Mass-radius relation for neutron stars in $f(R)=R+\alpha R^2$ gravity: A comparison between purely metric and torsion formulations

Within the framework of $f(R)=R+\alpha R^2$ gravity, we study realistic models of neutron stars, using equations of state compatible with the LIGO constraints, i.e., APR4, MPA1, SLy, and WW1. By numerically solving modified Tolman-Oppenheimer-Volkoff equations, we investigate the mass-radius relation in both metric and torsional $f(R)=R+\alpha R^2$ gravity models. In particular, we observe that torsion effects decrease the compactness and total mass of neutron star with respect to the general relativity predictions, therefore mimicking the effects of a repulsive massive field. The opposite occurs in the metric theory, where mass and compactness increase with $\alpha$, thus inducing an excess of mass that overtakes the standard general relativity limit. We also find that the sign of $\alpha$ must be reversed whether one considers the metric theory (positive) or torsion (negative) to avoid blowing up solutions. This could draw an easy test to either confirm or discard one or the other theory by determining the sign of parameter $\alpha$.

Figure 1

Solutions of the TOV equations for GR (blue line) and purely metric $R+\alpha R^2$ with $\alpha =0.05$ (orange line), using the SLy EOS. All the plotted quantities show small deviations with respect to GR. Note the asymptotic decay of the metric potentials $\lambda$ and $\psi$ as $r\to \infty$. Our choice of $\alpha$ explains the oscillatory behavior as reported in [76].

Figure 2

Profiles for the pressure $P$ (left) and the Ricci scalar $R$ (right) corresponding to $R_c=\{R_{G{R}_c},0.2R_{G{R}_c},2R_{G{R}_c}\}$ for the $f(R)=R+\alpha R^2$ model with $\alpha =0.1$. In the enlarged plot for the pressure, the grid lines fix two possible values for the radius of the star ${\mathcal{R}}_S$ that depend on to the accuracy chosen in defining its position as $p({\mathcal{R}}_{\mathcal{S}})/pc\le \{{10}^{-9},{10}^{-10}\}$ providing a relative difference of about 4%. Complementary, on the right-hand side plot we show the $R=0$ point for different choices of the central value $R_c$. Notice that on the latter the effects of choosing one or another $R_c$ contribute in total about the $\sim 2\%$ between $0.2R_{G{R}_c}$ and $2R_{G{R}_c}$ choices, thus this error being smaller than our error estimate in defining ${\mathcal{R}}_S$.

Figure 3

Results of our analysis with $\alpha =0.05$ for $\lambda$ and $\psi$ (left plots) and the derivatives for $R^{\prime }$ and ${\psi }^{\prime }$ (right plots) for the exact numerical solution (blue line); the Schwarzschild solution (orange line) with mass $\mathcal{M}=1.43{\mathcal{M}}_{\odot }$; a Schwarzschild fit (green line) to the numerical data outside the star, that is, with $\mathcal{R}>11.6\mathrm{km}$. We note that, for $\alpha$ small and averaging out all the oscillations, all physical quantities reproduce rather well the Schwarzschild solution outside the star, while matching as well the junction conditions (26). From the fitted results we get $\mathcal{M}=1.40{\mathcal{M}}_{\odot }$, thus very close to the theoretical one.

Figure 4

$\mathcal{M}\text{−}\mathcal{R}$ relations obtained within the purely metric formalism with $\alpha =\{0,0.001,0.01,0.05,0.1\}$ for the four EOS considered in this work. Note the general increase of the total mass as the quadratic term takes larger values, thus favoring the formation of more massive objects than in standard GR.

Figure 5

Analogous $\mathcal{M}\text{−}\mathcal{R}$ relations to those of Fig. 4 but here obtained within the torsional formalism. The effect of the torsion tends to decrease the total mass of the NS, contrary to what occurs in the purely metric case. This is dominantly caused by a sign flip on the $\alpha$-dependent part of Eq. (25) with respect to Eq. (22), which actually acts as a repulsive term.

Figure 6

Results of the analysis in the torsional case with $\alpha =0.05$. We show the metric potentials $\lambda$ and $\psi$ (left plots) and the derivatives for ${\psi }^{\prime }$ and $R^{\prime }$ (right plots) for the exact numerical solution (blue line), the Schwarzschild solution (orange line) and a Schwarzschild fit (green line) to the numerical data outside the star, that is with $\mathcal{R}>11.6\mathrm{km}$. Notice that once the oscillations are averaged out, all the distributions satisfy (up to numerical accuracy) the junction conditions.

Figure 7

$\mathcal{M}\text{−}\mathcal{R}$ relations for $\alpha =0.1$ in GR (blue line), metric (green line) and torsion (orange line) for the four EOS considered in this work. The torsion contributions tend to decrease the total mass of the system.