Constraining energy-momentum-squared gravity by binary pulsar observations

In this paper, we introduce the post-Minkowskian approximation of energy-momentum-squared gravity (EMSG). This approximation is used to study the gravitational energy flux in the context of EMSG. As an application of our results, we investigate the EMSG effect on the first time derivative of the orbital period of the binary pulsars. Utilizing this post-Keplerian parameter, the free parameter of the EMSG theory, $f_0^{\prime }$, is estimated for six known binary pulsars. Taking the binaries that have the most accurate observations, it turns out that $-6\times {10}^{-37}\mathrm{m}{\mathrm{s}}^2{\mathrm{kg}}^{-1}<f_0^{\prime }<+{10}^{-36}\mathrm{m}{\mathrm{s}}^2{\mathrm{kg}}^{-1}$. This bound is in agreement with the precedent studies.

Figure 1

The scaled flux of energy $\mathcal{P}(5/32)(G/{\eta }^2)(cl/Gm{)}^5$ in terms of the scaled retarded time $\tau /P$ where $P$ is the orbital period. Here, we assume that $e=0.7$. In this figure, $\alpha$ shows the EMSG correction. We see that near the pericenter, even for a rather small value of $\alpha$, the flux of energy changes considerably comparing with the GR case with $\alpha =0$.

Figure 2

The orbital period variation of the double neutron star binary pulsar $\mathrm{J}2129+1210\mathrm{C}$. In this figure, the green dot-dashed line shows ${\dot{P}}_{\mathrm{EMSG}}$ in terms of $f_0^{\prime }$. cf. Eq. (83). We also insert ${\dot{P}}_{\mathrm{GR}}$ and ${\dot{P}}_{\mathrm{obs}}$ in this plot. Where the ${\dot{P}}_{\mathrm{EMSG}}$ line intersects the blue solid and red dashed lines indicates $f_0^{\prime }$ and ${f_0^{\prime }}_{\mathrm{GR}}$, respectively. Moreover, the upper and lower limits of $f_0^{\prime }$ are obtained from the intersection of the green dot-dashed and blue dotted lines. The numerical values of these are given in Table 2.

Figure 3

The free parameter of the EMSG theory in terms of the density of the neutron star. Here, we consider that the density can change in the interval $({10}^{17}-8\times {10}^{17})\mathrm{kg}{\mathrm{m}}^{-3}$. Here, $f_0^{\prime }$ is displayed for the introduced relativistic binary pulsars. The colored solid and dashed curves represent the solutions (84) and (85), respectively. Here, we show the range of $f_0^{\prime }$ where most of the systems behave similarly. We find that the binary pulsars $\mathrm{J}1537+1155$ and $\mathrm{J}1738+0333$ have no solution in this range. Also, the dashed curves belonging to J1141-6545 do not enter this range.

Figure 4

The absolute value of the free parameter of the EMSG theory in terms of the percent error $|({\dot{P}}_{\mathrm{obs}}-{\dot{P}}_{\mathrm{GR}})/{\dot{P}}_{\mathrm{GR}}|\times 100$. For the sake of convenience, the horizontal axis is scaled logarithmically. Here, except for J1141-6545*, we consider that the density of the neutron star is of the order ${10}^{17}\mathrm{kg}{\mathrm{m}}^{-3}$. For J1141-6545*, we assume that the density of the pulsar is of the order $6\times {10}^{17}\mathrm{kg}{\mathrm{m}}^{-3}$. The horizontal lines, from top to bottom, represent the percent error 5% and 1%, respectively. Moreover, the colored area shows the allowed region for $f_0^{\prime }$ introduced in [40]. Four cases J0737-3039A, $\mathrm{J}2129+1210\mathrm{C}$, $\mathrm{J}1915+1606$, and J1141-6545* are well located in this region. These systems are displayed by the black circle. The rest systems where the percent error is more than 5% are indicated by the red square.

Figure 5

The value of $f_0^{\prime }$ and its error bar for the systems with percent error $<5\%$. The dashed lines show the estimated bound for $f_0^{\prime }$. Here, we choose the worse error to estimate this bound. The blue dashed lines represent the region deduced from the four systems and the red ones exhibit the bound found from the binary pulsars $\mathrm{J}2129+1210\mathrm{C}$, $\mathrm{J}1915+1606$, and J0737-3039A.