In Section 5 we discussed the existence of thin-shell solutions in metric f (R) gravity in the Minkowski background, i.e., without the backreaction of metric perturbations. For the f (R) dark energy models (4.83) and (4.84), Frolov [266] anticipated that the curvature singularity at (shown in Figure 3) can be accessed in a strong gravitational background such as neutron stars. Kobayashi and Maeda [349, 350] studied spherically symmetric solutions for a constant density star with a vacuum exterior and claimed the difficulty of obtaining thin-shell solutions in the presence of the backreaction of metric perturbations. In [594] thin-shell solutions were derived analytically in the Einstein frame of BD theory (including f (R) gravity) under the linear expansion of the gravitational potential at the surface of the body (valid for ). In fact, the existence of such solutions was numerically confirmed for the inverse power-law potential [594].

For the f (R) models (4.83) and (4.84), it was numerically shown that thin-shell solutions exist for by the analysis in the Jordan frame [43, 600, 42] (see also [167]). In particular Babichev and Langlois [43, 42] constructed static relativistic stars both for constant energy density configurations and for a polytropic equation of state, provided that the pressure does not exceed one third of the energy density. Since the relativistic pressure tends to be stronger around the center of the spherically symmetric body for larger , the boundary conditions at the center of the body need to be carefully chosen to obtain thin-shell solutions numerically. In this sense the analytic estimation of thin-shell solutions carried out in [594] can be useful to show the existence of static star configurations, although such analytic solutions have been so far derived only for a constant density star.

In the following we shall discuss spherically symmetric solutions in a strong gravitational background with for BD theory with the action (10.10). This analysis covers metric f (R) gravity as a special case (the scalar-field degree of freedom defined in Eq. (2.31) with ). While field equations will be derived in the Einstein frame, we can transform back to the Jordan frame to find the corresponding equations (as in the analysis of Babichev and Langlois [42]). In addition to the papers mentioned above, there are also a number of works about spherically symmetric solutions for some equation state of matter [330, 332, 443, 444, 300, 533].

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