Gravitational wave asteroseismology of neutron and strange stars in $R^2$ gravity

We study the oscillations of neutron and strange stars in $R^2$ gravity. More precisely, the nonradial $f$ modes are examined and the differences with pure general relativity are investigated. Using these results, we build several gravitational wave asteroseismology relations. Our goal is to determine up to what extent these relations are equation of state independent and whether they deviate enough from general relativity in order to produce an observable effect. The results show that the differences coming from $R^2$ gravity are up to 10%, which will be difficult to observe in the near future. On the other hand, the small deviations in some of the asteroseismology relations show that they are not only equation of state independent, but they are also quite insensitive to the gravitational theory. That is why solving the inverse problem can give us quite robust estimates of the neutron star parameters.

Figure 1

The mass-of-radius relations for all EOS in pure GR. It could be seen that they cover a wide range of stiffnesses.

Figure 2

Results for one hadronic EOS, namely, SLy, and one quark EOS, namely, SQS B60. Both have maximum mass around two solar masses. In the left panel, the normalized frequency as a function of the neutron star mass is presented, while in the right panel—the oscillation frequency in kHz as a function of the average density of the star in ${\mathrm{km}}^{-1}$.

Figure 3

The change of the frequency with respect to the full GR case as a function of the parameter $a$. The results are for three different values of the average density.

Figure 4

In the left panel, we present the frequency as a function of the average density for all examined EOS for the case of GR and for $R^2$ gravity with parameter $a={10}^4$, which induces a deviation close to the maximum possible. It could be seen that the results for the stiffest EOS, namely, MS1 and SQS B40, seriously differ from all the other results. However, the two groups of results form two narrow bands. In the right panel, we plot the relation $M\omega (M/R)$. The presented results are again for all EOS for GR and $f(R)$ theory with $a={10}^4$. As one can see, the results for hadronic and quark EOS are only shifted from each other, but there is quite good universality for both cases independently. The separation between the GR and $f(R)$ dependencies is negligible.

Figure 5

The relation $M\omega (\eta )$ is presented in the graphs. In the right panel, we investigate the deviations of this relation from GR for EOS SLy and several different values of the parameter $a$. In the left panel, results for all EOS are presented.The relation for both the neutron and the strange stars are quite EOS independent, and a visible difference between the GR and the $a={10}^4$ cases exists. The maximum observed deviation is close to 10%.