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

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 $\Phi \sim 0.3$ 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.

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 ${\rho }_{\mathrm{c}}$, as functions of the radial coordinate $r$ (in units of $M_P{\tilde{\rho }}_{\mathrm{c}}^{-1/2}$).

Figure 2

The rescaled bare potentials (19), shown by the solid blue line, the matter part of the effective potential, shown by the dashed red line, and the effective potential for the chameleon model (20), shown by the dotted black line. The parameters are chosen as follows: $Q=1$, $v_0=0.05$, ${ϵ}_0=0.05$, ${ϵ}_1=1$, and $\widehat{\rho }-3\widehat{P}=0.14$.

Figure 3

Profiles of the scalar field $\varphi$ (in Planck units) as a function of the radius of the star (in units of $r_0=M_P{\widehat{\rho }}_{\mathrm{c}}\tilde{\rho }_{\mathrm{c}}{}^{1/2}$) for constant-density stars with the equation of state (23). The parameters are chosen as follows: $\sigma =0.01$, $Q={ϵ}_1=1$, ${ϵ}_0=v_0=0.01$, and the rescaled densities of the star are (from light gray to black), ${\widehat{\rho }}_{\mathrm{c}}=0.01$, 0.02, 0.05, 1. The values of the gravitational potential at the surface of the star are, respectively, 0.001 68, 0.003 34, 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 $\varphi$ (in Planck units) as a function of radius of the star (in units of $r_0=M_P{\widehat{\rho }}_{\mathrm{c}}\tilde{\rho }_{\mathrm{c}}{}^{1/2}$) for stars with $\tilde{\rho }-3\tilde{P}<0$ in the central region. The rescaled densities of the star (from light gray to black): ${\widehat{\rho }}_{\mathrm{c}}=1$, 1.7, 1.8, 1.9. The other parameters are the same as in Fig. 3. 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 $Q_{\mathrm{eff}}/Q$ (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 ${ϵ}_0={10}^{-2}$, $v_0={10}^{-8}$, and ${\xi }_{*}=1$.

Figure 6

Profile for the scalar field $\varphi$ (in Planck units), shown by the solid (blue) line, as a function of the radius (in units of $r_0=M_P{\tilde{\rho }}_{\mathrm{c}}^{-1/2}$), for the equation of state (26) with ${ϵ}_0=0.01$, $v_0=0.01$, and ${\widehat{\rho }}_{\infty }={10}^{-4}$. The value of the gravitational potential, $\widehat{\mathcal{M}}(\xi )/\xi$, at the surface of the star is 0.211. The value ${\varphi }_{\min }$ for the minimum of the effective potential is plotted by the dashed (gray) line.

Figure 7

Potential $V$ (in units of $M_P^2{R}_0$) as a function of $\varphi$ (in Planck units) for $n=1$ and $x_{\infty }=3.6$. The lower black dot corresponds to the de Sitter attractor while the upper-right dot shows the curvature singularity.

Figure 8

Profile of the scalar field $\varphi$ (in Planck units), shown by the solid (blue) line, as a function of the radius (in units of $M_P{\tilde{\rho }}_{\mathrm{c}}^{-1/2}$), for the model (30) with $n=1$, $x_{\infty }=3.6$ (see Appendix for the definition of $x_{\infty }$) and $v_0={10}^{-4}$. The value ${\varphi }_{\min }$ for the minimum of the effective potential is plotted by the dashed (gray) line.

Figure 9

Potential $V$ (in units of $M_P^2{R}_0$) as a function of $\varphi$ (in Planck units) for $n=1$ and $x_{\infty }=3.6$. The blue curve corresponds to the original Starobinsky cosmological model with the curvature singularity at $\varphi =0$. The red curve represents the cured Starobinsky model with $\sigma ={10}^{-3}$.

Figure 10

Profile of the scalar field in the model (34) with the parameters $\sigma =2\times {10}^{-6}$, $c=0.4$, $b=4$, $v_0={10}^{-3}$.