Modified Gravity with Galactic dynamics

The lack of detection of supersymmetric particles is leading to look at alternative avenues for explaining dark matter's effects. Among them, modified theories of gravity may play an important role accounting even for both dark components needed in the standard cosmological model.

Scalar-tensor-vector gravity theory has been proposed to resolve the dark matter puzzle. Such a modified gravity model introduces, in its weak field limit, a Yukawa-like correction to the Newtonian potential, and is capable to explain most of the phenomenology related to dark matter at scale of galaxies and galaxy clusters. Nevertheless, some inconsistencies appear when studying systems that are supposed to be dark matter dominated such as dwarf galaxies. In this sense, Antlia II, an extremely diffuse galaxy which has been recently discovered in Gaia's second data release, may serve to probe the aforementioned theory against the need for invoking dark matter. Our analysis shows several inconsistencies and leads to argue that MOdified Gravity may not be able to shed light on the intriguing nature of dark matter [1].

In the case of the Model A, we are able to fit the dispersion velocity profile emulating, in the framework of MOG, the extended dark matter core needed in standard gravity. However, our best fit value of the two MOG's parameters differs from the values found using spiral galaixes at more than 3 and 1.5σ for  α and µ, respectively, indicating strong difficulties of MOG to self-consistently account for the dynamics of stars in dark matter dominated systems such as dwarf galaxies.  Second, the best fit value of the anisotropic parameter β differs from zero (isotropy) at more than 1.5σ, and it is strongly negative. This results is rather controversial. It means that to fit the dispersion velocity profile we need the tangential velocity to be much higher then the radial component, against the obervational insights hypothesis. Nevertheless, having such a tangential bias would requires that this dwarf galaxy is populated by mainly stars in quasi-circular orbits, losing ergodicity and, hence,lowering the dispersion velocity profile toward the center, against the central observed flatness of the dispersion velocity.  In the model B, We set β=0, forcing the model to be isotropic, but allowing the Plummer scale to vary. First, the results still show a discrepancy of  α and µ with the values obtained using spiral galaxies, but this is reduced to 1σ level in both parameters, while the huge tension with other independent analysis of dwarfs is confirmed. Second, the value of the Plummer scale is at 1.5σ  from the observed one and, hence,  changing the enclosed mass. Finally, and most importantly, we are not able to well match the dispersion velocity profile. As shown in the bottom panel of the Figure, under those assumptions, we never emulate the effect of a dark matter core. If the dispersion velocity profile get flatter below 0.5 kpc then it decrease quickly at outermost radii. On the contrary, when the outer data points are fitted, the dispersion velocity is more cuspy in the innermost region till reaching about 25 km/s, which is a factor 5 times higher than the measured values. 

We use the kinematic data of the stars in eight dwarf spheroidal galaxies to assess whether f(R) gravity can fit the observed profiles of the line-of-sight velocity dispersion of these systems without resorting to dark matter. Our model assumes that each galaxy is spherically symmetric and has a constant velocity anisotropy parameter β and constant mass-to-light ratio consistent with stellar population synthesis models. We solve the spherical Jeans equation that includes the Yukawa-like gravitational potential appearing in the weak field limit of f(R) gravity, and a Plummer density profile for the stellar distribution. The f(R) velocity dispersion profiles depend on two parameters: the scale length ξ-1, below which the Yukawa term is negligible, and the boost of the gravitational field δ > -1. δ and ξ are not universal parameters, but their variation within the same class of objects is expected to be limited.  The f(R) velocity dispersion profiles fit the data with a value  1/ξ=1.2(+18.6)(−0.9)  Mpc for the entire galaxy sample. On the contrary, the values of δ show a bimodal distribution that picks at δ=−0.986(±0.002) and δ=−0.92(±0.01). These two values disagree at 6σ and suggest a severe tension for f(R) gravity. It remains to be seen whether an improved model of the dwarf galaxies or additional constraints provided by the proper motions of stars measured by future astrometric space missions can return consistent δ's for the entire sample and remove this tension [2]. 

Our analysis returns a length scale of the Yukawa potential, 1/ξ=1.2(+18.6)(−0.9)  Mpc for all the galaxies. This result is reassuring for two reasons: (1) a single value of 1/𝜉 is valid for the entire dSph sample, as expected in 𝑓(𝑅) gravity within the same class of objects; (2) the value of 1/𝜉 guarantees that the extra-degrees of freedom do not play any relevant role in the dynamics of self-gravitating systems on the scale below 1 Mpc, where the effects of the accelerated expansion of the Universe or the effects of a fifth force are lacking. Our result confirms previous analyses. For instance, De Martino (2016) found 1/𝜉 ~1 Mpc by fitting the SZ temperature anisotropy profile of the Coma cluster. Capozziello et al. (2009) and Napolitano et al. (2012) also found similar values of 1/𝜉 by modelling the mass profile of 12 X-ray galaxy clusters and the stellar kinematics of elliptical galaxies, respectively. Actually, the estimated values of 1/𝜉 in Capozziello et al. (2009) range from 100 kpc to 10 Mpc depending on the systems. However, these fluctuations of 1/𝜉 may be due to simplistic modelling based on the assumption of a phenomenological X-ray gas density whose parameters are not fitted together with the 𝑓(𝑅)-gravity parameters. On the other hand, our result on 𝛿 suggests a possible tension for 𝑓(𝑅) gravity. The parameter 𝛿 controls the intensity of the gravitational field. In Eq. (14) for the modified gravitational potential of an extended mass distribution, the term 1/(1+𝛿) multiplies the mass density: 𝛿 can thus mimic an increasing dark matter content by assuming values increasingly close to 1. The values of 𝛿 we find for the dSph sample have a bimodal distribution peaking around two values δ=−0.986(±0.002) and δ=−0.92(±0.01). These values differ by 6-𝜎. Therefore, unlike 𝜉, 𝛿 does not assume a single value which is valid for all the dSphs in the sample. Our analysis rather suggests that different values of 𝛿 are required for different objects within the same class, at odds with the expectations from 𝑓(𝑅) gravity. Different values of 𝛿 are expected in different classes of objects but not within the same class of objects. For example, for gravitational systems where dark matter is not required in standard gravity, like stellar systems, 𝛿 is expected to be sufficiently close to zero to guarantee negligible departures, if any, from the standard Newtonian dynamics. Indeed, De Martino et al. (2021) showed that reproducing the orbital motion of the S2 star around the supermassive black hole in the centre of the Galaxy requires 𝛿 =-0•01 (+0•61)(-0•14), which is compatible with zero at 1-𝜎. On the contrary, for systems dominated by dark matter in the standard model, like clusters of galaxies, 𝛿 is close to -1. For example, Capozziello et al. (2009) found 𝛿 in the range [-0•95,–-0•84] for the X-ray clusters mentioned above. The disagreement we find for the two values of 𝛿 for the dSphs suggests a severe tension for 𝑓(𝑅) gravity. In principle, more sophisticated extended theories of gravity might properly describe the kinematics of dwarf galaxies and reduce, or completely eliminate, the tension we find here. Alternatively, improved models of the dwarf galaxies, where our simplifying assumptions are dropped, may be sufficient to remove the tension for 𝑓(𝑅) gravity. 

Figure: MCMC posterior distributions of the parameters θ = {log α, μ, β, M∗/L} for Carina Draco, Fornax, and Leo I. The blue-shaded areas with decreasing darkness depict the 68%, 95%, and 99% confidence regions, respectively. On top of each column, we report the median values of the posterior distributions with their 68% confidence intervals. The red shaded areas correspond to the best-fit values and the 1σ uncertainties of the velocity anisotropy parameter reported in Walker et al. (2009), and the expected values of M∗/L.

Our results pointed out some tensions on the α parameter within the data set, while comparison with previous analysis shows the effectiveness of STVG in replacing dark matter with extra massive fields. Further improvements will require more sophisticated modelling of the line-of-sight velocity dispersion which will be possible as soon as high-precision astrometric data in dwarf spheroidals will become available.

TBA

Bibliography:

1. I. De Martino,’Giant low-surface-brightness dwarf galaxy as a test bench for MOdified Gravity.’, 2020, Mon. Not. R. Astron. Soc., 493, 2373-2376

2. I. De Martino, A. Diaferio, L. Ostorero, "Dynamics of dwarf galaxies in f(R) gravity", 2023, Mon. Not. R. Astron. Soc., 519, 4424-4433

3. I. De Martino, "Dynamics of Dwarf Galaxies in Scalar-Tensor-Vector-Gravity", 2023, Physical Review D, 108, 044074