Dark Matter in Fractional Gravity III: Dwarf Galaxies Kinematics

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 $\Lambda$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 dig deeper to probe FG via high-quality data of individual dwarf galaxies, by exploiting the rotation velocity profiles inferred from stellar and gas kinematic measurements in $8$ dwarf irregulars, and the projected velocity dispersion profiles inferred from the observed dynamics of stellar tracers in $7$ 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 analysed galaxies, performing in most instances significantly better than the standard Newtonian setup.


Introduction
The standard ΛCDM 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 from N−body simulations to follow an approximately universal shape, well described by the classic Navarro-Frenk-White [3] profile ρ ∝ (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 ≡ R 200 /r s , with R 200 being the radius where the average DM density is 200 times that of a critical Universe ρ crit .In fact, recent zoom-in N−body simulations [5] have demonstrated that c 200 correlates very well with the mass M 200 ≡ (4π/3) 200 ρ crit R 3 200 over the astonishingly extended range from M 200 ∼ 10 −5 M ⊙ to 10 15 M ⊙ .
Although on large scales observational data undoubtedly confirm the above picture, in the realm of dwarf galaxies with total masses ≲ 10 11 M ⊙ 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], 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 [30,32].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], 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 nonlocal interaction mediated by gravity.Specifically, in such a framework the gravitational potential associated to 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 which 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, since the cuspy NFW density profile of ΛCDM originates in FG a dynamics very similar to a cored profile in the standard Newtonian setting.
In [45,46] we tested FG over an extended mass range M 200 ∼ 10 9 − 10 15 M ⊙ by ex- ploiting 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, so implying that FG can substantially alleviate the small-scale issues of the standard ΛCDM paradigm, while retaining its successes on large cosmological scales.
In this paper we aim at digging deeper into 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 probe 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 (dwSph) by exploiting the velocity dispersion profile inferred from the dynamics of stellar tracers.
The structure of the paper is the following: 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 ΛCDM cosmology with rounded parameter values [50]: matter density Ω m ≈ 0.3, baryon density Ω b ≈ 0.05, Hubble constant H 0 = 100 h km s −1 Mpc −1 with h ≈ 0.7.

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.

Dark Matter in Fractional Gravity
N-body simulations in the standard ΛCDM cosmology indicate that virialized halos of DM particles follow an approximately universal density profile, routinely described via the Navarro-Frenk-White [3] shape ρ(r) = ρ s r 3 s /r (r + r s ) 2 in terms of a scale radius r s and of a characteristic density ρ s .
In the standard Newtonian theory, the potential Φ N (r) associated to a given density distribution ρ(r) can be computed from the Poisson equation: where ∆ 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 with it is easy to verify that |dΦ N /dr| = G M(< r)/r 2 , as a direct consequence of Birkhoff's theorem.
In the FG framework, the potential Φ F (r) is instead derived from the modified Poisson equation [40,45] (−∆) s Φ F (r) = −4πG ℓ 2−2s ρ(r) (3) where (−∆) s is the fractional Laplacian operator (see the excellent textbook [51] for details), s ∈ [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 ℓ 2−2s .Therefore in the following, without loss of generality, we will set it to ℓ ≈ r s that 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 fractionalize; this is a procedure often followed in the mathematical-physics literature [52][53][54] to insert fractional dynamics in a system avoiding to add a dimensional parameter of ambiguous interpretation and problematic estimation.
with Γ(s) = ∞ 0 dx x s−1 e −x being the Euler Gamma function and 2 F 1 (a, b, c; x) = ∑ ∞ k=0 (a) k (b) k x k /(c) k k! being the ordinary hypergeometric function in terms of the Pochammer symbols (q) k defined as (q) 0 = 1 and (q) k = q (q + 1) . . .(q + k − 1); plainly, Φ F (r) for s = 1 coincides with the usual expression Φ N (r) of Equation (2).For the limiting case s = 3/2, the computation requires some principal-value regularization and the solution reads where Li 2 (x) = ∑ ∞ k=1 x k /k 2 is the dilogarithm function.Being a nonlocal framework, in FG the Birkhoff theorem does not hold, but one can insist in writing |dΦ F /dr| = 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 ρ F (r) = (1/4π r 2 ) × dM F /dr.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 depends on the NFW scale radius r s and density ρ s or equivalently the mass M s = 4π ρ s r 3 s ; however, in the following, it is convenient to trade off these quantities for the mass M 200 and the concentration c 200 ≡ R 200 /r s at the reference radius R 200 where the average density is 200 times the critical density ρ 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 ΛCDM cosmology derived from zoom-in N−body simulations by [5] and spanning twenty orders of magnitude in DM mass within the range M 200 ∼ 10 −5 − 10 15 M ⊙ .

Dynamical modeling
For a rotation-dominated system, the crucial quantity to be compared with the data is the total rotation velocity, which is given by where v 2 halo = G M F (< r)/r is the contribution from the DM halo, v 2 gas is the gas contribution from HI measurements, and v 2 disk 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] where β ≡ 1 − σ 2 θ /σ 2 r is the anisotropy parameter (hereafter assumed constant with radius) and is the surface mass density of the tracers (e.g., stars) in terms of the volume one ρ ⋆ (r); in addition, the radial velocity dispersion is obtained by solving the Jeans equation, which yields in terms of the tracer mass Typically, for the dispersiondominated galaxies considered in this work, stellar tracers are exploited, with a density distribution following the Plummer's model 1 [57].In such a case ρ ⋆ (r , with r 1/2 the 2D half-light radius; the normalization is derived from the observed surface luminosity profile via a global mass-to-light ratio M ⋆ /L.The overall dispersion velocity depends on four parameters, i.e., the mass-to-light ratio M ⋆ /L, the anisotropy parameter β, the mass of the system M 200 and the fractional index s.

Data and Bayesian Analysis
We probe the FG framework by exploiting the rotation velocity profiles of dwIrr galaxies and the l.o.s.dispersion velocity profiles of dwSph.
For rotation-dominated systems, we rely the SPARC database [58,59], which provides the stellar and gas contribution to the rotation velocity as found by numerically solving 1 For DragonFly 44 we actually exploit a Sersic surface density profile, as detailed in [63].
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).Top half of the table includes rotation-dominated dwarfs, for which the listed quantities refer to the luminosity at 3.6 µm; bottom half of the Table includes dispersion-dominated dwarfs, for which the listed quantities refer to the V−band luminosity.the standard Poisson equation for the observed surface brigthness profile at 3.6 µm (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 consider 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, 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 consider 7 Milky-Way dwSph for which high quality determination of the spatially-resolved dispersion velocity has been obtained via stellar tracers ( [60,61]; see also data collection by [62] and references therein) and for which tidal effects are not appreciably influencing the inner kinematics: Carina, Leo I, Leo II, Sculptor, Draco, Sextans, and Fornax.To these we add the dispersion-dominated ultradiffuse galaxy DragonFly 44 [63].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 old stellar population in passive evolution [63,68].DragonFly 44 belongs to this last category, and being dispersion-dominated has been treated here along with the dwSph sample.In addition, it is a particularly interesting object since 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 an useful laboratory to test the nature of DM and gravity [69,70].
In Table 1 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 µm band for dwIrr and to the V−band for dwSph.The third column lists the stellar mass-to-light-ratio expected from stellar population synthesis models [71][72][73]: for disk-dominated dwIrr values M ⋆ /L ≈ 0.5 applies with little uncertainties at 3.6 µm; for dwSph the M ⋆ /L values are estimated from the V − I color index and thus are more dispersed and uncertain [62,69].
For our Bayesian analysis, we consider the parameter set θ ≡ (M ⋆ /L, M 200 , s) for rotation-dominated dwarfs and θ ≡ (M ⋆ /L, β, M 200 , s) for dispersion-dominated ones.To estimate these parameters, we adopted a Bayesian framework and built the Gaussian log-likelihood log L(θ where the chi-square ) is obtained by comparing our empirical model expectations M(θ, r i ) to the data values D(r i ) with their uncertainties σ D (r i ), summing over the different radial coordinates r i of the data.
We adopt flat priors π(θ) on s ∈ [1, 3/2] and on log M 200 [M ⊙ ] ∈ [6,13].Moreover, we assume a lognormal prior on log M ⋆ /L with average and dispersion as expected from stellar population synthesis models (see Table 1).As to β, since by definition it varies in the range (−∞, 1], we actually prefer to perform inference on the symmetrized version β sym ≡ β/(2 − β) that maps the original quantity in a compact domain β sym ∈ (−1, 1]; a flat prior on β sym within this range is used.Finally, to help robustly break any possible degeneracy between the halo and stellar masses, we follow [59] and add as a ΛCDM prior the stellar mass vs. halo mass relation derived from multi-epoch abundance matching by [74], which is also consistent with independent observational determinations from satellite kinematics [75], rotation curve modeling [76], and weak lensing analysis [77]. We then sample the parameter posterior distributions P (θ) ∝ L(θ) π(θ) via the MCMC Python package emcee [78], running it with 10 4 steps and 100 walkers; each walker is initialized with a random position extracted from the priors discussed above.To speed up convergence, we adopt a mixture of differential evolution and snooker moves of the walkers, in proportion of 0.8 and 0.2 respectively, that emulates a parallel tempering algorithm.After checking the auto-correlation time, we remove the first 30% of the flattened chain to ensure burn-in; the typical acceptance fractions of the various runs are around 30%.

Results
The results of our Bayesian analysis for rotation-dominated dwIrr galaxies are displayed in Figures 2, 3 and in Table 2. Specifically, in the top panel of Figure 2 we illustrate the MCMC posterior distributions for two representative dwIrr in the sample, namely NGC3741 (the one with the best and most extended data) and DDO64.Red lines/contours refer to the outcomes for FG, and green ones 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 2σ 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 bestfit to the total velocity is reported in green.In Figure 3 the bestfits in FG and in the Newtonian case are illustrated for the other 6 dwIrr 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).Columns report the median values and the 1σ 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 χ2 r of the fit, and the Bayesian inference criterion (BIC) for model comparison 2 are also reported.
The FG fits are always excellent, and comparable or better then those in Newtonian gravity.In particular, for D631-7, DDO64, DDO161, UGC5005, UGC5414, and NGC3741 there is a clear preference for FG both in terms of χ 2 r and of the BIC.In such cases, the fractional index takes on typical values s ≈ 1.2 − 1.3, the stellar mass-to-light ratios M ⋆ /L are 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 Table 2. Marginalized posterior estimates (mean and 1σ confidence intervals are reported) for the parameters from the MCMC analysis of the individual rotation-dominated dwIrr in fractional and Newtonian gravity (marked by s = 1).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 χ 2 r for the overall fit, and of the Bayesian inference criterion (BIC) for model comparison.are appreciably larger of factors a few than in the Newtonian fit.In the other cases, namely UGC731 and UGC7608, s is close to 1, the estimates of the fitting parameters in FG and in the Newtonian setting are consistent to within 1σ, and the overall quality of the fits are comparable.We have 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.Maybe an interesting evidence comes from the kinematics of their HI disks, which appear asymmetric and disturbed by a past or ongoing gravitational interaction [79]; this could possibly alters the shape of the outer rotation curve and originate a variant outcome when modeling it in the FG framework.An extended sample of dwIrr with high quality rotation curve and environmental characterization would be needed to investigate further the issue in a statistically sound manner.

Galaxy log M
As it can be seen in the top panel of Figure 2 for the representative cases of NGC3741 and DDO64, in FG there is no strong degeneracy in the fitting parameters, besides 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 dominates the rotation curve, with the baryonic contribution being relevant only in the innermost region within a few r 1/2 ; this situation is shared by all dwIrr 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 a NFW profile in standard gravity, while the task can be easily achieved in the FG framework.Colored contours/lines refer to the standard Newtonian (green) and to the FG framework (red).The contours show 1 − 2 − 3σ confidence intervals, with the bestfit values in FG identified by white crosses.The marginalized distributions are in arbitrary units (normalized to 1 at their maximum value).Bottom panel: Fits to the rotation curve with the Newtonian (green) and the FG (red) framework for the galaxy NGC3741 (left) and DDO64 (right).Solid lines refer to the total rotation velocity, while (for clarity only in FG) the dashed line highlights the halo contribution and the dotted line the baryonic one.
Solid lines illustrate the median, and the shaded areas show the 2σ credible interval from sampling the posterior distribution.The value of the reduced χ 2 r of the fit for FG is also reported.Circles represent data from the SPARC database [58] for the total rotation curve, while the contributions from the stellar (for M ⋆ /L = 1) and gaseous disk are highlighted by the starred and squared symbols, respectively.The results for dispersion-dominated galaxies are displayed in Figures 5 and in Table 3. 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, red lines/contours refer to the outcomes for FG, and green ones 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 2σ 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 6 dwSph in the sample.In Table 3 we summarize the marginalized posterior estimates of the parameters for dwSph, both in FG and in the standard Newtonian case Table 3. Marginalized posterior estimates (mean and 1σ 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.Columns report the values of the stellar mass-to-light ratio M ⋆ /L, of the symmetrized anisotropy parameter β sym , of the DM mass M 200 , of the fractional index s, of the reduced χ 2 r for the overall fit, and of the Bayesian inference criterion (BIC) for model comparison.(marked with s = 1).Columns report the median values and the 1σ credible intervals of the stellar mass-to-light ratio M ⋆ /L, of the symmetrized anisotropy parameter β sym , of the DM mass M 200 , and of the fractional index s; the reduced χ 2 r of the fit, and the BIC are also reported.

Galaxy
The FG fits to dispersion-dominated galaxies are very good, and in several instances appreciably better than in Newtonian gravity.In particular, for Carina, Leo I, Sculptor, and Sextans there is a clear preference for FG both in terms of χ 2 r and in terms of the BIC.In such cases, the fractional index takes on typical values s ≳ 1.2, the stellar mass-to-light ratios M ⋆ /L are 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 are substantially larger of factors several with respect to the Newtonian fits.In other cases, namely Leo II and Draco, the index s ≲ 1.1 is smaller, the estimates of the fitting parameters in FG and in the Newtonian setting are consistent to with 3σ, and the overall quality of the fits are comparable.Finally, in the cases of Fornax and DragonFly 44 there is a clear preference for large values of s ≈ 1.5, but the improvement in the FG fits with respect to the Newtonian case, albeit clear at a visual inspection, is not statistically significant enough to make definite conclusions.r of the fit for FG is also reported.Circles represent data from [60] for Sculptor and from [63] for DragonFly 44.As it 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 β 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 shows preference for almost isotropic orbits, while the Newtonian fit favors tangentially-dominated motions.The bottom panel of the same Figure illustrates visually the quality of the FG fit for Sculptor and DragonFly 44, which is excellent within the scatter of the datapoints.FG performs definitely better than the Newtonian case for Sculptor, while for DragonFly 44 the evidence is made barely significant in terms of reduced χ 2 r and of the BIC.In this respect, however, it is also interesting to look at the inset where the excess kurtosis ∆κ (the kurtosis is related to the fourth velocity moments of the stellar tracers, and the excess is respect to the value 3 for a reference Gaussian velocity distribution) is illustrated.Although the measured value is largely uncertain, there is clear a tendency for a definite positive ∆κ; qualitatively, this is consistent with the FG result at 2σ, while being highly discordant (more than 3σ) with the Newtonian fit.It is worth mentioning that the estimated M 200 ≲ 10 11 M ⊙ from our analysis in FG is consistent with the recent determination from the abundance of globular clusters in DragonFly 44 by [80].Blue circles refer to dwIrr and red ones to dwSph; the cyan line and shaded areas illustrate the bestfit relation and 1 − 2 − 3σ dispersion from the analysis of stacked rotation curves for disc-dominated galaxies by [45].Right: the Radial Acceleration Relation.Blue symbols refer to dwIrr and red ones to dwSph; the orange line and shaded areas illustrate the bestfit relation and 1 − 2 − 3σ dispersion from the determination by [15].
By inspecting Tables 2 and 3 overall one can conclude that the evidence in favor of FG is less compelling in dwSph with respect to dwIrr.This is due to several reasons.First, the main observable for dwIrr is the rotation velocity, which is a direct probe of the mass profile; contrariwise, in dwSph the l.o.s.dispersion profile encases the mass profile in an integrated way, weighted by a kernel that depends on the tracer profiles and on the anisotropy parameter.In addition, the priors on the stellar mass-to-light ratio from population synthesis model are looser for dwSph than for dwIrr (especially when considering 3.6 µm luminosities for the latter).Finally, the observed l.o.s.dispersion profiles are more scattered and featureless with respect to the rotation curves.Thus it should not be surprising that the constraints from dwSph are less statistically significant.Nevertheless, these systems may offer an environment where any evidence in favor of FG is more robust, since 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 Fig. 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 for a few exceptions (objects with s ≲ 1.1), the values from our analysis of individual dwarf galaxies in FG are consistent to within 2 − 3σ with the expectation from the relationship by [45], that has been 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 tot and the baryonic one g bar .This is an empirical relationship known to hold for different kind of galaxies, whose average and 1 − 2 − 3σ dispersion is plotted as an orange line surrounded by shaded areas.For rotationally supported systems we compute g bar = [v 2 gas (r 1/2 ) + (M ⋆ /L) × v 2 disk (r 1/2 )]/r 1/2 and g tot = v 2 tot (r 1/2 )/r 1/2 in terms of Equation (6).For dispersion-dominated systems we instead estimate g bar = G (M ⋆ /L) × L ⋆ /2 r 2 1/2 and g tot = 3 σ 2 r (r 1/2 )/r 1/2 in terms of Equation (9).For the sake of simplicity we compute the accelerations at r 1/2 , by using the bestfit values of M ⋆ /L, M 200 and s from our analysis in FG.Reassuringly, almost all our estimated accelerations are consistent within 2 − 3σ with the RAR by [15], with the dwIrr galaxies clustering around the value of g bar where the relation starts to flatten, and with some dwSph tracing the flat portion of the RAR and its scatter.

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 to 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 which 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 ΛCDM 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) 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, since in FG the cuspy NFW density profile of ΛCDM originates a dynamics 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 have dug deeper to probe FG via high-quality data of individual dwarf galaxies, by exploiting the rotation velocity profiles inferred from stellar and gas kinematic measurements in 8 dwarf irregulars, and the projected velocity dispersion profiles inferred from the observed dynamics of stellar tracers in 7 dwarf spheroidals and in the ultra-diffuse galaxy DragonFly 44.We have found that FG reproduces extremely well the rotation and dispersion curves of all the analysed 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 of a factor a few, and (for dispersion-dominated systems) more isotropic orbits.We have stressed that our bestfit 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 − 3σ with the relationship by [45], that has been derived from fitting stacked rotation curves of rotation-dominated galaxies.We have also highlighted that our findings are consistent with the Radial Acceleration Relation by [15].
We have pointed out that the evidence in favor of FG is less compelling in dwSph with respect to dwIrr; this is because various reasons: the l.o.s.velocity dispersion is less sensitive than the rotation velocity to the mass profile; uncertainties on the stellar massto-light ratio from stellar models (used as priors) are larger for dwSph than for dwIrr; the uncertainty in dispersion profile measurements are larger than in the rotation curve data.However, it should be considered that dwSph and ultra-diffuse galaxies could potentially provide a more robust environment to test FG, since they are strongly DM dominated also in the innermost regions, and thus should not have suffered from baryonic feedback processes or baryon-induced modification of the density profile.Future observations by astrometric space mission aimed at precision determination of the dispersion profiles in dwSph 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.All in all, these have demonstrated that the FG framework works can solve the small scales issues of the standard ΛCDM, 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 to explain the physical origin of the nonlocal effects subtended by the FG framework, and investigate to what extent the theory can be generalized in a fully relativistic setting.

Figure 1 .
Figure 1.Radial profiles of effective mass (middle) and density (right) in the FG framework for different values of the fractional index s (color-coded); for reference, the dotted lines refer to the maximal value s = 3/2.

Figure 2 .
Figure 2. Top panel: MCMC posterior distributions of the stellar mass-to-light ratio M ⋆ /L, the DM mass M 200 and the fractional index s for the galaxy NGC3741 (left) and DDO64 (right).Colored contours/lines refer to the standard Newtonian (green) and to the FG framework (red).The contours show 1 − 2 − 3σ confidence intervals, with the bestfit values in FG identified by white crosses.The marginalized distributions are in arbitrary units (normalized to 1 at their maximum value).Bottom panel: Fits to the rotation curve with the Newtonian (green) and the FG (red) framework for the galaxy NGC3741 (left) and DDO64 (right).Solid lines refer to the total rotation velocity, while (for clarity only in FG) the dashed line highlights the halo contribution and the dotted line the baryonic one.Solid lines illustrate the median, and the shaded areas show the 2σ credible interval from sampling the posterior distribution.The value of the reduced χ 2 r of the fit for FG is also reported.Circles represent data from the SPARC database[58] for the total rotation curve, while the contributions from the stellar (for M ⋆ /L = 1) and gaseous disk are highlighted by the starred and squared symbols, respectively.

Figure 3 .
Figure 3. Same as bottom panel in the previous figures for other 6 dwIrr galaxies, as labeled.For clarity, data and models only for the total rotation curves are shown.

Figure 4 .
Figure 4. Top panel: MCMC posterior distributions of the mass-to-light ratio M ⋆ /L, the symmetrized anisotropy parameter β sym , the mass M 200 and the fractional index s for the dwSph galaxy Sculptor (left) and the ultra-diffuse galaxy DragonFly 44 (right).Colored contours/lines refer to the standard Newtonian (green) and to the FG framework (red).The contours show 1 − 2 − 3σ confidence intervals, with the bestfit values in FG identified by white crosses.The marginalized distributions are in arbitrary units (normalized to 1 at their maximum value).Bottom panel: Fits to the l.o.s.dispersion profile with the Newtonian (green) and the FG (red) framework for the dwSph galaxy Sculptor (left) and DragonFly 44 (right).The inset on the right bottom panel illustrates the excess kurtosis ∆κ with respect to a Gaussian velocity distribution.Solid lines illustrate the median, and the shaded areas show the 2σ credible interval from sampling the posterior distribution.The value of the reduced χ 2r of the fit for FG is also reported.Circles represent data from[60] for Sculptor and from[63] for DragonFly 44.

Figure 5 .
Figure 5. Same as bottom panel in the previous figure for 6 dwSph galaxies, as labeled.

Figure 6 .
Figure 6.Left: fractional index as a function of DM mass from the outcomes of our Bayesian analysis.Blue circles refer to dwIrr and red ones to dwSph; the cyan line and shaded areas illustrate the bestfit relation and 1 − 2 − 3σ dispersion from the analysis of stacked rotation curves for disc-dominated galaxies by[45].Right: the Radial Acceleration Relation.Blue symbols refer to dwIrr and red ones to dwSph; the orange line and shaded areas illustrate the bestfit relation and 1 − 2 − 3σ dispersion from the determination by[15].