In the present paper, we consider the extended scalar-tensor Gauss-Bonnet gravity with a massive scalar field. We prove numerically the existence of Gauss-Bonnet black holes for three different forms of the coupling function including the case of spontaneous scalarization. We have performed a systematic study of the black hole characteristics such as the area of the horizon, the entropy, and the temperature for these coupling functions and compared them to the Schwarzschild solutions. The introduction of scalar field mass leads to a suppression of the scalar field, and the increase of this mass brings the black holes closer to the Schwarzschild case. For linear and exponential coupling, a nonzero scalar field mass expands the domain of existence of black hole solutions. Larger deviations from the Schwarzschild solution are observed only for small masses, and these differences decrease with the increase of the scalar field mass. In the case of a coupling function which leads to scalarization, the scalar field mass has a significant influence on the bifurcation points where the scalarized black holes branch out of the Schwarzschild solution. The largest deviation from the case with a massless scalar field are observed for black hole masses close to the bifurcation point.

We study how rapid rotation influences the relation between the normalized moment of inertia $\overline{I}$ and quadrupole moment $\overline{Q}$ for scalarized neutron stars. The questions one has to answer are whether the equation of state universality is preserved in this regime and what are the deviations from general relativity. Our results show that the $\overline{I}-\overline{Q}$ relation is nearly equation of state independent for scalarized rapidly rotating stars, but the differences with pure Einstein’s theory increase compared to the slowly rotating case. In general, smaller negative values of the scalar field coupling parameters $\beta$ lead to larger deviations, but these deviations are below the expected accuracy of the future astrophysical observations if one considers values of $\beta$ in agreement with the current observational constraints. An important remark is that although the normalized $\overline{I}-\overline{Q}$ relation is quite similar for scalar-tensor theories and general relativity, the unnormalized moment of inertia and quadrupole moment can be very different in the two theories. This demonstrates that although the $\overline{I}-\overline{Q}$ relations are potentially very useful for some purposes, they might not serve us well when trying to distinguish between different theories of gravity.

Binary pulsar observations and gravitational wave detections seriously constrained scalar–tensor theories with massless scalar field allowing only sm

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.