  Citation: Della Monica, R.; de Martino, I.; de Laurentis, M. Constraining MOdified Gravity with with the S2 Star. Universe 2022, 8, 137. https://doi.org/10.3390/ universe8020137 Academic Editors: Mariusz P. D˛ abrowski, Adam Balcerzak and Vincenzo Salzano Received: 24 January 2022 Accepted: 16 February 2022 Published: 21 February 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). universe Communication Constraining MOdified Gravity with the S2 Star Riccardo Della Monica 1, * , Ivan de Martino 1 and Mariafelicia de Laurentis 2,3 1 Departamento de Fisica Fundamental, Universidad de Salamanca, P. de la Merced, 37008 Salamanca, Spain; ivan.demartino@usal.es 2 Dipartimento di Fisica, Universitá di Napoli “Federico II”, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 Napoli, Italy; mariafelicia.delaurentis@unina.it 3 INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 Napoli, Italy * Correspondence: rdellamonica@usal.es Abstract: We have used publicly available kinematic data for the S2 star to constrain the parameter space of MOdified Gravity. Integrating geodesics and using a Markov Chain Monte Carlo algorithm, we have provided the first constraint on the scales of the Galactic Centre for the parameter α of the theory, which represents the fractional increment of the gravitational constant G with respect to its Newtonian value. Namely, α  0.662 at 99.7% confidence level (where α = 0 reduces the theory to General Relativity). Keywords: alternative gravity; galactic centre; black holes 1. Introduction Scalar-Tensor-Vector Gravity (STVG), also referred to in the literature as MOdified Gravity (MOG), is a theory of gravity firstly proposed in [1] as an alternative to Einstein’s theory of General Relativity (GR). It introduces extra fields in the description of the gravita- tional interaction, allowing for correct predictions on galactic and extragalactic scales [2–6], without resorting to dark matter [7]. The gravitational action in MOG presents additional terms along the classical Hilbert–Einstein action, depending on the metric tensor g αβ of spacetime. More specifically, a massive vector field ϕ α is introduced and its mass, μ, is treated as a scalar field. Furthermore, Newton’s gravitational constant G N is also elevated to a scalar field G. The motion of test particles in MOG is affected by the presence of the vector field ϕ α which acts as a fifth force, whose repulsive character counteracts the increased attraction due to the scalar field nature of G. The fractional increment of G, with respect to its Newtonian value, G N , is given by a new parameter of the theory, α =(G − G N )/G N . A distinctive feature in the motion of test massive bodies in MOG is that Keplerian orbits in a central potential are characterized by an increased value of the rate of orbital precession [8,9]. This is given by: Δω MOG = Δω GR  1 + 5 6 α  , (1) where Δω GR is the usual expression of the periastron advance in GR, Δω GR = 6πG N M ac 2 (1 − e 2 ) , (2) related to semi-major axis, a, and eccentricity, e, of the orbiting body. Here, we will summarize the extended work carried out in [8], where we used publicly available data for the S2 star from [10], along with the measurement of its orbital precession from [11] to constrain the parameter space of MOG. Universe 2022, 8, 137. https://doi.org/10.3390/universe8020137 https://www.mdpi.com/journal/universe