) for the coefficients of the CPL parametrisation $w_0$ and $w_a$. We computed this value by following Wang (2008):

$$ \text{FoM}_{w_0, w_a} = \frac{1}{\sqrt{\det \text{Cov}(w_0, w_a)}}, \qquad (18) $$

where $\text{Cov}(w_0, w_a)$ represents the covariance matrix of the DE equation of state parameters. We notice that this definition is in agreement with the one adopted in Euclid Collaboration (2020). We underline that, contrary to other constraint accuracy indicators (e.g. the relative or percentage errors), the higher the FoM value, the better the precision on the analysed cosmological parameters. We report this quantity in the last column of Table 3.

As a first exploration of the cosmic void statistics combined power, we now compare the forecasts from the void size function provided in this work with other Euclid forecasts. We present as a first comparison the results of the $\Omega_{de}-w$ confidence contour with the model-calibrated forecasts presented in Hamaus et al. (2022). The latter are computed by modelling the observable distortions of average shapes in redshift space via RSD and the Alcock-Paczyński effect, for voids to be measured in the Euclid spectroscopic galaxy distribution. Contrary to the model-independent case, in the presented approach the nuisance parameters of the model have been calibrated by means of Flagship data. In this comparison we consider the wCDM scenario with fixed neutrino mass and we focus on the $\Omega_{de}-w$ parameter space. Given the assumption of flat spatial geometry, to compute the corresponding $\Omega_{de}$ forecasts, we converted the $\Omega_m$ obtained in the MCMC analysis as $\Omega_{de} = 1 - \Omega_m$.

As a second comparison we take the results of Fisher analysis reported by the inter-science taskforce for forecasting (IST:F, Euclid Collaboration 2020) obtained in the optimistic setting for the single probes weak lensing and galaxy clustering. We considered in this case the $w_0w_a$ CDM scenario with fixed neutrino mass and we focus on the $\Omega_m-\sigma_8$ degeneracy. To compute the IST confidence contour we made use of the publicly available<fnref n="8" /> Fisher matrices and we marginalised on the parameters not reported in the plot with the code CosmicFish (Raveri et al. 2016). We recall that amplitude of density fluctuations at $z=0$, $\sigma_8$, is computed as a derived parameter in our analysis and its variation is given by the modifications caused by the free cosmological parameters of the model to the total matter power spectrum. We also stress the fact that a larger set of cosmological parameters is used in IST forecasts. This includes in particular the baryon matter energy density, $\Omega_b$, the dimensionless Hubble parameter, $h$, and the spectral index of the primordial density power spectrum, $n_s$. The impact on forecasts when including these parameters in the model will be tested in future work.

We show the presented comparisons in Fig. 6, representing in blue the forecasts obtained in this work considering a void size function model with fixed calibrated parameters. In the left panel we compare our results with the $\Omega_{de}-w$ confidence contour computed with the model-calibrated forecasts presented Hamaus et al. (2022; in magenta). In the right panel we show instead the comparison of $\Omega_m-\sigma_8$ confidence contour provided by IST forecasts considering the optimistic setting for weak lensing (in orange) and galaxy clustering (in purple). See Appendix B for

an analogous comparison considering the cosmological forecasts presented above but with less optimistic modelling approaches.

In both panels we can appreciate the comparable extension of the presented contours and in the latter we can notice in particular the strong complementarity of the void size function forecasts with those of the Euclid standard probes. While a more accurate analysis would require proper accounting of covariance between analysed cosmological constraints, Fig. 6 shows how the presented probes explore the parameter space differently and motivates investigation on combination to be performed in future works.

## 5. Conclusions and discussion

In this work we presented state-of-the-art forecasts for cosmological constraints from the void size function to be expected from the Euclid mission. We measured the void number counts from the Flagship mock galaxy spectroscopic catalogue in redshift bins and matched the measurements with the theoretical definition given by the Vdn model (Sheth & van de Weygaert 2004; Jennings et al. 2013). We employed an extension of the Vdn model that conservatively accounts for the effects of the galaxy large-scale bias, $b_{eff}$, on the void effective radii. With this method, we parametrised the Vdn model's characteristic threshold $\delta_v^L$ according to the prescriptions of Contarini et al. (2019), also verifying the calibration of the function $\mathcal{F}(b_{eff})$. The parametrisation method further allowed us to account for the modifications on the void sizes caused by the volume change of cosmic voids in redshift space.

We showed that the extended Vdn calibrated on Flagship data is effective in predicting the measured void number counts both in real and redshift space. Indeed we obtained a remarkable agreement between the measured and predicted void size functions, for all the redshift bins and all the spatial scales considered in our analysis. We also performed a MCMC analysis, estimating the constraints from void number counts on two main cosmological models: assuming in one case a scenario characterised by a constant equation-of-state parameter (wCDM) and in the other case a scenario with a dynamical DE component described by the CPL parametrisation ($w_0w_a$ CDM). For each scenario we presented the Euclid cosmological forecasts considering both approaches: by fixing the extended Vdn model parameters and by relaxing their boundaries to those provided by the calibration with Flagship mock catalogues. The former represents the ideal situation in which the simulations used to calibrate the void size function model allow us to have no uncertainties nor systematic errors on the calibration parameters; or alternatively, the case in which the value of the tracer bias inside voids is fully determined thanks to theoretical modelling.

In the wCDM scenario we forecasted relative percentage errors on the constant DE component, $w$, below the 10% for each analysed case. In the $w_0w_a$ CDM scenario, with the optimistic approach of fixing the model calibration parameters, we computed a $\text{FoM}_{w_0,w_a}$ equal to 4.9 or 17, in the case of leaving $\Omega_m$ or $M_\nu$, respectively, as additional free cosmological parameters of the model. As a reference, the corresponding FoM values computed by the IST:F for spectroscopic galaxy clustering and weak lensing are 55 and 44, respectively (see Table 13 in Euclid Collaboration 2020, for flat-$w_0w_a$ CDM cosmology and optimistic scenario).

The marginalised constraints on the derived parameter $\sigma_8$ are lower than 5% in every analysed case, while the relative errors on $\Omega_m$ are of the order of 2% in the wCDM scenario and of 3% in the $w_0w_a$ CDM scenario. The $1\sigma$ upper limit on $M_\nu$ is instead of

<footnote n="8">⁸ See https://github.com/euclidist-forecasting/fisher_for_public.</footnote>
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Comparison between the 68% and 95% confidence levels computed in this work with the void size function and different *Euclid* forecasts. **Left:** cosmological constraints on the $\Omega_{de}-w$ plane computed in this work (in blue) considering a wCDM scenario with fixed calibration parameters and in Hamaus et al. (2022; in magenta), modelling the void-galaxy cross-correlation function in redshift space, with a model-calibrated approach. **Right:** cosmological constraints on the $\Omega_m-\sigma_8$ plane computed in this work considering a $w_0w_a$CDM scenario (in blue) with fixed calibration parameters and the marginalised IST Fisher forecasts computed in the optimistic setting with spectroscopic galaxy clustering (in purple) and weak lensing (in orange).</figcaption></figure>

0.03 eV in the most optimistic case of the wCDM scenario and of 0.08 eV in the $w_0w_a$CDM scenario. We recall that, in the cosmological models with free neutrino mass, the total matter energy density was fixed to the Flagship simulation true value, therefore the degeneracy of $\Omega_v$ with $\Omega_m$ is not considered in the results.

Our analysis showcases the impressive constraining power of the void size function from the *Euclid* survey, strongly complementing the *Euclid* primary probes. This complementarity will make the combination powerful in particular for weak lensing and galaxy clustering, additionally enhancing robustness to systematic effects in both cases.

In this work we considered extremely conservative assumptions when analysing the void sample. Such conservative assumptions dramatically reduce the statistical power of our void catalogues, to ensure strong reliability: in the future, modelling improvements will allow a more efficient void selection, critically enhancing results while maintaining full robustness.

Among the conservative choices in modelling the void size function and in building the likelihood we recall the treatment of both the threshold value and of the minimum void radius accepted for the analysis. In particular, we selected a low value of the underdensity threshold ($\delta_{v,\text{tr}}^{\text{NL}} = -0.7$) to avoid the shallowest voids in the sample, characterised by higher Poissonian noise contamination. We also strictly restricted the range of considered radii to avoid modelling poorly sampled voids of the Flagship galaxy catalogue, in order to prevent the inclusion of spatial scales affected by a loss of void counts. This conservative approach allows us to have a sample of voids composed by a limited number of objects but characterised by high purity. Different techniques will be tested in the future to better model the scales affected by numerical incompleteness (see e.g. Cousinou et al. 2019) and include them in the analysis, safely obtaining access to much larger statistics. A better modelling of these effects will lead to further improvements in the constraining power of the void size function.

Further future prospects to expand this work include exploiting void number count forecasts to predict constraints from other void applications (such as the stacked void-galaxy cross-correlation function, see Hamaus et al. 2022, void lensing, see Bonici et al. 2022, the void-void correlation function, see Kreisch et al. 2022), and subsequently combine joint constraints from voids with other *Euclid* probes (primary and not, e.g. galaxy clustering, galaxy weak lensing, cluster counts and clustering, baryonic acoustic oscillations, supernova distance measurements, CMB cross-correlations, etc.).

Other areas to explore include considering other cosmological parameters for the likelihood modelling, a more realistic treatment of observational effects (a more complex survey mask, a more realistic $\sigma_z$ and further survey-related systematic effects). Moreover, to prepare the application to real-survey data, the large-scale effective bias can be recovered from redshift-space catalogues. To this purpose, a number of methods are available (see e.g. Chan et al. 2012; Sheth et al. 2013; Lazeyras et al. 2016; Sánchez et al. 2017a) and Desjacques et al. 2018 for an extensive review), which would allow us to reliably model linear bias as a function of redshift, $b(z)$. These methods can be exploited in the context of the *Euclid* mission, deriving tight constraints on $b(z)$, especially from the cross-correlation between galaxy clustering and weak lensing, capable of breaking the degeneracies with cosmological parameters (Tutusaus et al. 2020).

Finally, a purely theoretical treatment of the impact of RSD and of the Alcock-Paczyński effect on the void size function, based on work from companion papers (Hamaus et al. 2022), as well as a comprehensive analysis of the systematic effects linked to these corrections is relevant to consider in future works.

For the future, a possible approach to model the measured *Euclid* void number counts would be to apply directly the relation $F(b_{\text{eff}})$ calibrated on the redshift-space void sample of the Flagship mock (see Eq. (16)). A close match of the galaxy properties in the mock catalogue with galaxies to be observed by
*Euclid* would make this approach particularly effective. Future
works will aim to explore these methodologies and their impact
on void constraints, along with testing the range of applicability
of the *F*(*b*<sub>eff</sub>) relation and its possible cosmological dependency
(expected however to be mild from Contarini et al. 2021).

This paper – with a first analysis on a full mock, the *Euclid*
Flagship simulation – shows the impressive constraining capa-
bility of void number counts to tackle the properties of DE and
neutrinos, demonstrating for the first time the feasibility of the
technique with an end-to-end data-like application, and setting
the ground for a robust use of the void size function for cosmol-
ogy with *Euclid*.

**Acknowledgements.** We acknowledge the grant ASI n.2018-23-HH.0. SC, FM and LM acknowledge the use of computational resources from the parallel computing cluster of the Open Physics Hub (https://site.unibo.it/openphysicshub/en) at the Physics and Astronomy Department in Bologna. GV is supported by Università degli Studi di Padova and in part by the project “Combining Cosmic Microwave Background and Large Scale Structure data: an Integrated Approach for Addressing Fundamental Questions in Cosmology”, funded by the MIUR Progetti di Rilevante Interesse Nazionale (PRIN) Bando 2017 – grant 2017YJYZAH. AP is supported by NASA ROSES grant 12-EUCLID12-0004, and NASA grant 15-WFIRST15-0008 to the Nancy Grace Roman Space Telescope Science Investigation Team “Cosmology with the High Latitude Survey”. NH is supported by the Excellence Cluster ORIGINS, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2094 – 390783311. MS acknowledges support by the P. E. Filén fellowship and a fellowship at the Swedish Collegium for Advanced Study (SCAS). LM acknowledges support from PRIN MIUR 2017 WSCC32 “Zooming into dark matter and proto-galaxies with massive lensing clusters”. AR acknowledges funding from Italian Ministry of Education, University and Research (MIUR) through the ‘Dipartimenti di ecceffenza’ project Science of the Universe. He is supported in part by the project “Combining Cosmic Microwave Background and Large Scale Structure data: an Integrated Approach for Addressing Fundamental Questions in Cosmology”, funded by the MIUR Progetti di Rilevante Interesse Nazionale (PRIN) Bando 2017 – grant 2017YJYZAH We acknowledge use of the Python libraries NumPy (Harris et al. 2020), Matplotlib (Hunter 2007) and ChainConsumer (Hinton 2016). This work has made use of Cosmo-Hub (Carretero et al. 2017; Tallada et al. 2020). CosmoHub has been developed by the Port d'Informació Científica (PIC), maintained through a collaboration ofthe Institut de Física d'Altes Energies (IFAE) and the Centro de Investigaciones Energéticas, Medioambientes y Tecnológicas (CIEMAT) and the Institute of Space Sciences (CSIC & IEEC), and was partially funded by the “Plan Estatalde Investigación Científica y Técnica y de Innovación” program of the Spanish government. 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 Nederlandse 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

Achitou, I. 2016, Phys. Rev. D, 94, 103524

Achitou, I. 2019, Phys. Rev. D, 100, 123513

Achitou, I., Blake, C., Carter, P., Koda, J., & Beutler, F. 2017, Phys. Rev. D, 95, 083502

Alam, S., Ata, M., Bailey, S., et al. 2017, MNRAS, 470, 2617

Alcock, C., & Paczynski, B. 1979, Nature, 281, 358

Aubert, M., Cousinou, M.-C., Escoffier, S., et al. 2022, MNRAS, 513, 186

Audren, B., Lesgourgues, J., Bird, S., Haehnelt, M. G., & Viel, M. 2013, J. Cosmol. Astropart. Phys., 2013, 026

Baccigalupi, C. 1999, Phys. Rev. D, 59, 123004

Baker, T., Clampitt, J., Jain, B., & Trodden, M. 2018, Phys. Rev. D, 98, 023511

Baldi, M. 2012, MNRAS, 422, 1028

Baldi, M., & Villaescusa-Navarro, F. 2018, MNRAS, 473, 3226

Ballinger, W. E., Peacock, J. A., & Heavens, A. F. 1996, MNRAS, 282, 877

Banerjee, A., & Dalal, N. 2016, J. Cosmol. Astropart. Phys., 2016, 015

Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2016, ApJ, 762, 109

Bennett, J. J., Bulden, G., De Salas, P. F., et al. 2021, J. Cosmol. Astropart. Phys., 04, 073

Bernardeau, F. 1994, ApJ, 427, 51

Beutler, F., Seo, H.-J., Saito, S., et al. 2017, MNRAS, 466, 2242

Biswas, R., Alizadeh, E., & Wandelt, B. D. 2010, Phys. Rev. D, 82, 023002

Blanton, M. R., Hogg, D. W., Bahcall, N. A., et al. 2003, ApJ, 592, 819

Blanton, M. R., Lupton, R. H., Schlegel, D. J., et al. 2005, ApJ, 631, 208

Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, AJ, 154, 28

Blumenthal, G. R., da Costa, L. N., Goldwirth, D. S., Lecar, M., & Piran, T. 1992, ApJ, 388, 234

Bond, J. R., Cole, S., Efstathiou, G., & Kaiser, N. 1991, ApJ, 379, 440

Bond, J. R., Kofman, L., & Pogosyan, D. 1996, Nature, 380, 603

Bonici, M., Carbone, C., Vielzeuf, P., et al. 2022, A&A, submitted [arXiv:2206.14211]

Brouwer, M. M., Demchenko, V., Harnois-Déraps, J., et al. 2018, MNRAS, 481, 5189

Cai, Y.-C., Cole, S., Jenkins, A., & Frenk, C. S. 2010, MNRAS, 407, 201

Cai, Y.-C., Neyrinck, M. C., Szapudi, I., Cole, S., & Frenk, C. S. 2014a, ApJ, 786, 110

Cai, Y.-C., Li, B., Cole, S., Frenk, C. S., & Neyrinck, M. 2014b, MNRAS, 439, 2978

Cai, Y.-C., Padilla, N., & Li, B. 2015, MNRAS, 451, 1036

Cai, Y.-C., Taylor, A., Peacock, J. A., & Padilla, N. 2016, MNRAS, 462, 2465

Cai, Y.-C., Neyrinck, M., Mao, Q., et al. 2017, MNRAS, 466, 3364

Carretero, J., Tallada, P., Casals, J., et al. 2017, in Proceedings of the European Physical Society Conference on High Energy Physics. 5–12 July, 488

Chan, K. C., & Hamaus, N. 2021, Phys. Rev. D, 103, 043502

Chan, K. C., Scoccimarro, R., & Sheth, R. K. 2012, Phys. Rev. D, 85, 121304

Chan, K. C., Hamaus, N., & Biagetti, M. 2019, Phys. Rev. D, 99, 083509

Chantavat, T., Sawangwit, U., Sutter, P. M., & Wandelt, B. D. 2016, Phys. Rev. D, 93, 043523

Chantavat, T., Sawangwit, U., & Wandelt, B. D. 2017, ApJ, 836, 156

Chevallier, M., & Polarski, D. 2001, Int. J. Mod. Phys. D, 10, 213

Chuang, C.-H., Kitaura, F.-S., Liang, Y., et al. 2017, Phys. Rev. D, 95, 063528

Clampitt, J., & Jain, B. 2015, MNRAS, 454, 3357

Clampitt, J., Cai, Y.-C., & Li, B. 2013, MNRAS, 431, 749

Cole, S. 1991, ApJ, 367, 45

Contarini, S., Ronconi, T., Marulli, F., et al. 2019, MNRAS, 488, 3526

Contarini, S., Marulli, F., Moscardini, L., et al. 2021, MNRAS, 504, 5021

Correa, C. M., Paz, D. J., Padilla, N. D., et al. 2019, MNRAS, 485, 5761

Correa, C. M., Paz, D. J., Sánchez, A. G., et al. 2021, MNRAS, 500, 911

Correa, C. M., Paz, D. J., Padilla, N. D., et al. 2022, MNRAS, 509, 1871

Cousinou, M. C., Pisani, A., Tilquin, A., et al. 2019, Astron. Comput., 27, 53

Dark Energy Survey Collaboration (Abbott, T., et al.) 2016, MNRAS, 460, 1270

Davies, C. T., Cautun, M., Giblin, B., et al. 2021, MNRAS, 507, 2267

Dawson, K. S., Kneib, J.-P., Percival, W. J., et al. 2016, AJ, 151, 44

de Bernardis, F., Martinelli, M., Melchiorri, A., Mena, O., & Cooray, A. 2011, Phys. Rev. D, 84, 023504

DESI Collaboration (Aghamousa, A., et al.) 2016, ArXiv eprints [arXiv:1611.00036]

Desjacques, V., Jeong, D., & Schmidt, F. 2018, Phys. Rep., 733, 1

Dong, F., Yu, Y., Zhang, J., Yang, X., & Zhang, P. 2021, MNRAS, 500, 3838

Doré, O., Werner, M. W., Ashby, M. L. N., et al. 2018, ArXiv eprints [arXiv:1805.05489]

Eisenstein, D. J., Zehavi, I., Hogg, D. W., et al. 2005, ApJ, 633, 560

Euclid Collaboration (Blanchard, A., et al.) 2020, A&A, 642, A191

Falck, B., Koyama, K., Zhao, G.-B., & Cautun, M. 2018, MNRAS, 475, 3262

Fang, Y., Hamaus, N., Jain, B., et al. 2019, MNRAS, 490, 3573

Flender, S., Hotchkiss, S., & Nadathur, S. 2013, J. Cosmol. Astropart. Phys., 2013, 013

Forero-Sánchez, D., Zhao, C., Tao, C., et al. 2022, MNRAS, 513, 5407

Froustej, J., Pitrou, C., & Volpe, M. C. 2020, J. Cosmol. Astropart. Phys., 2020, 015

Furlanetto, S. R., & Piran, T. 2006, MNRAS, 366, 467

Granett, B. R., Neyrinck, M. C., & Szapudi, I. 2008, ApJ, 683, L99

Gregory, S. A., & Thompson, L. A. 1978, ApJ, 222, 784

Gruen, D., & Dark Energy Survey Collaboration 2016, Am. Astron. Soc. Meeting Abstracts, 228, 40304

Guzzo, L., Scodeggio, M., Garilli, B., et al. 2014, A&A, 566, A108

Hamann, J., Hannestad, S., & Wong, Y. Y. Y. 2012, J. Cosmol. Astropart. Phys., 2012, 052

Hamaus, N., Sutter, P. M., Lavaux, G., & Wandelt, B. D. 2014a, J. Cosmol. Astropart. Phys., 2014, 013

Hamaus, N., Sutter, P. M., & Wandelt, B. D. 2014b, Phys. Rev. Lett., 112, 251302

Hamaus, N., Sutter, P. M., Lavaux, G., & Wandelt, B. D. 2015, J. Cosmol. Astropart. Phys., 2015, 036
Hamaus, N., Pisani, A., Sutter, P. M., et al. 2016, Phys. Rev. Lett., 117, 091302

Hamaus, N., Cousinou, M.-C., Pisani, A., et al. 2017, J. Cosmol. Astropart. Phys., 2017, 014

Hamaus, N., Pisani, A., Choi, J.-A., et al. 2020, J. Cosmol. Astropart. Phys., 2020, 023

Hamaus, N., Aubert, M., Pisani, A., et al. 2022, A&A, 658, A20

Hang, Q., Alam, S., Cai, Y.-C., & Peacock, J. A. 2021, MNRAS, 507, 510

Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357

Hawken, A. J., Granett, B. R., Iovino, A., et al. 2017, A&A, 607, A54

Hawken, A. J., Aubert, M., Pisani, A., et al. 2020, J. Cosmol. Astropart. Phys., 2020, 012

Hinton, S. R. 2016, J. Open Source Softw., 1, 00045

Hu, W., & Sawicki, I. 2007, Phys. Rev. D, 76, 064004

Hunter, J. D. 2007, Comput. Sci. Eng., 9, 90

Icke, V. 1984, MNRAS, 206, 1P

Ilić, S., Langer, M., & Douspis, M. 2013, A&A, 556, A51

Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111

Jennings, E., Li, Y., & Hu, W. 2013, MNRAS, 434, 2167

Jones, D. H., Read, M. A., Saunders, W., et al. 2009, MNRAS, 399, 683

Kendall, M. G., Stuart, A., & Ord, J. K. 1987, Kendall's Advanced Theory of Statistics (USA: Oxford University Press, Inc.)

Khededkar, S., & Majumdar, S. 2013, J. Cosmol. Astropart. Phys., 2013, 030

Khoraminezhad, H., Viel, M., Baccigalupi, C., & Archidiacono, M. 2020, J. Cosmol. Astropart. Phys., 2020, 039

Khoraminezhad, H., Vielzeuf, P., Lazeyras, T., Baccigalupi, C., & Viel, M. 2022, MNRAS, 511, 4333

Kitaura, F.-S., Chuang, C.-H., Liang, Y., et al. 2016, Phys. Rev. Lett., 116, 171301

Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18

Kovács, A. 2018, MNRAS, 475, 1777

Kovács, A., Sánchez, C., García-Bellido, J., et al. 2017, MNRAS, 465, 4166

Kovács, A., Sánchez, C., García-Bellido, J., et al. 2019, MNRAS, 484, 5267

Kovács, A., Beck, R., Smith, A., et al. 2022a, MNRAS, 513, 15

Kovács, A., Jeffrey, N., Gatti, M., et al. 2022b, MNRAS, 510, 216

Kreisch, C. D., Pisani, A., Carbone, C., et al. 2019, MNRAS, 488, 4413

Kreisch, C. D., Pisani, A., Villaescusa-Navarro, F., et al. 2022, ApJ, 935, 100

Lahav, O., Kiakotou, A., Abdalla, F. B., & Blake, C. 2010, MNRAS, 405, 168

Landy, S. D., & Szalay, A. S. 1993, ApJ, 412, 64

Lavaux, G., & Wandelt, B. D. 2012, ApJ, 754, 109

Lazeyras, T., Wagner, C., Baldauf, T., & Schmidt, F. 2016, J. Cosmol. Astropart. Phys., 2016, 018

Leclercq, F., Jasche, J., Sutter, P. M., Hamaus, N., & Wandelt, B. 2015, J. Cosmol. Astropart. Phys., 2015, 047

Lee, J., & Park, D. 2009, ApJ, 696, L10

Lewis, A., Challinor, A., & Lasenby, A. 2000, ApJ, 538, 473

Li, B., & Efstathiou, G. 2012, MNRAS, 421, 1431

Liang, Y., Zhao, C., Chuang, C.-H., Kitaura, F.-S., & Tao, C. 2016, MNRAS, 459, 4020

Linder, E. V. 2003, Phys. Rev. Lett., 90, 091301

Mangano, G., Miele, G., Pastor, S., et al. 2005, Nucl. Phys. B, 729, 221

Mao, Q., Berlind, A. A., Scherrer, R. J., et al. 2017a, ApJ, 835, 160

Mao, Q., Berlind, A. A., Scherrer, R. J., et al. 2017b, ApJ, 835, 161

Martinelli, M., Calabrese, E., de Bernardis, F., et al. 2011, Phys. Rev. D, 83, 023012

Marulli, F., Bolzonella, M., Branchini, E., et al. 2013, A&A, 557, A17

Marulli, F., Veropalumbo, A., & Moresco, M. 2016, Astron. Comput., 14, 35

Marulli, F., Veropalumbo, A., Sereno, M., et al. 2018, A&A, 620, A1

Massara, E., & Sheth, R. K. 2018, ArXiv eprints [arXiv:1811.03132]

Massara, E., Villaescusa-Navarro, F., Viel, M., & Sutter, P. M. 2015, J. Cosmol. Astropart. Phys., 2015, 018

Melchior, P., Sutter, P. M., Sheldon, E. S., Krause, E., & Wandelt, B. D. 2014, MNRAS, 440, 2922

Mo, H. J., & White, S. D. M. 1996, MNRAS, 282, 347

Monaco, P. 1995, ApJ, 447, 23

Moresco, M., Amati, L., Amendola, L., et al. 2022, ArXiv eprints [arXiv:2201.07241]

Nadathur, S. 2016, MNRAS, 461, 358

Nadathur, S., & Crittenden, R. 2016, ApJ, 830, L19

Nadathur, S., & Percival, W. J. 2019, MNRAS, 483, 3472

Nadathur, S., Hotchkiss, S., & Sarkar, S. 2012, J. Cosmol. Astropart. Phys., 2012, 042

Nadathur, S., Carter, P. M., Percival, W. J., Winther, H. A., & Bautista, J. E. 2019, Phys. Rev. D, 100, 023504

Nadathur, S., Woodfinden, A., Percival, W. J., et al. 2020, MNRAS, 499, 4140

Neyrinck, M. C. 2008, MNRAS, 386, 2101

Neyrinck, M. C., Aragón-Calvo, M. A., Jeong, D., & Wang, X. 2014, MNRAS, 441, 646

Pace, F., Meyer, S., & Bartelmann, M. 2017, J. Cosmol. Astropart. Phys., 2017, 040

Paillas, E., Cautun, M., Li, B., et al. 2019, MNRAS, 484, 1149

Pápai, P., & Szapudi, I. 2010, ApJ, 725, 2078

Paranjape, A., Lam, T. Y., & Sheth, R. K. 2012, MNRAS, 420, 1648

Paz, D., Lares, M., Ceccarelli, L., Padilla, N., & Lambas, D. G. 2013, MNRAS, 436, 3480

Peacock, J. A., & Heavens, A. F. 1990, MNRAS, 243, 133

Peebles, P. J. E. 2001, ApJ, 557, 495

Perico, E. L. D., Voivodic, R., Lima, M., & Mota, D. F. 2019, A&A, 632, A52

Perotto, L., Lesgourgues, J., Hannestad, S., Tu, H., & Wong, Y. 2006, J. Cosmol. Astropart. Phys., 2006, 013

Pezzotta, A., de la Torre, S., Bel, J., et al. 2017, A&A, 604, A33

Pisani, A., Sutter, P. M., Hamaus, N., et al. 2015a, Phys. Rev. D, 92, 083531

Pisani, A., Sutter, P. M., & Wandelt, B. D. 2015b, ArXiv eprints [arXiv:1506.07982]

Pisani, A., Massara, E., Spergel, D. N., et al. 2019, BAAS, 51, 40

Planck Collaboration VI. 2020, A&A, 641, A6

Platen, E., van de Weygaert, R., & Jones, B. J. T. 2007, MNRAS, 380, 551

Pollina, G., Baldi, M., Marulli, F., & Moscardini, L. 2016, MNRAS, 455, 3075

Pollina, G., Hamaus, N., Dolag, K., et al. 2017, MNRAS, 469, 787

Pollina, G., Hamaus, N., Paech, K., et al. 2019, MNRAS, 487, 2836

Potter, D., Stadel, J., & Teyssier, R. 2017, Comput. Astrophys. Cosmol., 4, 2

Raveri, M., Martinelli, M., Zhao, G., & Wang, Y. 2016, ArXiv eprints [arXiv:1606.06268]

Reed, D. S., Schneider, A., Smith, R. E., et al. 2015, MNRAS, 451, 4413

Roerdink, J. B. T. M., & Meijster, A. 2000, Fundam. Inform., 41, 187

Ronconi, T., & Marulli, F. 2017, A&A, 607, A24

Ronconi, T., Contarini, S., Marulli, F., Baldi, M., & Moscardini, L. 2019, MNRAS, 488, 5075

Sahlén, M. 2019, Phys. Rev. D, 99, 063525

Sahlén, M., & Silk, J. 2018, Phys. Rev. D, 97, 103504

Sahlén, M., Zubeldía, I., & Silk, J. 2016, ApJ, 820, L7

Sánchez, A. G., Scoccimarro, R., Crocce, M., et al. 2017a, MNRAS, 464, 1640

Sánchez, C., Clampitt, J., Kovacs, A., et al. 2017b, MNRAS, 465, 746

Schuster, N., Hamaus, N., Pisani, A., et al. 2019, J. Cosmol. Astropart. Phys., 2019, 055

Scoccimarro, R. 2004, Phys. Rev. D, 70, 083007

Sellentin, E., Quartin, M., & Amendola, L. 2014, MNRAS, 441, 1831

Sheth, R. K., & Tormen, G. 2002, MNRAS, 329, 61

Sheth, R. K., & van de Weygaert, R. 2004, MNRAS, 350, 517

Sheth, R. K., Chan, K. C., & Scoccimarro, R. 2013, Phys. Rev. D, 87, 083002

Spergel, D., Gehrels, N., Baltay, C., et al. 2015, ArXiv eprints [arXiv:1503.03757]

Spolyar, D., Sahlén, M., & Silk, J. 2013, Phys. Rev. Lett., 111, 241103

Sutter, P. M., Lavaux, G., Wandelt, B. D., & Weinberg, D. H. 2012, ApJ, 761, 187

Sutter, P. M., Pisani, A., Wandelt, B. D., & Weinberg, D. H. 2014a, MNRAS, 443, 2983

Sutter, P. M., Lavaux, G., Hamaus, N., et al. 2014b, MNRAS, 442, 462

Sutter, P. M., Lavaux, G., Hamaus, N., et al. 2015, Astron. Comput., 9, 1

Szapudi, I., Kovács, A., Granett, B. R., et al. 2015, MNRAS, 450, 288

Tallada, P., Carrettero, J., Casals, J., et al. 2020, Astron. Comput., 32, 100391

Tamura, N., Takato, N., Shimono, A., et al. 2016, SPIE Conf. Ser., 9908, 99081M

Taruya, A., Nishimichi, T., & Saito, S. 2010, Phys. Rev. D, 82, 063522

Thompson, L. A., & Gregory, S. A. 2011, ArXiv eprints [arXiv:1109.1268]

Tikhonov, A. V., & Karachentsev, I. D. 2006, ApJ, 653, 969

Tutusaus, I., Martinelli, M., Cardone, V. F., et al. 2020, A&A, 643, A70

Verza, G., Pisani, A., Carbone, C., Hamaus, N., & Guzzo, L. 2019, J. Cosmol. Astropart. Phys., 2019, 040

Vielzeuf, P., Kovács, A., Demirbozan, U., et al. 2021, MNRAS, 500, 464

Wang, X., Chen, X., Zheng, Z., et al. 2009, MNRAS, 394, 1775

Wang, Y. 2008, Phys. Rev. D, 77, 123525

Wolz, L., Kilbinger, M., Weller, J., & Giannantonio, T. 2012, J. Cosmol. Astropart. Phys., 2012, 009

Xu, X., Cuesta, A. J., Padmanabhan, N., Eisenstein, D. J., & McBride, C. K. 2013, MNRAS, 431, 2834

Yang, L. F., Neyrinck, M. C., Aragón-Calvo, M. A., Falck, B., & Silk, J. 2015, MNRAS, 451, 3606

Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, ApJ, 736, 59

Zeldovich, I. B., Einasto, J., & Shandarin, S. F. 1982, Nature, 300, 407

Zhao, C., Tao, C., Liang, Y., Kitaura, F.-S., & Chuang, C.-H. 2016, MNRAS, 459, 2670

Zivick, P., Sutter, P. M., Wandelt, B. D., Li, B., & Lam, T. Y. 2015, MNRAS, 451, 4215
¹ Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, Via Piero Gobetti 93/2, 40129 Bologna, Italy

² INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy

³ INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy

⁴ INFN-Padova, Via Marzolo 8, 35131 Padova, Italy

⁵ Dipartimento di Fisica e Astronomia “G.Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy

⁶ Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA

⁷ Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010 New York, NY, USA

⁸ The Cooper Union for the Advancement of Science and Art, 41 Cooper Square, New York, NY 10003, USA

⁹ Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany

¹⁰ Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 515, 751 20 Uppsala, Sweden

¹¹ Swedish Collegium for Advanced Study, Thunbergsvägen 2, 752 38 Uppsala, Sweden

¹² INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy

¹³ University of Lyon, UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, France

¹⁴ Aix-Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France

¹⁵ Instituto de Astrofísica de Canarias (IAC); Departamento de Astrofísica, Universidad de La Laguna (ULL), 38200 La Laguna, Tenerife, Spain

¹⁶ Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain

¹⁷ Institut d'Astrophysique de Paris, UMR 7095, CNRS et Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France

¹⁸ Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

¹⁹ Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

²⁰ Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK

²¹ SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, TS, Italy

²² IFPU, Institute for Fundamental Physics of the Universe, Via Beirut 2, 34151 Trieste, Italy

²³ Univ. Lyon, Univ. Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 69622 Villeurbanne, France

²⁴ Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Édouard Belin, 31400 Toulouse, France

²⁵ Université St Joseph; Faculty of Sciences, Beirut, Lebanon

²⁶ Department of Mathematics and Physics, Roma Tre University, Via della Vasca Navale 84, 00146 Rome, Italy

²⁷ Institut d'Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France

²⁸ Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy

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³¹ INFN-Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy

³² INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy

³³ Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

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³⁵ INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

³⁶ Port d'Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra, Barcelona, Spain

³⁷ Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra, Barcelona, Spain

³⁸ INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy

³⁹ INFN section of Naples, Via Cinthia 6, 80126 Napoli, Italy

⁴⁰ Department of Physics “E. Pancini”, University Federico II, Via Cinthia 6, 80126 Napoli, Italy

⁴¹ Institut national de physique nucléaire et de physique des particules, 3 rue Michel-Ange, 75794 Paris Cedex 16, France

⁴² Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK

⁴³ Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK

⁴⁴ ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain

⁴⁵ European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Roma, Italy

⁴⁶ Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland

⁴⁷ Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK

⁴⁸ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande 1749-016, Lisboa, Portugal

⁴⁹ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande 1749-016 Lisboa, Portugal

⁵⁰ Department of Astronomy, University of Geneva, ch. d'Écogia 16, 1290 Versoix, Switzerland

⁵¹ Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK

⁵² AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, 91191 Gif-sur-Yvette, France

⁵³ Institut d'Estudis Espacials de Catalunya (IEEC), Carrer Gran Capità 2-4, 08034 Barcelona, Spain

⁵⁴ Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain

⁵⁵ INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy

⁵⁶ Istituto Nazionale di Astrofisica (INAF) – Osservatorio di Astrofisica e Scienza dello Spazio (OAS), Via Gobetti 93/3, 40127 Bologna, Italy

⁵⁷ Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irne-rio 46, 40126 Bologna, Italy

⁵⁸ INAF-Osservatorio Astronomico di Padova, Via dell'Osservatorio 5, 35122 Padova, Italy

⁵⁹ Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany

⁶⁰ Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy

⁶¹ INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy

⁶² INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy

⁶³ Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway

⁶⁴ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

⁶⁵ von Hoerner & Sulger GmbH, SchloßPlatz 8, 68723 Schwetzingen, Germany

⁶⁶ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
67 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, Astro-
physique, Instrumentation et Modélisation Paris-Saclay, 91191 Gif-
sur-Yvette, France

68 Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland

69 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland

70 European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

71 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

72 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany

73 Department of Physics, Institute for Computational Cosmology, Durham University, South Road, DH1 3LE, UK

74 University of Applied Sciences and Arts of Northwestern Switzerland, School of Engineering, 5210 Windisch, Switzerland

75 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, 8000 Aarhus C, Denmark

76 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

77 Space Science Data Center, Italian Space Agency, Via del Politecnico snc, 00133 Roma, Italy

78 Centre National d'Études Spatiales, Toulouse, France

79 Institute of Space Science, Bucharest 077125, Romania

80 Aix-Marseille Univ, CNRS, CNES, LAM, Marseille, France

81 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain

82 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal

83 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, 30202 Cartagena, Spain

84 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands

85 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA

86 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France
## Appendix A: Void count measures

Table A.1 provides the number of voids identified in the redshift-space distribution of galaxies, in different redshift bins. We note that, conversely to Table 1, it relies on generic equi-spaced $\Delta z = 0.1$ bins, to facilitate the use for future forecasts of different void applications and combination with other probes. Here we report void number counts obtained both before

and after the application of the cleaning procedure. Moreover, we show the number of cosmic voids considering both an optimistic and a pessimistic cut on smaller void radii, that is voids with radius over 1 time the MGS, and voids over 2 times the MGS, respectively. The lowering of void counts in the outermost bins is caused by survey mask effects at redshift boundaries of the simulated light-cone.

<table><caption>**Table A.1.** Void counts in nine equi-spaced bins in redshift, measured in the redshift-space mock galaxy catalogue, provided as a reference for future forecast analyses.</caption><thead><tr><th rowspan="2">z range</th><th rowspan="2">shell volume [(h<sup>-1</sup> Gpc)<sup>3</sup>]</th><th rowspan="2">MGS [h<sup>-1</sup> Mpc]</th><th colspan="2">all voids</th><th colspan="2">voids after cleaning</th></tr><tr><th>R > MGS</th><th>> 2 MGS</th><th>R<sub>eff</sub> > MGS</th><th>> 2 MGS</th></tr></thead><tbody><tr><td>0.9 – 1.0</td><td>1.308</td><td>10.28</td><td>8928</td><td>6032</td><td>4845</td><td>726</td></tr><tr><td>1.0 – 1.1</td><td>1.427</td><td>11.02</td><td>8987</td><td>6637</td><td>5253</td><td>840</td></tr><tr><td>1.1 – 1.2</td><td>1.531</td><td>11.74</td><td>7735</td><td>5824</td><td>4690</td><td>699</td></tr><tr><td>1.2 – 1.3</td><td>1.622</td><td>12.63</td><td>7167</td><td>5237</td><td>4140</td><td>500</td></tr><tr><td>1.3 – 1.4</td><td>1.700</td><td>13.51</td><td>6575</td><td>4756</td><td>3703</td><td>321</td></tr><tr><td>1.4 – 1.5</td><td>1.766</td><td>14.45</td><td>5636</td><td>4078</td><td>3152</td><td>249</td></tr><tr><td>1.5 – 1.6</td><td>1.821</td><td>15.45</td><td>5132</td><td>3624</td><td>2719</td><td>160</td></tr><tr><td>1.6 – 1.7</td><td>1.867</td><td>16.48</td><td>4389</td><td>3049</td><td>2286</td><td>94</td></tr><tr><td>1.7 – 1.8</td><td>1.904</td><td>17.63</td><td>2248</td><td>934</td><td>851</td><td>4</td></tr><tr><td>0.9 – 1.8</td><td>14.95</td><td>13.69</td><td>56 797</td><td>40 171</td><td>31 639</td><td>3593</td></tr></tbody></table>

**Notes.** The first column represents the minimum and the maximum redshift values for each bin, while the second and the third columns provide the volume in units of $(h^{-1} \text{ Gpc})^3$ corresponding to each shell of the sky octant, and the MGS, respectively. The next two columns show the number of voids identified by the VIDE void finder, selected with an effective radius greater than 1 and 2 times the MGS, respectively. In the last two columns we provide the void number counts obtained after the cleaning procedure, to be modelled using the void size function theory. The latter are reported with the same radius selections as described before. In the last row we show the total shells' volume, the mean MGS and the total void counts corresponding to the entire range of redshifts.
## Appendix B: Different setting forecasts

We show in Fig. B.1 the same forecast comparison presented in Fig. 6 but using less optimistic settings for the analyses. In particular, we report here the *Euclid* forecasts on a flat $w$CDM and $w_0w_a$CDM cosmology with massless neutrinos, in the left and right panels respectively, indicating the void size function constraints as light-blue contours with dashed borders.

In the left panel of Fig. B.1 we compare our relaxed-calibration results (see Sect. 3.4 and Sect. 4.1), with the *Euclid* model-independent forecasts of Hamaus et al. (2022), represented in magenta with solid borders. The latter are computed by means of the void-galaxy cross-correlation function in redshift space, with a calibration-independent approach: the two nuisance parameters of the model are left free to vary and therefore are constrained by the data directly (instead of being fixed to the values calibrated with mock catalogues, i.e. the Flagship light-cone). The authors emphasize that the calibration-independent

approach is to be preferred, as fixing the nuisance parameters to the mock values may introduce a prior dependence on the model parameters assumed in the mocks, possibly yielding biased cosmological constraints and underestimated relative uncertainties.

Finally, in the right panel of Fig. B.1 we show the comparison of our relaxed-calibration constraints with the Fisher forecasts computed with the pessimistic setting described in *Euclid Collaboration (2020)* for the probes weak lensing (WL, in orange) and spectroscopic galaxy clustering ($GC_s$, in purple). According to the authors, the pessimistic configuration used for these constraints differs from the optimistic one mainly for a stronger cut of the maximum angular mode for the weak lensing angular power spectrum, $\ell_{max}(\text{WL}) = 1500$ (5000 in the optimistic setting), and of the power spectrum maximum scale, $k_{max}(\text{GC}_s) = 0.25 \, h^{-1} \, \text{Mpc}^{-1}$ (0.3 $h^{-1} \, \text{Mpc}^{-1}$ in the optimistic setting). Analogously to Fig. 6, the presented confidence contours marginalised on the analysed parameter space are comparable and partially complementary.

<figure><img src="image_9.png" /><figcaption>**Fig. B.1.** Same as Fig. 6 but for different forecast settings. In this case the confidence contours obtained in this work from the void size function model (light-blue contours with dashed lines) are computed relaxing the constraints given by calibration parameters. The *Euclid* forecasts derived with void cross-correlation are computed with a model-independent approach, while IST forecasts are computed with the pessimistic setting described (Euclid Collaboration 2020).</figcaption></figure>