For the star with an equation of state , the effective potential of the field (in the presence of a matter coupling) does not have an extremum, see Eq. (11.7). In those cases the analytic results in Section 11.2 are no longer valid. For the equation of state there is a tachyonic instability that tends to prevent the existence of a static star configuration [42]. For realistic neutron stars, however, the equation of state proposed in the literature satisfies the condition throughout the star.

Babichev and Langlois [43, 42] chose a polytropic equation of state for the energy density and the pressure in the Jordan frame:

where kg, fm, and . Solving the continuity equation coupled with Einstein equations, [43, 42] showed that can remain smaller than for realistic neutron stars. Note that the energy density is a decreasing function with respect to the distance from the center of star. Even for such a varying energy density, static star configurations have been shown to exist [43, 42].The ratio between the central density and the cosmological density at infinity is parameterized by the quantity . Realistic values of are extremely small and it is a challenging to perform precise numerical simulations in such cases. We also note that the field mass in the relativistic star is very much larger than its cosmological mass and hence a very high accuracy is required for solving the field equation numerically [600, 581]. The authors in [43, 600, 42] carried out numerical simulations for the values of of the order of . Figure 11 illustrates an example of the thin-shell field profile for the polytropic equation of state (11.39) in the model (4.84) with and [43]. In the regime the field is nearly frozen around the extremum of the effective potential, but it starts to evolve toward its asymptotic value for .

Although the above analysis is based on the f (R) models (4.83) and (4.84) having a curvature singularity at , such a singularity can be cured by adding the term [350]. The presence of the term has an advantage of realizing inflation in the early universe. However, the f (R) models (4.83) and (4.84) plus the term cannot relate the epoch of two accelerations smoothly [37]. An example of viable models that can allow a smooth transition without a curvature singularity is [37]

where , (), , and are constants. In [42] a static field profile was numerically obtained even for the model (11.40).Although we have focused on the stellar configuration with , there are also works of finding static or rotating black hole solutions in f (R) gravity [193, 497]. Cruz-Dombriz et al. [193] derived static and spherically symmetric solutions by imposing that the curvature is constant. They also used a perturbative approach around the Einstein–Hilbert action and found that only solutions of the Schwarzschild–Anti de Sitter type are present up to second order in perturbations. The existence of general black hole solutions in f (R) gravity certainly deserves for further detailed study. It will be also of interest to study the transition from neutron stars to a strong-scalar-field state in f (R) gravity [464]. While such an analysis was carried out for a massless field in scalar-tensor theory, we need to take into account the field mass in the region of high density for realistic models of f (R) gravity.

Pun et al. [498] studied physical properties of matter forming an accretion disk in the spherically symmetric metric in f (R) models and found that specific signatures of modified gravity can appear in the electromagnetic spectrum. In [92] the virial theorem for galaxy clustering in metric f (R) gravity was derived by using the collisionless Boltzmann equation. In [398] the construction of traversable wormhole geometries was discussed in metric f (R) gravity. It was found that the choice of specific shape functions and several equations of state gives rise to some exact solutions for f (R).

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