Void Number Counts as a Cosmological Probe for the Large-Scale Structure

Cosmic void, which is characterized by low density, large volume and linear evolution, is an important component of the cosmic large-scale structure (LSS). A large number of void samples can be obtained by galaxy surveys for studying the formation and evolution of the LSS and properties of dark energy and dark matter. A variety of cosmological probes associated with voids have been proven to be very effective, such as the galaxy-void cross-correlation in redshift space (e.g. Hamaus et al., 2017; Hamaus et al., 2022; Correa et al., 2022) and the void size function (VSF, e.g. Pisani et al., 2015; Verza et al., 2019; Contarini et al., 2022, 2023; Pelliciari et al., 2023; Verza et al., 2023, 2024). Cosmic voids are suitable for studying modified gravity and massive neutrinos due to their large volume and low density (e.g. Cai et al., 2015; Pisani et al., 2015; Zivick et al., 2015; Pollina et al., 2016; 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; Kreisch et al., 2019; Schuster et al., 2019; Contarini et al., 2021; Kreisch et al., 2022; Mauland et al., 2023; Verza et al., 2023; Vielzeuf et al., 2023), and can also be used to measure baryonic acoustic oscillations (BAO) (e.g. Chan & Hamaus, 2021; Forero-Sánchez et al., 2022; Khoraminezhad et al., 2022). The great potential of voids has been shown in these existing studies, which can promote our research on the LSS of the Universe.

Here we propose to use void number counts (VNC) to explore the LSS and constraining the cosmological parameters. The VNC is the integral of the VSF over the void size at a given redshift. The VSF has been proven to be an effective method for constraining cosmological models in spectroscopic galaxy surveys, e.g. BOSS DR12 (Contarini et al., 2023). As a function representing the number density of voids at different scales at a given redshift, the VSF can illustrate the features and evolution of voids. The theoretical models of the VSF are also developed in current relevant researches, that usually assume the voids have spherical shape. One of the popular theoretical VSF models is represented by the Sheth and van de Weygaert model (SvdW, Sheth & van de Weygaert, 2004), and it assumes that large voids grow from isolated small voids. And the SvdW model was later extended to the volume conserving model ($V{\rm d}n$, Jennings et al., 2013), which assumes that large voids merge from small voids rather than evolve independently, and it is now one of the most widely used VSF model.

However, the shape of voids can be arbitrary and irregular, and the current void finders are usually based on the watershed algorithm (Platen et al., 2007), without any assumption on the void shape. So it is necessary to trim the void catalogs for selecting the voids with spherical shape to match the theoretical model in current VSF studies, and recent works have trimmed void catalog in various degrees or improve existing theoretical models (e.g. Contarini et al., 2023; Pelliciari et al., 2023; Verza et al., 2023; Song et al., 2024; Verza et al., 2024). This obviously will exclude large numbers of voids, and dramatically lose statistical significance. On the other hand, if we consider the VNC, in principle, we do not need to trim the void catalog significantly, since the VNC is not as sensitive as the VSF to the void shape by integrating over the void size, and the theoretical model can be modified easily and may explain the data better. This will retain more voids in the analysis, and could improve the accuracy of cosmological constraint.

In order to investigate the feasibility of this method, we create mock galaxy and void catalogs based on simulations, and assume the China Space Station Telescope (CSST, Zhan, 2011, 2021; Gong et al., 2019; Miao et al., 2023) as the instrument to perform the spectroscopic survey. We use the widely used watershed algorithm of the void finder and obtain the void effective radius, ellipticity, and volume weighted center needed in our analysis. Then we generate the mock data of the VNC in six redshift bins from $z=0.3$ to $1.3$, and discuss two void samples with employing two selection criteria, i.e. applying empirical void size cut-off and the void ellipticity cut-off, respectively. We constrain the cosmological and void parameters by using the Markov Chain Monte Carlo (MCMC) method. We also compare the result to that from the VSF method given by Song et al. (2024) based on the same galaxy catalog, and demonstrate the feasibility of the VNC method.

The paper is organized as follows: In Section 2, we introduce the simulations for generating the mock galaxy and void catalogs; In Section 3, we discuss the computation of the theoretical model and generation of the mock data for the VNC; In Section 4, we show the constraint results of the cosmological and void parameters; The summary and conclusion is given in Section 5.

Theoretically, the VNC are obtained by integrating the VSF over the void radius in a survey volume $V_{\rm S}$ at a given redshift, and it can be calculated by

By assuming that the volume fraction $V{\rm d}n$ is constant from Lagrangian space to Eulerian space (Jennings et al., 2013), the VSF can be estimated as

where $R_{\text{L}}$ is the Lagrangian void radius. The peak height $\nu$ is decided by the void linear underdensity threshold of void formation $\delta_{\text{v}}$ and the root-mean-squared density fluctuation at different redshifts, which is given by $\nu=|\delta_{\text{v}}|/\sigma_{M}(z)$, where $\sigma_{M}(z)=\sigma_{0}(R_{\rm L})D(z)$ and $D(z)$ is the linear growth factor. We obtain $\sigma_{M}(z)$ using CAMB in our work (Lewis et al., 2000). For the void radius ranges we choose, $\mathcal{F}(\nu)$ can be approximated very well by (Sheth & van de Weygaert, 2004)

Here $\mathcal{D}=|\delta_{\text{v}}|/(\delta_{\text{c}}+|\delta_{\text{v}}|)$, and $\delta_{\text{c}}=1.686$. $\delta_{\rm v}$ usually can be derived theoretically assuming spherical evolution, and it is found to be $-2.731$ in the $\Lambda$CDM model when the void matter desity $\rho_{\rm v}=0.2\bar{\rho}_{\rm m}$ (Jennings et al., 2013). However, since the voids identified in our catalog do not assume any shape, we set $\delta_{\rm v}$ as a free parameter at a given redshift in the model fitting process, and it can be seen as an effective void linear underdensity threshold in our model. Then the relation between $R_{\text{L}}$ and $R_{\text{v}}$ is given by

where $c_{\rm v}=1.594$ (Bernardeau, 1994; Jennings et al., 2013).

In addition, we also consider the RSD and Alcock-Paczyński (AP, Alcock & Paczynski, 1979) effects in the model by two factors, for relating the void radius in observational redshift and real spaces. Following Correa et al. (2021), we have

Here $f_{\rm RSD}=1-(1/6)\beta\Delta(R_{\rm v})$, where $\Delta(R_{\rm v})=(1-\delta_{\rm v}/c_{\rm v})^{-c_{\rm v}}-1$ (Bernardeau, 1994), $\beta=f/b$, $f$ is the growth rate, and $b$ is the tracer bias. We set $\beta$ as free parameters at different redshift bins in the fitting process. $f_{\rm AP}=\alpha_{\parallel}^{1/3}\alpha_{\perp}^{2/3}$, where $\alpha_{\parallel}=H^{\rm fid}(z)/H(z)$ and $\alpha_{\perp}=D_{\rm A}(z)/D_{\rm A}^{\rm fid}(z)$, which are the ratios for Hubble parameters and angular diameter distances in the fiducial and real cosmologies.

In Figure 2, we show the VNC mock data and best-fit theoretical curves for Case 1 (red) and Case 2 (green). We use jackknife method to derive the error bar of each data point. We can find that, as expected, the number of voids decreases as the redshift increases, and the void number in Case 1 is lower than Case 2 since more voids are excluded in Case 1 by applying narrower void radius ranges.

We adopt $\chi^{2}$ for constraining the model parameters, which takes the form as

where $N_{\text{v}}^{\text{data}}(z_{i})$ and $N_{\text{v}}^{\text{th}}(z_{i})$ are the data and theoretical model of the VNC, respectively, and $\sigma^{i}_{\text{v}}$ is the data error in the $i$th redshift bin. The likelihood function can be calculated by $\mathcal{L}$ $\propto$ exp($-\chi^{2}$/2).

We constrain the free parameters using the Markov Chain Monte Carlo (MCMC) method with emcee (Foreman-Mackey et al., 2013). We choose 112 walkers and obtain 20000 steps for each chain. The first 10 percent of steps are discarded as the burn-in process. In our model we totally have 18 free parameters. The cosmological free parameters are the dark energy equation of state $w$, reduced Hubble constant h, the total matter density parameter $\Omega_{\text{m}}$, baryon density parameter $\Omega_{\rm b}$, spectral index $n_{\rm s}$, and amplitude of initial power spectrum $A_{\rm s}$. The free parameters about void $\delta_{\text{v}}^{i}$ is the threshold for void formation from the six redshift bins from $z=0.3$ to 1.3. And we also set the RSD parameter $\beta^{i}$ in the six redshift bins. The flat priors of the free parameters are $\Omega_{\rm m}\in(0.1,0.5)$, $\Omega_{\rm b}\in(0.02,0.08)$, $A_{\text{s}}/10^{-9}\in(1.0,3.0)$, $n_{\rm s}\in(0.7,1.2)$, $w\in(-1.8,-0.2)$, $h\in(0.5,0.9)$, and $\delta_{\text{v}}^{i}\in(-2,0)$ and $\beta^{i}\in(0,0.5)$ in the six redshift bins. And we set the fiducial value of cosmological parameter with $w=-1$, $h=0.6766$, $\Omega_{\text{m}}=0.3111$, $\Omega_{\text{b}}=0.049$, $n_{\text{s}}=0.9665$, and $A_{\rm s}/10^{9}=2.1$. Note that $\delta_{\text{v}}^{i}$ and $\beta^{i}$ are not the input parameters in the simulation, and they do not have the fiducial values.

In Figure 3, we show the one-dimensional (1D) probability distribution functions (PDFs) and contour maps of the six cosmological parameters constrained by the VNC for Case 1 and Case 2 at 68% and 95% confidence levels (CL). The details of the constraint result of all the free parameters are shown in Table 2, containing the best-fit values, 1$\sigma$ errors, and relative accuracies. We can find that the fitting results of the cosmological parameters are consistent with the corresponding fiducial values within 1$\sigma$ CL. This means that our method of the VNC can derive the cosmological information correctly.

Besides, the constraint powers in Case 1 and Case 2 are similar for the cosmological parameters, which give the constraint accuracies as $\Omega_{\rm m}\sim 30\%$, $\Omega_{\rm b}\sim 50\%$, $A_{\text{s}}/10^{-9}\sim 20\%$, $n_{\rm s}\sim 20\%$, $w\sim 27\%$, and $h\sim 10\%$. Note that the constraint accuracy can be improved by about one order of magnitude, if considering the full CSST spectroscopic survey covering 17500 deg². We can find that this result is comparable to that using the VSF method based on the same simulation and galaxy catalog as shown in Song et al. (2024). This is probably because that the method of the VNC is not as sensitive as the VSF method to the void shape. Hence, we can adopt simpler selection criteria, keep more voids in the cosmological analysis and include more redshift bins, i.e. six redshift bins from $z=0.3$ to 1.3 in this work while only four bins from $z=0.5$ to 1.1 in Song et al. (2024), which can effectively improve the statistical significance. Besides, our theoretical modeling without fixing $\delta_{\rm v}$ also could match the data better, since detailed void features are integrated over in the VNC. All these advantages can make the VNC obtain stringent and accurate constraints on the cosmological parameters. Here we do not show the constraint result of $\beta$, since we find that the VNC is not very sensitive to $\beta$ and there is no stringent constraint on it (with an accuracy $>60\%$) for the current mock data we use.

In Figure 4, we plot the best-fit values and 1$\sigma$ errors of $\delta_{\rm v}$ in the six redshift bins for Case 1 and Case 2, and the results from the VSF method given by Song et al. (2024) are also shown for comparison. The contour maps and and 1D PDFs of $\delta_{\rm v}^{i}$ are shown in Appendix A. We can find that the values of $\delta_{\rm v}$ in all the three cases have significant discrepancy compared to the theoretical values $\delta_{\rm v}\simeq-2.7$, assuming the spherical evolution and simulation particles as tracers (Jennings et al., 2013). Since our voids are identified by the watershed algorithm without any assumption of void shape and adopting galaxies as tracers, the current result is reasonable.

We also notice that the values of $\delta_{\rm v}$ at different redshifts in the three cases have a similar trend, which are approaching to 0 when redshift increases. This means that, as expected in the linear evolution, the Eulerian void size $R_{\rm v}$ will be close to the Lagrangian void size $R_{\rm L}$ at high redshifts (see Eq. (11)). In addition, the result of Case 1 is more consistent with that from the VSF method in $z=0.5-1.1$, since the two analyses select similar ranges of void radius. On the other hand, the values of $\delta_{\rm v}$ in Case 2 are always lower than the other two cases, especially at $z<0.6$. This is because the value of $\delta_{\rm v}$ can reflect the merging of small voids into large voids in the $V{\rm d}n$ model. Typically, a smaller $\delta_{\rm v}$ means the average void radius in a sample is smaller (see e.g. Russell, 2013). Since the lower limit of the void radius range in Case 2 is smaller than that in Case 1, Case 2 contains more small voids used in the analysis. This will suppress the average void radius in Case 2 and lead to a smaller $\delta_{\rm v}$ compared to Case 1, especially at low redshifts. If choosing similar radius ranges for Case 1 and Case 2, we find the values of $\delta_{\rm v}$ will be closer in these two cases.

In this work, we propose to use the VNC as a cosmological probe for studying the LSS. To check the feasibility, we generate the galaxy mock catalog based on Jiutian simulations and the CSST spectroscopic galaxy survey, and identify voids by $\tt VIDE$ without assuming void shape. The mock void catalog and data of the VNC are then derived at the six redshift bins from $z=0.3$ to $1.3$ in two cases, i.e. Case 1 and Case 2, by using empirical void radius ranges and considering the void ellipticity, respectively. We also set $\delta_{\rm v}$ as a free parameter at a given redshift in the theoretical model for better fitting the mock data and studying its redshift dependency and evolution. The RSD and AP effects are also considered in the analysis. Then we perform a joint fit of the cosmological, void and RSD parameters using the mock VNC data by the MCMC method.

We find that both Case 1 and Case 2 with different selection criteria lead to similar results. For the constraints on the cosmological parameters, the VNC can correctly derive the cosmological information, that the constraint power is comparable to the VSF, and can provide a few percentage level constraints on the cosmological parameters in the CSST spectroscopic survey. This is due to that the VNC is insensitive to void shape, and more voids and redshift bins can be kept by simpler selection criteria in the analysis, which could effectively improve the statistical significance. The theoretical model of the VNC also can be effectively modified by assuming a free $\delta_{\rm v}$ at a given redshift, and can match the data better than the VSF by integrating over the void size. For the constraint on the void linear underdensity threshold $\delta_{\rm v}$, the results at different redshift bins from the VNC method have a similar trend as that from the VSF method, i.e. it becomes larger and larger and close to zero when redshift increases. The value of $\delta_{\rm v}$ is also dependent on the chosen void radius range at a given redshift. All of these indicate that the VNC can be a feasible and effective probe in cosmological studies.

YS and YG acknowledge the support from National Key R&D Program of China grant Nos. 2022YFF0503404, 2020SKA0110402, and the CAS Project for Young Scientists in Basic Research (No. YSBR-092). KCC acknowledges the support the National Science Foundation of China under the grant number 12273121. XLC acknowledges the support of the National Natural Science Foundation of China through Grant Nos. 11473044 and 11973047, and the Chinese Academy of Science grants ZDKYYQ20200008, QYZDJ-SSW-SLH017, XDB 23040100, and XDA15020200. QG acknowledges the support from the National Natural Science Foundation of China (NSFC No.12033008). This work is also supported by science research grants from the China Manned Space Project with Grant Nos. CMS- CSST-2021-B01 and CMS-CSST-2021-A01.

The data that support the findings of this study are available from the corresponding author upon reasonable request.