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Having in mind the general scheme described in S-star section, we carried out the following analysis
Testing f(R)-gravity at the Galactic Center with S2 stars
We compute mock orbits using the geodesic equations in the spherically symmetric metric in [1]
where the gravitational potential is
We left free to vary the following model parameters:
Here, the first two parameters describe the mass and the distance from Earth of the source of the gravitational potential in which the star moves, the seven Keplerian elements provide with initial conditions for the numerical integration of the geodesic equations, and the Thiele-Innes elements needed to project the predicted orbit in the observer's reference frame. Then, five additional parameters account for the zero-point offset and drift of the reference frame with respect to the mass centroid, and the last two parameters, constrain the f(R)-gravity model. Thus, we have 16 parameters that entirely describe the model. We explored the parameter space using a Markov Chain Monte Carlo (MCMC) algorithm implemented in emcee. Finally, our results are shown in the figure below.
No deviations from GR are revealed at the scale of λ<6300 AU with the strength of the Yukawa potential restricted to δ=-0.01(-0.14; +0.61).
Testing MOG at the Galactic Center with S2 stars
We have obtained the first constraint of the parameter space of Scalar-Tensor-Vector-Gravity using the motion of the S2-star around the supermassive black hole at the centre of the Milky Way, and we did not find any serious tension with General Relativity. We used the Schwarzschild-like metric of Scalar-Tensor-Vector-Gravity to predict the orbital motion of S2-star, and to compare it with the publicly available astrometric data, which include 145 measurements of the positions, 44 measurements of the radial velocities of S2-star along its orbit, and only the inferred rate of precession, as the latest GRAVITY data are not yet public. We employed a Monte Carlo Markov Chain algorithm to explore the parameter space, and constrained the only one additional parameter of Scalar-Tensor-Vector-Gravity to α<0.662 at 99,7% confidence level, where α=0 reduces this modified theory of gravity to General Relativity [2].
We started from a spherically symmetric metric tensor, a Schwarzschild-like black hole solution:
where
We compute mock orbits using the geodesic equations [2], and compare them with the data leaving free to vary the following model parameters:
Thus, we have 15 parameters that entirely describe the model. We explored the parameter space using a Markov Chain Monte Carlo (MCMC) algorithm implemented in emcee. Finally, our results are shown in the figure below.
This represents the first constraints on the dimensionless parameter of Scalar-Tensor-Vector-Gravity: α<0.662 at 99,7% confidence level.
Testing Horndeski theory at the Galactic Center with S2 stars
We have explored a completely new and alternative way to restrict the parameter space of Horndeski theory of gravity. Using its Newtonian limit, it is possible to test the theory at a regime where, given its complexity and the small magnitude of the expected effects, it is poorly probed. At Newtonian and Post-Newtonian level, it gives rise to a generalized Yukawa-like Newtonian potential which we have tested using S2 star orbit data. Our model adds five parameters to the General Relativity model, and the analysis constrains two of them with unprecedented precision to these energy scales, while only gives an exclusion region for the remaining parameters. We have shown the potential of weak-field tests to constrain Horndeski gravity opening, as a matter of fact, a new avenue that deserves to be further, and deeply, explored near in the future [3].
The starting point of our study is the weak-filed limit of the Horndeski's gravity calculated through a perturbative expansion of the field equations around a Minkowski background, and a constant cosmological background value Φ of the scalar field:
For a mass point source the solution at Newtonian order is (for more details see [4]):
where
We compute mock orbits using the geodesic equations, and compare them with the data leaving free to vary the following model parameters:
We explored the parameter space using a Markov Chain Monte Carlo (MCMC) algorithm implemented in emcee. Finally, our results are shown in the figures below.
Figure: Focus on the posterior distribution of the Taylor coefficients of Horndeski gravity.
Figure: 95% confidence level exclusion regions of the G2(0,1) -G3(1,0) slice of the parameter space from our posterior analysis.
This represents the first constraints of weak-field limit of the Horndeski's gravity. No assumptions on the functions Gi have been made.
Measuring 1PPN parameters using observations of S2 and S62
The Parameterized Post-Newtonian (PPN) formalism offers an agnostic framework for evaluating theories of gravity that extend beyond General Relativity. Departures from General Relativity are represented by a set of dimensionless parameters that, at the first order in the expansion, reduce to two parameters, which describe deviations in spatial curvature and non-linear superposition effects of gravity, respectively. When expressed in the standard Schwarzschild coordinates, and truncating the expansion at 1PN order, this space-time metric takes the expression
being the metric coefficients
with M the mass of the central object. We exploit future observations of stars at the Galactic Center, orbiting the supermassive black hole Sagittarius A*, to forecast the ability to constrain the first-order PPN parameters. We have generated a mock catalog of astrometric and spectroscopic data for S2, based on the Schwarzschild metric, simulating observations over multiple orbital periods with the GRAVITY and SINFONI instruments. Our analysis includes the effects of relativistic orbital precession and line-of-sight (LOS) velocity gravitational redshift. Since future data for S2 can provide constraints only on a linear combination of the PPN parameters we also analyzed the impact of future observations of the gravitational lensing for stars that pass closer in the sky to Sgr A*, like the known star S62, which can potentially provide tight constraints on the parameter $\gamma$, that alone regulates the amplitude of the astrometric deviations due to lensing. When combining lensing observations for S62, and the precise orbital tracking of S2, one obtains independent constraints on both parameters, providing a precision test of General Relativity and its extensions.
Bibliography:
De Martino, I., Della Monica, R., De Laurentis, M., "f(R)-gravity after the detection of the orbital precession of the S2 star around the Galactic Center massive black hole", 2021, Phys. Rev. D, 104, L101502
Della Monica, R., de Martino, I., de Laurentis, M., "Orbital precession of the S2 star in Scalar-Tensor-Vector-Gravity", 2021, Mon. Not. R. Astron. Soc., 510, 4757-4766
Della Monica, R., de Martino, I., Vernieri, D., de Laurentis, M., "Testing Horndeski gravity with S2 star orbit", 2023, Mon. Not. R. Astron. Soc, 519, 1981-1988
Hohmann, M., "Parameterized post-Newtonian limit of Horndeski's gravity theory", 2015, Phys. Rev. D 92, 064019 (2015)
de Mora Losada, V.; Della Monica, R. ; de Martino, I. ; De Laurentis, M., "Future prospects for measuring 1PPN parameters using observations of S2 and S62 at the Galactic center", 2025, Astronomy & Astrophysics, 694, A280, 9
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