Scaling relations for dark matter core density and radius from Chandra X-ray cluster sample
Introduction
The current concordance (CDM) cosmological model consisting of 25% cold dark matter and 70% dark energy, agrees very well with Planck CMB and large scale structure observations [1]. However, at scales smaller than about 1 Mpc, the cold dark matter paradigm runs into a number of problems such as the core/cusp problem, missing satellite problem (although see [2]), too big to fail problem, satellites plane problem etc. (See Refs. [3], [4], [5] for recent reviews on this subject). We also note that some of these problem can be ameliorated using various baryonic physics effects [6], [7], [8], [9]. At a more fundamental level, another issue with the CDM model is that there is no laboratory evidence for any cold dark matter candidate [10]. Furthermore, LHC or other particle physics experiments have yet to find experimental evidence for theories beyond the Standard Model of Particle Physics, which predict such cold dark matter candidates [10]. Therefore, a large number of theoretical alternatives to CDM model have been proposed, and a variety of observational tests devised to test these myriad alternatives.
An intriguing observational result discovered more than a decade ago is that the dark matter halo surface density is constant, for a wide variety of systems spanning over 18 orders in blue magnitude for a diverse suite of galaxies, such as spiral galaxies, low surface brightness galaxies, dwarf spheroidal satellites of Milky way [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] etc. See however Refs. [21], [22], [23], [24], [25], [26], [27] and references therein, which dispute these claims and argue for a mild dependence of the dark matter surface density with halo mass and other galaxy properties. These results for a constant dark matter surface density were obtained by fitting the dark matter distribution in these systems to a cored profile, either Burkert [28], pseudo-isothermal profile [11], or a simple isothermal sphere [29]. All these cored profiles can be parameterized by a central density () and core radius (); and the halo surface density is defined as the product of and . The existence of a constant dark matter surface density was found to be independent of which cored profile was used [12]. Alternately, some groups have also calculated a variant of the above dark matter halo density, which has been referred to as the dark matter column density [21], [24],1 whose value remains roughly invariant with respect to the choice of the dark matter profile. This column density is equivalent to the product of and for a Burkert profile [24], and provides a more precise value of the surface density for non-cored profiles, such as the widely used NFW profile [30]. The best-fit values for the dark matter surface density for single galaxy systems using the latest observational data is given by [20].
A large number of theoretical explanations have been proposed to explain the constancy of dark matter halo density. Within the standard CDM model, some explanations include: transformation of cusps to cores due to dynamical feedback processes [31], self-similar secondary infall model [21], [24], [26], [32], dark matter–baryon interactions [33], non-violent relaxation of galactic haloes [34]. Some explanations beyond CDM include ultralight scalar dark matter [35], super-fluid dark matter [36], self-interacting dark matter [5], [37], [38], [39], MOND [40], etc. A constant halo surface density is also in tension with fuzzy dark matter models [41].
It behooves us to test the same relation for galaxy clusters. Galaxy clusters are the most massive collapsed objects in the universe and are a wonderful laboratory for a wide range of topics from cosmology to galaxy evolution [42], [43]. In the last two decades a large number of new galaxy clusters have been discovered through dedicated optical, X-ray, and SZ surveys, which have provided a wealth of information on Astrophysics and Cosmology. However, tests of the constancy of dark matter surface density for galaxy clusters have been very few.
The first such study for galaxy clusters was done by Boyarsky et al. [44], who used the dark matter profiles from literature for 130 galaxy clusters and showed that the dark matter column density () goes as , where is the mass within the density contrast at [45], where the density contrast () is defined with respect to the critical density. Hartwick [16] used the generalized NFW profile [30] fits in Ref. [46] (using strong and weak lensing data) for the Abell 611 cluster, and found that . This is about twenty times larger than the corresponding value obtained for galaxies [20]. Lin and Loeb [37] estimated for the Phoenix cluster, using multi-wavelength data obtained by the SPT collaboration [47]. Using a model for self-interacting dark matter including annihilations, they also predicted the following relation between the surface density and [37] Del Popolo et al. also predicted [24], [26] a similar relation between the dark matter column density and , within the context of a secondary infall model [32] valid for cluster scale haloes
The first systematic study of the correlation between and for an X-ray selected cluster sample, and without explicitly assuming any dark matter density model, was done by Chan [48] (C14, hereafter). C14 first considered the X-ray selected HIFLUGCS cluster catalog consisting of ASCA and ROSAT observations [49]. They considered 106 relaxed clusters from this catalog. From the hydrostatic equilibrium equation and parametric models for the gas density and temperature profiles, the total mass () was obtained as a function of radius. The total density as a function of radius () was then obtained from the total mass, assuming spherical symmetry.
One premise in C14 is that the total mass is dominated by the dark matter contribution, while the stellar and gas mass can be ignored. was obtained from extrapolating the dark matter density distribution to . The core radius was obtained by finding the radius () at which . This emulates the definition of in the Burkert profile [28]. Therefore, the estimate of core density and radius was done in C14 without explicitly positing any dark matter profile. We note that from weak and strong lensing observations, galaxy clusters are estimated to have cored or shallower than cuspy NFW dark matter profiles [50], [51]. However, these results have been disputed [52], and some works have also found evidence for cuspy haloes in clusters [53]. Therefore, there is no consensus on this issue [54]. Nevertheless, no explicit assumptions about the dark matter profile was made in C14, while obtaining the dark matter core density and radius, although the gas density models used for deducing this, were initially motivated from assuming isothermality and a King profile for the total mass distribution in galaxy clusters. We also note that in some models, for example the cusp to core transformation model [31] or the self-interacting dark matter with annihilations [37], the product of the core density and core radius for the cored profile is same as the product of scale density and scale radius of cuspy NFW-like profiles.
In their analysis, C14 used two different density profiles (single- and double- model) for the gas density. They also did separate fits for both the cool-core and the non cool-core clusters. Using the double- model, they obtained for the HIFLUGCS sample. Results from fits with other profiles for the same sample as well as other samples can be found in C14. Therefore, their result shows that the dark matter surface density is not constant for clusters. C14 also carried out a similar analysis on the LOCUSS cluster sample analyzed in Shan et al. [55] and found that . Therefore, these results indicate that unlike single-galaxy systems, the dark matter surface density is not constant for galaxy clusters and is about an order of magnitude larger than for single galaxy systems.
We now implement the procedure recommended in C14 to determine and for a catalog of 12 galaxy clusters, selected using pointed X-ray and archival ROSAT observations by Vikhlinin et al. [56] (V06, hereafter). Detailed parametric profiles for gas density and temperature profiles have been compiled by V06. This cluster sample has been used to constrain a plethora of modified gravity theories and also to test non-standard alternatives to CDM model [57], [58], [59], [60], [61]. We have also previously used this sample to constrain the graviton mass [62] as well as to assess the importance of relativistic corrections to hydrostatic mass estimates [63]. Our work improves upon C14 in that, we account for the baryonic mass distribution while estimating the dark matter halo properties.
The outline of this manuscript is as follows. We describe the V06 cluster sample and associated models for the density and temperature profile in Section 2. Our analysis and results for the relation between core radius and density can be found in Section 3. Comparison with various theoretical scenarios is discussed in Section 4. We also test for dependence vs in Section 5. We conclude in Section 6.
Section snippets
Details of Chandra X-ray sample
V06 (See also Ref. [64]) derived density and temperature profiles for a total of 13 nearby relaxed galaxy clusters (A133, A262, A383, A478, A907, A1413, A1795, A1991, A2029, A2390, MKW4, RXJ1159+55531, USGC 2152) using measurements obtained from pointed observations with the Chandra X-ray satellite. The redshifts of these clusters range approximately up to . These measurements extended up to very large radii of about for some of the clusters. For lower redshift clusters, because of
Determination of scaling relations
The resulting values of and along with error bars for each of the 12 clusters estimated using the procedure outlined in Section 2.3 can be found in Table 1. We note that our values for and are of the same order of magnitude as for other galaxy cluster systems estimated in C14. Our estimated dark matter surface density is about an order of magnitude larger than that found for galaxy systems [12], [20].
Fig. 1 shows the log versus log plot, and we observe a tight scaling
Comparison with theoretical models
Our results from the previous section show that we see deviations from a constant halo surface density only at about 1.4. However, the resulting value of our halo surface density is about ten times larger than for galaxies. We briefly discuss some theoretical models which are consistent with such a scenario.
The problems with CDM at small scales can be solved by self-interacting dark matter with a velocity-dependent cross-section ranging from on galaxy scales to on
Dependence on
We now use our results for and to check for correlation with , as suggested in some works [24], [37]. The first step in doing this is to estimate from . In V06, the masses () and concentration parameters () for the overdensity (with respect to the critical density) level and its corresponding radius () have already been derived, where was obtained using the mass–concentration relations in [93]. We have determined the values using same prescription
Conclusions
A large number of studies in the past decade have found that the dark matter surface density, given by the product of dark matter core radius () and core density () is constant for a wide range of galaxy systems from dwarf galaxies to giant galaxies over 18 orders in blue magnitude. This cannot be trivially predicted by the vanilla CDM model, but it can be easily accommodated in various alternatives to CDM or by invoking various feedback mechanisms in CDM.
However, there have been very
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We are grateful to Man-Ho Chan and Antonio Del Popolo for useful correspondence, and Alexey Vikhlinin for providing us the data in V06. We are also thankful to the anonymous referee for several constructive feedback on our manuscript.
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