Power spectrum of the CODEX clusters

The clustering of galaxy clusters is sensitive to the geometry and the growth of structure in the Universe. Because of this, it has been an active field of cosmological research (see e.g., Bahcall, 1995; Borgani et al., 1999; Moscardini et al., 2000; Estrada et al., 2009; Sánchez et al., 2005; Allen et al., 2011; Marulli et al., 2018, 2021; Moresco et al., 2021; Garrel et al., 2022; Lesci, G. F. et al., 2022; Euclid Collaboration et al., 2024). X-ray selected galaxy clusters, in particular, have been shown to possess great potential as a cosmological probe (see e.g., Guzzo et al., 1999; Schuecker et al., 2003; Schuecker, 2005; Pratt et al., 2009; Guzzo et al., 2009; Pillepich et al., 2012; Böhringer et al., 2014; Mantz et al., 2014; Schellenberger & Reiprich, 2017; Käfer et al., 2019a; Clerc & Finoguenov, 2023; Ghirardini et al., 2024). This paper is another contribution to this field of research.

The dataset we analyze in this paper is the Constrain Dark Energy with X-ray (CODEX) galaxy cluster catalog (Finoguenov et al., 2020). This is a large catalog of X-ray sources identified as galaxy clusters by detecting an optical counterpart. The combined X-ray and optical observations allow for an accurate characterization of cluster properties, such as their masses. We give a description of the CODEX catalog and our sample selection in Section 2.

This paper aims to present the measurements of the power spectrum of the CODEX sample and infer cosmological information in the form of constraints on the cosmological parameters. Previous measurements (Schuecker et al., 2001; Hütsi, 2010; Balaguera-Antolínez et al., 2011) have characterized the cluster power spectrum as a scaled version of the underlying dark matter power spectrum, and show how the cluster clustering amplitude depends on observed properties such as the X-ray luminosity and richness, which paves the way toward a joint astrophysical and cosmological analysis (Allen et al., 2011; Evrard et al., 2014; Balaguera-Antol´ınez, 2014). In Section 3 we describe the respective methods to measure the cluster power spectrum and to model it. We present our measurements of the CODEX power spectrum and the results of our cosmological analysis in Section 4. Section 5 provides the conclusion of our analysis and results.

This paper is a continuation of the work presented in Lindholm et al. (2021) (L21 hereafter), where the clustering of CODEX clusters was studied using the two-point correlation function. Theoretically, the two-point correlation function is simply the Fourier transform of the power spectrum, so the quantities are equivalent. In practice, however, the two quantities rely on completely different estimators and are thus complementary. In addition to measuring different clustering statistics, this work includes a few improvements to the L21 analysis. The most important ones are a more accurate covariance matrix (Section 3.4) and a proper handling of uncertainty in the cluster mass estimates (Section 4).

The CODEX galaxy cluster survey is constructed by applying the red-sequence matched-filter probabilistic percolation (redMaPPer) algorithm (Rykoff et al., 2014) to the Sloan Digital Sky Survey (SDSS, Blanton et al., 2017) photometry inside the $10\ 000$ square degree area of the Baryon Oscillation Spectroscopic Survey (BOSS, Dawson et al., 2013) footprint and identifying faint X-ray sources detected in the ROentgen SATellite (ROSAT) All-Sky Survey (RASS) (Truemper, 1993; Voges et al., 1999). A detailed description of the survey and the catalog is presented in Finoguenov et al. (2020).

In what follows, we refer to several probability distributions that are logarithmically normal. Thus, we define the following quantities: $r_{c}\equiv\ln(R_{c}/\rm{kpc})$ (core radius of the X-ray surface brightness), $l\equiv\ln(L_{x}/\rm{ergs}/s)$ (rest-frame X-ray luminosity in the 0.1–2.4 keV band), $\mu\equiv\ln(M_{200c}/M_{\odot})$ (total mass measured within the overdensity of 200 with respect to the critical density), $\lambda\equiv\ln(\mathrm{SDSS\;Richness})$ (defined at the optical peak, with a detailed description provided in Rykoff et al., 2014).

To perform the clustering analysis, we selected the part of the CODEX catalog characterized by a low probability of chance cluster identification. Following previous CODEX studies, we applied the following cut, based on the measurement of SDSS richness,

where $z$ is the redshift of the cluster. Further discussion of cleaning is presented in Klein et al. (2019). We describe the effect of this cleaning as a $P^{\mathrm{RASS}}(I|\lambda,z)$ term in the modeling. As well as applying a redshift-dependent richness cut, we also excluded all clusters with a richness below 25.

We applied the BOSS stellar mask to remove the areas in which stars affect optical cluster detection. We assume that the optical completeness of the CODEX catalog, above the applied richness cut, is constant across the BOSS area and model it using

This is obtained using the tabulations of (Rykoff et al., 2014). We use an error function with the mean of $\lambda_{50\%}(z)$ and a $\sigma=0.2$, which reproduces the 75% and 90% quantiles of the distribution tabulated in (Rykoff et al., 2014). The probability of optical detection of a cluster in SDSS data is modeled as

The RASS survey coverage is highly inhomogeneous, with the limiting flux varying by an order of magnitude. To properly account for the variations in the cluster distribution caused by this, we generate a random catalog, with a total number of objects six times that of the data catalog. We partition the survey area into 100 zones of equal sensitivity, $S$, with each consecutive zone having a 12% difference in flux sensitivity. We denote the sky area of these zones as $\Delta\Omega_{S}$. The probability of cluster detection is computed as

where $\eta$ denotes the X-ray count (superscript “true” stands for predicted, “ob” for detected), $\nu\equiv\frac{\lambda-\langle\lambda|\mu,z\rangle}{\sigma_{\lambda|\mu}}$, and all the probability distributions are described in detail in Finoguenov et al. (2020). Equation 4 takes into account how the X-ray shapes ($r_{c}$, $\beta$) of the clusters and the RASS sensitivity affect the selection of the clusters, and it predicts changes in the distribution of X-ray shapes based on the measured covariance of the properties of the cluster (Cavaliere & Fusco-Femiano, 1976; Mulroy et al., 2019; Farahi et al., 2019; Käfer et al., 2019b). This approach allows us to account for the anticorrelation between X-ray luminosity scatter and optical richness, as well as the anticorrelation between galaxy cluster core radii and their luminosity scatter. Finally, we estimate the expected number of clusters in each redshift bin as

where

The sample selection in terms of richness and the construction of the random catalog described above is the same as in L21. However, our final sample differs from L21 in two ways. First, we selected a subset of the full CODEX footprint to have the sky area fully covered by simulated catalogs. The CODEX catalog consists of two disjoint patches, one of which has slightly larger angular coverage than the mock catalogs available to us (see Section 3.4). We thus selected a subset of the larger patch to be able to cover the whole footprint with one set of mock catalogs. Second, we also used a slightly narrower redshift range, namely $z=0.12-0.30$ (as opposed to $z=0.1-0.5$), to be compatible with the simulations. The final cluster sample we used for the power spectrum estimation contains 615 objects and the corresponding random sample is 11160 objects. The sky footprint of each sample can be seen in Fig. 1 and the corresponding redshift and richness distributions in Fig. 2 (along with those of the simulated mock catalogs). We also show the standard deviation of the mock catalogs as error bars. Clearly, the differences in the distributions are within the estimated variance, so the three types of catalogs (CODEX, random, and mocks) are compatible with being drawn from the same ensemble.

Figure 6 shows the power spectrum multipoles measured from the CODEX sample, along with the theoretical predictions from Eqs. 7, 8 and 9. We show predictions for the Planck 2018 cosmology, the best-fit cosmology obtained by fitting all the CODEX power spectrum multipoles, and the best-fit cosmology obtained by fitting the monopole alone to highlight the importance of the higher multipoles in our cosmological analysis (see the following paragraphs for more details). In Fig. 7 we show a comparison of the power spectrum monopole measured from our CODEX sample (same as the solid blue line in the top panel of Fig. 6) and the isotropic power spectrum measured from the REFLEX II sample (Balaguera-Antolínez et al., 2011). In both figures, the error bars are computed as the square root of the diagonal elements of the covariance matrix of Eq. 13. Our measurement of the CODEX power spectrum is in agreement with the Planck 2018 cosmology prediction within the error bars. A qualitative agreement with results from the REFLEX II samples is also seen, with differences likely being due to the redshift range and cluster mass cut used in each sample.

The power spectrum measurements described above can be used to obtain constraints on cosmological parameters. To do this, we ran a Markov chain Monte Carlo (MCMC) sampling of the present-day matter density parameter $\Omega_{\rm m}$ and the power spectrum amplitude $\sigma_{8}$ to obtain their posterior. The other $\Lambda$CDM parameters were fixed to the Planck 2018 values. We expect the main uncertainty in our modeling to be associated with the cluster mass estimates. To fully account for this, we also sampled the two richness-to-mass scaling relation parameters $\alpha$ and $e^{\beta}$. We used Gaussian approximations to the posteriors presented in K21 as priors for these parameters. To include the scatter in the richness-to-mass conversion, we generated a Gaussian noise vector with $\sigma=0.28$ (corresponding to the scatter determined in K21) and added this to the logarithmic mass of each cluster at each likelihood evaluation. The sampling was implemented using the emcee⁵⁵5https://github.com/dfm/emcee Python library (Foreman-Mackey et al., 2013).

We combine all the power spectrum multipoles into a single data vector, denoted with $\vec{\hat{P}}$. The same goes for the covariance matrix, denoted with $\vec{C}$. These are taken to be independent of the cosmology, even though estimating $\vec{\hat{P}}$ and $\vec{C}$ requires assuming a cosmology, to transform angles and redshifts into distances. Instead, we include the geometric effects of changing the cosmology in our modeling of the power spectrum multipoles following Gil-Mar´ın et al. (2020). The method consists of two steps:

1. 1.

   Transforming the wave vector magnitude and the angle with the line of sight direction according to changes in the Hubble distance $D_{\rm H}(z)\equiv c/H(z)$ and the angular diameter distance due to differences in cosmology.

2. 2.

   Renormalizing all the multipoles with a single factor to account for the isotropic volume rescaling caused by the differences in cosmology.

In our case, the reference cosmology is the one used to compute the CODEX power spectra and the covariance matrix (Planck 2018) and the comparison cosmology is the one used at each likelihood evaluation. Both transformations 1 & 2 are redshift-dependent. Here we used the mean redshift of our sample, $z=0.20$. We assume the likelihood to be Gaussian in $\vec{\hat{P}}$:

Here vector $\vec{\Theta}$ denotes the sampled parameters and $\vec{P}(\vec{\Theta})$ are the predicted power spectrum multipoles. The vector $\vec{P}(\vec{\Theta})$ depends on $\Omega_{\rm m}$ and $\sigma_{8}$ through both the linear matter power spectrum prediction and the mean bias of Eq 11. The dependence on the scaling parameters $\alpha$ and $e^{\beta}$ comes from the masses entering the computation of the mean bias.

We used the estimated power spectrum multipoles over scales of $k<0.2h{\rm Mpc}^{-1}$. Figure 6 suggests that the Planck 2018 cosmology model and the measurements agree up to the largest $k=0.35h{\rm Mpc}^{-1}$. We notice that, since our mock measurements exhibit a weaker monopole for $k\gtrsim 0.2h{\rm Mpc}^{-1}$, the covariance matrix estimate might not be reliable on these scales. However, we did run a cosmological parameter fit including all scales up to $k=0.35h{\rm Mpc}^{-1}$ to assess their effect on the obtained parameter constraints. This is shown in Fig. 11 and the discussion at the end of this section.

In Fig.8 we show the 2D posterior distributions for $\Omega_{\rm m}$ and $\sigma_{8}$ in the case where $\ell=0$, $\ell=0,2$, and $\ell=0,2,4$ are included in the data vector. For the monopole alone, we found the distribution to be bimodal, with one peak at around canonical values of $\sigma_{8}\sim 1$ and another one with significantly larger values of $\sigma_{8}\gtrsim 3$. Adding the quadrupole removes the latter peak due to the large amplitude of the model $P_{2}(k)$ in this region (see Fig. 6), which is inconsistent with the measurement. This effect is also shown in Fig. 9, in which we show the posterior distribution for $\Omega_{\rm m}$ and $\sigma_{8}$ when we only use the quadrupole. In this case, the peak at $\sigma_{8}\gtrsim 3$ is excluded with high significance. Including the hexadecapole has a significantly smaller effect, but still decreases the area of the $68\%$ and $95\%$ confidence regions by $13\%$ and $6\%$, respectively.

Figure 10 shows the posterior distributions for the two cosmological parameters $\Omega_{\rm m}$ and $\sigma_{8}$, along with the two scaling relation parameters, $\alpha$ and $e^{\beta}$, when we include all the three multipoles in the sampling. The plot was made using the corner⁶⁶6https://github.com/dfm/corner.py Python library (Foreman-Mackey, 2016). From these, we obtained parameter constraints of $\Omega_{\rm m}=0.24_{-0.04}^{+0.06}$ and $\sigma_{8}=1.13_{-0.24}^{+0.43}$, which correspond to the 16%, 50%, and 84% quantiles of the marginalized posterior distributions. These can be compared, for example, to the Planck 2018 results, $\Omega_{\rm m}=0.315\pm 0.007$ and $\sigma_{8}=0.811\pm 0.006$. Our marginalized constraints would suggest a larger than $1\sigma$ difference compared to the Planck 2018 results, but in Fig 10 the Planck 2018 value is indeed contained within the 68% confidence region in the $(\sigma_{8},\Omega_{\rm m})$ plane. Regarding the richness-mass scaling relation, we obtained marginalized constraints of $\alpha=0.98\pm 0.09$ and $e^{\beta}=75.5_{-17.8}^{+18.9}$. So, in essence, we reproduced the results of K21: $\alpha=0.98\pm 0.09$ and $e^{\beta}=74.4_{-18.2}^{+21.4}$.

Figure 11 shows the effect of including wavenumbers up to $0.35h{\rm Mpc}^{-1}$ in obtaining the posterior distribution for $\Omega_{\rm m}$ and $\sigma_{8}$. The results are statistically consistent with the $k<0.2h{\rm Mpc}^{-1}$ case. Going to smaller scales shrinks the $68\%$ and $95\%$ confidence regions by $36\%$ and $39\%$, respectively. Clearly, the additional data points at $0.2h{\rm Mpc}^{-1}<k<0.35h{\rm Mpc}^{-1}$ increase the constraining power of the power spectrum measurement. However, due to potential problems with the covariance matrix estimate in this region (as demonstrated by Fig. 4), we consider $k<0.2h{\rm Mpc}^{-1}$ our conservative baseline case.

We presented power spectrum measurements for a subset of CODEX galaxy clusters and used these measurements to constrain cosmological parameters. This is a continuation of the work in L21, in which the cluster two-point correlation function was implemented. We have made some improvements to the L21 analysis methods, most notably regarding the covariance matrix estimation and the handling of the mass estimate uncertainties.

We found that the measured power spectrum multipoles are compatible with theoretical predictions that combine the linear matter power spectrum, the Kaiser approximation for redshift space distortions, and a large-scale bias estimated from the observed cluster richnesses. We constructed a likelihood function using the measured and predicted multipole signal and used this to generate constraints on the cosmological parameters $\Omega_{\rm m}$ and $\sigma_{8}$, which led to $\Omega_{\rm m}=0.24_{-0.04}^{+0.06}$ and $\sigma_{8}=1.13_{-0.24}^{+0.43}$. These suggest slightly larger than $1\sigma$ deviations from Planck 2018 cosmology, for example. However, taking the degeneracy of these parameters into account shows that the 68% confidence region of our posterior distribution does indeed contain the Planck 2018 cosmology.

We see a few possible extensions as future work prospects. The full CODEX catalog consists of two disjoint patches. In this work, we included only one of them to be able to cover the sky region with mock catalogs. To expand our sample, we would need to use a set of mock catalogs that span larger cosmological volumes to cover the full CODEX footprint, or to cover the excluded patch with another set of mock catalog realizations drawn from the same ensemble as the ones used in this work. The latter option could cause an under-representation of large-scale correlations between the two patches, the importance of which should be verified. Another improvement would be to include the halo mass function in our MCMC sampling. Also, improving the modeling of the mock measurements would allow us to use the measured spectra up to larger wavenumbers in the cosmological analysis and most likely significantly tighten the obtained parameter constraints.