Constraining the equation of state in modified gravity via universal relations

Modern multimessenger astronomical observations and heavy ion experiments provide new insights into the structure of compact objects. Nevertheless, much ambiguity remains when it comes to superdense matter above the nuclear saturation density such as that found within neutron stars. Recently, equation of state (EOS)-independent universal relations between the physical parameters of static neutron stars at the maximum-mass point have been derived within General Relativity (GR), and they were used to constrain EOS candidates. In the present paper, we generalize these relations to neutron stars in massive scalar-tensor theories. More specifically, we explore 53 different EOS candidates and prove that the resulting relations for scalarized neutron stars are indeed EOS independent up to a large extent and they can be significantly different compared to GR. As a result, multiple constraints on the EOS are derived within scalar-tensor theories, and it turns out that the allowed set of EOS can differ a lot with respect to Einstein’s theory. This demonstrates the importance of modified gravity effects when imposing constraints on the EOS. The derived universal relations can be potentially used as well to test the theory of gravity in an EOS-independent way. We examine also the effects coming from nonbaryonic EOS both in GR and in scalar-tensor theories and show that, even though they spoil significantly the EOS universality, as expected, the effects coming from the modification of the theory of gravity can be significantly larger.

Figure 1

Mass-radius relations of the ALF2 EOS for GR and $\beta =-5,-6,-7$. Massless and $m_{\phi }={10}^{-13}\mathrm{eV}$ cases for each $\beta$ value are shown in solid green and dashed red lines, respectively.

Figure 2

$x_{*}(R_{\mathrm{mix},x_{*}}^{*})$ fits of the form (9) going from left to right: GR (solid line), $\beta =-5$ (dotted line), $\beta =-6$ (dashed line), and $\beta =-7$ (dot-dashed line) for the scalar-tensor theory. The lower plots indicate the relative percentage error between the fit and the actual result, labeled as $\delta$. (a.) Pressure ${\rho }_{c*}$ fits, (b.) Density ${\rho }_{c*}$ fits, and (c.) Speed of sound $c_{cs*}$ fits.

Figure 3

$x_{*}(R_{\mathrm{mix},x_{*}}^{*})$ fits of the form (9) for all EOS at $\beta =-6$. Baryonic points are shown as black circles, and nonbaryonic points are shown as gray triangles. The lower plots indicate the relative percentage error between the fit and the actual result, labeled as $\delta$ with the same color scheme.

Figure 4

$y_{*}({\chi }_{\mathrm{mix},y_{*})}$ fits of the form (11) going from left to right: GR (solid line), $\beta =-5$ (dotted line), $\beta =-6$ (dashed line), and $\beta =-7$ (dot-dashed line) for the scalar-tensor theory. The lower plots indicate the relative percentage error between the fit and the actual result, labeled as $\delta$. (a.) Mass $M_{*}$ fits, (b.) Radius $R_{*}$ fits, and (c.) Speed of sound $c_{cs*}$ fits.

Figure 5

Final constraints on maximum mass-radius and central pressure-density relations using only baryonic EOS for GR (top panels) and STT with $\beta =-5$ (bottom panels), where the final allowed range of EOS is shaded in gray. The green region corresponds to the causality limit (with the corresponding errors) obtained from (13) and (14) on the $P_{c*}-{\rho }_{c*}$ and $M_{*}-R_{*}$ planes, respectively. The blue curves divide the regions between maximum mass lower or higher than $M=1.97M_{\odot }$ with the latter being where the allowed EOS should reside. In the $M_{*}-R_{*}$ plane (right panels), this is naturally a single line, while in the $P_{c*}-{\rho }_{c*}$ plane (left panels), the constraint is obtained from (15), and therefore it is a region due to the fitting errors.

Figure 6

Comparison between the maximum-mass points plotted on the $M(R)$ plane for GR and scalar-tensor theory with $\beta$ values of $-5$ and $-7$ demonstrating the reduced scatter of the points for smaller $\beta$ values.