Cosmological Constraints on Ω_m and σ₈ from Cluster Abundances Using the GalWCat19 Optical-spectroscopic SDSS Catalog

The ${\mathtt{GalWCat}}{\mathtt{19}}$ catalog is incomplete in the distribution of clusters with respect to comoving distance (redshift), and in the distribution of clusters with respect to mass. In this section, we discuss such incompleteness and how to make corrections.

The completeness in comoving volume (redshift) of the ${\mathtt{GalWCat}}{\mathtt{19}}$ catalog can be investigated by calculating the abundance of clusters predicted by a theoretical model and comparing it with the abundance of ${\mathtt{GalWCat}}{\mathtt{19}}$ clusters. We adopt the functional form of Tinker et al. (2008; hereafter Tinker08) to calculate the halo mass function (HMF;⁹ see Section 4.1 for more details) and consequently the predicted abundance of clusters.

The integrated abundance of clusters as a function of redshift for the ${\mathtt{GalWCat}}{\mathtt{19}}$ sample, $N(\lt z)$, is presented in the upper left panel of Figure 1. Note that N (<z) is calculated for the clusters with redshift z ≥ 0.04 to remove the effect of nearby regions where the cosmic variance has a large effect due to the small volume. The plot shows that the catalog is matched with the prediction of Tinker08 for z ≲ 0.09. Also, the fractional error of N (<z) relative to the expectation of Tinker08, ${(N{(\lt z)}_{\mathrm{obs}}-N{(\lt z)}_{\mathrm{model}})/N(\lt z)}_{\mathrm{model}}$, for each model and the expected Poisson noise (gray shaded area) are presented in the lower left panel. The plot shows that the scatter relative to each model is nearly constant (around zero) for z ≲ 0.09 before it blows up after this redshift limit. This indicates that ${\mathtt{GalWCat}}{\mathtt{19}}$ is approximately complete in volume for z ≲ 0.09 (or equivalently comoving distance of D ≲  265 ${h}^{-1}\,$Mpc for the ΛCMD universe with ${{\rm{\Omega }}}_{m}$ = 0.3). We call this volume-complete subsample ${\mathtt{NoSelFVC}}$.

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Similarly, the integrated abundance of clusters as a function of cluster mass, N (>M), is presented in the upper right panel of Figure 1 in comparison to five Tinker08 models and the scatter is presented in the lower right panel. The plot shows that the data is matched with the models of ${{\rm{\Omega }}}_{m}$ = [0.20, 0.305, 0.40] with ${\sigma }_{8}$ = 0.825 better than the models of ${{\rm{\Omega }}}_{m}$ = 0.305 and ${\sigma }_{8}$ = [0.725, 0.925]. Even though it is not an easy task to specifically determine the mass threshold at which the catalog is complete, the three matched models indicate that ${\mathtt{GalWCat}}{\mathtt{19}}$ is approximately complete for log(M) ≳ 13.9 ${h}^{-1}\ {M}_{\odot }$. We discuss the systematics of adapting this mass threshold on our analysis in Section 5.1. The large scatter at the high-mass end is due to the small number of massive clusters, while the large scatter at the low mass end comes from the incompleteness of ${\mathtt{GalWCat}}{\mathtt{19}}$.

In order to correct for the incompleteness in volume of ${\mathtt{GalWCat}}{\mathtt{19}}$ each cluster should be weighed by ${ \mathcal S }(D)$, where ${ \mathcal S }$ is the selection function at a distance D. Figure 2 introduces the normalized number density ${{ \mathcal N }}_{n}(D)$, defined as the cluster number density normalized by the average number density calculated for clusters within comoving distance D < 265 ${h}^{-1}\,$Mpc, for all clusters and for five mass bins as described in Table 1. The distribution of points in Figure 2 can be described by an exponential function that represents the selection function ${ \mathcal S }(D)$. It has the form

$$\begin{eqnarray}&&{ \mathcal S }(D)=a\exp \left[-{\left(\displaystyle \frac{D}{b}\right)}^{\gamma }\right].\end{eqnarray} \tag{ 1 }$$

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Table 1.  The Cluster Average Number Density for Different Mass Bins

Mass	Number of	Average	Color
Bin	Clusters	Number Density
(${h}^{-1}\ {M}_{\odot }$)		$({10}^{-5}\,{h}^{3}$ Mpc−3)
13.6–15.2	1800	5.6	Black
13.6–13.8	527	2.2	Blue
13.8–14.0	461	1.5	Green
14.0–14.2	411	1.0	Red
14.2–14.5	326	0.7	Cyan
14.5–15.2	75	0.2	Magenta

Note. Columns: (1) the mass bin in units of $\mathrm{log}M$ (${h}^{-1}\ {M}_{\odot }$); (2) the number of clusters in each mass bin; (3) the average number density calculated for clusters within comoving distance D < 265 ${h}^{-1}\,$Mpc in each mass bin; (4) the color of the number density profile as shown in the right panel of Figure 1.

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The parameters a, b, and γ are determined by applying the chi-squared algorithm using the Curve Fitting MatLab Toolbox. The best-fit values of these parameters are a = 1.07 ± 0.12, b = 293.4 ± 20.7 ${h}^{-1}\,$Mpc, and γ = 2.97 ± 0.90 with an rms error of 0.15. Note that the normalization a is greater than unity because of the scatter and the effect of the cosmic variance. However, we apply the selection function with the condition that ${ \mathcal S }(D)\leqslant 1$.

We should be cautious when using ${ \mathcal S }(D)$ at large distances. This is because ${S}$(D ≳ 500) ${h}^{-1}\,$Mpc drops to ≳0.01 as demonstrated in Figure 2, which means that a distant cluster would be weighted at least 100 times as a nearby cluster. This will overestimate or overcorrect the number of clusters at large distances, and consequently the estimated CMF will be noisy. Thus, in order to avoid the overcorrection and the noisiness of CMF we restrict our sample to a maximum comoving distance of D ≤ 365 (or z ≤ 0.125) for which ${ \mathcal S }(D)\lesssim 0.2$.

It is well-known that the cluster number density of a given mass decreases with redshift for a 100% complete sample because of the HMF evolution effect. Thus, the CMF should be scaled or corrected by an evolution function, ${{ \mathcal S }}_{\mathrm{evo}}(D)$. For a sample with a broad range of redshifts, the only way to take the evolution into account is to calculate this function. However, the disadvantage of this approach is that the correction is model dependent: the measured HMF (i.e., CMF) is a convolution of the true HMF and theoretical estimate of ${{ \mathcal S }}_{\mathrm{evo}}(D)$. However, for a sample with a narrow range of redshifts (as in our case) we show in Appendix A that the evolution effect is less than 3% for clusters in the redshift range of 0.045 ≤ z ≤ 0.125. In Appendix B, we discuss the effect of adopting this redshift interval in our results.

In this section, we compute the CMF, dn(M)/dlog(M), and its corresponding cumulative mass function, n (>M), which are estimated for a ΛCDM cosmology with ${{\rm{\Omega }}}_{m}$ = 0.3 and Ω_Λ = 0.7. The CMF is defined as the number density of clusters per logarithmic cluster mass interval. Also, the cumulative CMF is defined as the number density of clusters more massive than a given mass M.

Mathematically, the CMF, weighted by the selection function ${ \mathcal S }$, is given by

$$\begin{eqnarray}&&\displaystyle \frac{{dn}(M)}{d\mathrm{log}M}=\displaystyle \frac{1}{d\mathrm{log}M}\displaystyle \sum _{i}\displaystyle \frac{1}{V}\displaystyle \frac{1}{{ \mathcal S }({D}_{i})},\end{eqnarray} \tag{ 2 }$$

where D_i is the comoving distance of a cluster i, and V is the comoving volume, which is given by

$$\begin{eqnarray}&&V=\displaystyle \frac{4\pi }{3}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{s}}{\rm{u}}{\rm{r}}}}{{{\rm{\Omega }}}_{{\rm{s}}{\rm{k}}{\rm{y}}}}({D}_{2}^{3}-{D}_{1}^{3}),\end{eqnarray} \tag{ 3 }$$

where Ω_sky = 41,253 deg² is the area of the sky, Ω_sur ≃  10,000 deg² is the area covered by GalWCat19, and D₁ and D₂ are the minimum and maximum comoving distances of the cluster sample.

Figure 3 introduces the cumulative CMF computed from ${\mathtt{GalWCat}}{\mathtt{19}}$. The black line is the CMF computed from the MDPL2 simulation (for the snapshot hlist_0.91520.list at z ∼ 0.09 or D ∼ 265; Klypin et al. 2016). The blue points introduce the CMF for ${\mathtt{NoSelFVC}}$ without the correction of ${ \mathcal S }(D)$, since this sample is already complete in volume (see Section 3.1 and Figure 1). The red points represent our CMF corrected by ${ \mathcal S }(D)$ for D ≤ 365 ${h}^{-1}\,$Mpc (z ∼ 0.125). Comparing the CMF estimated by the ${\mathtt{NoSelFVC}}$ subsample with that derived from the MDPL2 simulation indicates that the sample is approximately complete in mass for $\mathrm{log}(M)\gtrsim 13.9$ ${h}^{-1}\ {M}_{\odot }$, while it drops lower than the CMF of MDPL2 at the low-mass end. Also, our CMF, corrected by ${ \mathcal S }(D\leqslant 365)$, is in good agreement with the CMF derived from ${\mathtt{NoSelFVC}}$ with a scatter of 0.026 dex. The mass completeness of ${\mathtt{GalWCat}}{\mathtt{19}}$ is discussed in Section 3.1 and Figure 1. In Appendix B, we show that the results of deriving the cosmological parameters from ${\mathtt{NoSelFVC}}$ are consistent with that derived from ${\mathtt{SelFMC}}$. This indicates that weighting each cluster in our sample by $S(D\leqslant 365)$ introduced in Section 3.1 and Equation (1) is sufficient to correct for the volume incompleteness of ${\mathtt{GalWCat}}{\mathtt{19}}$.

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Therefore, our final subsample, corrected by S(D) is restricted by log(M) ≥ 13.9 ${h}^{-1}\ {M}_{\odot }$ and 0.045 ≤ z ≤ 0.125. The number of clusters of this subsample is 756, which represents ∼42% of the ${\mathtt{GalWCat}}{\mathtt{19}}$ sample. We use this subsample to constrain ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$  and call it a fiducial ${\mathtt{SelFMC}}$ sample.

The number of dark matter halos per unit mass per unit comoving volume of the universe, HMF, is given by

$$\begin{eqnarray}&&\displaystyle \frac{{dn}}{d\mathrm{ln}M}=f(\sigma )\displaystyle \frac{{\rho }_{0}}{M}\left|\displaystyle \frac{d\mathrm{ln}\sigma }{d\mathrm{ln}M}\right|;\end{eqnarray} \tag{ 4 }$$

here ρ₀ is the mean density of the universe, σ is the rms mass variance on a scale of radius R that contains mass $M=4\pi {\rho }_{0}{R}^{3}/3$, and f(σ) represents the functional form that defines a particular HMF fit.

Assuming a Gaussian distribution of mass fluctuation, Press & Schechter (1974) used a linear theory to derive the first theoretical model (hereafter PS) of HMF. While fairly successful in matching the results of N-body simulations, the PS formalism tends to predict too many low-mass clusters and too few high-mass clusters. More recently proposed theoretical models provide better approximations to the output from N-body simulations (e.g., Jenkins et al. 2001; Sheth et al. 2001; Warren et al. 2006; Tinker & Wetzel 2010; Bhattacharya et al. 2011; Behroozi et al. 2013a).

In this paper, we adopt the functional form proposed by Tinker et al. (2008) (hereafter Tinker08) as our form of the HMF. This approach assumes universality of the HMF across the cosmological parameter space considered in this work, and uses a fitting function that was calibrated against N-body simulations. The Tinker08 model is formally accurate to better than 5% for the cosmologies close to the ΛCDM cosmology and for the mass and redshift range of interest in our study (e.g., Vikhlinin et al. 2009). Although the formula has been calibrated using dissipationless N-body simulations (i.e., without the effect of baryons), hydrodynamic simulations suggest that these have negligible impact for clusters with masses as high as those considered here (e.g., Rudd et al. 2008; Velliscig et al. 2014; Bocquet et al. 2016). Finally, note that the Tinker08 model is defined in spherical apertures enclosing overdensities similar to the mass we derive for the ${\mathtt{GalWCat}}{\mathtt{19}}$ observed clusters.

$$\begin{eqnarray}&&f(\sigma ,z)=A\left[{\left(\displaystyle \frac{\sigma }{b}\right)}^{-a}+1\right]\exp (-c/{\sigma }^{2})\end{eqnarray} \tag{ 5 }$$

where $A=0.186{\left(1+z\right)}^{-0.14}$, $a=1.47{\left(1+z\right)}^{-0.06}$, $b\,=2.57{\left(1+z\right)}^{-\alpha }$, c = 1.19, and $\mathrm{ln}\alpha ({{\rm{\Delta }}}_{\mathrm{vir}})={\left[75/\left(\mathrm{ln}({{\rm{\Delta }}}_{\mathrm{vir}}/75)\right)\right]}^{1.2}$, and σ² is the mass variance defined as

$$\begin{eqnarray}&&{\sigma }^{2}(M,z)=\displaystyle \frac{g(z)}{2\pi }\int P(k){W}^{2}\left({kR}\right){k}^{2}{dk}\end{eqnarray} \tag{ 6 }$$

P(k) is the current linear matter power spectrum (at z = 0) as a function of wavenumber k, $W({kR})\,={3\left[\sin ({kR})-{kR}\cos ({kR})\right])/({kR})}^{3}$ is the Fourier transform of the real-space top-hat window function of radius R, and g(z) = σ₈(z)/σ₈(0) is the growth factor of linear perturbations at scales of 8 ${h}^{-1}\,$Mpc, normalized to unity at z = 0.

The current linear power spectrum P(k) is defined as P(k) = B kⁿ T²(k), where T(k) is the transfer function, B is the normalization constant and n is the spectral index. Usually the normalization B is calculated from the cosmological parameter ${\sigma }_{8}$ (e.g., Reiprich & Böhringer 2002; Murray et al. 2013). The function kⁿ imprints the primordial power spectrum during the epoch of inflation. The transfer function T(k) quantifies how this primordial form is evolved with time to the current linear power spectrum on different scales. The transfer function T(k) is calculated using the public Code for Anisotropies in the Microwave Background (CAMB;¹⁰ Lewis et al. 2000). The quantities ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ are the main cosmological parameters that define the HMF. The other parameters do not strongly affect the HMF and thus we fix them during the calculation of the HMF as described below (e.g., Reiprich & Böhringer 2002; Bahcall et al. 2003; Wen et al. 2010).

The HMF is calculated using the publicly available HMFcalc¹¹ code (Murray et al. 2013). The code provides about 20 fitting functions that can be used to calculate the HMF. In this paper, in order to constrain ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$, we use Tinker08 (Equation (5)) as discussed above. We calculate the HMF by allowing ${{\rm{\Omega }}}_{m}$  to range between [0.1, 0.6] and ${\sigma }_{8}$ between [0.6, 1.2], both in steps of 0.005. We keep the following cosmological parameters fixed: the CMB temperature ${T}_{\mathrm{cmb}}=2.725\,{\rm{K}}$, baryonic density Ω_b = 0.0486, and spectral index n = 0.967 (Planck Collaboration et al. 2014), at redshift z = 0.089 (the mean redshift of ${\mathtt{GalWCat}}{\mathtt{19}}$).

In order to determine the best-fit mass function and constrain ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ we use a standard χ² procedure

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i=1}^{N}\left(\displaystyle \frac{{\left[{y}_{o,i}-{y}_{m,i}\right]}^{2}}{{\sigma }_{i}^{2}}\right),\end{eqnarray} \tag{ 7 }$$

where the likelihood, ${ \mathcal L }(y| {\sigma }_{8},{{\rm{\Omega }}}_{m})$, of data (CMF) given a model (HMF) is

$$\begin{eqnarray}&&{ \mathcal L }(y| {\sigma }_{8},{{\rm{\Omega }}}_{m})\propto \exp \left(\displaystyle \frac{-{\chi }^{2}(y| {\sigma }_{8},{{\rm{\Omega }}}_{m})}{2}\right)\end{eqnarray} \tag{ 8 }$$

y_o and y_m are the data and model cumulative mass functions at a given mass and σ is the statistical uncertainty of the data.

Using the fiducial SelFMC sample of 756 clusters with $\mathrm{log}(M)\geqslant 13.9$ and 0.045 ≤ z ≤ 0.125, the best-fit parameters for the minimum value of χ² are ${{\rm{\Omega }}}_{m}={0.310}_{-0.029}^{+0.025}$ and ${\sigma }_{8}={0.810}_{-0.034}^{+0.039}$ for Tinker08 at redshift z = 0.085. In Section 5.1 we discuss the systematics of cluster mass uncertainty, mass threshold, and selection function.

The banana shape in Figure 4 shows the well-known degeneracy between ${\sigma }_{8}$ and ${{\rm{\Omega }}}_{m}$. The relationship between ${\sigma }_{8}$ and ${{\rm{\Omega }}}_{m}$  is often expressed as

$$\begin{eqnarray}&&{\sigma }_{8}=\alpha \,{{\rm{\Omega }}}_{m}^{\beta }.\end{eqnarray} \tag{ 9 }$$

The parameters α, β, and δ are determined by applying the χ² algorithm using the Curve Fitting MatLab. The best-fit values of these parameters are α = 0.425 ± 0.006 and β = −0.550 ± 0.007 with an rms error of 0.005 for the Tinker08 model.

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We now ask the question—how do ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ contribute individually to the HMF? In other words, why do cluster abundance studies introduce a degeneracy between ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$? The degeneracy occurs because a low abundance of massive clusters could be caused either by a small amount of matter in the universe (a low value of ${{\rm{\Omega }}}_{m}$) or small fluctuations in the density field (a low value of ${\sigma }_{8}$). Similarly, a high abundance of massive clusters could be caused either by a large amount of matter in the universe (a high value of ${{\rm{\Omega }}}_{m}$) or large fluctuations in the density field (a high value of ${\sigma }_{8}$). Therefore, it is possible to obtain the same abundance of massive clusters by fixing one parameter and varying the other one. Figure 5 introduces two sets of HMFs calculated by Tinker08. The first set is shown on the left panel for five different values of ${{\rm{\Omega }}}_{m}$  = [0.1 0.2 0.3 0.4 0.5], while fixing ${\sigma }_{8}$ = 0.8. The second set is shown on the right panel for five different values of ${\sigma }_{8}$ = [0.6 0.7 0.8 0.9 1.0] while fixing ${{\rm{\Omega }}}_{m}$  = 0.3. As expected, increasing the matter density of the universe increases the number of clusters of all masses. But increasing the rms mass fluctuation increases the number of high-mass clusters more dramatically than the number of low-mass clusters. In other words, ${\sigma }_{8}$ is very sensitive to the high-mass end of the HMF.

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In constraining ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ in Section 4.2, we only account for the statistical uncertainty of the estimated cumulative CMF using the fiducial ${\mathtt{SelFMC}}$ sample. In this section, we discuss the systematics due to mass uncertainty, mass threshold, and parameterization of the selection function.

5.1.1. Mass Uncertainty

The first uncertainty comes from the difficulty of calculating cluster masses accurately. Generally, masses that are estimated using scaling relations, such as luminosity, richness, temperature, and dispersion velocity–mass relations, introduce large scatter and consequently large systematic uncertainties (e.g., Mantz et al. 2016; Mulroy et al. 2019). Masses that are computed by dynamical estimators are subject to systematic uncertainties (e.g., Wojtak & Łokas 2007; Rozo et al. 2010; Old et al. 2018). However, using the virial theorem, corrected for the surface pressure term, provides a relatively unbiased estimation of cluster masses (e.g., Rines et al. 2010; Ruel et al. 2014), particularly when using a sophisticated interloper rejection technique such as GalWeight (Abdullah+18). Also, the virial mass estimator calculates the total cluster mass including baryonic (gas and galaxies) and dark matter regardless of the internal complex physical processes associated with the baryonic component in clusters. However, the virial mass estimator still introduces scatter in estimating cluster masses (see Section 2). Abdullah+20 showed that the application of the virial mass estimator on two mock catalogs (HOD2 and SAM2) recalled from Old et al. (2015) returned intrinsic scatter of ∼0.23 dex in the recovered mass relative to the fiducial cluster mass. Also, the ${\mathtt{GalWCat}}{\mathtt{19}}$ catalog introduced the fractional uncertainty (see Section 2) of each cluster mass.

Assuming a normal distribution, we investigate the systematics of the mass uncertainty by generating ∼8000 estimate for each cluster mass using both the fractional uncertainty for each cluster and the intrinsic scatter for the entire sample. In other words, we reanalyze ${\mathtt{SelFMC}}$ ∼8000 times and refit for ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ for each time. The left panel of Figure 6 introduces the effect of cluster mass uncertainty on the constraints on ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$. Using the fractional uncertainty, we obtain ${{\rm{\Omega }}}_{m}$ = 0.305 ± 0.014 and ${\sigma }_{8}$ = 0.816 ± 0.021, where the red ellipse represents 68% CL for the distribution of the reestimated 8000 pairs of ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$. Using the intrinsic scatter (blue ellipse), we find ${{\rm{\Omega }}}_{m}$ = 0.309 ± 0.014 and ${\sigma }_{8}$ = 0.815 ± 0.022. Both results indicate that the cluster mass uncertainty (fractional or intrinsic) does not affect our constraints on ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ using ${\mathtt{SelFMC}}$.

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5.1.2. Mass Threshold

5.1.3. Selection Function Parameterization

The left panel of Figure 7 introduces the 68% CL derived from ${\mathtt{SelFMC}}$ in comparison to the results obtained from other cluster abundance studies. Samples of galaxy clusters constructed from galaxy surveys include optical photometric (e.g., Kirby et al. 2019), X-ray (e.g., Mantz et al. 2015), and SZ (e.g., Zubeldia & Challinor 2019) catalogs as listed in Table 2. The figure shows that the CLs of all cluster abundance studies introduce a degeneracy between ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ as we discussed in Section 4.2. Also, the CL derived from ${\mathtt{SelFMC}}$ overlaps the CLs obtained from all other results as shown in the figure. Regardless of this overlapping, the right panel of Figure 7 shows that the constraints on ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ from cluster abundance studies are in tension with each other, even for the studies that use the same type of cluster sample. Specifically, the X-ray independent studies listed in Table 2 introduce different values of ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$, which  vary from ∼0.22 to 0.40 and 0.71 to 0.89, respectively. Also, the independent studies that use SZ-cluster samples show that ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ vary from ∼0.25 to 0.31 and 0.77 to 0.98, respectively.

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Table 2.  Comparison of Constraints on Cosmological Parameters ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ Derived from Clusters Abundances (CMF) and from Other Cosmological Probes, Including Cosmic Shear, Galaxy–Galaxy Lensing, Angular Clustering, BAO, Supernovae, and CMB

Sample	Mass Estimation	${{\rm{\Omega }}}_{m}$	${\sigma }_{8}$	${S}_{8}$ a	${{\rm{\Delta }}}_{\mathrm{pl}}$ b	Reference
Spectroscopically selected catalogs—cluster abundance
${\mathtt{GalWCat}}{\mathtt{19}}$	Virial theorem	${0.305}_{-0.042}^{+0.037}$	${0.810}_{-0.056}^{+0.053}$	0.817	0.032	This work
Optical photometrically selected catalogs—cluster abundance
MaxBCG	Richness-mass +WL	${0.281}_{-0.066}^{+0.066}$	${0.804}_{-0.073}^{+0.073}$	0.779	0.108	Rozo et al. (2010)
RedMaPPer	Richness-mass +WL	${0.220}_{-0.040}^{+0.050}$	${0.910}_{-0.100}^{+0.110}$	0.778	0.325	Costanzi et al. (2019)
RedMaPPer	Richness-mass +X-ray	${0.250}_{-0.040}^{+0.040}$	${0.850}_{-0.08}^{+0.06}$	0.776	0.212	Kirby et al. (2019)
X-ray catalogs—cluster abundance
REFLEX	Luminosity-mass	${0.341}_{-0.029}^{+0.030}$	${0.711}_{-0.031}^{+0.039}$	0.758	0.148	Schuecker et al. (2003)
Chandra-ROSAT	Luminosity-mass	${0.255}_{-0.043}^{+0.043}$	${0.820}_{-0.013}^{+0.013}$	0.757	0.191	Vikhlinin et al. (2009)
ROSAT (WtG)c	Luminosity-mass	${0.260}_{-0.030}^{+0.030}$	${0.830}_{-0.04}^{+0.04}$	0.773	0.176	Mantz et al. (2015)
ROSAT - HIFLUGCS	Luminosity-mass	${0.217}_{-0.054}^{+0.073}$	${0.893}_{-0.095}^{+0.098}$	0.760	0.327	Schellenberger & Reiprich (2017)
XMM-XXL	Temperature-mass	${0.399}_{-0.094}^{+0.094}$	${0.721}_{-0.071}^{+0.071}$	0.832	0.289	Pacaud et al. (2018)
SZ catalogs—cluster abundance
ACT, [BBN+H0+ACTcl(B12)]	SZ-mass	${0.252}_{-0.047}^{+0.047}$	${0.872}_{-0.065}^{+0.065}$	0.799	0.214	Hasselfield et al. (2013)
ACT, [BBN+H0+ACTcl(Dyn)]	SZ-mass	${0.301}_{-0.082}^{+0.082}$	${0.975}_{-0.108}^{+0.108}$	0.977	0.207	Hasselfield et al. (2013)
SPT	SZ-mass	${0.276}_{-0.047}^{+0.047}$	${0.781}_{-0.037}^{+0.037}$	0.776	0.129	Bocquet et al. (2019)
HECS-SZ	SZ-mass	⋯	⋯	0.751	⋯	Ntampaka et al. (2019)
Planck18	SZ-mass	${0.310}_{-0.020}^{+0.020}$	${0.770}_{-0.040}^{+0.040}$	0.783	0.138	Zubeldia & Challinor (2019)
Other cosmological probes
DES-Y1	CS+GGL+ACd	${0.270}_{-0.040}^{+0.041}$	${0.820}_{-0.036}^{+0.038}$	0.778	0.143	Abbott et al. (2018b)
KiDS+GAMA	CS+GGL+AC	${0.315}_{-0.092}^{+0.068}$	${0.785}_{-0.117}^{+0.111}$	0.804	0.032	van Uitert et al. (2018)
Pantheon	SNe	${0.307}_{-0.012}^{+0.012}$	⋯	⋯	⋯	Scolnic et al. (2018)
6dF+DR7+BOSSe	BAO	${0.346}_{-0.045}^{+0.045}$	⋯	⋯	⋯	Alam et al. (2017)
WMAP9	CMB only	${0.280}_{-0.040}^{+0.041}$	${0.820}_{-0.036}^{+0.038}$	0.792	0.112	Hinshaw et al. (2013)
Planck18	CMB only	${0.315}_{-0.007}^{+0.007}$	${0.811}_{-0.006}^{+0.006}$	0.832	0.000	Planck Collaboration et al. (2018)

Notes.

^aThe cluster normalization condition parameter, S₈, is defined as ${S}_{8}={\sigma }_{8}{({{\rm{\Omega }}}_{m}/0.3)}^{0.5}$ as used in the literature. ^b ${{\rm{\Delta }}}_{\mathrm{pl}}=\sqrt{{\left[({{\rm{\Omega }}}_{m,\mathrm{ref}}-{{\rm{\Omega }}}_{m,\mathrm{pl}})/{{\rm{\Omega }}}_{m,\mathrm{pl}}\right]}^{2}+{\left[({\sigma }_{8,\mathrm{ref}}-{\sigma }_{8,\mathrm{pl}})/{\sigma }_{8,\mathrm{pl}}\right]}^{2}}$ is the scatter of ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ obtained from each method listed the table relative to that obtained from Planck18 (Planck Collaboration et al. 2018). ^cMantz et al. (2015) used the combination of luminosity, temperature, gas mass, and lensing mass to estimate cluster masses, which was referred to as Weighting the Giant (WtG). ^dCC = cosmic shear, GGL = galaxy–galaxy lensing, AC = angular clustering. ^e6dF = Six Degree Field Galaxy Survey (Beutler et al. 2011), DR7 = SDSS data release 7 (Ross et al. 2015), BOSS = Baryon Oscillation Spectroscopic Survey (Alam et al. 2017).

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The question now is why are the cosmological constraints derived from many of the cluster abundance techniques in tension with each other? All cluster samples constructed from photometric surveys or detected by the SZ effect do not return an estimate of each cluster's mass directly. For such samples the cluster mass has to be inferred indirectly from other observables, which scale tightly with cluster mass. Among these mass proxies are X-ray luminosity, temperature, the product of X-ray temperature and gas mass (e.g., Vikhlinin et al. 2009; Mantz et al. 2016), richness (e.g., Yee & Ellingson 2003; Simet et al. 2017), and SZ signal (e.g., Bocquet et al. 2019). To estimate cluster masses for the clusters in these samples it is necessary to follow-up a subset of clusters and calculate their masses using, e.g., weak lensing or X-ray observations. Then, an observable–mass relation can be calibrated for these subsamples. Finally, the mass of each cluster in the sample can be estimated from this scaling relation. However, this reliance on observable–mass proxies introduces significant systematic uncertainties, which is the dominant source of error (e.g., Henry et al. 2009; Mantz et al. 2015) for the reasons explained in the next paragraph.

First, the masses obtained for the follow-up subsample of clusters are often biased. For example, it is known that X-ray mass estimates are typically biased low and so a mass bias factor, (1-β), needs to be introduced and calibrated. Second, the size of the subsample used for calibration is usually small (tens of clusters), which introduces large uncertainties in both the slope and the normalization of the scaling relation. Third, many cluster catalogs span a large redshift range, so evolution (due to both the evolution of the universe and the physical processes of baryons in clusters) in the the scaling relations used to estimate the masses needs to be carefully handled, introducing another source of uncertainty. All of the aforementioned assumptions can introduce large uncertainties in the estimates of cluster mass and consequently the constraints on cosmological parameters. For instance, ${\sigma }_{8}$ is specifically very sensitive to the high-mass end of the CMF and any offset of cluster true masses leads to biased estimation of ${\sigma }_{8}$. Other observational systematics that introduce additional uncertainties are photometric redshift errors and cluster miscentering.

By using the ${\mathtt{GalWCat}}{\mathtt{19}}$ cluster catalog and deriving cluster masses using the virial theorem, we were able to avoid most of the complexities described above. First, we were able to identify clusters, assign membership, and determine cluster centers and redshifts with high accuracy from the high-quality SDSS spectroscopic data set. Second, cluster membership was determined by the GalWeight technique, which has been shown to be ∼98% accurate in assigning cluster membership (Abdullah+18). Third, a mass for each cluster was determined directly using the virial theorem. Therefore, we were able to recover a total (dark plus baryonic) mass for each cluster and circumvent having to make any assumptions about the complicated physical processes associated with the baryons. It has been suggested that cluster masses estimated via the virial theorem are overestimated by 20%. But we note that we have applied a correction for the surface pressure term, which we believe decreases this bias, especially when applied in combination with our GalWeight membership technique (Abdullah+18). Abdullah+20 showed that the virial mass estimator performed well in comparison to the other mass estimators described in Old et al. (2015), and resulted in a relatively low bias and scatter when applied to two semianalytical simulations (see Figure 3 in Abdullah+20). Fourth, since ${\mathtt{GalWCat}}{\mathtt{19}}$ is a low-redshift cluster catalog it eliminates the need to make any assumptions about evolution in clusters themselves and evolution in cosmological parameters. Finally, because of the large size of the ${\mathtt{GalWCat}}{\mathtt{19}}$ we are able to determine the CMF well and consequently constrain the cosmological parameters ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ with high precision.

Cosmological parameters can be estimated from different cosmological probes rather than cluster abundance studies. We use measurements of primary CMB anisotropies from both WMAP (9 yr data; Hinshaw et al. 2013) and Planck satellites focused on the TT+lowTEB data combination from the 2018 analyses (Planck Collaboration et al. 2018). We also use angular diameter distances as probed by BAO including the 6dF Galaxy Survey (Beutler et al. 2011), the SDSS Data Release 7 (Ross et al. 2015), and the BOSS Data Release 12 (Alam et al. 2017). Furthermore, we use measurements of luminosity distances from Type Ia supernovae from the Pantheon sample (Scolnic et al. 2018). Finally, we use the measurements from a joint analysis of three cosmological probes: cosmic shear, galaxy–galaxy lensing, and angular clustering, including the results of the Kilo Degree Survey and the Galaxies And Mass Assembly survey (KiDS+GAMA; van Uitert et al. 2018) and the first year of the Dark Energy Survey (DES Y1; Abbott et al. 2018b; see Table 2). The left panel of Figure 8 introduces the 68% CL derived from ${\mathtt{SelFMC}}$ in comparison to the those obtained from the aforementioned cosmological probes. As shown, the CL derived from ${\mathtt{SelFMC}}$ overlaps the CLs obtained from all non-cluster abundance probes.

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We define the scatter

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{\mathrm{pl}}=\sqrt{{\left(\displaystyle \frac{{{\rm{\Omega }}}_{m,\mathrm{ref}}-{{\rm{\Omega }}}_{m,\mathrm{pl}}}{{{\rm{\Omega }}}_{m,\mathrm{pl}}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{8,\mathrm{ref}}-{\sigma }_{8,\mathrm{pl}}}{{\sigma }_{8,\mathrm{pl}}}\right)}^{2}},\end{eqnarray} \tag{ 10 }$$

to compare the constraints on ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ obtained from all cosmological probes, which are listed in Table 2 with that obtained from Planck18 (Planck Collaboration et al. 2018). Note that the constraints on ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ derived from most of the cluster abundance studies independently introduce a relatively large scatter compared to the CMB experiment of Planck18. However, our constraints on ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ are very comparable and competitive with Planck18 with a minimum value of Δ_pl = 0.018. Moreover, our constraint on ${{\rm{\Omega }}}_{m}$  is in excellent agreement with the results of the BAO and Pantheon, separately. This remarkable consistency demonstrates that our derived cluster catalog at low redshift and calculating cluster masses, using a spectroscopic database of galaxy surveys, is essential to obtain robust cosmological parameters. These results also emphasize the need to construct accurate cluster catalogs at high redshifts using the ongoing and upcoming galaxy surveys and to perform similar analyses as introduced in this work.

As discussed above there is a degeneracy between ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ derived from the CMF at low redshift. We combine our 68% CL with those obtained from Planck18+BAO+Pantheon, to eliminate the degeneracy of the our likelihood and to remarkably shrink the uncertainties of the cosmological parameters. The joint analysis gives ${{\rm{\Omega }}}_{m}={0.315}_{-0.011}^{+0.013}$ and ${\sigma }_{8}={0.810}_{-0.01}^{+0.011}$.

In this paper, we derived the CMF and the cosmological parameters ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ using a mass-complete subsample of 756 clusters (${\mathtt{SelFMC}}$) obtained from the ${\mathtt{GalWCat}}{\mathtt{19}}$ cluster catalog, which was constructed from the SDSS-DR13 spectroscopic data set. The advantages of using these catalogs are (i) we were able to identify clusters, assign membership, and determine cluster centers and redshifts with high accuracy from the high-quality SDSS spectroscopic data set; (ii) cluster membership was determined by the GalWeight technique, which has been shown to be ∼98% accurate in assigning cluster membership (Abdullah+18); (iii) the cluster masses were calculated individually using the virial theorem and corrected for the surface pressure term; (iv) ${\mathtt{GalWCat}}{\mathtt{19}}$ is a low-redshift cluster catalog that eliminates the need to make any assumptions about evolution in clusters themselves and evolution in cosmological parameters; (v) ${\mathtt{GalWCat}}{\mathtt{19}}$ is one of the largest available spectroscopic samples in order to be a fair representation of the cluster population.

Our CMF closely matches predictions from MultiDark Planck N-body simulations (snapshot hlist_0.91520.list¹² , with z ∼ 0.09) for log(M) ≳ 13.9 ${h}^{-1}\ {M}_{\odot }$. Assuming a flat ΛCDM cosmology, we used the publicly available HMFcalc¹³ code (Murray et al. 2013) to estimate HMFs for the Tinker08 model (Equation (5)). Then, using a standard χ² procedure, we compared our cumulative mass function to HMFs to determine the best-fit mass function and constrain ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$. We measured ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ to be ${{\rm{\Omega }}}_{m}={0.310}_{-0.027}^{+0.023}\pm 0.041$ (systematic) and ${\sigma }_{8}={0.810}_{-0.036}^{+0.031}\pm 0.035$ (systematic), with a cluster normalization relation of ${\sigma }_{8}=0.43{{\rm{\Omega }}}_{m}^{-0.55}$.

The cosmological constraints we derived are very competitive with those recently derived using both cluster abundance studies and other cosmological probes. In particular, our constraints on ${{\rm{\Omega }}}_{m}$  and ${\sigma }_{8}$ are consistent with Planck18+BAO+Pantheon constraints. This remarkable consistency highlights the potential of using ${\mathtt{GalWCat}}{\mathtt{19}}$ and its subsample ${\mathtt{SelFMC}}$, which are derived from the SDSS-DR13 spectroscopic data set utilizing the application of GalWeight to produce precision constraints on cosmological parameters. The joint analysis of our cluster data with Planck18+BAO+Pantheon gives ${{\rm{\Omega }}}_{m}={0.315}_{-0.013}^{+0.011}$ and ${\sigma }_{8}={0.810}_{-0.010}^{+0.011}$.