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Numerical simulations of Fuzzy Dark Matter halos have extensively studied the properties of the stable configurations that the system assumes, e.g. the generation of coherent standing waves of dark matter in the center of gravitationally-bound systems. These prominent dark matter solitonic cores are surrounded by wave interference patterns whose azimuthal average follows a Navarro-Frenk-White (NFW) profile, while the radial profile of the solitonic core is well approximated by
where
The core radius, on turn, can be related to the virial mass of the entire halo via a scaling relation
Forecasts with GRAVITY data
We use the orbital motion of the star S2 around the supermassive black hole at the center of the Galaxy to narrow the range allowed for the mass of an ultralight boson. It is well known that ultralight bosons form a solitonic dark matter core in the innermost part of the halo. The scale length of such a soliton depends on the inverse of the mass of the boson. On the other hand, the orbital motion of stars in the Galactic center depends on the distribution of matter whether baryonic or dark. Thus, we predict that future astrometric and spectroscopic observations of S2 will place an upper limit on the mass of the boson which in fact, once complementary constraints are considered, will help to restrict the allowed range of the boson mass [1].
Since we are concerned with the motion of test particles around the Galactic Center SMBH happening on scale of ~1000 AU, we can safely assume that the only contribution of the dark matter distribution to the orbital dynamics of stars is given by the innermost region described by the solitonic profile. This contribution results in an additional acceleration term, provided by the dark matter mass enclosed in the orbit of a test particle (see Appendix in [1] for more details):
Since we are working in the weak field limit of GR, this acceleration term can be linearly added to the 1PN acceleration term experienced by the particle due to the gravitational field of a Schwarzschild black hole %. In particular, we rely on the 1PN term of the geodesic equations around a Schwarzschild BH, in which the acceleration of a massive-test particle is given by
The total acceleration is thus given by
We have explored the 9-dimensional parameter space of the orbital model by applying a Markov Chain Monte Carlo (MCMC) algorithm implemented in emcee. The parameter space is composed by our distance from the Galactic Centre; the 7 keplerian elements, and the boson mass
Our analysis showed the capability of shutting the allowed range of the boson mass using forthcoming observations of S2 made by GRAVITY.
Figure: Marginalized posterior distribution of the boson mass in logarithmic scale. The red vertical line corresponds to the 95\% upper limit of the parameter resulting from our posterior analysis.
Figure: We compare the allowed range for the boson mass from our analysis with other constraints in literature coming from analysis at different scales.
Constraining the boson mass with publicly available data
When the wave DM density profile is introduced in the energy-momentum tensors in the Einstein's field equation, it leads to a stationary and spherically symmetric BH solutions embedded in a wave DM halo:
being F an hypergeometric function of the radial coordinate r, given by
The positive root of the function F(r) identifies the radial coordinate rH of the event horizon. We derive rH numerically as a function of the wave DM boson mass mψ in the range of interest of ultralight axions, resulting in a subpercent variation with respect to the Schwarzschild horizon radius as shown in the left panel of Figure.
Figure: Left panel: Dependence of the orbital precession for the S2 star from the boson mass mψ. The pink horizontal bands report the 1σ, 2σ, and 3σ experimental bounds imposed by the Gravity Collaboration from analysis of the orbital data for S2. Right panel: Enclosed mass Ms(r) of the solitonic wave DM distribution as a function of radius in units of solar masses, for different values of the boson mass mψ as reported in the corresponding labels. The yellow shaded strip highlights the extent of the S2 star orbit. Blue dotted lines report values of the enclosed mass corresponding to 1%, 0.1%, and 0.01% of Sgr A* mass M, respectively. The inset plot shows the solitonic density profiles in Eq. (1) for the same different values of the boson mass mψ
In order to impose constraints on the boson mass mψ we fit the predicted orbits for the S2 star to the publicly available astronomical data [2]. Such a data set consists of astrometric sky-projected positions (right ascension, α, and declination, δ) recorded in the near-infrared at Np = 145 epochs over the course of the last three decades (from ∼1992 to ∼2017), referred to the “GC radio-to-infrared reference system” and of Nv = 44 spectroscopically derived measurements of the line-of-sight velocity (vLOS) covering approximately the same temporal period. Thus, we are able to place an upper limit on the boson mass as shown in the following Figure.
Figure: Posterior distribution for the boson mass mψ from our MCMC analysis. The vertical dashed line, corresponding to the 95% confidence level upper limit for the boson mass from our analysis.
Quite remarkably, our analysis resulted in an upper limit for the boson mass, that, combined with the other astrophysical tests, constrains the values of mψ to an unprecedented level, by narrowing the range of allowed mass to only one order of magnitude. In particular, combining our result with the one from the analysis of the Lyman-α forest, results in an allowed mass range at 95% confidence level of
Bibliography:
Della Monica, R., de Martino, I., "Narrowing the allowed mass range of the ultralight bosons with S2 star", 2023, Astron. Astroph., L4, 8
Della Monica, R., de Martino, I., "Bounding the mass of ultralight bosonic Dark Matter particles with the motion of the S2 star around Sgr A*", 2023, Phys. Re. D, 108, L101303
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