Dark Matter in Fractional Gravity III: Dwarf Galaxies Kinematics

Abstract

Recently, we put forward a framework where the dark matter (DM) component within virialized halos is subject to a non-local interaction originated by fractional gravity (FG) effects. In previous works, we demonstrated that such a framework can substantially alleviate the small-scale issues of the standard $\mathsf{\Lambda }$CDM paradigm, without altering the DM mass profile predicted by N-body simulations, and retaining its successes on large cosmological scales. In this paper, we investigate further, to probe FG via the high-quality data of individual dwarf galaxies, by exploiting the rotation velocity profiles inferred from stellar and gas kinematic measurements in eight dwarf irregulars, and the projected velocity dispersion profiles inferred from the observed dynamics of stellar tracers in seven dwarf spheroidals and in the ultra-diffuse galaxy DragonFly 44. We find that FG can reproduce extremely well the rotation and dispersion curves of the analyzed galaxies, performing in most instances significantly better than the standard Newtonian setup.

1. Introduction

The standard $\mathsf{\Lambda }$CDM cosmology envisages galaxies to be hosted in virialized halos of dark matter (DM), which largely dominate the total mass and, hence, mostly determine the overall gravitational potential well and the dynamical properties of the baryons [1,2]. Remarkably, the density distribution of such halos is predicted by N-body simulations to follow an approximately universal shape, well described by the classic Navarro–Frenk–White [3] profile $\rho \propto {(r/r_s)}^{-1}{(1+r/r_s)}^{-2}$, with $r_s$ being a characteristic scale radius where the logarithmic slope equals $-2$.

Only a minor deviation from such a scale-invariant behavior is expected, which amounts to a relationship between $r_s$ and the total DM mass [4]. This is often expressed in terms of the concentration parameter $c_{200}\equiv R_{200}/r_s$, with $R_{200}$ being the radius where the average DM density is 200 times that of a critical Universe ${\rho }_{\mathrm{crit}}$. In fact, recent zoom-in N-body simulations [5] have demonstrated that $c_{200}$ correlates very well with the mass $M_{200}\equiv (4\pi /3)200{\rho }_{\mathrm{crit}}{R}_{200}^3$ over a remarkably extended range from $M_{200}\sim {10}^{-5}{M}_{\odot }$ to ${10}^{15}{M}_{\odot }$.

Although on large scales observational data undoubtedly confirms the above picture, in the realm of dwarf galaxies with total masses $\lesssim {10}^{11}{M}_{\odot }$ the situation becomes more uncertain. The most relevant issue for the present context emerges from galaxy kinematics and/or gravitational lensing data, which seem to indicate a much flatter density profile in the inner regions (i.e., a core), with respect to the cuspy NFW behavior; this occurrence is often referred to as the cusp–core problem [6,7,8,9].

In addition, there are other well-known issues associated with small galaxy scales that are worth mentioning [10]: the missing satellites problem [11,12] concerns the observed satellites in Milky-Way-sized galaxies that are found to be much less numerous than the bound DM halos in N-body simulations; the too-big-to-fail problem [13] concerns the halos hosting dwarf galaxies, which from kinematical measurements are found to be less massive than expected; the radial acceleration relation [14,15], the universal core surface density [16], and the core radius vs. the disk scale length scaling [17] all constitute tight empirical relationships between the properties of the DM and of the baryons that are extremely puzzling in the standard paradigm.

There are various viable solutions to these issues. The most obvious claims a misinterpretation of the data, due to poor resolution effects or other complex features in the DM distribution [18]. Another one invokes the impact of ordinary matter physics on the DM profile via stellar feedback [19,20] or transfer of energy/angular momentum to the DM via dynamical friction [21,22]. Another possibility involves nonstandard particle candidates, such as warm or sterile neutrino DM [23,24], fuzzy or particle-wave DM [25,26], self-interacting DM [27], and dark-photon DM [28,29], that by various processes (e.g., free streaming, quantum pressure effects, and/or dark sector interactions) can avoid the formation or later erase the inner cusp [30,31,32]. Finally, the observed galaxy kinematics can be explained with or without DM by modified gravity theories [33,34,35], such as MOND [36,37], fractional-dimensional gravity [38,39,40,41,42], and emergent entropic gravity [43,44].

Recently, in [45,46], we put forward a fractional gravity (FG) framework that strikes an intermediate course between a modified gravity theory and an exotic DM scenario (in this respect, similar to the dynamical non-minimally-coupled DM model explored by our team in [47,48,49]). FG envisages the DM component to be present though subject to a non-local interaction mediated by gravity. Specifically, in such a framework, the gravitational potential associated with a given DM density distribution (e.g., the NFW one) is determined by a modified Poisson equation including fractional derivatives (i.e., derivatives of noninteger type) that are aimed at describing non-locality. Very interestingly, it can be shown that FG can be reformulated in terms of the standard Poisson equation, but with an effective density distribution that is flatter in the inner region, with respect to the true one. This is actually the density behavior that an observer would infer by looking at the kinematic data and interpreting them in terms of standard Newtonian theory. Thus, in FG the cusp–core problem is basically solved at its root, as the cuspy NFW density profile of $\mathsf{\Lambda }$CDM originates in FG a dynamic very similar to a cored profile in the standard Newtonian setting.

In [45,46] we tested FG over an extended mass range $M_{200}\sim {10}^9–{10}^{15}{M}_{\odot }$ by exploiting stacked rotation curves of spiral galaxies and joint X-ray/Sunyaev-Zel’dovich observations of galaxy clusters. We found that FG performs extremely well in reproducing the data in all these systems. Moreover, our analysis highlighted that the strengths of FG effects tend to weaken toward more massive systems, thus implying that FG can substantially alleviate the small-scale issues of the standard $\mathsf{\Lambda }$CDM paradigm, while retaining its successes on large cosmological scales.

For this paper, we aimed at investigating more deeply the regime where FG effects are expected to be more relevant, focusing on individual dwarf galaxies. In these objects, the cusp–core problem is observationally very pressing, and its solution via baryonic effects is difficult to be envisaged, given the paucity of baryons. Specifically, we probed FG, both in irregular dwarf (dwIrr) galaxies, by exploiting the rotation velocity profile inferred from stellar and gas kinematical measurements, and in dispersion-dominated dwarf spheroidals (dwSphs), by exploiting the velocity dispersion profile inferred from the dynamics of stellar tracers.

The structure of the paper is as follows: in Section 2, we describe our methods and data analysis; in Section 3, we present and discuss our results; in Section 4, we summarize our findings and outlook future perspectives. Throughout the work, we adopt the standard, flat $\mathsf{\Lambda }$CDM cosmology with rounded parameter values [50]: matter density ${\mathsf{\Omega }}_m\approx 0.3$, baryon density ${\mathsf{\Omega }}_b\approx 0.05$, and Hubble constant $H_0=100h$ km s${}^{-1}$ Mpc${}^{-1}$ with $h\approx 0.7$.

2. Methods

In this Section, we recall our basics framework, with particular focus on the basic kinematic observables in dwarfs. We then discuss the data and the Bayesian analysis exploited to probe such a scenario.

2.1. Dark Matter in Fractional Gravity

N-body simulations in the standard $\mathsf{\Lambda }$CDM cosmology indicate that virialized halos of DM particles follow an approximately universal density profile, routinely described via the Navarro–Frenk–White [3] shape $\rho \left(r\right)={\rho }_s{r}_s^3/r{(r+r_s)}^2$, in terms of a scale radius $r_s$ and of a characteristic density ${\rho }_s$.

In the standard Newtonian theory, the potential ${\mathsf{\Phi }}_{\mathrm{N}}\left(r\right)$ associated with a given density distribution $\rho \left(r\right)$ can be computed from the Poisson equation:

$$\Delta {\mathsf{\Phi }}_{\mathrm{N}}\left(\mathbf{r}\right)=4\pi G\rho \left(\mathbf{r}\right),$$

(1)

where $\Delta$ is the Laplacian operator; this is an inherently local equation, in that the potential at a point depends only on the value of the density there. For the spherically symmetric NFW profile, one easily finds that

$${\mathsf{\Phi }}_{\mathrm{N}}\left(r\right)=-\frac{G{M}_s}{r}ln\left(1+\frac{r}{r_s}\right),$$

(2)

with $M_s\equiv 4\pi {\rho }_s{r}_s^3$. Computing the NFW mass $M(<r)=4\pi {\int }_0^r\mathrm{d}{r}^{\prime }{r}^{\prime 2}\rho \left(r^{\prime }\right)=M_s[ln(1+r/r_s)-r/(r+r_s)]$, it is easy to verify that $|\mathrm{d}{\mathsf{\Phi }}_{\mathrm{N}}/\mathrm{d}r|=GM(<r)/r^2$, as a direct consequence of Birkhoff’s theorem.

In the FG framework, the potential ${\mathsf{\Phi }}_{\mathrm{F}}\left(r\right)$ is instead derived from the modified Poisson equation [40,45],

$${(-\Delta )}^s{\mathsf{\Phi }}_{\mathrm{F}}\left(\mathbf{r}\right)=-4\pi G{\mathcal{l}}^{2-2s}\rho \left(\mathbf{r}\right),$$

(3)

where ${(-\Delta )}^s$ is the fractional Laplacian operator (see the excellent textbook [51] for details), $s\in [1,3/2]$ is the fractional index (this range of values for s is required to avoid divergences; see Appendix A in [45]), and ℓ is a fractional length scale that must be introduced for dimensional reasons. At variance with the standard case, the fractional Laplacian is inherently nonlocal; the index s measures the strength of this nonlocality, while the length scale ℓ can be interpreted as the typical size below which gravitational effects are somewhat reduced and above which they are instead amplified by nonlocality.

In [45,46], ℓ was left as a free parameter to, be fitted by comparison with data. However, such a quantity enters only in the normalization of the potential but does not modify its radial shape; as such, it is strongly degenerate with the total mass, and very difficult to be constrained via pure kinematical data; essentially, one can only infer the combination $M_s{\mathcal{l}}^{2-2s}$. Therefore, in the following, without loss of generality, we set it to $\mathcal{l}\approx r_s$, which is the relevant spatial scale in the NFW density. This position is equivalent to making the original Poisson equation non-dimensional, in terms of quantities at $r_s$, and then fractionalized; this is a procedure often followed in the mathematical–physics literature [52,53,54], to insert fractional dynamics in a system, avoiding the addition of a dimensional parameter of ambiguous interpretation and problematic estimation.

For $s\in [1,3/2)$, the solution reads [45]

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_{\mathrm{F}}\left(r\right)=& -\frac{G{M}_s}{r_s}\frac{1}{2^{2s}\sqrt{\pi }}\frac{\mathsf{\Gamma }\left(\frac{3}{2}-s\right)}{\mathsf{\Gamma }(s+1)}\frac{r_s}{r}\left\{\frac{2\pi s}{sin\left(2\pi s\right)}\left[{\left(1+\frac{r}{r_s}\right)}^{2s-2}\right.\right.\hfill \\ \hfill & \\ \hfill & -\left.{\left(1-\frac{r}{r_s}\right)}^{2s-2}\right]+\frac{{(r/r_s)}^{2s}}{1-{(r/r_s)}^2}\left[\left(1+\frac{r}{r_s}\right){}_2{F}_1\left(1,1,2s+1,\frac{r}{r_s}\right)\right.\hfill \\ \hfill & \\ \hfill & +\left.\left.\left(1-\frac{r}{r_s}\right){}_2{F}_1\left(1,1,2s+1,-\frac{r}{r_s}\right)-\frac{4s}{2s-1}\right]\right\},s\in [1,3/2),\hfill \end{array}$$

(4)

with $\mathsf{\Gamma }\left(s\right)={\int }_0^{\infty }\mathrm{d}x{x}^{s-1}{e}^{-x}$ being the Euler Gamma function and ${}_2{F}_1(a,b,c;x)={\sum }_{k=0}^{\infty }$${\left(a\right)}_k{\left(b\right)}_k{x}^k/{\left(c\right)}_{k}k!$ being the ordinary hypergeometric function, in terms of the Pochammer symbols ${\left(q\right)}_k$ defined as ${\left(q\right)}_0=1$ and ${\left(q\right)}_k=q(q+1)\dots (q+k-1)$; plainly, ${\mathsf{\Phi }}_{\mathrm{F}}\left(r\right)$ for $s=1$ coincides with the usual expression ${\mathsf{\Phi }}_{\mathrm{N}}\left(r\right)$ of Equation (2). For the limiting case $s=3/2$, the computation requires some principal value regularization, and the solution reads

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_{\mathrm{F}}\left(r\right)=& -\frac{G{M}_s}{r}\frac{1}{\pi }\left\{2\frac{r}{r_s}\left[log\left(\frac{r}{r_s}\right)-1\right]-\left(1+\frac{r}{r_s}\right)log\left(\frac{r}{r_s}\right)log\left(1+\frac{r}{r_s}\right)\right.\hfill \\ \hfill & \\ \hfill & +\left.\left(\frac{r}{r_s}-1\right){\mathrm{Li}}_2\left(1-\frac{r}{r_s}\right)-\left(1+\frac{r}{r_s}\right){\mathrm{Li}}_2\left(-\frac{r}{r_s}\right)+\frac{{\pi }^2}{6}\right\},s=3/2,\hfill \end{array}$$

(5)

where ${\mathrm{Li}}_2\left(x\right)={\sum }_{k=1}^{\infty }{x}^k/k^2$ is the dilogarithm function.

Being a nonlocal framework, in FG the Birkhoff theorem does not hold, but one can insist on writing $|\mathrm{d}{\mathsf{\Phi }}_{\mathrm{F}}/\mathrm{d}r|=G{M}_F(<r)/r^2$, in terms of an effective mass $M_F(<r)$; then, one can differentiate the latter, to obtain an effective density profile ${\rho }_{\mathrm{F}}\left(r\right)=(1/4\pi r^2)\times \mathrm{d}{M}_{\mathrm{F}}/\mathrm{d}r$. These are actually the mass and density profiles that one would infer by looking at the dynamical observables and interpreting them in terms of Newtonian gravity. We illustrate the effective mass and density profiles for different values of s in Figure 1. With s increasing from unity (Newtonian case), the mass profile steepens and the density profile flattens; in the inner region, a uniform sphere behavior (corresponding to a cored density profile) tends to be progressively enforced.

These relevant profiles depend on the NFW scale radius $r_s$ and density ${\rho }_s$ or equivalently the mass $M_s=4\pi {\rho }_s{r}_s^3$; however, in the following, it is convenient to trade off these quantities for the mass $M_{200}$ and the concentration $c_{200}\equiv R_{200}/r_s$ at the reference radius $R_{200}$, where the average density is 200 times the critical density ${\rho }_{\mathrm{c}}$. The conversion between these variables can be performed easily from the definition of $R_{200}$ and from the NFW mass distribution. Furthermore, we adopt the relationship $c(M_{200},z)$ in the $\mathsf{\Lambda }$CDM cosmology derived from zoom-in N-body simulations by [5] and spanning twenty orders of magnitude in DM mass within the range $M_{200}\sim {10}^{-5}-{10}^{15}{M}_{\odot }$.

2.2. Dynamical Modeling

For a rotation-dominated system, the crucial quantity to be compared to the data is the total rotation velocity, which is given by

$$v_{\mathrm{rot}}^2\left(r\right)=v_{\mathrm{halo}}^2\left(r\right)+\frac{M_{★}}{L}{v}_{\mathrm{disk}}^2\left(r\right)+v_{\mathrm{gas}}^2\left(r\right),$$

(6)

where $v_{\mathrm{halo}}^2=G{M}_{\mathrm{F}}(<r)/r$ is the contribution from the DM halo, $v_{\mathrm{gas}}^2$ is the gas contribution from HI measurements, and $v_{\mathrm{disk}}^2$ is the contribution from the disk starlight appropriately converted into the stellar mass one via a global mass-to-light ratio $M_{★}/L$. The overall rotation velocity depends on three parameters: namely, the stellar mass-to-light ratio $M_{★}/L$, the total mass of the system $M_{200}$, and the fractional index s.

For a dispersion-dominated system, the crucial observable is the velocity dispersion projected along the line-of-sight (l.o.s.), which is given by [55,56]

$${\sigma }_{\mathrm{los}}^2\left(R\right)=\frac{2}{{\Sigma }_{★}\left(R\right)}{\int }_R^{\infty }\mathrm{d}r\left[1-\beta \frac{R^2}{r^2}\right]\frac{{\rho }_{★}\left(r\right){\sigma }_r^2\left(r\right)r}{\sqrt{r^2-R^2}},$$

(7)

where $\beta \equiv 1-{\sigma }_{\theta }^2/{\sigma }_r^2$ is the anisotropy parameter (hereafter assumed to be constant with the radius) and

$${\Sigma }_{★}\left(R\right)=2{\int }_R^{\infty }\mathrm{d}r\frac{{\rho }_{★}\left(r\right)r}{\sqrt{r^2-R^2}}$$

(8)

is the surface mass density of the tracers (e.g., stars), in terms of the volume one ${\rho }_{★}\left(r\right)$; in addition, the radial velocity dispersion is obtained by solving the Jeans equation, which yields

$${\sigma }_r^2\left(r\right)=\frac{1}{r^{2\beta }{\rho }_{★}\left(r\right)}{\int }_r^{\infty }\mathrm{d}{r}^{\prime }{r}^{\prime 2\beta }{\rho }_{★}\left(r^{\prime }\right)\frac{G[M_{\mathrm{F}}(<r^{\prime })+M_{★}(<r^{\prime })]}{r^{\prime 2}},$$

(9)

in terms of the tracer mass $M_{★}(<r)=4\pi {\int }_0^r\mathrm{d}{r}^{\prime }{r}^{\prime 2}{\rho }_{★}\left(r^{\prime }\right)$. Typically, for the dispersion-dominated galaxies considered in this work, stellar tracers are exploited, with a density distribution following the Plummer’s [57] model (for DragonFly 44, we actually exploit a Sersic surface density profile, as detailed in [58]). A computationally efficient strategy to compute Equation (7) is detailed in Appendix A.

2.3. Data and Bayesian Analysis

We probed the FG framework by exploiting the rotation velocity profiles of dwIrr galaxies and the l.o.s. dispersion velocity profiles of dwSphs.

For rotation-dominated systems, we relied on the SPARC database [59,60], which provided the stellar and gas contribution to the rotation velocity, as found by numerically solving the standard Poisson equation for the observed surface brightness profile at $3.6\mathsf{\mu }$m (with a reference stellar mass-to-light ratio $M_{★}/L=1$, so that this must be rescaled in building the total rotation velocity, as in Equation (6)), and the HI surface density profile. Specifically, we considered the 8 galaxies in SPARC that are classified as dwIrr and are flagged as having high-quality data on the rotation curve: D631-7, DDO64, DDO161, UGC731, UGC5005, UGC5414, UGC7608, and NGC3741. The latter is the galaxy with the best dataset, and actually constitutes the object with the most extended rotation curve (relative to the half-light radius) measured to date.

For dispersion-dominated systems, we considered 7 Milky-Way dwSphs for which high-quality determination of the spatially-resolved dispersion velocity had been obtained via stellar tracers ([61,62]; see also, data collection by [63] and references therein) and for which tidal effects were not appreciably influencing the inner kinematics: Carina, Leo I, Leo II, Sculptor, Draco, Sextans, and Fornax. To these, we added the dispersion-dominated ultra-diffuse galaxy DragonFly 44 [58]. Note that ultra-diffuse galaxies are a mixed bag of objects with very different properties: some feature large angular momentum, a rich gas reservoir with ongoing active star-formation activity [64,65,66,67]; others show no signs of rotation, a poor gas content, and an old stellar population in passive evolution [58,68]. DragonFly 44 belongs to this last category and, being dispersion-dominated, was treated here along with the dwSph sample. In addition, it is a particularly interesting object, as it features a very small stellar mass, with respect to its large size; in fact, the DM mass is expected to dominate even at small radii. Therefore, DragonFly 44 has been exploited as a useful laboratory for testing the nature of DM and gravity [69,70].

In

Table 1. Properties of the galaxy sample considered in this work: half-light radius, total luminosity, and mass-to-light ratio estimated from stellar population synthesis models (used in building the priors of our Bayesian analysis, see Section 2.3). The top half of the table includes rotation-dominated dwarfs, for which the listed quantities refer to the luminosity at $3.6\mathsf{\mu }$m; the bottom half of the Table includes dispersion-dominated dwarfs, for which the listed quantities refer to the V-band luminosity.

Galaxy	${\mathit{r}}_{\mathbf{1}/\mathbf{2}}$ [kpc]	$log\mathit{L}\left[{\mathit{L}}_{\odot }\right]$	${\mathit{M}}_{\mathbf{★}}/\mathit{L}[{\mathit{M}}_{\odot }/{\mathit{L}}_{\odot }]$
D631-7	$1.22$	$8.28$	$0.5\pm 0.1$
DDO64	$1.22$	$8.18$	$0.5\pm 0.1$
DDO161	$2.04$	$8.74$	$0.5\pm 0.1$
UGC731	$1.40$	$8.51$	$0.5\pm 0.1$
UGC5005	$5.0$	$9.61$	$0.5\pm 0.1$
UGC5414	$2.33$	$9.05$	$0.5\pm 0.1$
UGC7608	$1.60$	$8.42$	$0.5\pm 0.1$
NGC3741	$0.32$	$7.45$	$0.5\pm 0.1$
Carina	$0.27$	$5.57$	$3.4\pm 2.9$
Leo I	$0.29$	$6.74$	$8.8\pm 5.6$
Leo II	$0.22$	$5.87$	$0.4\pm 0.4$
Sculptor	$0.31$	$6.36$	$3.6\pm 2.0$
Draco	$0.24$	$5.45$	$11.1\pm 4.7$
Sextans	$0.75$	$5.64$	$8.5\pm 3.3$
Fornax	$0.79$	$7.31$	$7.1\pm 6.0$
DragonFly 44	$3.87$	$8.37$	$1.5\pm 0.4$

, we report some relevant properties of the galaxies considered in our analysis. Specifically, the first two columns list the circularized half-light radius of the projected surface brightness profile and the total luminosity, as determined from photometric observations (uncertainties are negligible for the purpose of this analysis); these quantities refer to the $3.6\mathsf{\mu }$m band for dwIrr galaxies and to the V-band for dwSphs. The third column lists the stellar mass-to-light-ratio expected from stellar population synthesis models [71,72,73]: for disk-dominated dwIrr galaxy values, $M_{★}/L\approx 0.5$ applies with few uncertainties at $3.6\mathsf{\mu }$m; for dwSphs, the $M_{★}/L$ values are estimated from the $V-I$ color index and thus are more dispersed and uncertain [63,69].

For our Bayesian analysis, we considered the parameter set $\theta \equiv (M_{★}/L,M_{200},s)$ for rotation-dominated dwarfs and $\theta \equiv (M_{★}/L,\beta ,M_{200},s)$ for dispersion-dominated ones. To estimate these parameters, we adopted a Bayesian framework and built the Gaussian log-likelihood,

$$log\mathcal{L}\left(\theta \right)=-{\chi }^2\left(\theta \right)/2,$$

(10)

where the chi-square ${\chi }^2\left(\theta \right)={\sum }_i{[\mathcal{M}(\theta ,r_i)-\mathcal{D}\left(r_i\right)]}^2/{\sigma }_{\mathcal{D}}^2\left(r_i\right)$ was obtained by comparing our empirical model expectations $\mathcal{M}(\theta ,r_i)$ to the data values $\mathcal{D}\left(r_i\right)$ with their uncertainties ${\sigma }_{\mathcal{D}}\left(r_i\right)$, summing over the different radial coordinates $r_i$ of the data.

We adopted flat priors $\pi \left(\theta \right)$ on $s\in [1,3/2]$ and on $log{M}_{200}\left[M_{\odot }\right]\in [6,13]$. Moreover, we assumed a lognormal prior on $log{M}_{★}/L$ with average and dispersion as expected from stellar population synthesis models (see Table 1). As to $\beta$, since by definition it varies in the range $(-\infty ,1]$, we actually preferred to perform inference on the symmetrized version ${\beta }_{\mathrm{sym}}\equiv \beta /(2-\beta )$ that mapped the original quantity in a compact domain ${\beta }_{\mathrm{sym}}\in (-1,1]$; a flat prior on ${\beta }_{\mathrm{sym}}$ within this range was used. Finally, to help robustly break any possible degeneracy between the halo and stellar masses, we followed [60] and added as a $\mathsf{\Lambda }$CDM prior the stellar mass vs. halo mass relation derived from multi-epoch abundance matching by [74], which was also consistent with independent observational determinations from satellite kinematics [75], rotation curve modeling [76], and weak lensing analysis [77].

We then sampled the parameter posterior distributions $\mathcal{P}\left(\theta \right)\propto \mathcal{L}\left(\theta \right)\pi \left(\theta \right)$ via the MCMC Python package emcee [78], running it with ${10}^4$ steps and 100 walkers; each walker was initialized with a random position extracted from the priors discussed above. To speed up convergence, we adopted a mixture of differential evolution and snooker moves of the walkers, in proportions of $0.8$ and $0.2$, respectively, which emulated a parallel tempering algorithm. After checking the auto-correlation time, we removed the first $30\%$ of the flattened chain, to ensure burn-in; the typical acceptance fractions of the various runs were around $30\%$.

3. Results

The results of our Bayesian analysis for rotation-dominated dwIrr galaxies are displayed in Figure 2 and Figure 3 and in

Table 2. Marginalized posterior estimates (mean and $1\sigma$ confidence intervals are reported) for the parameters from the MCMC analysis of the individual rotation-dominated dwIrr galaxies in fractional and Newtonian gravity (marked by $s=1$). The columns report the values of the stellar mass-to-light ratio $M_{★}/L$, of the DM mass $M_{200}$, of the fractional index s, of the reduced ${\chi }_r^2$ for the overall fit, and of the Bayesian inference criterion (BIC) for model comparison.

Galaxy	$log{\mathit{M}}_{\mathbf{★}}/\mathit{L}$ [${\mathit{M}}_{\odot }/{\mathit{L}}_{\odot }$]	$log{\mathit{M}}_{\mathbf{200}}$$\left[{\mathit{M}}_{\odot }\right]$	s	${\mathit{\chi }}_{\mathit{r}}^{\mathbf{2}}$	BIC
D631-7	$-0.39{}_{-0.07}^{+0.07}$	$10.59{}_{-0.06}^{+0.06}$	$>1.47$	$4.25$	63
	$-0.77{}_{-0.07}^{+0.07}$	$10.15{}_{-0.03}^{+0.03}$	1	$19.50$	279
DDO64	$-0.28{}_{-0.09}^{+0.09}$	$10.55{}_{-0.07}^{+0.07}$	$1.16{}_{-0.04}^{+0.04}$	$0.56$	14
	$-0.45{}_{-0.08}^{+0.08}$	$10.36{}_{-0.06}^{+0.06}$	1	$1.66$	25
DDO161	$-0.76{}_{-0.07}^{+0.07}$	$10.26{}_{-0.02}^{+0.03}$	$1.35{}_{-0.02}^{+0.02}$	$0.75$	31
	$-0.82{}_{-0.06}^{+0.06}$	$10.44{}_{-0.01}^{+0.01}$	1	$15.27$	450
UGC731	$-0.26{}_{-0.08}^{+0.08}$	$10.70{}_{-0.03}^{+0.03}$	$1.05{}_{-0.03}^{+0.02}$	$0.30$	12
	$-0.29{}_{-0.08}^{+0.08}$	$10.67{}_{-0.02}^{+0.02}$	1	$0.66$	14
UGC5005	$-0.44{}_{-0.08}^{+0.08}$	$11.00{}_{-0.06}^{+0.06}$	$1.29{}_{-0.06}^{+0.06}$	$0.64$	15
	$-0.47{}_{-0.08}^{+0.08}$	$10.98{}_{-0.05}^{+0.05}$	1	$3.13$	35
UGC5414	$-0.28{}_{-0.09}^{+0.09}$	$10.91{}_{-0.07}^{+0.07}$	$1.28{}_{-0.06}^{+0.04}$	$0.42$	8
	$-0.67{}_{-0.07}^{+0.07}$	$10.54{}_{-0.04}^{+0.04}$	1	$7.37$	34
UGC7608	$-0.25{}_{-0.09}^{+0.09}$	$10.68{}_{-0.07}^{+0.07}$	$1.08{}_{-0.06}^{+0.03}$	$1.26$	14
	$-0.29{}_{-0.09}^{+0.09}$	$10.63{}_{-0.06}^{+0.06}$	1	$1.35$	14
NGC3741	$-0.21{}_{-0.08}^{+0.08}$	$10.29{}_{-0.04}^{+0.04}$	$1.23{}_{-0.03}^{+0.03}$	$0.47$	20
	$-0.49{}_{-0.07}^{+0.07}$	$10.09{}_{-0.02}^{+0.02}$	1	$4.65$	97

. Specifically, in the top panel of Figure 2, we illustrate the MCMC posterior distributions for two representative dwIrr galaxies in the sample: namely, NGC3741 (the one with the best and most extended data) and DDO64. The red lines/contours refer to the outcomes for FG, and the green ones to the outcomes for Newtonian gravity; the white crosses mark the best fit value of the parameters in FG. In the bottom panel, the best fit (solid lines) and the $2\sigma$ credible intervals (shaded areas) sampled from the posteriors are shown. The solid line is for the total rotation velocity, while the dashed and dotted lines show the halo and disk contribution, respectively; for comparison, the Newtonian best fit to the total velocity is reported in green. In Figure 3, the best fits in FG and in the Newtonian case are illustrated for the other six dwIrr galaxies in the sample. In Table 2, we summarize the marginalized posterior estimates of the parameters, both in FG and in the standard Newtonian case (marked with $s=1$). The columns report the median values and the $1\sigma$ credible intervals of the stellar mass-to-light ratio $M_{★}/L$, of the DM mass $M_{200}$, and of the fractional index s, the reduced ${\chi }_r^2$ of the fit, and the Bayesian inference criterion (BIC) for model comparison. The BIC is defined as BIC $=2ln{\mathcal{L}}_{\max }+N_{\mathrm{par}}ln{N}_{\mathrm{data}}$, in terms of the maximum likelihood estimate ${\mathcal{L}}_{\max }$ of the number of parameters $N_{\mathrm{par}}$ and the number of data points $N_{\mathrm{data}}$. The BIC comes from approximating the Bayes factor, which gives the posterior odds of one model against another, presuming that the models are equally favored a priori. Note that what matters is the relative value of the BIC among different models; in particular, a difference of around 10 or more indicates evidence in favor of the model with the smaller value.

The FG fits were always excellent, and comparable to or better than those in Newtonian gravity. In particular, for D631-7, DDO64, DDO161, UGC5005, UGC5414, and NGC3741, there was a clear preference for FG, both in terms of ${\chi }_r^2$ and of the BIC. In such cases, the fractional index took on typical values $s\approx 1.2–1.3$, the stellar mass-to-light ratios $M_{★}/L$ were slightly larger than for the Newtonian case and more in line with the value around $0.5$ expected from stellar population synthesis models, and the DM masses $M_{200}$ were appreciably larger, by a few factors, than in the Newtonian fit. In the other cases, namely UGC731 and UGC7608, s was close to 1, the estimates of the fitting parameters in FG and in the Newtonian setting were consistent to within $1\sigma$, and the overall quality of the fits was comparable. We looked for some property of these two galaxies that could correlate with their smaller values of s, but were unable to reach a definite conclusion. Potentially interesting evidence came from the kinematics of their HI disks, which appeared asymmetric and disturbed by a past or ongoing gravitational interaction [79]; this could possibly have altered the shape of the outer rotation curve and originated a variant outcome when modeling it in the FG framework. An extended sample of a dwIrr galaxy with a high-quality rotation curve and environmental characterization would be needed, to investigate the issue further, in a statistically sound manner.

As can be seen in the top panel of Figure 2 for the representative cases of NGC3741 and DDO64, in FG there was no strong degeneracy in the fitting parameters, aside from a weak direct dependence between s and both $M_{★}/L$ and $M_{200}$. The bottom panel of the same Figure shows that for NGC3741 and DDO64 the halo component largely dominated the rotation curve, with the baryonic contribution being relevant only in the innermost region within a few $r_{1/2}$; this situation was shared by all the dwIrr galaxies in the analyzed sample. Therefore, the shape of the rotation curve strongly constrains the halo mass/density profile; in particular, the rising trend of the rotation velocity out to large radii is difficult to be reproduced with an NFW profile in standard gravity, while the task can be easily achieved in the FG framework.

The results for dispersion-dominated galaxies are displayed in Figure 4 and Figure 5 and in

Table 3. Marginalized posterior estimates (mean and $1\sigma$ confidence intervals are reported) for the parameters from the MCMC analysis of the individual dispersion-dominated dwarf spheroidal in fractional (first lines) and Newtonian (second lines, with $s=1$) gravity. The columns report the values of the stellar mass-to-light ratio $M_{★}/L$, of the symmetrized anisotropy parameter ${\beta }_{\mathrm{sym}}$, of the DM mass $M_{200}$, of the fractional index s, of the reduced ${\chi }_r^2$ for the overall fit, and of the Bayesian inference criterion (BIC) for model comparison.

Galaxy	$log{\mathit{M}}_{\mathbf{★}}/\mathit{L}$$[{\mathit{M}}_{\odot }/{\mathit{L}}_{\odot }]$	${\mathit{\beta }}_{\mathbf{sym}}$	$log{\mathit{M}}_{200}$$\left[{\mathit{M}}_{\odot }\right]$	s	${\mathit{\chi }}_{\mathit{r}}^2$	BIC
Carina	$+0.54{}_{-0.26}^{+0.26}$	$+0.19{}_{-0.11}^{+0.11}$	$9.77{}_{-0.12}^{+0.12}$	$1.37{}_{-0.05}^{+0.04}$	$1.67$	48
	$-0.75{}_{-0.16}^{+0.28}$	$-0.81{}_{-0.12}^{+0.013}$	$9.06{}_{-0.06}^{+0.13}$	1	$3.49$	87
Leo I	$+0.74{}_{-0.07}^{+0.12}$	$-0.08{}_{-0.16}^{+0.16}$	$10.35{}_{-0.07}^{+0.08}$	$1.37{}_{-0.06}^{+0.11}$	$0.64$	17
	$+0.19{}_{-0.12}^{+0.16}$	$-0.80{}_{-0.07}^{+0.05}$	$10.03{}_{-0.07}^{+0.09}$	1	$2.64$	39
Leo II	$-0.40{}_{-0.28}^{+0.28}$	$+0.37{}_{-0.25}^{+0.25}$	$9.50{}_{-0.13}^{+0.13}$	$1.11{}_{-0.05}^{+0.04}$	$0.37$	11
	$-0.74{}_{-0.24}^{+0.24}$	$+0.09{}_{-0.27}^{+0.27}$	$9.32{}_{-0.11}^{+0.11}$	1	$0.42$	9
Sculptor	$+0.60{}_{-0.19}^{+0.19}$	$-0.04{}_{-0.06}^{+0.06}$	$10.13{}_{-0.09}^{+0.09}$	$1.23{}_{-0.05}^{+0.03}$	$1.32$	54
	$-0.37{}_{-0.13}^{+0.13}$	$-0.32{}_{-0.06}^{+0.06}$	$9.51{}_{-0.06}^{+0.06}$	1	$1.65$	65
Draco	$+1.05{}_{-0.14}^{+0.14}$	$+0.01{}_{-0.17}^{+0.17}$	$9.93{}_{-0.09}^{+0.09}$	$1.13{}_{-0.03}^{+0.03}$	$0.27$	14
	$+0.75{}_{-0.12}^{+0.12}$	$-0.32{}_{-0.17}^{+0.24}$	$9.71{}_{-0.07}^{+0.07}$	1	$0.56$	15
Sextans	$+0.90{}_{-0.13}^{+0.13}$	$-0.34{}_{-0.05}^{+0.07}$	$9.94{}_{-0.08}^{+0.08}$	$>1.45$	$1.54$	24
	$+0.54{}_{-0.12}^{+0.12}$	$-0.89{}_{-0.07}^{+0.03}$	$9.62{}_{-0.07}^{+0.07}$	1	$3.07$	38
Fornax	$-0.36{}_{-0.12}^{+0.15}$	$+0.16{}_{-0.03}^{+0.03}$	$10.03{}_{-0.08}^{+0.1}$	$>1.49$	$1.30$	69
	$<-1.84$	$-0.16{}_{-0.04}^{+0.04}$	$9.08{}_{-0.03}^{+0.03}$	1	$1.15$	62
DragonFly 44	$+0.17{}_{-0.09}^{+0.09}$	$-0.12{}_{-0.07}^{+0.11}$	$10.79{}_{-0.07}^{+0.07}$	$>1.31$	$0.46$	12
	$+0.09{}_{-0.09}^{+0.09}$	$-0.48{}_{-0.08}^{+0.08}$	$10.69{}_{-0.06}^{+0.06}$	1	$1.27$	16

. Figure 4 focuses on the representative dwSph Sculptor and on the ultra-diffuse galaxy DragonFly 44. In the top panel of Figure 4, we illustrate the MCMC posterior distributions for Sculptor and DragonFly 44. As above, the red lines/contours refer to the outcomes for FG, the green ones to the outcomes for Newtonian gravity, with the white crosses marking the best fit value of the parameters in FG. In the bottom panel, the best fit (solid lines) and the $2\sigma$ credible intervals (shaded areas) sampled from the posteriors are shown, with the reference Newtonian fit in green. In Figure 5, the fits in FG and in the Newtonian case are illustrated for the other six dwSphs in the sample. In Table 3, we summarize the marginalized posterior estimates of the parameters for the dwSphs, both in FG and in the standard Newtonian case (marked with $s=1$). The columns report the median values and the $1\sigma$ credible intervals of the stellar mass-to-light ratio $M_{★}/L$, of the symmetrized anisotropy parameter ${\beta }_{\mathrm{sym}}$, of the DM mass $M_{200}$, and of the fractional index s; the reduced ${\chi }_r^2$ of the fit and the BIC are also reported.

The FG fits to dispersion-dominated galaxies were very good, and in several instances appreciably better than in Newtonian gravity. In particular, for Carina, Leo I, Sculptor, and Sextans there was a clear preference for FG, both in terms of ${\chi }_r^2$ and in terms of the BIC. In such cases, the fractional index took on typical values $s\gtrsim 1.2$, the stellar mass-to-light ratios $M_{★}/L$ were appreciably larger than for the Newtonian case and more in line with the prior from stellar population synthesis models, and the DM masses $M_{200}$ were substantially larger by several factors, with respect to the Newtonian fits. In other cases, namely Leo II and Draco, the index $s\lesssim 1.1$ was smaller, the estimates of the fitting parameters in FG and in the Newtonian setting were consistent with $3\sigma$, and the overall quality of the fits was comparable. Finally, in the cases of Fornax and DragonFly 44 there was a clear preference for large values of $s\approx 1.5$, but the improvement in the FG fits with respect to the Newtonian case, while clear on a visual inspection, was not statistically significant enough to make definite conclusions.

As can be seen in the top panel of Figure 4 for Sculptor and DragonFly 44, the most relevant degeneracy between the fitting parameters involves s and the ${\beta }_{\mathrm{sym}}$, in such a way that FG models with larger s tend to be more isotropic. In fact, for Sculptor and DragonFly 44 the FG fit showed preference for almost isotropic orbits, while the Newtonian fit favored tangentially dominated motions. The bottom panel of the same Figure illustrates visually the quality of the FG fit for Sculptor and DragonFly 44, which was excellent within the scatter of the datapoints. FG performed definitively better than the Newtonian case for Sculptor, while for DragonFly 44 the evidence was rendered barely significant, in terms of reduced ${\chi }_r^2$ and of the BIC. In this respect, however, it is also interesting to look at the inset, where the excess kurtosis $\Delta \kappa$ is illustrated (the kurtosis is related to the fourth velocity moments of the stellar tracers, and the excess is with respect to the value 3 for a reference Gaussian velocity distribution). Although the measured value was largely uncertain, there was clearly a tendency towards a definite positive $\Delta \kappa$; qualitatively, this was consistent with the FG result at $2\sigma$, while being highly discordant (more than $3\sigma$) with the Newtonian fit. It is worth mentioning that the estimated $M_{200}\lesssim {10}^{11}{M}_{\odot }$ from our analysis in FG is consistent with the recent determination from the abundance of globular clusters in DragonFly 44 by [80].

By inspecting Table 2 and Table 3, overall one can conclude that the evidence in favor of FG is less compelling in the dwSphs than in the dwIrr galaxies. This is for several reasons. First, the main observable for the dwIrr galaxies was the rotation velocity, which was a direct probe of the mass profile; contrariwise, in the dwSphs, the l.o.s. dispersion profile encased the mass profile in an integrated way, weighted by a kernel that depended on the tracer profiles and on the anisotropy parameter. In addition, the priors on the stellar mass-to-light ratio from the population synthesis model were looser for the dwSphs than for the dwIrr galaxies (especially when considering $3.6\mathsf{\mu }$m luminosities for the latter). Finally, the observed l.o.s. dispersion profiles were more scattered and featureless, with respect to the rotation curves. Thus, it should not be surprising that the constraints from the dwSphs were less statistically significant. Nevertheless, these systems may offer an environment where any evidence in favor of FG is more robust, as the lack of baryons even in the innermost regions does not allow us to rely on different interpretations related to baryonic-induced modification of the DM profile.

Finally, in Figure 6 we illustrate two interesting scaling relations that constitute relevant crosschecks of our results. The diagram on the left panel displays the fractional index s as a function of the DM mass $M_{200}$. Apart from a few exceptions (objects with $s\lesssim 1.1$), the values from our analysis of individual dwarf galaxies in FG are consistent to within $2-3\sigma$ with the expectation from the relationship by [45] that was derived from fitting stacked rotation curves of rotation-dominated galaxies.

The diagram on the right reports the radial acceleration relation or RAR [14,15] between the total acceleration $g_{\mathrm{tot}}$ and the baryonic one $g_{\mathrm{bar}}$. This is an empirical relationship known to hold for different kind of galaxies, whose average and 1–2–3$\sigma$ dispersion is plotted as an orange line surrounded by shaded areas. For rotationally supported systems, we computed $g_{\mathrm{bar}}=[v_{\mathrm{gas}}^2\left(r_{1/2}\right)+(M_{★}/L)\times v_{\mathrm{disk}}^2\left(r_{1/2}\right)]/r_{1/2}$ and $g_{\mathrm{tot}}=v_{\mathrm{tot}}^2\left(r_{1/2}\right)/r_{1/2}$, in terms of Equation (6). For dispersion-dominated systems, we instead estimated $g_{\mathrm{bar}}=G(M_{★}/L)\times L_{★}/2r_{1/2}^2$ and $g_{\mathrm{tot}}=3{\sigma }_r^2\left(r_{1/2}\right)/r_{1/2}$, in terms of Equation (9). For the sake of simplicity, we computed the accelerations at $r_{1/2}$ by using the best fit values of $M_{★}/L$, $M_{200}$, and s from our analysis in FG. Reassuringly, almost all our estimated accelerations were consistent within $2-3\sigma$ with the RAR by [15], with the dwIrr galaxies clustering around the value of $g_{\mathrm{bar}}$ where the relation started to flatten, and with some dwSphs tracing the flat portion of the RAR and its scatter.

4. Summary

Dark matter (DM) in fractional gravity (FG) constitutes a framework that strikes an intermediate course between a modified gravity theory and an exotic DM scenario. It envisages the DM component in virialized cosmic structures to be affected by a non-local interaction mediated by gravity. Specifically, in such a framework the gravitational potential associated with a given DM density distribution is determined by a modified Poisson equation, including fractional derivatives, that are aimed at describing non-locality.

Remarkably, FG can be reformulated in terms of the standard Poisson equation, but with an effective density distribution that is flatter in the inner region, with respect to the true one. Therefore, FG offers a straightforward solution to the core–cusp problem of the standard $\mathsf{\Lambda }$CDM model without altering the NFW density profile indicated by N-body simulations. An observer trying to interpret the kinematic data (e.g., rotation curves in dwIrr galaxies) in terms of the canonical (instead of the fractional) Poisson equation would claim the need for a cored density distribution. However, this is only apparent, as in FG the cuspy NFW density profile of $\mathsf{\Lambda }$CDM originates a dynamic very similar to a cored profile in the standard Newtonian setting. In previous works [45,46], we tested our FG framework by exploiting stacked rotation curves of galaxies with different masses and joint X-ray/Sunyaev-Zel’dovich observations of galaxy clusters; our analysis highlighted that the strengths of FG effects tend to weaken toward more massive systems, so implying that dwarf galaxies constitute the best environment to constrain such a scenario.

In this paper, we investigated further, to probe FG via the high-quality data of individual dwarf galaxies, by exploiting the rotation velocity profiles inferred from stellar and gas kinematic measurements in eight dwarf irregulars, and the projected velocity dispersion profiles inferred from the observed dynamics of stellar tracers in seven dwarf spheroidals and in the ultra-diffuse galaxy DragonFly 44. We found that FG reproduces extremely well the rotation and dispersion curves of all the analyzed galaxies, performing in most instances significantly better than the standard Newtonian gravity. With respect to the latter, the FG fits imply slightly larger stellar mass-to-light ratios $M_{★}/L$ (more in line with the values expected from galaxy colors and stellar population synthesis models), appreciably larger DM masses $M_{200}$ by a few factors, and (for dispersion-dominated systems) more isotropic orbits. We stressed that our best fit determinations of the fractional index s and of the DM masses $M_{200}$ from the kinematics of individual dwarf galaxies are consistent to within 2–3$\sigma$ with the relationship by [45] that was derived from fitting stacked rotation curves of rotation-dominated galaxies. We also highlighted that our findings are consistent with the radial acceleration relation by [15].

We pointed out that the evidence in favor of FG was less compelling for dwSphs than for dwIrr galaxies. This was for various reasons: the l.o.s. velocity dispersion was less sensitive than the rotation velocity to the mass profile; the uncertainties of the stellar mass-to-light ratio from stellar models (used as priors) were larger for the dwSphs than for the dwIrr galaxies; the uncertainty in dispersion profile measurements was larger than in the rotation curve data. However, it should be considered that dwSphs and ultra-diffuse galaxies could potentially provide a more robust environment to test FG, as they are strongly DM-dominated also in the innermost regions, and thus should not suffer from baryonic feedback processes or baryon-induced modification of the density profile. Future observations by astrometric space missions aimed at precision determination of the dispersion profiles in dwSphs and ultra-diffuse galaxies will be extremely helpful, to robustly strengthen the constraints on the FG framework presented here.

This work concludes a series of papers aimed at testing FG on different astrophysical scales, from dwarf galaxies to galaxy clusters. Overall, these have demonstrated that the FG framework can solve the small-scale issues of the standard $\mathsf{\Lambda }$CDM, by reconciling with data the DM density distribution expected from N-body simulations, and saving its successes on large cosmological scales. Our future efforts will be directed at explaining the physical origin of the nonlocal effects subtended by the FG framework, and at investigating to what extent the theory can be generalized in a fully relativistic setting.