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

TL;DR

This work numerically evaluates the exact, non-perturbative Post-Newtonian parameters $\beta_{\text{exact}}$, $\gamma_{\text{exact}}$, and $\delta_{\text{exact}}$ in Brans–Dicke and Entangled Relativity theories. It develops two independent methods—direct Tolman–Oppenheimer–Volkoff integration and exterior matching to the Janis–Newman–Winicour solution—to incorporate the internal structure of gravitating bodies and obtain structure-dependent PN parameters. The results show sizable deviations from standard PN predictions for compact objects in Brans–Dicke theory and reveal an even stronger influence of the matter Lagrangian in Entangled Relativity, with potential observable consequences in dipolar gravitational radiation for certain Lagrangian choices. The findings imply that strong-field tests, including binary pulsar and solar-system observations, can substantially constrain these theories, particularly if the on-shell matter Lagrangian is taken as $\mathcal{L}_m=-\rho$.

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 Farèse non-pertubative parameter $α_{DEF}$. 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 $\mathcal{L}_m=-ρ$, where $ρ$ 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.

Figures (8)

• Figure 1: Plot of $1-\gamma_\textrm{exact}$ for 150 different values of $\omega$ and of central densities. The values of $\omega$ ranges from $10^{-1}$ to $10^6$ meanwhile the values of central density ranges from $100$ MeV/fm$^3$ to 1570 MeV/fm$^3$. This limit comes from the fact that we want the speed of sound in the neutron star to be less that the conservative limit considered of $c/\sqrt 3$bedaque:2015pr. The deviation of $\gamma_\textrm{exact}$ from unity is represented in colour shades.
• Figure 2: Plot of the relative difference between the exact and standard post-Newtonian parameters $\gamma$ for 150 different values of $\omega$ and central densities. The values of $\omega$ ranges from $10^{-1}$ to $10^6$ meanwhile the values of central density ranges from $100$ MeV/fm$^3$ to 1570 MeV/fm$^3$. This limit comes from the fact that we want the speed of sound in the neutron star to be less that the conservative limit considered of $c/\sqrt 3$bedaque:2015pr. In color shades is represented the relative deviation in percent of both $\gamma$ definitions.
• Figure 3: Plot of $1-\delta_\textrm{exact}$ for 150 different values of $\omega$ and central densities. The values of $\omega$ ranges from $10^{-1}$ to $10^6$ meanwhile the values of central density ranges from $100$ MeV/fm$^3$ to 1570 MeV/fm$^3$. This limit comes from the fact that we want the speed of sound in the neutron star to be less that the conservative limit considered of $c/\sqrt 3$bedaque:2015pr. In color shades is represented $\delta_\textrm{exact}$.
• Figure 4: Plot of the relative difference between the exact and standard post-Newtonian parameters $\delta$ for 150 different values of $\omega$ and central densities. The values of $\omega$ ranges from $10^{-1}$ to $10^6$ meanwhile the values of central density ranges from $100$ MeV/fm$^3$ to 1570 MeV/fm$^3$. This limit comes from the fact that we want the speed of sound in the neutron star to be less that the conservative limit considered of $c/\sqrt 3$bedaque:2015pr. In color shades is represented the relative deviation in percent of both $\delta$ definitions.
• Figure 5: Plot of $1-\gamma_\textrm{exact}$ versus the central density of a Neutron Star. The values of central density ranges from $100$ MeV/fm$^3$ to 1578 MeV/fm$^3$. This limit comes from the fact that we want the speed of sound to be less that the conservative limit considered of $c/\sqrt 3$bedaque:2015pr.