Balkar Yildirim, Alan Albert Coley

We construct a perturbative scalar-tensor solution describing a central inhomogeneity embedded in an evolving cosmological background, with the aim of studying black hole persistence through a nonsingular bounce. Scalar-tensor gravity provides a natural framework for realizing bouncing cosmologies, while the inclusion of a localized inhomogeneity makes the field equations substantially more difficult to solve. We therefore adopt a perturbative scheme, with perturbative parameter $ε$, in which the leading-order equations are solved by a spatially flat bouncing FLRW spacetime sourced by a radiation perfect fluid. At next order, a central inhomogeneity is introduced through a generalized McVittie geometry, with the perturbations encoded in the corresponding first-order metric and scalar-field functions. We first allow an anisotropic fluid with radial and tangential pressures, whose diagonal components solve the diagonal field equations. The field equations are solved as a series expansion up to $\mathcal{O}(η^4)$ near the bounce at $η=0$. The resulting perfect fluid solution contains three arbitrary functions which are constrained by requiring the spacetime to asymptote to FLRW as $r\to\infty$. With suitable initial conditions preserving the parabolic structure of the bounce, the integration constant $d_0$ emerges as the true perturbative parameter: all perturbations vanish as $d_0\to0$. Finally, we find a small evolving horizon, $r_h\sim d_0$, which we interpret as the horizon of the central inhomogeneity. Its persistence through the bounce supports the interpretation of a black hole surviving the cosmological transition, and its evolution is not symmetric about $η=0$.