On the numerical evaluation of the `exact' Post-Newtonian parameters in Brans-Dicke and Entangled Relativity theories

TL;DR

This paper develops numerical methods to accurately compute the 'exact' post-Newtonian parameters in scalar-tensor gravity theories, revealing significant deviations from standard parameters in certain astrophysical scenarios.

Contribution

It introduces two new numerical techniques for calculating exact parameters in Brans-Dicke and Entangled Relativity theories, linking them to existing non-perturbative parameters and exploring observational constraints.

Findings

Exact parameters can differ by over 80% from standard post-Newtonian values.

Methodology applied to Sun, Earth, and neutron stars to derive specific parameters.

Current experiments may constrain these theories assuming specific matter Lagrangian forms.

Abstract

In context of Brans-Dicke scalar-tensor theories of gravity, it has recently been obtained that the post-Newtonian parameters should be generalized in the context of strongly gravitating bodies, and that its generalization -- the so-called $\textit{exact parameters}$ -- actually depends on the pressure and energy density of a considered celestial body. Here we develop two new methods to numerically obtain the $\textit{exact parameters}$ by means of usual Tolman-Oppenheimer-Volkoff computation, and find that the difference with the value of standard post-Newtonian parameters can be more than 80% in some situations. We also provide the connection with the Damour-Esposito Far\`ese non-pertubative parameter $\alpha_{DEF}$. We then apply the methodology to the case of Entangled Relativity, and derive these exact parameters for the Sun and the Earth, as well as for neutron stars. We argue that current and foreseeable experiments are likely able to constrain the theory under the assumption that $\mathcal{L}_m=-\rho$, where $\rho$ is the total energy density. If $\mathcal{L}_m=T$ instead, as often advocated in the literature, then there is no deviation with respect to General Relativity and the prospects of testing Entangled Relativity become much more remote in time, as only compact objects with extreme electric or magnetic fields could lead to some deviation from General Relativity.