A test of the standard dark matter density evolution law using galaxy clusters and cosmic chronometers

In this paper, we implement a test of the standard law for the dark matter density evolution as a function of redshift. For this purpose, only a flat universe and the validity of the FRW metric are assumed. A deformed dark matter density evolution law is considered, given by ρc(z)∝(1+z)3+ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _c(z) \propto (1+z)^{3+\epsilon }$$\end{document}, and constraints on ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} are obtained by combining the galaxy cluster gas mass fractions with cosmic chronometers measurements. We find that ϵ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon =0$$\end{document} within 2σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} c.l., in full agreement with other recent analyses.


Introduction
The standard model of Cosmology [1], or the concordance model, consists of 5% of ordinary baryonic matter, formed by all kinds of matter and radiation that we can see directly and/or detect by several methods, about 25% consisting of non-interacting cold dark matter (CDM), which can be measured just by its gravitational interactions and finally the remaining 70% of dark energy (DE), consisting of a smooth fluid whose net stress energy tensor leads to gravitational repulsive, and can attributed to a constant vacuum energy density [2]. Such a three component model is in good agreement with recent Planck CMB observations [3]. Although this is a very robust model at the background level, there are also important peculiarities of the standard model that deserve further evaluation, such as the composition and failure to detect CDM in the laboratory, the core-cusp problem, Radial acceleration relation in galaxies, the missa e-mail: ph18resch11003@iith.ac.in b e-mail: holandarfl@fisica.ufrn.br c e-mail: shntn05@gmail.com (corresponding author) d e-mail: s.pereira@unesp.br ing satellite problems at small scales, problems with Big-Bang Nucleosynthesis related to Lithium-7 abundance, the theoretical discrepancy related to cosmological constant, the measure of present Hubble constant and σ 8 tension, low-l anomalies in the Cosmic Microwave Background, the cosmic coincidence problem [4][5][6][7][8][9][10][11][12][13][14]. The cosmic coincidence problem refers to why we live at special moment of evolution. This special moment is when the CDM and DE densities are exactly of the same order of magnitude. An uptodate review of all the tensions within the standard CDM model of Cosmology can be found in Perivolaropoulos and Skara [15]. Therefore, for all these reasons, several alternative models to the CDM have been proposed recently to deal with some of these issues [16][17][18].
Specifically considering the cosmic coincidence problem [14], the introduction of an interaction term between the dark sectors of the universe has been shown to be an interesting alternative to solve or alleviate the problem, as extensively discussed in [19] and more recently in [20]. The possibility of a coupling between dark matter and dark energy has been recently investigated in a number of works [21][22][23][24][25]. In [26], it was shown that a model of dynamical dark energy and interacting DE-DM approaches become indistinguishable at the background and linear perturbation level. There are also studies involving the dark sector interaction using both model-dependent and model-independent approaches, some of which are also related to the Hubble tension [27][28][29][30][31][32][33][34][35].
As a direct consequence of the interacting DE-DM models, an investigation on possible departures for the evolution of the dark matter density with respect to the usual a −3 scaling was done in [36] by combining Type Ia supernovae (SNe Ia) observations and gas mass fraction (GMF) measurements in galaxy clusters. This data could neither confirm nor rule out such an interaction. More recently, a new method to explore a possible departure from the standard time evolution law for the dark matter density was proposed [37]. By using a deformed evolution law proportional to (1 + z) 3+ , where = 0 represents the violation of standard model, the aforementioned work has shown that is consistent with zero within 1σ c.l. The dataset used for this analysis consisted of Strong Gravitational Lensing (SGL) data and gas mass fraction measurements of galaxy clusters. The lens profiles in the SGL systems can be segregated into three sub-samples, based on their stellar velocity dispersion values. The combined analyses with GMF data showed a negligible departure of the standard law ( = −0.088 ± 0.11).
In this paper, we propose and implement another test to probe a possible evolution of the dark matter density law (ρ c (z) ∝ (1 + z) 3 ) using cosmic chronometers along with X-ray gas mass fraction data from Mantz et al. [38], to complement our previous works [36,37]. This search for a possible departure from the standard evolution law is carried out by adding an ad-hoc term ( ), which is a function of the cosmic scale factor i.e. (a), such as ρ c (z) ∝ (1+z) 3+ [39,40]. Such an evolution could arise from a non-gravitational interaction between the dark sectors. In the first part of our analyses, we posit the standard dark matter density evolution law ( = 0) and verify if the gas depletion factor (γ (z)), i.e. the ratio by which the gas mass fraction of galaxy clusters is depleted with respect to the universal mean of baryon fraction, is constant with redshift considering γ (z) = γ 0 (1+γ 1 z) and the data set. This is the simplest extension one can consider. Since we obtain γ 1 = 0 within 1σ c.l., in the second part of this work we propose a test to probe a possible evolution of dark matter density law. Our main conclusion is that = 0 within 2σ c.l., in full agreement with other recent results.
This manuscript is organized as follows. In Sect. 2, we briefly present the data sample used for this analysis. Section 3 describes the methodology followed in this work. Our analysis and results are discussed in Sect. 4. Our conclusions are presented in Sect. 5.

Gas mass fraction
The Chandra X-ray sample used for this analysis, consists of 40 galaxy clusters [38], spanning the redshift range 0.078 ≤ z ≤ 1.063 identified through a comprehensive search of the Chandra archive for hot (kT ≥ 5 keV), massive and morphologically relaxed systems. The choice of relaxed systems minimize the systematic biases in the hydrostatic masses. The gas mass fraction f gas is calculated in the spherical shell 0.8 ≤ r/r 2500 ≤ 1.2 rather than the cumulative fraction integrated over all radii (< r 2500 ). It excludes the innermost regions of the clusters, since these regions contain complex structures even in the most relaxed clusters. A more detailed discussion about the data can be found in [38] 1 The gas mass fraction data comprises of five clusters at z < 0.16. These low redshift data have been used to obtain a bound on the quantity: h 3/2 b0 c0 + b0 = 0.089 ± 0.012, where h, c0 , and b0 are the Hubble parameter, dark matter density, and baryon density at z = 0 in units of critical density, respectively. Using h = 0.732 ± 0.013 from the latest SH0ES results [42] and 100 b0 h 2 = 2.235 ± 0.033 from Big Bang Nucleosynthesis [43] along with the aforementioned constraint, we get: 3 and ρ m0 = 29.53 ± 4.34(×10 −31 g/cm 3 ). We shall use these values in our analysis outlined in Sect. 4. Furthermore, we exclude the five gas mass fraction data at z < 0.16, and thereby use only the remaining 35 f gas data for the analysis.

Cosmic chronometers
Cosmic chronometers (CC) are one of the most widely used probes in Cosmology for deducing the observational value of the Hubble parameter at different redshifts (see [44] and references therein). According to Jimenez and Loeb [45], if passively evolving galaxies at different redshifts are considered, then computing the age difference of the galaxies yields the Hubble parameter based on the following equation: The derivative term dz/dt in Eq. (1) is obtained with respect to the cosmic time. This method is agnostic with respect to the choice of the cosmological model used, given that the only assumption for the CC is the stellar population model. We use 31 cosmic chronometer H (z) data from Li et al. [46] in the redshift range 0.07 z 1.965 in order to derive the angular diameter distance to the cluster (see next section).

Methodology
In a first part of our analyses, we shall check if the gas depletion factor (γ (z)) is constant with redshift, by considering a linear evolution law: γ (z) = γ 0 (1 + γ 1 z). In the second part of this work, using these results for γ , we implement a test of a possible evolution of the dark matter density law using galaxy clusters and cosmic chronometers.

Gas depletion factor
The matter content of galaxy clusters should approximately match the matter content of the Universe as a whole [47]. Therefore the gas mass fraction which is defined as the ratio of baryonic matter density b to the total matter density m of the universe should be a constant over the cosmic time for massive and relaxed clusters. Therefore, this quantity can be used as a cosmological probe [48,49]. The gas mass fraction is given by [50], Here, K (z) is the calibration constant, which accounts for any bias in gas mass due to bulk motion and non-thermal pressure in cluster gas [38,[50][51][52]. This value has been obtained selfconsistently from the data by comparing with clusters having weak lensing masses from Weighing the Giants sample [53], and is equal to 0.9 ± 0.09. A(z) represents the angular correction factor representing the change in angle subtended at r 2500 as the cosmology is varied [50]. Since this factor is close to unity [50], we do not incorporate it in our analysis. and D A is the angular diameter distance to each cluster. The asterisk in Eq. (2) denotes the corresponding quantity when a flat CDM fiducial cosmology (H 0 = 70 km/s/Mpc and m = 0.3) is assumed. The most relevant quantity for this work is the gas depletion factor γ (z), which is a measure of how much baryonic gas is depleted compared to the cosmic mean. Hence from Eq. (2), it can be re-written as, Simulations of hot, massive, and dynamically relaxed galaxy clusters (M 500 > 10 14 M ) evolving with different physical processes have been used to verify the evolution of γ with cluster redshift and no significant trend of γ (z) as a function of redshift has been found [51,52]. Recent works have estimated the depletion factor using only cosmological observations, such as: galaxy cluster gas mass fraction (calculated at r 2500 ), SNe Ia and strong gravitational lensing (SGL) systems [54,55]. In particular, the analysis involving galaxy cluster and SGL systems showed a mild evolution for γ (z) [56]. On the other hand, constraints on a possible evolution of γ (z) using gas mass fraction measurements at r 500 have also been obtained through a joint analysis with cosmic chronometers [57,58]. In these works, it was pointed out that one cannot use f gas values at r 500 as a stand-alone probe for any model-independent cosmological tests since these studies have found a non-negligible γ (z) evolution. In this way, before we performed our test on a possible evolution of dark matter density law using galaxy clusters (calculated at r 2500 ) with cosmic chronometers, it is necessary to verify if these data support a constant γ or point to an evolving γ with red-shift. We note that Mantz et al. [38] also looked for a similar redshift variation of γ using only the gas mass fraction measurements and found negligible variation. Here, we do this test in a model-independent method using chronometers, as discussed below. In order to derive the angular diameter distance to each galaxy cluster, we use 31 cosmic chronometer H (z) data from Li et al. [46] in the redshift range 0.07 z 1.965. For this purpose, we choose Gaussian Processes Regression [59] to reconstruct the angular diameter distance at each cluster's redshift(for more details, see [58,60,61]). The reconstructed angular diameter distance is obtained via, where H (z ) is the non-parametric reconstruction of Hubble parameter using Gaussian Processes. Now, we adopt the linear parametric form to model the gas depletion factor γ (z) (which is widely used in our previous works [56,58]) as follows, where γ 0 is the normalization constant and γ 1 shows the redshift dependent part of the gas depletion factor.

Testing a possible evolution of dark matter density law
The total matter density of the universe is the summation of baryonic and dark matter density, and so Eq. (2) can be rewritten as, If we posit putative dark matter interactions with the remaining dark sectors, then an ad-hoc term can be added to the standard dark matter density evolution law, and the deformed evolution law can then be written as: ρ c = ρ c0 (1 + z) 3+ where represents the interaction term arises due to DM-DE interactions [37,39,40]. So Eq. (6) can be recast as, where ρ b0 and ρ c0 are the baryonic matter density and dark matter density respectively at the current epoch. This is the key equation that will be used to constrain the parameter.

Constraints on the gas depletion factor
To constrain the parameters γ 0 and γ 1 of Eq. (5), we need to maximize the likelihood equation which is given as follows, Here, σ 2 i includes the error in f gas , K (z), m (z), b (z), D A and D * A which is calculated by error propagation of aforementioned quantities.

Constraints on parameter
Similarly, the constraints on the parameter present in Eq. (7) can be obtained by maximizing the likelihood distribution function, L given by Here, σ 2 i denotes the observational errors which is obtained by propagating the errors in ρ b0 , ρ c0 , K (z), γ (z), f gas (z), D * A and D A .

Results
The next step is to estimate the free parameters viz. γ 0 and γ 1 by maximizing the likelihood function (Eq. (8)) using the emcee MCMC sampler [63]. The outcome of this maximization is shown in Fig. 1. The diagonal histograms represent the one dimensional marginalized posterior distributions for each parameter and the off-diagonal entries show the 68%, 95%, and 99% two-dimensional marginalized confidence intervals. We find that γ 0 = 0.891 ± 0.087 and γ 1 = −0.012 +0.231 −0.195 . Our result imply no evolution for the gas depletion factor γ (z) as a function of redshift within 1σ at R 2500 .
Similarly, we maximize the likelihood function (Eq. (9)) to estimate the free parameter present in Eq. (7). Since our resulting γ 1 from our first analysis is consistent with zero at 1σ , we set γ (z) = γ 0 in Eq. (9). The allowed region for  is shown in Fig. 2. We obtain: = −0.234 +0.159 −0.171 which shows no interaction of dark matter with the dark sectors to within 1.4σ . The evolution of the dark matter density ρ c (z) as a function of redshift along with the standard dark matter Fig. 3 The evolution law for the dark matter density as a function of redshift. The light blue shaded area shows the 2σ allowed region for the evolution law as found in this work. The standard evolution law for the dark matter density is indicated by the solid blue line density evolution law ( = 0) is displayed in Fig. 3. Table 1 summarizes our results and other recent estimates on presented in literature.

Conclusions
The aim of this work is twofold: First, we studied a possible evolution of the gas depletion factor using 35 X-ray gas mass fraction data from Mantz et al. [38] in the redshift range 0.078 ≤ z ≤ 1.063 at near R 2500 , in a model-independent manner. The key equation used for this analysis is Eq. (3). In order to derive the angular diameter distance to each cluster, we incorporated the cosmic chronometer data from [46] spanning the redshift range 0.07 z 1.965 and then applied Gaussian Processes to estimate the angular diameter distance for each cluster redshift, as given by Eq. (4). Furthermore, we adopted a linear parametric form for the gas depletion factor, γ (z) = γ 0 (1 + γ 1 z). We constrained the γ 0 and γ 1 parameters by maximizing the log-likelihood given by Eq. (8). The one dimensional marginalized likelihoods for each parameter along with two dimensional confidence intervals are shown in Fig. 1. We found no evolution of the gas depletion factor as a function of redshift within 1σ at near R 2500 .
In the second part, we performed a test to study a possible departure of the standard dark matter density evolution law, ρ c = ρ c0 (1+z) 3 , as a followup to our previous studies which did this test using combination of gas mass fraction measurements and Type 1a supernova [36] as well as strong lensing systems [37]. For this purpose, we added an ad-hoc term to the evolution law: ρ c = ρ c0 (1 + z) 3+ where represents the interaction term arises due to DM-DE interactions. To constrain the parameter presented in Eq. (7), we maximized the log-likelihood in Eq. (9). Note that we used a constant value of γ (z), obtained from the first analysis of this work. Figure 2 shows the one dimensional likelihood for parameter. Our final result from this analysis is: = −0.234 +0.159 −0.171 indicating no interaction of DM-DE at 1.4σ . This is shown in Fig. 3 along with the standard DM evolution law( = 0) as a function of redshift. Constraints on the obtained in recent studies along with our result are summarized in Table 1. The results in this work are in agreement with those from previous analyses.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .