Implications of the Intriguing Constant Inner Mass Surface Density Observed in Dark Matter Halos

Abstract

It has long been known that the observed mass surface density of cored dark matter (DM) halos is approximately constant, independently of the galaxy mass (i.e., ${\rho }_c{r}_c\simeq \mathrm{constant}$, with ${\rho }_c$ and $r_c$ being the central volume density and the radius of the core, respectively). Here, we review the evidence supporting this empirical fact as well as its theoretical interpretation. It seems to be an emergent law resulting from the concentration–halo mass relation predicted by the current cosmological model, where the DM is made of collisionless cold DM particles (CDM). We argue that the prediction ${\rho }_c{r}_c\simeq \mathrm{constant}$ is not specific to this particular model of DM but holds for any other DM model (e.g., self-interacting) or process (e.g., stellar or AGN feedback) that redistributes the DM within halos conserving its CDM mass. In addition, the fact that ${\rho }_c{r}_c\simeq \mathrm{constant}$ is shown to allow the estimate of the core DM mass and baryon fraction from stellar photometry alone is particularly useful when the observationally expensive conventional spectroscopic techniques are unfeasible.

1. Introduction

The shape of the dark matter (DM) halos hosting galaxies can be inferred from rotation curves or other kinematical measurements, e.g., [1,2,3]. The resulting DM radial profiles often show an inner plateau or core characterized by a central mass density ${\rho }_c$ and a core radius $r_c$ which, when combined, happen to yield a surface density approximately constant,

$${\rho }_c{r}_c\simeq \mathrm{constant},$$

(1)

a property observed to hold in a wide range of halo masses $M_h$, between ${10}^9$ and ${10}^{12}{M}_{\odot }$ [4,5,6,7,8,9,10,11,12] (actual values and details will be given in Section 2 and Appendix A). Originally, it was a rather surprising result [4], but currently it is interpreted in the literature as an emergent law caused by the well-known relation between halo mass and concentration arising in collisionless cold dark matter (CDM) numerical simulations [13,14,15]. In CDM-only simulations, the CDM halos do not have cores. They follow the canonical NFW profiles [16] or the Einasto profiles [17], with a pronounced inner cusp where the density grows continuously toward the center of the halo. Thus, an additional physical process must operate to transform the cuspy CDM halos into cored halos, conserving the original DM mass. This transformation is usually assumed to be driven by baryon processes like star-formation feedback, AGN feedback, or galaxy mergers, which shuffle around the baryonic mass, thus changing the overall gravitational potential and affecting the distribution of CDM. CDM cores appear in model galaxies formed in full hydrodynamical cosmological numerical simulations, e.g., [18,19,20]. Thus, Equation (1) is often regarded as a support for CDM; [15] and references therein. However, the formation of cores in DM halos can be driven by any physical process that thermalizes the DM distribution [21,22]. They will also render Equation (1), provided the process just redistributes the available mass, not much changing the relation between the halo mass and concentration set by the cosmological initial conditions (Section 5.1).

The purpose of this work is to review the observational evidence for Equation (1) as well as the theory behind it. The interpretation can be pinned down to the relation between the mass of a DM halo and its age of formation (Section 5.1), which is set by cosmology and to a lesser extent by details on the nature of DM. As a spin-off, we demonstrate how Equation (1) can be used to estimate the mass in the DM halo of a galaxy based solely on the distribution of its stars. The approach is based on the fact that dwarf galaxies also tend to show a central plateau or core in the stellar distribution, e.g., [23,24]. The radii of the stellar and the DM cores are expected to scale with each other [25,26]. We worked out the relation between the core radius of the stellar distribution and the DM mass.

The paper is organized as follows: Section 2 collects observational evidence for Equation (1). Section 3 works out the explanation of Equation (1) within CDM. Section 4 compares the observations in Section 2 with the theory in Section 3. Based on Equation (4), Section 5 writes down a semi-empirical relation between the stellar core radius and DM halo mass. It also shows that the stellar mass surface density is a proxy for the baryon fraction in the center of a galaxy. Ready-to-use relations are given in Equations (25) and (26). Section 6 summarizes the main conclusions in the work.

2. Observations Supporting Equation (1)

As we point out in Section 1, the product ${\rho }_c{r}_c$ is approximately constant over a large range in galaxy mass. To emphasize the existing evidence, we have compiled a number of relations between ${\rho }_c{r}_c$ and $M_h$ from the literature. They are based on uneven measurements prone to bias, including the determination of the DM halo mass of a galaxy and the definition of core radius. However, the conclusion is clear, with the different independent determinations agreeing within error bars. The result of the compilation is shown in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5. Details of how the individual works were interpreted to construct the figures are given in Appendix A. In particular, here and throughout the paper, we assume the core radius to be the radius where the density drops to half the central value,

$$\rho \left(r_c\right)={\rho }_c/2,$$

(2)

with ${\rho }_c=\rho \left(0\right)$. This definition is not universally used and so the radii quoted in the original reference often have to be transformed to our definition, as detailed in Appendix A.

Figure 1 gives the scatter plot of ${\rho }_c{r}_c$ versus $M_h$. The extreme values are likely unreliable but it is clear that the product ${\rho }_c{r}_c$ tends to be constant, at least for $M_h<{10}^{11}{M}_{\odot }$. This fact is better appreciated in Figure A1, which is identical to Figure 1 but with the vertical axis spanning the same eight orders of magnitude of the horizontal axis corresponding to the DM halo masses. Histograms with the values of ${\rho }_c{r}_c$ in Figure 1 are shown in Figure 2. They include all the observed values (the blue line), when $M_h<{10}^{12}{M}_{\odot }$ (the red line), and when $M_h<{10}^{11}{M}_{\odot }$ (the green line). An inset in the figure also gives the median and the 1-sigma percentiles of the distributions (i.e., 50%, 15.9%, and 84.1%) which correspond to

$${\rho }_c{r}_c={44}_{-21}^{+44}{M}_{\odot }{\mathrm{pc}}^{-2},$$

(3)

when $M_h<{10}^{11}{M}_{\odot }$, a limit representative of dwarf galaxies. We note that the used $r_c$, as set by Equation (2), is typically a factor of two smaller than the core radii commonly defined in the literature (e.g., b when the Schuster–Plummer profile in Equation (5) is used). Thus, the surface density in Equation (3) is fully consistent with a value around $100M_{\odot }{\mathrm{pc}}^{-2}$ often quoted in the literature (see, e.g., [5,14]). As we explain in Appendix A, the estimate of $M_h$ used in Figure 1 relies on the observed absolute magnitude of the galaxies, assuming a mass-to-light ratio and a relation between stellar mass and DM halo mass as inferred from abundance matching [27]. However, the trend for ${\rho }_c{r}_c$ to become constant in dwarf galaxies is already present in the original data; see Figure 3, where the abscissa are given by the measured absolute magnitude of the galaxy.

Figure 4 gives ${\rho }_c{r}_c$ (top panel) and ${\rho }_c{r}_c^3$ (bottom panel) versus $r_c$. Note that the latter gives the DM mass in the core and it scales as $r_c^2$ following Equation (3), which is represented in the figure by the gray dashed line. These relations are independent of the uncertainties in $M_h$.

Figure 5 gives the relation of $r_c$ with $M_h$ (top panel) and ${\rho }_c$ with $M_h$ (bottom panel). The correlation happens to be very clear in both cases. The larger the mass, the larger the radius and the smaller the density. In order to guide the eye, the figure includes power laws as $r_c\propto M_h^{0.4}$ (top panel) and ${\rho }_c\propto M_h^{-0.4}$ (bottom panel), which approximately describe the observed trends. Note that combined, these power laws render Equation (1).

3. Theory: Cores Resulting from Redistributing Collisionless Cold Dark Matter Halos

If the DM was collisionless CDM and if there were no baryons, then the distribution of DM within each halo would approximately follow the iconic NFW profile [16],

$${\rho }_{\mathrm{NFW}}\left(r\right)={\displaystyle \frac{{\rho }_s}{(r/r_s){(1+r/r_s)}^2}},$$

(4)

describing the variation with radius r of the DM volume density ${\rho }_{\mathrm{NFW}}\left(r\right)$. The parameters $r_s$ and ${\rho }_s$ stand for a scaling radius and a scaling density, respectively. The mass available to form any DM halo today is provided by the initial conditions set by cosmology (see Section 5.1). It would be the same independently of whether a physical process redistributes this mass in a different mass density profile. Probably, the most general of such a process is the thermalization the DM distribution. In this case, one expects the formation of a core with a generic polytropic shape, characteristic of self-gravitating systems reaching thermodynamic equilibrium [21,22,28]. For analytic simplicity, we assume $m=5$ polytrope (best known as the Schuster–Plummer profile), but the core of all the polytropes has virtually the same shape, e.g., [28]. In this case,

$${\rho }_5\left(r\right)={\displaystyle \frac{{\rho }_c}{{\left[1+{(r/b)}^2\right]}^{5/2}}},$$

(5)

with ${\rho }_c$ the central density and b a length scale setting the core radius defined as in Equation (1),

$$r_c=b\times \sqrt{2^{2/5}-1}\simeq b\times 0.56525\dots .$$

(6)

Thus, the new density profile resulting from the core formation is a piecewise function defined as Equation (5) in the core, Equation (4) in the outskirts, and continuous in the matching radius $r_m$,

$$\rho \left(r\right)=\left\{\begin{array}{cc}{\rho }_5\left(r\right),\hfill & \mathrm{when}r<r_m,\hfill \\ {\rho }_5\left(r_m\right)={\rho }_{\mathrm{NFW}}\left(r_m\right),\hfill & \mathrm{when}r=r_m,\hfill \\ {\rho }_{\mathrm{NFW}}\left(r\right),\hfill & \mathrm{when}r>r_m.\hfill \end{array}\right.$$

(7)

In addition, to conserve mass,

$${\int }_0^{\infty }\rho \left(r\right)r^{2}dr={\int }_0^{\infty }{\rho }_{\mathrm{NFW}}\left(r\right)r^{2}dr,$$

(8)

which, considering Equation (7), renders

$${\int }_0^{r_m}{\rho }_5\left(r\right)r^{2}dr={\int }_0^{r_m}{\rho }_{\mathrm{NFW}}\left(r\right)r^{2}dr.$$

(9)

Examples of these cored DM profiles with NFW outskirts are given in Figure 6. This kind of piecewise shape has already been used in the literature, e.g., [29,30,31].

Equations (7) and (9) provide a mapping between the parameters of the NFW profile (${\rho }_s$ and $r_s$) and the parameters defining the core (${\rho }_c$ and b). The continuity at $r_m$ forces

$${\displaystyle \frac{{\rho }_c}{{(1+{(r_m/b)}^2)}^{5/2}}}={\displaystyle \frac{{\rho }_s}{(r_m/r_s){(1+r_m/r_s)}^2}},$$

(10)

whereas mass conservation, Equation (9), leads to

$${\rho }_c{r}_m^3{\displaystyle \frac{1}{3{[1+{(r_m/b)}^2]}^{3/2}}}={\rho }_s{r}_s^3\left[ln(1+{\displaystyle \frac{r_m}{r_s}})-{\displaystyle \frac{r_m/r_s}{1+r_m/r_s}}\right].$$

(11)

After some algrebra, Equations (10) and (11) render,

$$1+{\left({\displaystyle \frac{r_m/r_s}{b/r_s}}\right)}^2={\displaystyle \frac{3{(1+r_m/r_s)}^2}{{(r_m/r_s)}^2}}\left[ln(1+{\displaystyle \frac{r_m}{r_s}})-{\displaystyle \frac{r_m/r_s}{1+r_m/r_s}}\right],$$

(12)

and

$${\displaystyle \frac{{\rho }_c{b}^3}{{\rho }_s{r}_s^3}}={\displaystyle \frac{3{[1+{(r_m/b)}^2]}^{3/2}}{{(r_m/b)}^3}}\left[ln(1+{\displaystyle \frac{r_m}{r_s}})-{\displaystyle \frac{r_m/r_s}{1+r_m/r_s}}\right].$$

(13)

We note that once $r_m/r_s$ is set (i.e., the radius of match in units of $r_s$; see Equation (7)), Equations (12) and (13) give the full density profile. Equation (12) provides $b/r_s$, which can be used in Equation (13) to compute ${\rho }_c/{\rho }_s$, and then $\rho \left(r\right)/{\rho }_s$. This is the procedure followed to compute the densities shown in Figure 6.

Figure 7 shows the dependence on $r_m/r_s$ for $b/r_s$, ${\rho }_c/{\rho }_s$, and ${\rho }_{c}b/\left({\rho }_s{r}_s\right)$. We note that for $r_m\lesssim r_s$, $b\sim r_s$ and ${\rho }_{c}b\sim {\rho }_s{r}_s$. These dependences are easy to distill from the above equations in the limit $r_m\ll r_s$. In this case,

$$ln(1+{\displaystyle \frac{r_m}{r_s}})-{\displaystyle \frac{r_m/r_s}{1+r_m/r_s}}\simeq {\displaystyle \frac{{(r_m/r_s)}^2}{2}},$$

(14)

so that Equation (12) renders,

$$b/r_s\simeq \sqrt{2}(r_m/r_s).$$

(15)

Similarly, Equation (13) plus Equation (15) render

$${\rho }_{c}b\simeq {\rho }_s{r}_s{(3/2)}^2\sqrt{3}={\rho }_s{r}_s\times 3.89711\dots .$$

(16)

When $r_m=r_s$ (i.e., when the matching radius coincides with the characteristic radius defining the NFW profile), then things simplify even further so that

$${\displaystyle \frac{{\rho }_c{r}_c}{{\rho }_s{r}_s}}\simeq 1.0068\dots ,$$

(17)

where we have used Equation (6) to transform b into $r_c$ (details in Appendix B).

The NFW halos are given settings ${\rho }_s$ and $r_s$. In the context of CDM, these two variables are often replaced by the concentration c and the halo mass $M_h$, so that

$${\rho }_s={\displaystyle \frac{200c^3{\rho }_{crit}}{3\left[ln(1+c)-c/(1+c)\right]}},$$

(18)

and

$$r_s^3={\displaystyle \frac{3M_h}{800\pi {\rho }_{crit}{c}^3}}.$$

(19)

The concentration c stands for $r_{200}/r_s$, with $r_{200}$ defined so that the mean enclosed density within $r_{200}$ equals 200 times the critical density ${\rho }_{crit}$ of the Universe. Pieced together, Equations (18) and (19) render the dependence of the product ${\rho }_s{r}_s$ on c and $M_h$,

$${\rho }_s{r}_s={\displaystyle \frac{10}{3}}{\left({\displaystyle \frac{30}{\pi }}\right)}^{1/3}{\displaystyle \frac{{\rho }_{crit}^{2/3}{c}^2{M}_h^{1/3}}{ln(1+c)-c/(1+c)}},$$

(20)

a relation that can be found already in the literature, e.g., [13].

The numerical simulations of CDM predict a relation between c and $M_h$, which varies with redshift and is quite tight for $M_h>{10}^{10}{M}_{\odot }$ to become looser at smaller halo mass [32,33,34]. Examples of this relation are given in Figure 8, where we note that the range of variation of c is quite moderate, changing only by a factor of three for halos varying by seven orders of magnitude in mass, from ${10}^7$ to ${10}^{14}{M}_{\odot }$; see the blue lines in Figure 8. Thus, considering c constant, the dependence of ${\rho }_s{r}_s$ on halo mass predicted by Equation (20) is quite mild as it scales as $M_h^{1/3}$. This fact, together with the approximate equivalence given by Equations (6) and (16), indicates that the predicted ${\rho }_c{r}_c$ is expected to vary little with halo mass,

$${\rho }_c{r}_c\propto {\rho }_{c}b\propto {\rho }_s{r}_s\propto M_h^{1/3},$$

(21)

as it is indeed observed (Section 2).

The equations above yield ${\rho }_c{r}_c$ as a function of $M_h$. The algorithm to compute it is (1) set $r_m/r_s$, (2) obtain c of $M_h$ from the literature (Figure 8), (3) obtain ${\rho }_s$ and $r_s$ as a function of $M_h$ from Equations (18) and (19), (4) obtain $b/r_s$ of $M_h$ from $r_s$ and Equation (12), (5) obtain ${\rho }_c/{\rho }_s$ of $M_h$ from $b/r_s$, $r_m/r_s$ and Equation (13), (6) obtain $r_c/b$ from Equation (6) and, finally, (7) compute

$${\rho }_c{r}_c={\rho }_s\times r_s\times {\displaystyle \frac{b}{r_s}}\times {\displaystyle \frac{{\rho }_c}{{\rho }_s}}\times {\displaystyle \frac{r_c}{b}}.$$

(22)

Figure 9 shows the predicted variation of ${\rho }_c{r}_c$ as a function of $M_h$ for various $r_m/r_s$ assuming the c–$M_h$ relation at redshift zero given in [32] (the solid lines). Qualitatively, the trends for other c–$M_h$ relations and redshifts look the same. The figure also includes the variation of $r_c$ (the dashed lines) and ${\rho }_c$ (the dashed dotted lines) separately. Note how the increase in $r_c$ with $M_h$ is partly balanced by the decrease in ${\rho }_c$, leaving a fairly constant ${\rho }_c{r}_c$.

4. Comparison Between Observations and Theory

Figure 10 shows the observed ${\rho }_c{r}_c$ (the symbols) compared with the prediction using the simple equations worked out in Section 3, where the DM cores are assumed to result from the redistribution of the mass of the CDM halos. The observed data points in Figure 10 are those in Figure 1 but shown in a range spanning the same eight orders of magnitude variation for both ${\rho }_c{r}_c$ and $M_h$. This particular scaling evidences how constant ${\rho }_c{r}_c$ is, with the range of values in Equation (3) highlighted as the pale green region. The colored lines represent the theoretical predictions and they agree well with the observation without any fine tuning. They even reproduce a slight increase in ${\rho }_c{r}_c$ with halo mass, which is probably too large in the theoretical model, although given the observational uncertainties one should not stress this fact further. Note that the prediction depends on the parameter $r_m/r_s$ and the redshift z from which the relation c–$M_h$ was taken. The best agreement with the observation corresponds to $r_m/r_s$ between 1 and 2 starting off from halos at $z=0$, and between 0.5 and 1 starting from halos a bit earlier at $z=1$. Figure 10 is based on the theoretical c–$M_h$ from [32], but the results are similar for the other theoretical c–$M_h$ analyzed in Section 2 and Figure 8 and Figure 9.

5. Discussion

Here, we analyze the implications of the fair agreement between the theory and observation presented in Section 4.

5.1. What Sets the c–$M_h$ Relation?

Note that so far, the answer to the question of what sets ${\rho }_c{r}_c\simeq$ constant is the existence of a c–$M_h$ relation for the DM halos produced in the $\Lambda$CDM cosmology (see Figure 8 and Figure 9). Thus, unless we understand in physical terms what sets the c–$M_h$ relation of the collisionless CDM halos, the above explanation of why ${\rho }_c{r}_c$ is constant sounds circular.

Correa et al. [33] describe the current understanding in detail, and give a number of relevant references. According to this view, the relation seems to be driven by the inside-out growth of the DM halos combined with the fact that low mass halos collapse first. The build-up of all halos generally consists of an early phase of fast accretion and a late phase where the accretion slows down [35,36]. During the early phase, halos are formed with low concentration, and then the concentration increases during the second phase as the outer halo grows and the mass accretion rate decreases. The concentration grows during this second phase because the virial radius setting the size of the whole halo increases while $r_s$ remains rather constant. Halos of all masses undergo these two phases, but low mass halos complete the first phase early on and so they show large concentrations at present, whereas the very massive ones are still in the first phase. This process gives rise to the variation predicted by the numerical simulations shown in Figure 8. Contrary to the low mass halos, the high mass halos show little evolution of the concentration with redshift (or, equivalently, with time). According to this scenario, the actual c–$M_h$ relation should depend significantly on the cosmological parameters, in particular, on ${\sigma }_8$ that parameterizes the amplitude of the matter density fluctuations in the early Universe, and on ${\Omega }_m$ that quantifies the total amount of matter. The larger ${\sigma }_8$ or ${\Omega }_m$, the earlier the halos assemble and the larger the resulting concentration [33].

5.2. Relation Between DM Core Mass and Stellar Core Radius

The DM halo mass within the visible stellar core is

$$M_{hc}={\displaystyle \frac{4\pi }{3}}{\rho }_c{r}_{★c}^3={\displaystyle \frac{4\pi g{\kappa }_c}{3}}{r}_{★c}^2,$$

(23)

with ${\kappa }_c$ the constant ${\rho }_c{r}_c$, $r_{★c}$ the stellar core radius, and $g=r_{★c}/r_c$. Provided $g\lesssim 1$, Equation (23) gives the DM mass within the observed stellar core. Even if this is a relationship between the core DM halo mass and the stellar radius, it is encouraging to note that a similar relation is observed to hold between the DM core mass and the DM core radius (Figure 4), and between the total DM halo mass and the core radius; see the dashed line in Figure 5, corresponding to $M_h\propto r_c^{2.5}$. The baryon fraction in the core, defined as

$$f_{bc}={\displaystyle \frac{M_{★c}}{M_{hc}}}={\displaystyle \frac{{\rho }_{★c}{r}_{★c}}{g{\kappa }_c}},$$

(24)

can be inferred from the observed stellar mass surface density, ${\rho }_{★c}{r}_{★c}$, provided g can be measured or estimated. Thus, if Equation (1) holds, from the stellar distribution alone one can estimate the DM core mass and the baryon fraction in the core. Using ${\kappa }_c$ from Equation (3), Equations (23) and (24) become

$$M_{hc}\simeq {1.7}_{-0.8}^{+1.7}\times {10}^5{M}_{\odot }{\left({\displaystyle \frac{r_{★c}}{30\mathrm{pc}}}\right)}^{2}g,$$

(25)

and

$$f_{bc}\simeq {2.2}_{-1.1}^{+2.1}\times {10}^{-3}{\displaystyle \frac{{\rho }_{★c}{r}_{★c}}{0.1M_{\odot }{\mathrm{pc}}^{-2}}}{g}^{-1},$$

(26)

respectively. The error bars just consider the scatter in ${\kappa }_c$.

In order to test the reliability of the above equations, we have used existing observations of ultra faint dwarfs (UFDs) and dwarf spheroidal galaxies (dSph) to compare for individual galaxies the values of $M_{hc}$ computed from velocities and from Equation (25). The dynamical mass of a galaxy within $r_{★c}$ can be computed from the observed velocity dispersion within the core radius, ${\sigma }_{★c}$, as

$$M_{dyn}={\displaystyle \frac{2ln2}{G}}{\sigma }_{★c}^2{r}_{★c},$$

(27)

with G being the gravitational constant. In DM-dominated systems,

$$M_{hc}\simeq M_{dyn}.$$

(28)

Equation (27) uses the definition in Equation (2) and assumes spherical symmetry as detailed by, e.g., [11]. It differs from similar expressions found in the literature by factors of the order of one [37]. Figure 11 shows the DM halo mass estimated from photometry (Equation (25)) versus the value estimated from velocity dispersion (Equations (27) and (28)). The agreement is quite remarkable; often within the error bars set by Equation (3). The UFDs have been included to show that the approximation works even in this extremely low mass regime, keeping in mind that part of the observed scatter away from the one-to-one relation is due to uncertainties in their dynamical mass estimate. The dynamical masses of UFDs are particularly uncertain because they are affected by the presence of stellar binaries, which may contribute to the velocity dispersion as much as the gravitational potential, e.g., [38]. The horizontal error bars in Figure 11 result from the statistical errors in ${\sigma }_{★c}$, which are probably underestimating the real ones since the effect of the binaries is not included. We have used $g=1$ for simplicity but the assumption $g\sim 1$ seems to be quite realistic [25,26] and, eventually, it could be relaxed and refined if needed.

Given the good agreement between the dynamical DM mass and the photometric DM mass represented in Figure 11, Equation (25) seems to be a new valuable tool for estimating the DM halo mass from photometry alone. Photometry is much cheaper observationally than the spectroscopy required to determine the dynamical mass. The validity of Equation (25) implies the validity of Equation (26), which also provides a new empirical way of estimating the baryon fraction in galaxies only from stellar photometry. Moreover, it tells us that the surface density of stars is a proxy for the baryon fraction in the inner parts of a galaxy.

The above estimate can be extended to the mass of the whole DM halo using a model to represent the DM halo beyond the core (e.g., the piecewise profile in Equation (7) and Figure 6). Thus, $M_{hc}$ can be used to estimate $M_h$. To have a first idea of the ratio between them, assume that the stellar core radius is not very different from the matching radius $r_m$ that separates the inner and outer parts of the piecewise profile (Figure 6), which is a quite common assumption in the literature, e.g., [13,39]. Then, the ratio of masses turns out to be

$$M_h/M_{hc}\simeq {\displaystyle \frac{ln(1+r_{★c}/r_s)-(r_{★c}/r_s)/(1+r_{★c}/r_s)}{ln(1+c)-c/(1+c)}},$$

(29)

which varies from a few to a factor of ten when the concentration varies as predicted, from $c\sim 5$ in high mass halos to $c\sim 20$ in low mass halos (Figure 8, the blue lines).

5.3. Constant DM Dynamical Pressure

The dynamical pressure in a fluid scales like the density times the square of the characteristic velocity. Thus, for the DM in the core, the effective DM dynamical pressure is

$$P_c\propto {\rho }_c{\sigma }_c^2,$$

(30)

with ${\sigma }_c$ being the velocity dispersion of the DM particles in the core. Assuming the DM cores are to be virialized (i.e., assuming that Equations (27) and (28) hold for the DM particles too), then

$$P_c\propto {\left({\rho }_c{r}_c\right)}^2\simeq \mathrm{constant},$$

(31)

so that Equation (1) implies that the dynamical pressure to be exerted by the DM particles if they could collide would be the same in all halos, independently of their total mass or size. However, collisionless CDM particles do not collide, and Equation (31) has to be interpreted as a property that emerges from the existence of the c–$M_h$ relation.

6. Conclusions

We reviewed the observational evidence for ${\rho }_c{r}_c\simeq$ constant (Equation (1); Section 2) and then put forward a simple version of the commonly accepted interpretation behind it (Section 3). Equation (1) requires the existence of a core in the DM distribution. Halos formed in DM-only CDM cosmological numerical simulations do not have inner cores but cusps (Equation (4)); however, if any physical process redistributes the DM particles of the expected CDM halos, then Equation (1) is satisfied automatically. It emerges from the relation between the concentration and DM halo mass expected in $\Lambda$CDM cosmological simulations. This relation is set by the time of halo formation, so that low mass halos form earlier and present larger concentrations (Section 5.1). The conventional explanation to understand how the original cuspy CDM halos become cored halos is stellar feedback. This term encapsulates all the baryon-driven processes that shuffle gas and mass around (e.g., supernova explosions or stellar winds), modifying the overall potential, including the distribution of DM particles in the center of the galaxies [18,19]. However, this transformation is not specific to stellar feedback, keeping in mind that any physical process that thermalizes a self-gravitating structure tends to form cores [21,22]. Thus, any other sensible physical process that redistributes matter without altering the original mass of the CDM halos is able to account for Equation (1). In other words, the property of ${\rho }_c{r}_c$ to be approximately constant is not specific to CDM but, rather, it is also expected in many alternative DM theories forming cores, e.g., [13,31,40]. Theories that only redistribute mass to produce cores have the advantage of leaving the large scale structure of the Universe unchanged, thus being in agreement with the standard $\Lambda$CDM.

The mathematical development in Section 3 parallels others existing in the literature, except that the core is modeled with a different expression, e.g., [13,15]. Here, we provide a full account of the derivation of the main equations for the sake of comprehensiveness, which help us to make the qualitative comparison with observations in Section 4. However, we could have started off by assuming the relevant Equations (12) and (13) and proceed from here. This loose dependence of the results on the actual shape of the core is consistent with the fact that other alternative forms of the piecewise profile with cores that we tried (top hat profiles) render qualitatively similar results.

The agreement between the simple theory and observations is notable, keeping in mind that there is no fitting or fine tuning in matching lines and points in Figure 10. Even more, the theory predicts a moderate increase in ${\rho }_c{r}_c$ with $M_h$, similarly to the one hinted at by the observations. However, the best fitting c–$M_h$ relations correspond to large cores (the green dashed line represents $r_m/r_s=1.8$) or $z\ne 0$ (the solid orange and green lines in Figure 10 correspond to $z=1$). The latter is a result that we do not understand; even if the transformation of cusps to cores requires time and starts at high redshift, the accretion of DM in the outskirts of the halos should continue all the way to the present, a process leading to the c–$M_h$ relation at $z=0$. As we discussed in Section 5.1, the c–$M_h$ depends on the cosmological parameters ${\sigma }_8$ and ${\Omega }_m$ since they set the assembly time of the DM halos. Varying them may improve the agreement when employing the theoretical c–$M_h$ relations at $z=0$, but we have not pursued this idea further.

As a byproduct of the effort to compile ${\rho }_c{r}_c$ values, we show that the fact that the product is constant can be used to estimate the mass in the DM halo of a galaxy from the distribution of stars alone. This possibility can be very useful for low stellar mass galaxies where the determination of their DM content using traditional kinematical measurements is technically difficult, whereas their photometry is doable. The same argument allows one to estimate the baryon fraction in the core of these systems. Dwarf galaxies also tend to show a core in the stellar distribution, e.g., [23,24], with the radii of the stellar and DM cores expected to scale with each other [25,26]. This idea plus Equation (3) allows us to propose specific relations between the observed stellar core radius and the DM core mass (Equation (25)) and between the observed stellar mass surface density and the baryon fraction in the core (Equation (26)). The latter tells us that the surface density of stars is a proxy for the baryon fraction in the inner parts of a galaxy. The proposed calibrations are in good agreement with DM masses estimated from dynamical measurements in low mass galaxies (Figure 11). Note that the numerical coefficients of the proposed scaling laws depend on the definition of the core radius, for which we adopted Equation (2). Other definitions can be trivially recalibrated.