Relativistic stars in f(R) and scalar-tensor theories

TL;DR

The paper investigates relativistic stars in scalar-tensor and $f(R)$ gravity where the chameleon mechanism suppresses fifth forces in high-density environments. It numerically constructs static, spherically symmetric star solutions for both constant-energy-density and polytropic equations of state using a relaxation method, and analyzes the role of the effective potential and screening. The authors show that stars can exist with substantial surface potentials (up to $\Phi \sim 0.3$) even when curvature singularities are present cosmologically, provided the central EOS keeps $\tilde{\rho}-3\tilde{P}>0$; when this quantity becomes negative in large cores, tachyonic instabilities can prevent static configurations. They also compare chameleon and $f(R)$ models, demonstrate thin-shell screening via the effective coupling $Q_{\rm eff}$ and the post-Newtonian parameter $\tilde{\gamma}$, and explore regularized ($"cured"$) $f(R)$ models that remove the singularity while preserving viable neutron-star solutions, with implications for astrophysical tests of modified gravity.

Abstract

We study relativistic stars in the context of scalar tensor theories of gravity that try to account for the observed cosmic acceleration and satisfy the local gravity constraints via the chameleon mechanism. More specifically, we consider two types of models: scalar tensor theories with an inverse power law potential and f(R) theories. Using a relaxation algorithm, we construct numerically static relativistic stars, both for constant energy density configurations and for a polytropic equation of state. We can reach a gravitational potential up to $Φ\sim 0.3$ at the surface of the star, even in f(R) theories with an "unprotected" curvature singularity. However, we find static configurations only if the pressure does not exceed one third of the energy density, except possibly in a limited region of the star (otherwise, one expects tachyonic instabilities to develop). This constraint is satisfied by realistic equations of state for neutron stars.

Figures (10)

• Figure 1: Energy density ${\tilde{\rho}}$ (solid blue line), pressure ${\tilde{P}}$ (dashed purple line) and the combination ${\tilde{\rho}}-3{\tilde{P}}$ (black dotted line), in units of the central density $\rho_c$, as functions of the radial coordinate $r$ (in units of $M_P\tilde{\rho}_c^{-1/2}$).
• Figure 2: The rescaled bare potentials (\ref{['pot-r']}), shown by solid blue, the matter part of the effective potential, shown by dashed red, and the effective potential for the chameleon model (\ref{['eff_pot-r']}), shown by dotted black line. The parameters are chosen as follows, $Q=1$, $v_0=0.05$, $\epsilon_0=0.05$, $\epsilon_1=1$ and ${\hat{\rho}}-3{\hat{P}}=0.14$.
• Figure 3: Profiles of the scalar field $\phi$ (in Planck units) as a function of radius of the star (in units of $r_0=M_P{\hat{\rho}_c \tilde{\rho}_c}^{1/2}$) for constant density stars with the equation of state (\ref{['eoscd']}). The parameters are chosen as follows, $\sigma=0.01$, $Q= \epsilon_1=1$, $\epsilon_0=v_0=0.01$, and the rescaled densities of the star are (from light gray to black), $\hat{\rho}_c=0.01$, $0.02$, $0.05$, $1$. The values of the gravitational potential at the surface of the star are, respectively, $0.00168$, $0.00334$, $0.0083$ and $0.165$. The increasing rescaled density corresponds to an increasing physical density while the physical radius of the star is being fixed.
• Figure 4: Profiles of the scalar field $\phi$ (in Planck units) as a function of radius of the star (in units of $r_0=M_P{\hat{\rho}_c \tilde{\rho}_c}^{1/2}$) for stars with ${\tilde{\rho}}-3{\tilde{P}}<0$ in the central region. The rescaled densities of the star (from light gray to black): $\hat{\rho}_c=1$, $1.7$, $1.8$, $1.9$. The other parameters are the same as in Fig. \ref{['constant_chameleon']}. The values of the gravitational potential at the surface of the star are, respectively, $0.165$, $0.280$, $0.298$ and $0.318$.
• Figure 5: Evolution of the numerical estimates for $Q_{\rm eff}/Q$ (red triangles) and for the post-newtonian parameter $\tilde{\gamma}$ (blue dots) as we increase the energy density of the star (constant energy density star). The "bare" coupling is here $Q=1/2$, which corresponds to $\tilde{\gamma}=1/3$ in the absence of screening. The other parameters are $\epsilon_0=10^{-2}$, $v_0=10^{-8}$ and $\xi_*=1$.