Compact stars in energy-momentum squared gravity

A simple generalization to Einstein’s general relativity (GR) was recently proposed which allows a correction term $T_{\alpha \beta }{T}^{\alpha \beta }$ in the action functional of the theory. This theory is called energy-momentum squared gravity (EMSG) and introduces a new coupling parameter $\eta$. EMSG resolves the big bang singularity and has a viable sequence of cosmological epochs in its thermal history. Interestingly, in the vacuum EMSG is equivalent to GR, and its effects appear only inside the matter-energy distribution. More specifically, its consequences appear in high curvature regime. Therefore it is natural to expect deviations form GR inside compact stars. In order to study spherically symmetric compact stars in EMSG, we find the relativistic governing equations. More specifically, we find the generalized version of the Tolman-Oppenheimer-Volkov equation in EMSG. Finally we present two analytical solutions, and two numerical solutions for the field equations. For obtaining the numerical solutions we use polytropic equation of state which is widely used to understand the internal structure of neutron stars in the literature. Eventually we find a mass-radius relation for neutron stars. Also, we found that EMSG, depending on the central pressure of the star and the magnitude of free parameter $\eta$, can lead to larger or smaller masses for neutron stars compared with GR. Existence of high-mass neutron stars with ordinary polytropic equation of state in EMSG is important in the sense that these stars exist in GR when equation of state is more complicated.

Figure 1

The mass-radius relation for different values of $\eta$. The blue solid curve belongs to GR. Curves terminate when there is no physical solution for the governing equations. The mass has been scaled in terms of solar mass $M_{\odot }$.

Figure 2

The mass of the star with respect to the central pressure $p_c$ (${\mathrm{km}}^{-2}$) for the same values of $\eta$ presented in Fig 1.

Figure 3

The radius of the star $R^{*}(\mathrm{km})$ with respect to the central pressure $p_c$ (${\mathrm{km}}^{-2}$). Different colors belong to the different values of $\eta$ presented in Fig. 1. The blue solid curve belongs to GR.

Figure 4

Relative difference between density parameter $C$ in EMSG and GR with respect to $pc$ (${\mathrm{km}}^{-2}$). Different colors belong to different values of $\eta$ used in previous figures.

Figure 5

Top panels: Relative difference between ${\rho }_{\mathrm{GR}}$ and ${\rho }_{\mathrm{eff}}$ for two different values of $p_c$, i.e. $p_c=0.0004$ and 0.002, with respect to distance from center of the star. Bottom panels: The corresponding relative difference between $p_{\mathrm{GR}}$ and $p_{\mathrm{eff}}$ with respect to $r$.

Figure 6

In the all panels the blue curve belongs to GR and other colors demonstrate different values for $\eta$ given in the top right panel. Top panels: the left panel shows the mass-radius relation of strange quark stars in EMSG, and the right panel shows the radius of the star with respect to $p_c$. Bottom panels: The left panel represents the radius of the star with respect to $p_c$, and the right panel shows the relative difference between the density parameter $C$ of quark stars in GR and EMSG.