The SRG/eROSITA All-Sky Survey

In the standard cosmological formalism, most of the energy content of the universe is formed by a non-relativistic component called Cold Dark Matter (Rubin & Ford 1970, $\mathrm{CDM}$ hereafter) and an unknown Dark Energy (Riess et al. 1998, DE) responsible for the accelerated expansion of the universe, and frequently represented by the cosmological constant $\Lambda$. The third ingredient is an early phase of exponential expansion called Inflation (Brout et al. 1978). Under the influence of these elements, quantum fluctuations evolved in the early universe to form the large-scale structure (LSS) that we observe today. This standard formalism, the $\Lambda$CDM model, has been successful at describing a broad range of observations, from the statistical distribution of the fluctuations of the Cosmic Microwave Background (CMB) to the spatial correlations of galaxies in the late universe (Planck Collaboration et al. 2020; Amon et al. 2022; DESI Collaboration et al. 2024). Despite its success, the standard concordance model, $\Lambda$CDM, is being challenged by several tensions, such as discrepancies reported in the measurements of the Hubble constant ($H_{\mathrm{0}}$). The Hubble constant measured by early time probes is in 4 to 6$\sigma$ statistical tension with the one directly measured by late-time distance ladders (Di Valentino et al. 2021b). Whether these differences in the expansion rate measurements originate from fundamental physics or systematic effects is still debated (Freedman et al. 2024). This tension is supplemented by numerous but less significant discordant observations, such as anomalies in the CMB anisotropies like a higher lensing amplitude than the one predicted by $\mathrm{\Lambda CDM}$ (Planck Collaboration et al. 2020), discrepancies in the abundance of galaxy satellites (Kanehisa et al. 2024; Bullock 2010), potential anisotropies in the large-scale distribution of galaxy clusters (Migkas et al. 2020), or the unlikely presence of high velocity colliding clusters (Asencio et al. 2021) (see Peebles (2022) and Perivolaropoulos & Skara (2022) for a review of the most common reported anomalies). Recently, DESI Collaboration et al. (2024) reported a significant preference for an evolving dark energy equation of state when measuring the cosmological parameters from the baryonic acoustic oscillations. The tension is reported as high as $3.9\sigma$ when BAO are combined with the CMB and supernovae data. Interestingly, this preference is consistent with earlier cluster observations like Mantz et al. (2015) for which the best-fit values suggested that $w_{\mathrm{a}}<0$, however still in excellent agreement with $\Lambda\mathrm{CDM}$. The same was observed for galaxy clustering and weak lensing data from the Dark Energy Survey collaboration Abbott et al. (2023a).

Another reported tension is found in the measurement of the growth of structures. The values reported by the CMB are higher than the ones measured by a number of late-time cosmological probes (see Di Valentino et al. 2021a, and references therein). This tension is best seen in the $\Omega_{\textrm{m}}-\sigma_{\textrm{8}}$ plane, where the CMB constraints lie above the degeneracy of the LSS probes, and is usually quantified in terms of the $S_{\textrm{8}}=\sigma_{\textrm{8}}(\Omega_{\textrm{m}}/0.3)^{0.5}$ parameter. Although statistically less significant than the $H_{\mathrm{0}}$ tension, various probes report the $S_{\textrm{8}}$ tension as a motivation to investigate a potential physical origin. Among the most prominent early time probes, Planck Collaboration et al. (2020) (Planck20 hereafter), measured $S_{\textrm{8}}=0.832\pm 0.013$ (TT+TE+EE+lensing). Similar ”early-time” measurements are reported in the results of the Wilkinson Microwave Anisotropy Probe (Hinshaw et al. 2013, WMAP). This value is confirmed by the results of the South Pole Telescope (SPT) data through the lensing of the CMB combined with baryon acoustic oscillation (Bianchini et al. 2020), who found $S_{\textrm{8}}=0.862\pm 0.040$, and the recent result of the sixth data release of the Atacama Cosmology Telescope (ACT) using the same method and who measured $S_{\textrm{8}}=0.840\pm 0.028$ (Madhavacheril et al. 2024). Since CMB lensing probes linear scales with most of the signal coming from a redshift range of $0.5<z<5$, the good agreement between the primary CMB and the CMB lensing compared to other late-time probes might suggest a change in the non-linear regime at low redshift. Conversely, cosmic shear surveys tend to find a lower amplitude of structure formation. For example, the Dark Energy Survey (Abbott et al. 2022) finds $S_{\textrm{8}}=0.776\pm 0.017$.

Cluster counts have also been utilized in the $S_{\textrm{8}}$ measurements in recent years. In the millimeter wavelength, Zubeldia & Challinor (2019) used Planck-detected galaxy clusters with CMB lensing mass calibration to find $S_{\textrm{8}}=0.803\pm 0.039$. Recently, the SPT collaboration used a sample of 1,005 galaxy clusters with weak gravitational lensing data from DES and the Hubble Space Telescope (HST) and found $S_{\textrm{8}}=0.795\pm 0.029$ (Bocquet et al. 2024). Similarly, Salvati et al. (2022) combined the Planck and SPT samples and reported $S_{\textrm{8}}=0.74\pm 0.05$. Aymerich et al. (2024) reanalyzed the same Planck cluster sample with external Chandra data and obtained $S_{\textrm{8}}=0.81\pm 0.02$. In the optical wavelength, a preliminary study on the clusters detected in the Sloan Digital Sky Survey DR8 produced an $S_{\textrm{8}}$ value of $0.78\pm 0.04$ (Costanzi et al. 2019), while the latest DES results combined with SPT find $S_{\textrm{8}}=0.736\pm 0.049$ (Costanzi et al. 2021). Furthermore, Sunayama et al. (2023) analyzed a sample of 8,379 clusters detected by the redMaPPer algorithm with HSC weak lensing data and found $S_{\textrm{8}}=0.816^{+0.041}_{-0.039}$. Additionally, an analysis of a sample of 3652 galaxy clusters detected in KIDS produced $S_{\textrm{8}}=0.78\pm 0.04$. The recent results for the first eROSITA All-Sky-Survey (eRASS1, Ghirardini et al. 2024; Artis et al. 2024) established cluster number counts as a precision cosmological probe, placing tight constraints on $S_{\textrm{8}}=0.86\pm 0.01$. Interestingly, the eROSITA, as a later-time probe analysis, has no tension with the Planck20 measurements. These measurements in the literature are compared in Figure 1.

As the cluster mass function is established as a reliable cosmological probe, and the samples have reached a sizable number of clusters to ensure statistical precision with the launch of SRG/eROSITA All-Sky Survey, we take our primary cosmology analysis one step further and constrain the structure growth over cosmic time. The overarching goal of this paper is to measure the parameterization of the growth factor using the cosmic linear growth index ($\gamma$), a late-time reduction of the power spectrum, and a model agnostic method where the sample is divided into five different redshift bins 6. The first approach aims at modifying the growth rate of structure formation $f(z)$ and defines it as a power-law of the matter density parameter, i.e., $\Omega_{\mathrm{m}}^{\gamma}$. This method has been widely discussed in the literature in the context of testing general relativity (GR) at large scales. For instance, Wang & Steinhardt (1998) shows that the non-linear theory of structure formation assuming general relativity yields a growth rate well modeled by this function when $\gamma\sim 0.55$. Thus, deviations from this value would indicate that structures do not evolve as predicted and, more generally, suggest a potential departure from GR-$\Lambda\mathrm{CDM}$. A higher $\gamma$ is equivalent to a reduction of structure formation. Previous galaxy cluster surveys have been utilized in GR studies. Rapetti et al. (2009) used a sample of 78 X-ray selected clusters to constrain the luminosity-mass scaling relation in combination with CMB, SNe Ia, and X-ray cluster gas-mass fractions ($f_{\mathrm{gas}}$) and found consistent results with GR. Mantz et al. (2015) used a sample of 224 X-ray clusters detected by ROSAT (Truemper 1993), with weak lensing (WL hereafter) mass measurements obtained from the Subaru telescope and the Canada-France Hawaii Telescope (von der Linden et al. 2014). This cosmological study combines cluster abundance with the cluster gas fraction, primordial nucleosynthesis, and $H_{\mathrm{0}}$ priors to constrain the background expansion and break the $\Omega_{\mathrm{m}}-\sigma_{\mathrm{8}}$ degeneracy and found no evidence for physics beyond GR. Bocquet et al. (2015) used the same approach on a sample of 100 clusters detected by their millimeter-wave signature, the so-called Sunyaev Zel’dovich (SZ) effect, through the South Pole Telescope (SPT), agreeing with the previous studies when clusters are combined with the CMB measurement from the Planck Collaboration et al. (2014), as well as BAO and supernoave type Ia (SNe Ia). However, recent cosmological studies based on the LSS find a higher value of the cosmic linear growth index for the GR prediction ($\gamma>0.55$). Studies based on the galaxy clustering of the Sloan Digital Sky Survey (SDSS) data (Samushia et al. 2013; Beutler et al. 2014; Gil-Mar´ın et al. 2017) found a lower structure growth rate than the one inferred from the CMB. This was later confirmed by DES data (Basilakos & Anagnostopoulos 2020), once again using galaxy clustering in combination with galaxy-galaxy lensing, implying a damping of structure formation at low redshifts. This recurrent finding is intriguing and may indicate that the theory of structure formation does not describe observations accurately. If so, it might hint at physics beyond GR and $\Lambda\mathrm{CDM}$, or an unknown systematic effect related, for example, to baryonic feedback.

Apart from the parametrization using the cosmic linear growth index $\gamma$, another method used in this work is to directly measure the evolution of the parameter $\sigma_{\mathrm{8}}$ across cosmic time. Surveys using galaxy clustering provide robust constraints on $f\sigma_{\mathrm{8}}(z)$ parameter (Zarrouk et al. 2018; de Mattia et al. 2021, among others). The measurements of the evolution of the parameter, $f\sigma_{\textrm{8}}(z)=f(z)\times\sigma_{\textrm{8}}(z)$, is probed by redshift space distortions (RSD) where $f$ is the linear growth rate defined as $f=\mathrm{d}\ln D/\mathrm{d}\ln a$. Recent surveys have used weak lensing and galaxy clustering to probe this evolution (Jullo et al. 2019; Garc´ıa-Garc´ıa et al. 2021; White et al. 2022; Abbott et al. 2023b). In the context of cosmology with cluster number counts, Bocquet et al. (2024) probed the growth of structures by using a non-parametric approach on their SPT-SZ sample of 1,005 clusters combined with sound horizon priors from Planck20. They divided their sample into five redshift bins, allowing them to obtain a quasi-direct measurement of $\sigma_{\mathrm{8}}(z)$, and finding good agreement with predictions from the primary CMB probes.

The paper is organized as follows: Section 2 describes the survey data employed, while Section 3 discusses modifying the structure growth framework. Section 4 shows our result on the cosmic linear growth index. Section 5 and Section 6 present the results on the power spectrum reduction and the model agnostic method. Section 7 demonstrates the robustness of the analysis. Discussions and relation to current cosmological tensions are provided in Section 8. Throughout this paper, we use the notation $\log\equiv\log_{\mathrm{10}}$, and $\ln\equiv\log_{\mathrm{e}}$. Reported uncertainties correspond to a 68% confidence level unless noted otherwise.

The data used in this work is identical to the catalogs presented in Bulbul et al. (2024); Kluge et al. (2024). The weak lensing follow-up for mass calibration uses the data utilized in (Grandis et al. 2024b; Kleinebreil et al. 2024). The cosmological pipeline and analysis framework is the same in (Ghirardini et al. 2024; Artis et al. 2024). We summarize the multi-wavelength data, including X-ray, optical, and weak lensing observations, utilized in this work in the following section.

Galaxy cluster number counts efficiently place constraints on the low-redshift and large-scale regime, which cannot be explored by every cosmological probe, such as the anisotropies of the CMB. In this section, we describe the first framework we explore in this paper, namely the linear growth parametrization scenario ($\gamma$-model hereafter). The power spectrum reduction model and the model agnostic method are described respectively in section 5 and section 6.

In this work, we aim to constrain the growth of structure utilizing the eRASS1 cosmology sample with weak lensing mass calibration from the DES, KIDS, and HSC surveys. The results presented here are obtained by combining cluster counts with priors from the sound horizon scale at recombination ($\theta^{*}$) indicated by the CMB measurements presented in Planck20, as this represents one of the most robust cosmological measurements to date. Details on the sound horizon priors used can be found in Appendix E. Additionally, differently than the G24 framework, we use a conservative $H_{\mathrm{0}}\sim\mathcal{N}(70,5^{2})$. A comprehensive description of the different priors can be found in Table 1.

Here, we present constraints on the cosmic linear growth index for the parametrizations described in Section 3. In particular, we focus on two different models: the canonical $\Lambda\mathrm{CDM}$ model with the cosmic linear growth index parameterized as in Equation (5) ($\gamma\Lambda\mathrm{CDM}$ model) and the $w\mathrm{CDM}$ model with the dark energy equation of state parameter, $w$, free ($\gamma w\mathrm{CDM}$ model). The standard $\gamma\Lambda\mathrm{CDM}$ is shown in Figure 3. Neutrinos suppress the amplitude of the power spectrum, introducing a scale dependence of the growth factor (Equation 6), which leaves imprints on the halo mass function. Although Costanzi et al. (2013) presents a method to account for the effect of neutrinos on the halo mass function by replacing the total matter power spectrum with the cold dark matter power spectrum, this method is incompatible with the growth parameterization using the cosmic linear growth index employed in this work. Therefore, we do not consider the impact of neutrinos and assume that neutrinos are massless throughout this work. At our current statistical constraining power, this modeling choice does not affect our results, as shown in section 8.

We also emphasize that a higher value of the cosmic linear growth index leads to a flattening of the growth factor, which in turn yields a lower measurement of $\sigma_{\mathrm{8}}\equiv\sigma_{\mathrm{8}}(z=0)$. This definition does not imply that we expect a lower $\sigma_{\mathrm{8}}(z)$ for all redshifts, as found with probes that include non-linear physics (Abbott et al. 2022; White et al. 2022).

Previous sections present hints for departures from the standard structure formation scenario and potential damping in the formation of halos indicated by the eRASS1 cluster counts in the redshift range $0.1<z<0.8$. It is thus interesting to explore if these findings are consistent with other models describing a reduction of the power spectrum at low redshift. Following the results, we implement the late-time power spectrum reduction model proposed by Lin et al. (2024) as an additional effective parameterization of the observed deviations from the standard structure formation scenario. We stress that this model is a new parameterization of the modification of the linear growth factor, and we do not use the framework described in Section 3. Here, we use the standard growth factor introduced in equation 7, and thus the cosmic linear growth index $\gamma$ is irrelevant in this section. However, both models describe a reduction of structure formation as described in Section 8, and are thus complementary. The modified matter power spectrum is defined as

Fitting the eRASS1 cluster number counts with the sound horizon prior to this model, we find

at 68% confidence level, and where we fixed $p=1$. We recall that the priors on all the parameters can be found in Table 1, and especially that the priors on $\beta$ are $\mathcal{U}(-5,5)$. Since we are measuring a non-zero $\beta$ parameter consistently with the findings of Lin et al. (2024) and Section 4, these results suggest that a significant reduction of structure formation happens at low redshift. Although significant, this finding must be interpreted in cluster count cosmology, where the density of dark matter halos hosting clusters is computed through the halo mass function. The parameters of the HMF are assumed to be universal in all the models considered (see Section 3.1). The impact of the change that we apply to the power spectrum on these parameters should be further investigated in the future. Additionally, another main result is that the best-fit value of the $S_{\textrm{8}}$ parameter decreases when we add the new degree of freedom to the power spectrum, consistent with the results presented in Section 4. The $S_{\textrm{8}}$ measurements become consistent with the cosmic shear inferred values. For example, (Li et al. 2023) reports a value of $S_{\textrm{8}}=0.776^{+0.029}_{-0.027}$ obtained from the two-point correlation functions from the HSC data.

Galaxy clusters probe the linear regime of structure formation and involve only the linear matter power spectrum. Remarkably, adding a degree of freedom to the matter power spectrum suggests a reduction of structure formation and decreases the $S_{\textrm{8}}$ value. This finding is discussed in Section 8. Overall, the parameter $\beta$ quantifies how the power spectrum could be reduced due to unknown physical processes or systematic effects at low redshift. Our results suggest that such reduction exists, although its nature remains to be identified.

The frameworks and results presented in Section 3, Section 4, and Section 5 assume a functional form of the growth rate presented in Equation (5), or the power spectrum in Equation (18). To independently probe the structure growth free of any functional parametrization and test the potential deviations from the standard structure formation scenario, we use a different method to measure the structure evolution using eRASS1 cluster counts.

In Section 4, we assume a value of the growth index $\gamma$, we then compute the associated growth factor $D_{\gamma}$ with equations 6 and 8 and obtain the evolution of the r.m.s density fluctuations through $\sigma_{\mathrm{8}}(z)=\sigma_{\mathrm{8}}D_{\gamma}(z)$. However, the cosmic linear growth index lacks an immediate physical interpretation. Indeed, even though a redshift evolution of the cosmic linear growth index can be expected in some theoretical models (Linder & Cahn 2007; Wen et al. 2023), the cosmic linear growth index remains an effective parameterization. The same issue exists with the power spectrum reduction model we used in Section 5, also a phenomenological parameterization. To avoid these issues, another method commonly used in the literature for probing the growth of structure is to divide the cluster sample into multiple bins and to compute the cosmological parameters directly in those subsamples employing the standard $\Lambda$CDM parametrization in each bin (e.g., Bocquet et al. 2024).

The fluctuations of the density field probed by clusters number count over comoving spheres with a radius of $8\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ ($\sigma_{8}(z)$) in several redshift bins traces the evolution of structure growth with redshift. This is possible since cluster abundance in each redshift bin is indeed a direct probe of the halo mass function, defined in Equation 1. Although we can express $\sigma(R,z)$ using its value at redshift zero multiplied by the normalized growth rate, the halo mass function depends only on the value of $\sigma$ at the redshift of interest, and therefore $\sigma(R,z)$ is probed quasi-directly. In practice, the difference in growth rate modeling between the $\gamma\Lambda\mathrm{CDM}$ model and the canonical $\Lambda\mathrm{CDM}$ model with the standard GR will act just as a normalization difference in $\sigma(R,z)$, given that the growth rate is a function of redshift. This method works independently of modeling assumptions as long as the growth rate is only a function of redshift, $D=D(z)$. Although this method is assumed to be model-independent compared to the $\gamma\Lambda\mathrm{CDM}$ model, it still relies on the assumptions of linear perturbation theory with a conventional growth within the corresponding redshift bins. For this reason, we consider both approaches (the $\gamma\Lambda\mathrm{CDM}$ model and the binning method in the redshift) jointly and test them using the eRASS1 observations. In the case of damped structure growth at lower redshifts, we expect that the normalization of the power spectrum is affected, creating discrepancies between our measurement and values inferred from the CMB measurements.

Our approach to measuring the redshift evolution in structure growth is based on binning the cluster number counts in redshift to measure the evolution of structure growth. The eRASS1 cosmology sample, comprising 5,259 ICM-selected and optically confirmed clusters of galaxies, is the largest ICM-selected sample to date. The cosmological parameter inference reaches a statistical precision similar to CMB experiments (see G24). We determine the number of clusters in each bin such that the matter density parameter $\Omega_{\mathrm{m}}$ is measured with at least $\sim$15% precision and has a similar number of clusters. With this requirement, the sample provides sufficient statistical power to generate five independent²²2Although the bins are never totally uncorrelated, they are large enough so that we can neglect the covariance. redshift bins (see Table 2). The first four bins have 1,052 clusters each with this binning scheme, leaving 1,051 clusters for the last highest redshift bin ($0.452<z<0.8$). The corresponding redshift ranges are $z=0.1-0.175,0.175-0.244,0.244-0.330,0.330-0.452$, and $0.452-0.8$. The X-ray count-rate distribution as a function of redshift and richness of the subsamples are displayed in Figure 7.

Our sample has sufficient statistical constraining power to measure the scaling relation parameters independently in each redshift bin jointly with the cosmological parameters shown in Table 1, unlike the previously published ICM-selected samples (e.g. Bocquet et al. 2024). Consequently, we fix the pivot redshift $z_{\mathrm{p}}$ of the count-rate to mass ($C_{\mathrm{R}}-M$) and richness to mass ($\lambda-M$) scaling relations as described in G24 to the mean redshift in each bin $z_{\mathrm{p},i}=0.138,0.209,0.288,0.387,0.569$. Particularly relevant measurements are the normalization of the power spectrum $A_{\mathrm{s},i}$, and the matter density $\Omega_{\mathrm{m},i}$. These parameters are combined to recover the $\sigma_{8,i}$ parameter in the redshift bin space specified above. We utilize the Boltzmann solver CAMB (Lewis & Challinor 2011) to compute the power spectrum for each redshift bin and infer the cosmological parameters. This process additionally provides a consistency check for our standard cosmology analysis presented in G24, also shown in Figures 8 and  11. For instance, our posteriors on the cosmological parameters, $\Omega_{M}$, $\sigma_{8}$, and $S_{\mathrm{8}}$, are consistent with each other within the five redshift bins, and the results in G24 at the $2\sigma$ confidence level. Our results are summarized in Table 2.

Finally, we can use these posteriors to recover the redshift evolution of the r.m.s of the density fluctuations, i.e., $\sigma_{8}$ parameter. Similarly, we present our results that combine the sound horizon priors with eRASS1 cluster number counts in data points in black in Figure 9. The eRASS1 cosmology sample, when divided into five redshift intervals, shows hints of a reduction of structure formation at low redshifts. Although all the data points are compatible with the combined analysis of Planck, BAO, and JLA results (see Planck20) in Figure 9, they all remain above the CMB expectations. Although not significantly significant, this trend is visible in Figure 9. Consequently, we are compatible with the results provided in Section 4 and Section 5.

Our measurements broadly agree with the cluster abundance analysis from the SPT SZ experiment combined with the same sound horizon priors used in this work (Bocquet et al. 2024). However, it is worth noting that, unlike our approach, the analysis presented in Bocquet et al. (2024) uses the same scaling relation parameters for all redshift bins, as well as a unique $\Omega_{\mathrm{m}}$ at all redshift. Although their sample size is significantly smaller (1005 clusters of galaxies as opposed to 5259 clusters used in this work), this assumption is reflected in the uncertainties in the quoted results. This might explain why they do not recover the power suppression trend found in this work, as their sample covers a larger and broader redshift range.

We also compare our result with the DES Y3 combined cosmic shear and galaxy clustering results (Abbott et al. 2023b), who provided results with the same approach using cosmic shear and galaxy clustering. They divide their sample in redshift bins in a range of $0<z<1.5$, similar to the redshift coverage of eRASS1 clusters. The observed trend of lower structure formation in DES-Y3 data reported in the low redshift regime is in tension with our and CMB measurements. The same trend is found in the DESI Legacy Imaging Survey Luminous Red Galaxies (LRG) combined with CMB lensing (White et al. 2022).

In this work, we first defined a reduction of structure formation as a higher value of the cosmic linear growth index $\gamma$ (see Section 3). Indeed, equation 4 and equation 5 show that a higher $\gamma$ leads to a lower $f(z)$, and thus, the clustering of matter with time occurs at a slower rate. Consequently, we show that the growth factor is flattened, as shown in Figure 2. All the models we consider show the same trend, including the last measurements through the sample binning method. As a consequence, the flattening of the growth factor leads to a lower measurement of $\sigma_{\mathrm{8}}\equiv\sigma_{\mathrm{8}}(z=0)$. We emphasize that this definition does not imply that we expect a lower $\sigma_{\mathrm{8}}(z)$ for all redshifts, as found with probes that include non-linear physics (Abbott et al. 2022; White et al. 2022).

Overall, our model agnostic method agrees well with the CMB extrapolated values while hinting at a reduction of structure growth at low-$z$. We note that the fifth bin ($0.452<z<0.8$) has a higher inferred $\sigma_{\mathrm{8}}(z)$ compared to the trend observed in the previous ones. We investigate the impact of this specific bin in our analysis and the origin of this systematic effect in Section 7.

In this section, we evaluate the effect of the possible systematic effects on the results presented in this work, scaling relations, and cosmological constraints in general, shown G24 and Artis et al. (2024). Among those is the impact of neutrinos (see Section 8) and the large scatter observed in the scaling relations between the X-ray observable, i.e., count-rate, in the high redshift bin used to infer cluster mass. We examine the redshift-dependent scatter (the count rates follow a log-normal probability distribution function with scatter $\sigma_{\mathrm{X}}$) in the count-rate mass scaling relation ($\sigma_{X}=0.98_{-0.04}^{+0.05}$) to understand the effect of it in the subsequent cosmology analyses using the binned analysis presented in Section 6.

In this work, we present results on the measurement of the growth of structure using X-ray detected eRASS1 clusters, in combination with optical surveys, such as LS DR10-South for optical confirmation and redshift measurements, the DES, KiDS, and HSC for weak lensing mass calibration. Specifically, the $\gamma$ parameterization of the growth factor presented in equation 5 modifies the structure growth rate (Section 4) in connection with the power spectrum reduction model (Section 5). Secondly, a model agnostic strategy aiming at measuring the cosmological parameters at different redshifts to reconstruct the history of the evolution in density fluctuations (Section 6).

We also probe the evolution of structure formation with the late-time power suppression presented in Section 5. The two approaches are thus related, and we expect any potential deviation from the standard scenario appearing in one to be reflected in the other. In all three cases, we combine our structure growth constraints with the sound horizon priors from the cosmic microwave background $\theta^{*}$ as it is one of the most robust cosmological measurements available. In all three cases, we find observations that can be interpreted as a low redshift reduction of structure formation. First, the $\gamma$ parameterization of the growth factor, while still compatible with the standard $\gamma=0.55$ value at $2\sigma$ both in $\gamma\Lambda\mathrm{CDM}$ and $\gamma w\mathrm{CDM}$, has a best-fit value that favors higher values of the cosmic linear growth index. Additionally, the model agnostic redshift splitting of the eRASS1 cosmological catalog shows a similar pattern. Indeed, Figure 2 shows how the cosmic linear growth index increases the growth factor at low redshift. Our binned cosmological results indicate a preference for the same trend while still being statistically consistent with CMB measurements. Finally, the power spectrum reduction model shows a best-fit value that significantly excludes an absence of power reduction.

These three results indicate a deviation from the widely accepted paradigm of structure formation. Interestingly, in the case of the cosmic linear growth index and power spectrum reduction models, we find that modifying the power spectrum leads to a decrease in the $\sigma_{\mathrm{8}}$ value, and thus our $S_{\textrm{8}}$ inferred value. Consequently, these observations may offer evidence that the $S_{\textrm{8}}$ tension could be accounted for by a modification of the matter power spectrum at the scales investigated by cosmic shear and other lensing studies (Nguyen et al. 2023), provided that the explanation for the tension as a result of systematic uncertainties related to baryonic effects is ruled out. Indeed, Figure 11 shows that the in all of the three models used in this work ($\gamma\Lambda\mathrm{CDM}$, $\gamma w\mathrm{CDM}$, and the power spectrum reduction model) the value of the parameter $S_{\mathrm{8}}$ is reduced from the value found in G24 in the $\Lambda\mathrm{CDM}$ case, $S_{\textrm{8}}=0.86\pm 0.01$ to

These values are compatible with the ones routinely inferred by cosmic shear studies but in $\mathrm{\Lambda CDM}$ and at small scales. Using KIDS data, Li et al. (2023) found $S_{\textrm{8}}=0.776^{+0.029}_{-0.027}$, close to the HSC results presented in Dalal et al. (2023) of $S_{\textrm{8}}=0.776^{+0.032}_{-0.033}$. Using DES data and the same cosmological probe, Amon et al. (2022) found $S_{\textrm{8}}=0.772^{+0.018}_{-0.017}$. Finally, combining DES and KIDS data, Dark Energy Survey and Kilo-Degree Survey Collaboration et al. (2023) found $S_{\textrm{8}}=0.790^{+0.018}_{-0.014}$. We also note that other probes affected by non-linear physical effects also find lower $S_{\textrm{8}}$ values. For example Palanque-Delabrouille et al. (2020) used the Ly$\alpha$ forest to infer $S_{\textrm{8}}=0.77\pm 0.04$. Given that cosmological probes sensitive to the non-linear regime yield a lower $S_{\textrm{8}}$, our findings may imply that this discrepancy may be simply due to non-linear effects influencing those probes. Although galaxy clusters are highly non-linear structures, their number density can be described using the linear matter power spectrum. Effects related to non-linear physics are restricted to the HMF parameterization, which is assumed to be universal. Thus, when inferring cosmology using galaxy cluster number counts, we obtain values closer to the one inferred by the CMB. Introducing an additional degree of freedom in the power spectrum analysis could suggest that the reduction in structure formation is scale-dependent, potentially resolving the $S_{\textrm{8}}$ tension.

Neutrinos smooth structure formation at small scales, thus reducing the power spectrum at high $k$. Therefore, their effect on the cosmological parameter estimation impacts the power spectrum reduction we are probing here. However, in this work, we have chosen to consider massless neutrinos. This modeling choice thus requires a very careful justification. We show that the impact of neutrinos is negligible and intrinsically different compared to the power spectrum reduction reported here. We leave the combined measurement of the neutrino masses and the power spectrum reduction parameterization to future works.

Three main reasons explain our consideration of massless neutrinos: the lack of theoretical background for considering neutrinos together with a power spectrum reduction at low redshift, the relative size of the shift in the cosmological parameter space, and the intrinsic nature of the power impression we are investigating.

First, since Costanzi et al. (2013), we know that the effect of neutrinos on the halo mass function is well captured when the cold dark matter only power spectrum $P_{\mathrm{cdm}}(k,z)$ is considered instead of the total matter power spectrum $P_{\mathrm{m}}(k,z)$ in the fitting function parameterized by Tinker et al. (2008). However, in this formalism, neutrinos also induce a scale dependence in the growth factor presented in equation 9. In other words, we have

as $D(z)\leftarrow D(z,k)$. Consequently, when using the cold dark matter spectrum, we need to compute the spectrum directly at the considered redshift. Analytical models exist for the linear growth factor in the presence of neutrinos (Kiakotou et al. 2008), but they are not compatible with the cosmic linear growth index. In general, we do not currently have a satisfying way of parameterizing the linear growth factor in the presence of neutrinos together with the cosmic linear growth index in the context of galaxy cluster number counts. This consideration highlights the difficulty of finding a model that fits both. The following two paragraphs, however, show that the effect of neutrinos can be neglected in this work.

Figure 14 shows how neutrinos impact cosmological parameter estimation. Indeed, the shift in the cosmological parameter space produced by neutrinos compared to a model when they are massless is of $0.5\sigma$ in the $\Omega_{\mathrm{m}}/\sigma_{\mathrm{8}}$ plane. This number must be compared with the $2.4\sigma$ shift produced by the power spectrum reduction models considered in this work. In other words, neutrinos alone cannot explain the reduction observed and presented here. Although their effect is indeed degenerate with the power spectrum reduction, it is negligible compared to our results for a catalog of clusters of the size of eRASS1.

Finally, this work investigates a potential redshift reduction of the power spectrum. Neutrinos induce a scale-dependent damping of the power spectrum in the redshift range probed by eRASS1 galaxy cluster count. Indeed, at a fixed scale $k$, the reduction induced by neutrinos on the matter power spectrum is constant in the redshift range $0.1<z<0.8$. Figure 15. On the other hand, the models we are probing only modify structure formation at low redshift but at all scales. Consequently, both effects are different and do not affect the result in a way that can be compared.

For the aforementioned reasons, our results are primarily not affected by the absence of massive neutrinos. However, more theoretical efforts will be necessary to probe the structure reduction we detect when larger and deeper eRASS catalogs are available. At this time, it will not be possible to neglect the effect of neutrinos anymore.

Recent studies show the current implementation of baryonic feedback in numerical simulations isn’t enough to account for the reduction in $S_{\textrm{8}}$ observed in surveys sensitive to small scales, as demonstrated in (Grandis et al. 2024a). However, if baryonic feedback is poorly understood and its effects are greater than anticipated, it could potentially reconcile the linear and non-linear regimes and explain the observed$S_{\textrm{8}}$ tension (Preston et al. 2023).

Our results agree with other studies that infer a reduction of the structure formation at low redshifts (Samushia et al. 2013; Beutler et al. 2014; Kazantzidis & Perivolaropoulos 2021; Lin et al. 2024). However, a robust conclusion can only be provided if every potential systematic effect is extensively studied through numerical simulations. We also investigate redshift-dependent potential systematic effects in this work and our $\Lambda$CDM analysis. Although statistically consistent, the fifth and highest redshift bin ($0.452<z<0.8$) has a higher $S_{\textrm{8}}$ value than the other four at lower redshifts. Although the scatter in this bin is significantly higher, it does not affect the results of this work. However, the reason for this scatter in the high redshift regime, e.g., increased AGN contamination or photometric redshift uncertainties, will be investigated in upcoming cosmology analyses.