The mass discrepancy-acceleration relation: a universal maximum dark matter acceleration and implications for the ultra-light scalar field dark matter model

TL;DR

This paper uncovers a universal maximum acceleration in galaxy halos, links it to a constant surface density in dark matter profiles, and explores implications for ultra-light scalar field dark matter models, consistent with observational data.

Contribution

It demonstrates a universal maximum acceleration and surface density in dark matter halos, and applies these findings to constrain ultra-light scalar field dark matter models.

Findings

Universal maximum acceleration for galaxy halos.

Constant surface density $ ho_s r_s$ across halos.

Scalar field particle mass around $10^{-21} ext{ eV}/c^2$ fits data.

Abstract

Recent analysis of the rotation curves of a large sample of galaxies with very diverse stellar properties reveal a relation between the radial acceleration purely due to the baryonic matter and the one inferred directly from the observed rotation curves. Assuming the dark matter (DM) exists, this acceleration relation is tantamount to an acceleration relation between DM and baryons. This leads us to a universal maximum acceleration for all halos. Using the latter in DM profiles that predict inner cores implies that the central surface density $\mu_{DM} = \rho_s r_s$ must be a universal constant, as suggested by previous studies in selected galaxies, revealing a strong correlation between the density $\rho_s$ and scale $r_s$ parameters in each profile. We then explore the consequences of the constancy of $\mu_{DM}$ in the context of the ultra-light scalar field dark matter model (SFDM). We find that for this model $\mu_{DM} = 648 \, M_\odot {\rm pc}^{-2}$, and that the so-called WaveDM soliton profile should be an universal feature of the DM halos. Comparing with data from the Milky Way and Andromeda satellites, we find that they are consistent with a boson mass of the scalar field particle of the order of $10^{-21} \, {\rm eV}/c^2$, which puts the SFDM model in agreement with recent cosmological constraints.

Tables

Table 1. Table 1: Row(1): DM models and their characteristic quantities. Row(2) Dimensionless density profiles. Row(3): Value of the dimensionless radius r ^ max = r max / r s subscript ^ 𝑟 max subscript 𝑟 max subscript 𝑟 𝑠 \hat{r}_{\rm max}=r_{\rm max}/r_{s} at which the DM acceleration is maximal. Row(4): Maximal value of the (dimensionless) radial acceleration g ^ max subscript ^ 𝑔 max \hat{g}_{\rm max} . Row(5): The surface density μ D ​ M = ρ s ​ r s subscript 𝜇 𝐷 𝑀 subscript 𝜌 𝑠 subscript 𝑟 𝑠 \mu_{DM}=\rho_{s}r_{s} (in units of M ⊙ ​ pc − 2 subscript 𝑀 direct-product superscript pc 2 M_{\odot}{\rm pc}^{-2} ) obtained from the constraint ( 2 ).

	Burkert	MultiState-SFDM	PI	Spano	WaveDM	NFW
$f\left(r/r_s\right)$	${\left[\left(1+r/r_s\right)\left(1+r^2/r_s^2\right)\right]}^{-1}$	${\sin }^2\left(r/r_s\right)/{\left(r/r_s\right)}^2$	${\left(1+r^2/r_s^2\right)}^{-1}$	${\left(1+r^2/r_s^2\right)}^{-3/2}$	${\left(1+r^2/r_s^2\right)}^{-8}$	${\left[\left(r/r_s\right){\left(1+r/r_s\right)}^2\right]}^{-1}$
${\hat{r}}_{\max }$	$0.96$	$1.57$	$1.51$	$1.03$	$0.36$	$0$
${\hat{g}}_{\max }$	$1.59$	$4.02$	$2.89$	$2.19$	$0.86$	$2\pi$
${\mu }_{DM}$	$580$	$355$	$369$	$541$	$648$	$89$

Equations

$$g_h=\frac{g_{\mathrm{bar}}}{e^{\sqrt{g_{\mathrm{bar}}/g^{†}}}-1},$$

$$\frac{g_{h,\max }}{10^{-11}\mathrm{m}{\mathrm{s}}^{-2}}=0.014\left(\frac{{\mu }_{DM}}{M_{\odot }{\mathrm{pc}}^{-2}}\right){\hat{g}}_{h,\max }.$$

$${\left(\frac{r_s}{\mathrm{pc}}\right)}^{-3}{\left(\frac{m_a}{10^{-23}\mathrm{eV}}\right)}^{-2}=4.1\times 10^{-15}\left(\frac{{\mu }_{DM}}{M_{\odot }{\mathrm{pc}}^{-2}}\right).$$

Full text

In the standard cold dark matter (CDM) paradigm, the dark matter (DM) is $\sim$ 22% of the total matter budget in the UniverseAghanim et al. (2016), it is assumed to be collisionless and non-relativistic after decoupling, forming structure hierarchically, i.e. small halos merge to form more massive systems. Several cosmological CDM simulations that exclude the luminous matter have confirmed this formation scenario showing that in all scales halos share a common density profile with a characteristic cusp (divergent density) near their centersNavarro et al. (1997); Springel et al. (2005); Dutton and Macciò (2014); Oñorbe et al. (2015). Assuming galaxies are formed in these halos allows the comparison with observations. However, detailed comparison with the dynamics of low-mass galaxies have led to some longstanding discrepancies, e.g. cusp-core problem, the satellite abundance, too-big-to-failde Blok (2010); Klypin et al. (1999); Boylan-Kolchin et al. (2012); Rodrigues et al. (2017). Galaxies are then one of the greatest challenges for the CDM paradigm.

One quantity that summarizes the properties of the rotation curves in galaxies is the so-called mass-discrepancy-acceleration relation (MDAR)McGaugh et al. (2016); Lelli et al. . The MDAR is observed for a diverse sample of galaxies, from high to low surface brightness galaxies and of different sizes and morphologies. This seemingly independence of the relation on the luminous matter suggests that if there is a common origin to the MDAR, it is probably not strongly tied to the baryonic matter. There are two straightforward approaches to explain the origin of the MDAR, one is to assume that the acceleration relation results from modifying the gravitational force as suggested by the MOND hypothesisMilgrom (2016) (but see alsoDiez-Tejedor et al. (2016)), and the second is that the MDAR is a direct consequence of the intrinsic properties of the DMSalucci (2016). The latter approach will be assumed for the purposes of this Letter.

Considering the equivalent DM halo to explain the MDAR, it can be shown that the gravitational acceleration of the DM can be found fromMcGaugh et al. (2016)

[TABLE]

where $g_{\rm bar}$ is the acceleration produced by the baryons in the galaxy and $g^{\dagger}=1.2\times 10^{-10}{\rm m}\,{\rm s}^{-2}$ is a characteristic acceleration obtained from fitting the data. Apart from the characteristic acceleration $g^{\dagger}$, there exists a maximal acceleration $g_{h,{\rm max}}$ that can be obtained from Eq. (1), or from any other MOND functionBrada and Milgrom (1999). Given that Eq. (1) describes various galaxies, it follows that any halo will have a maximum acceleration. A straightforward calculation shows that the maximal acceleration provided by any DM halo must be $g_{h,{\rm max}}=0.65g^{\dagger}=7.8\times 10^{-11}{\rm m}\,{\rm s}^{-2}$. The existence of this universal maximal acceleration (UMA) can provide constraints on the surface density of some of the most common DM profiles in the literature, and in particular on the properties of scalar field (wave) dark matter (SFDM) modelMatos and Ureña López (2000, 2001, 2004); Hu et al. (2000); Schive et al. (2014a); Magaña et al. (2012); Suárez et al. (2014); Marsh (2016); Hui et al. (2016).111A similar approach has been pursued in Refs.Karukes and Salucci (2016); Hayashi and Chiba (2015); Burkert (2015); Ogiya et al. (2014); Del Popolo et al. (2012); Milgrom (2009); Donato et al. (2009), and their results are in agreement with ours once the appropriate conversions between physical quantities are taken into account.

For purposes of generality, let us assume that the DM density profile is spherically symmetric and given in the form $\rho(r)=\rho_{s}f(r/r_{s})$, where $\rho_{s}$ and $r_{s}$ are the characteristic density and scale radius of the profile, respectively, and $f(r/r_{s})$ is any given function of its argument. Notice that in the case of profiles with a core we expect that $f(0)=1$ and therefore $\rho_{s}$ is the central density. The magnitude of the (radial) gravitational acceleration produced by the DM halo at a radius $r$ can be calculated from $g_{h}(r)=GM(r)/r^{2}$, where $G$ is Newton’s constant and $M(r)$ is the enclosed mass inside a sphere of radius $r$. Then, given the general form of the density profile $\rho(r)$, the gravitational acceleration can be written as $g_{h}(r)=G\mu_{DM}\,\hat{g}_{h}(\hat{r})$, where $\hat{g}_{h}(\hat{r})=(4\pi/\hat{r}^{2})\int\limits^{\hat{r}}_{0}f(x)x^{2}\,dx$ is a dimensionless quantity, and the radial coordinate has been normalized to the characteristic radius as $\hat{r}=r/r_{s}$.

We can see then that the gravitational acceleration at any given radius is proportional to the DM central surface density, which we define simply as $\mu_{DM}=\rho_{s}r_{s}$. Furthermore, for any density profile the derived maximal acceleration is given by

[TABLE]

The value of the dimensionless maximal acceleration $\hat{g}_{h,\rm max}$ can be readily calculated for any density profile $f(\hat{r})$, and then Eq. (2) directly becomes a constraint equation for the DM surface density $\mu_{DM}$.

We have selected DM density profiles of common use in the literature, and derived the expected value of their central DM surface density $\mu_{DM}$ by imposing that each profile satisfies the UMA Eq. (2) at their corresponding point of maximal acceleration $\hat{g}_{h,\rm max}$. The left hand side of Eq. (2) is the result of a mean behavior of various galaxies, then the derived values of $\mu_{DM}$ in Table 1 will represent the expected overall behavior that the best-fit parameters of individual galaxies should follow. Our predicted values are in agreement with those reported in previous works for the BurkertDonato et al. (2009); Robles and Matos (2012); Rodrigues et al. (2017), MultiState SFDMMartinez-Medina et al. (2015), pseudo-isothermal (PI)Robles and Matos (2012) and that of Spano et alSpano et al. (2008) profiles. It can be seen that the standard CDM profile, also known as the Navarro-Frenk-White (NFW) profileNavarro et al. (1997), shows an acceleration that converges to its maximum at the center. That is, the maximum DM acceleration of the NFW profile is predicted to happen at the center of each galaxy, which is also noticed in CDM simulationsNavarro et al. (2016). In contrast, we find that for all core profiles the accelerations will reach their maximal value near their scale radius $r_{s}$ and then drop to zero for smaller radii.

Although the UMA obtained from the MDAR is a universal quantity independent of the DM profile, this is not the case for the surface density $\mu_{DM}$ or the dimensionless maximal value $\hat{g}_{h,\rm max}$, both dependent on the chosen DM profile. Nonetheless, once $\hat{g}_{h,\rm max}$ is calculated for a given density profile, its associated value of $\mu_{DM}$ will be fixed for all halos modeled using the same profile, which then implies the correlation of the two free parameters in the density profile $\rho_{s}$ and $r_{s}$; the latter are allowed to vary from galaxy to galaxy as long as their product $\rho_{s}r_{s}$ remains a constant. If the UMA in the DM is valid independently of the baryonic matter content in a galaxy, it implies that all DM profiles in Table 1 have only one free parameter to fit the rotation curve of any individual galaxy.

Empirical core profiles have parameters that are not necessarily tied to fundamental properties of the DM; however, the profile parameters in models that are theoretically motivated can be related to intrinsic quantities of the model under study. One particularly interesting case that falls in the latter category is that of ultra-light SFDM, which assumes that the DM particle is a scalar field of very small mass whose quantum properties appear at galactic scalesMatos and Ureña López (2000, 2001); Hu et al. (2000); Suárez et al. (2014); Marsh (2016); Hui et al. (2016). Although the relativistic theory may be complicatedHlozek et al. (2015); Ureña López and Gonzalez-Morales (2016), the properties of the halo density profile are dictated by those of the so-called Schrodinger-Poisson (SP) system of equations, seeGuzman and Ureña López (2004, 2006) and references therein. The soliton profile actually corresponds to the ground state solution of the SP system, and we refer to it as the WaveDM profile to distinguish it from other more general solutions of the SFDM model,e.g., considering multiple states of the fieldRobles and Matos (2013) whose analytical profile is also included in Table 1 (MultiState-SFDM).

In empirical profiles the two parameters $\rho_{s}$ and $r_{s}$ are treated as independent, and they are not linked to any particular DM property. For the WaveDM profile, however, the parameters $\rho_{s}$ and $r_{s}$ are predicted to have the following scaling property: $\rho_{s}=\lambda^{4}m^{2}_{a}\,m^{2}_{\rm Pl}/4\pi$ and $r_{s}=(0.23\,\lambda\,m_{a})^{-1}$, where $m_{\rm Pl}$ is the Planck mass, $m_{a}$ is the mass of the boson particle and $\lambda$ is a scaling parameterGuzman and Ureña López (2004); Marsh and Pop (2015). By the elimination of the scaling parameter $\lambda$, we then find the following expression for the surface density $\mu_{DM}$ in terms of the mass $m_{a}$ and the soliton radius $r_{s}$,

[TABLE]

The existence of a universal value of the surface density, namely $\mu_{DM}=648\,M_{\odot}\,{\rm pc}^{-2}$ (see Table 1), implies, similarly to other core profiles, a close correlation between $m_{a}$ and $r_{s}$.444Notice that the universality of $\mu_{DM}$ in Eq. (3) implies for the WaveDM profile the correlation $m_{a}\propto r^{-3/2}_{s}$, which is also reported in Fig. 2 of Ref.Chen et al. (2016), although it is erroneously attributed there to a constant core density $\rho_{s}$. See also Fig. 7 in Ref.Bernal et al. (2017). Moreover, within the WaveDM paradigm the boson mass $m_{a}$ is a fundamental physical parameter and as such it cannot vary from galaxy to galaxy. If we now consider the universality of the surface density implied by the MDAR, Eq. (3) must also fix the value of the scale radius $r_{s}$. In contrast to other empirical profiles, we then find that the MDAR in Eq. (1) ultimately implies the existence of an universal soliton (core) profile within the WaveDM paradigm.

Notably, if we neglect the assumption that the boson mass $m_{a}$ is fundamental and treat it as a free parameter, then the best-fit parameters modeling the rotation curves of individual galaxies are expected to satisfy Eq. (3). However, this fitting procedure will inevitably lead to a large dispersion in the mass which is simply a consequence of the diversity in galaxy sizes. Thus, as long as $m_{a}$ is treated as a free fitting parameter to describe a diverse sample of galaxies, we cannot derive a meaningful constraint of its value. The left panel in Fig. 3 illustrates this point, where we show the values of individual galaxy fits obtained in previous worksChen et al. (2016); Gonzáles-Morales et al. (2016); Bernal et al. (2017) in which $m_{a}$ and $r_{s}$ were treated as independent fitting parameters. In general, the fits lie closely along the line of constant surface density indicated by Eq. (3), which is the expected behavior from the universality of the MDAR. The large scatter in $m_{a}$ is the reflection that the fitting method in Chen et al. (2016); Gonzáles-Morales et al. (2016); Bernal et al. (2017) cannot provide a reliable determination of the boson mass $m_{a}$, and that the most they can do is to test the reliability of the constraint (3).

Nonetheless, as we shall show, Eq. (3), along with the assumptions that $m_{a}$ is constant and that all halos are described by the WaveDM profile, are enough to derive a simple estimate of the boson mass, we only require an independent estimate of $r_{s}$. We will use data from dwarf spheroidal (dSph) galaxies for our estimate. Being DM dominated systems, we expect the properties of dSphs to be similar, albeit with some possible scatter associated to their formation historiesFitts et al. (2016).

In the right panel of Fig. 3 we show the (theoretical) gravity profile $\hat{g}(r/r_{s})$ of the WaveDM model, together with the acceleration values reported in Ref.Fattahi et al. (2016) for the Milky Way (MW) dSphs. The data correspond to the enclosed mass $M_{1/2}$ within the half-light radius $r_{1/2}$, which were then converted to a gravitational acceleration at the same radius through the equation $g(r_{1/2})=GM_{1/2}/r^{2}_{1/2}$. The latter was then normalized as $g/G\mu_{DM}$ for a proper comparison with $\hat{g}$ using the value of $\mu_{DM}=648\,M_{\odot}\,{\rm pc}^{-2}$ found from the MDAR for the WaveDM profile. Hence, the only free parameter to adjust is $r_{s}$, which we use to normalize the half-light radius $r_{1/2}$ of each of the classical MW dSphs. We chose three different values, namely $r_{s}/{\rm kpc=0.3,2,10}$, which then correspond to three relative positions of the data points with respect to the theoretical curve. It can be seen that the best option is $r_{s}=0.3\,{\rm kpc}$ (red errorboxes in Fig. 3), which puts the data points on the right hand side of the point of maximal acceleration, where they even seem to follow the downward trend of the theoretical curve at large radii.

Using the same value of $r_{s}=0.3\,{\rm kpc}$ we scale a bigger sample of satellite galaxiesCollins et al. (2014); Lelli et al. , in Fig. 3 we include only the dwarf galaxies with high-quality observations selected inLelli et al. . Surprisingly, we find that the data follows the theoretical curve reasonably well after the point of maximal acceleration, the largest scatter coming from the galaxies with the large observational uncertainties. Even more, we see that satellites AndXIX, and AndXXI and AndXXV, which were labeled as outliers in the study of Ref.Collins et al. (2014), mostly because of their low mass as compared to their size, seem to vindicate the trend of the theoretical curve of the WaveDM profile at the lowest values of the gravitational acceleration.

For our estimated $r_{s}=0.3\,{\rm kpc}$, the constraint equation (3) implies the boson mass $m_{a}=1.2\times 10^{-21}\,{\rm eV}$, and the soliton mass $M_{s}=1.8\times 10^{7}\,M_{\odot}$, which is consistent with the uniformity of the mass estimates within $300\,{\rm pc}$ made in Ref.Strigari et al. (2008). Notice that the boson mass is somehow unexpected, as it is much larger than the values commonly reported in the literature for dwarf galaxiesSchive et al. (2014a); Chen et al. (2016); Marsh and Pop (2015); Gonzáles-Morales et al. (2016). However, this new and larger value of the boson mass is constant for all halos, as demanded by the hypothesis of the SFDM model, and avoids the stringent constraints coming from cosmological observationsHui et al. (2016); Corasaniti et al. (2016); Schive et al. (2016); Menci et al. (2017); Banik et al. (2017); Gonzáles-Morales et al. (2016); Sarkar et al. (2017).

Because the MDAR implies a single value of the scale radius $r_{s}$ in the WaveDM profile, there should be an universal soliton profile present in all galaxy halos, but due to the diversity of galaxy sizes this implies the WaveDM profile alone will prove unable to describe all DM halos, in particular large onesMartinez-Medina et al. (2015). This does not rule out the SFDM model, but it does rule out the possibility that the soliton profile represents all the DM halo; hence, a more general profile than the WaveDM profile is required that extends to larger radii.

One natural approach that has been proposed is to account for the superposition of excited states in the SFDM halo, which mostly affect the outer parts of the halo leaving the characteristic imprint of wiggles or oscillations in the density and rotation curve profiles (see the MultiState profile in Table 1)Robles and Matos (2013); Martinez-Medina et al. (2015); Matos and Ureña López (2007). A more ad hoc approach deals with smoothly matching the soliton to a NFW profile describing the outer part of the halo which adds a second parameter to the fitting density profileMarsh and Pop (2015); Gonzáles-Morales et al. (2016); Bernal et al. (2017), this is motivated by the results of numerical simulations of the SP systemSchive et al. (2014a, b); Schwabe et al. (2016); Du et al. (2016a, b).

We have performed numerical calculations on this more general profile (following the prescription in Ref.Bernal et al. (2017)) that models the full SFDM halo and applied to it the MDAR constraint Eq. (2). The resultant SFDM surface density is found to lie in the range $\mu_{DM}=(575-648)\,M_{\odot}{\rm pc}^{-2}$; these values are of the same order of magnitude as in the case of the soliton profile alone, and they would similarly imply that $r_{s}\sim 0.3\,{\rm kpc}$. We find that considering this general profile leaves the main results about the unique soliton and the estimated boson mass roughly unchanged and opens the possibility of fitting a diverse sample of galaxies to their outermost radii with a single boson mass. At the same time it also strengthens the possibility of an universal soliton profile with a total mass of $10^{7}\,M_{\odot}$ in the center of all galaxy halos.Strigari et al. (2008); Okayasu and Chiba (2016).

Summarizing, we have shown that the MDAR implies in general a universal value of the surface density for any given DM density profile, which translates into a strong correlation between their central density $\rho_{s}$ and scale radius $r_{s}$. We have explored the consequences of such correlation in the case of the SFDM model, which leads to the conclusion that all galaxy halos should have a similar central core structure, formed due to the Heisenberg uncertainty principle. Comparison with data of satellite galaxies of the MW and Andromeda suggests a boson mass in the SFDM model of $10^{-21}\,{\rm eV}/c^{2}$, which is in agreement with current cosmological and astrophysical constraints, while still having a distinguishable history of structure formation and halo density distribution from the standard CDM modelSchive et al. (2016); Du et al. (2016a); Corasaniti et al. (2016).