Black Hole Persistence in Scalar Tensor Theories

Technical Summary: Black Hole Persistence in Scalar Tensor Theories

Problem Statement
The paper addresses the theoretical question of whether black holes can persist through a non-singular cosmological bounce, transitioning from a contracting phase to an expanding phase. While bouncing cosmologies offer alternatives to the standard $\Lambda$CDM Big Bang model, the dynamics of localized inhomogeneities (such as black holes) within these evolving backgrounds are complex. Previous studies, such as those by Perez and Romero [24], utilized generalized McVittie metrics but often relied on ad hoc assumptions regarding the scale factor or imposed metric evolution to bypass under-determined field equations. The authors aim to construct a rigorous perturbative solution within a scalar-tensor theory of gravity to study the persistence of a central inhomogeneity through a bounce without imposing arbitrary ansatzes on the metric evolution.

Methodology
The authors employ a perturbative scheme within a generic scalar-tensor theory defined by the action $S = \int d^4x\sqrt{-g} [F(\phi)R - Z(\phi)\nabla_a\phi\nabla^a\phi - V(\phi)]$. The approach involves the following steps:

1. Background Solution: The zeroth-order ($O(\epsilon^0)$) solution is a spatially flat, bouncing Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime sourced by a radiation perfect fluid. The authors utilize known exact solutions from Generalized Brans-Dicke (GBD) theory [34, 35] where the scalar field $\phi$ and scale factor $a(\eta)$ exhibit a bounce at conformal time $\eta=0$.
2. Perturbative Ansatz: A central inhomogeneity is introduced via a generalized McVittie metric. The metric functions ($A$, $M$), energy density ($\rho$), pressures ($p_r, p_t$), and scalar field ($\Phi$) are expanded in terms of a perturbative parameter $\epsilon$. The leading order recovers the FLRW background, while the first-order ($O(\epsilon)$) terms encode the inhomogeneity.
3. Field Equations: The authors solve the linearized field equations derived from the scalar-tensor action. They initially allow for an anisotropic fluid (radial and tangential pressures) to satisfy the diagonal field equations. To close the system, they impose the condition that the perturbations represent a perfect fluid ($\tilde{p}_r = \tilde{p}_t$), which algebraically relates the metric perturbation $\tilde{a}$ to the scalar field perturbation $\tilde{\phi}$.
4. Series Expansion: The remaining coupled partial differential equations for the mass function $\tilde{m}$ and scalar perturbation $\tilde{\phi}$ are solved as a truncated series expansion up to $O(\eta^4)$ near the bounce ($\eta=0$).
5. Boundary and Initial Conditions:
   • Boundary: Asymptotic flatness to the FLRW limit is enforced as $r \to \infty$.
   • Initial Conditions: Conditions are imposed to preserve the parabolic nature of the bounce (vanishing first derivative of the scale factor perturbation at $\eta=0$). This constrains integration constants, revealing that the integration constant $d_0$ acts as the true perturbative parameter, replacing the artificial $\epsilon$.

Key Contributions and Results

• Perturbative Solution Construction: The paper derives a consistent first-order perturbative solution describing a central inhomogeneity embedded in a scalar-tensor bouncing cosmology. The solution is valid in the vicinity of the bounce and far from the central mass.
• Identification of the Perturbative Parameter: Through the imposition of initial conditions that maintain the bounce's parabolic structure, the authors demonstrate that the integration constant $d_0$ governs the magnitude of all perturbations. As $d_0 \to 0$, the inhomogeneity vanishes, and the spacetime returns to the pure FLRW background.
• Horizon Analysis: The authors calculate the location of the local horizon ($r_h$) by solving the condition $\theta(l)\theta(n) = 0$ for the expansion scalars of outgoing and ingoing null vectors.
  • Two distinct horizon branches emerge: a large cosmological horizon ($r_h^{(2)}$) and a small horizon ($r_h^{(1)}$) associated with the central inhomogeneity.
  • The small horizon scales as $r_h^{(1)} \sim d_0$, confirming its association with the localized mass.
  • The evolution of this small horizon is found to be asymmetric about the bounce time $\eta=0$, driven by a linear term in $\eta$.
• Persistence: The existence of a non-vanishing, evolving horizon for the central inhomogeneity through the bounce supports the interpretation that a black hole can survive the cosmological transition from contraction to expansion.

Significance and Claims
The paper claims to provide a mathematically justified framework for studying black hole persistence in modified gravity theories, specifically scalar-tensor theories, without relying on ad hoc metric prescriptions. By deriving the bounce and the inhomogeneity simultaneously from the field equations (rather than imposing the scale factor), the authors offer a more rigorous test of the "pre-big-bang black hole" hypothesis.

The results suggest that black holes formed or present during a contracting phase can persist through a non-singular bounce. The persistence is characterized by the survival of a small, evolving horizon that couples to the background cosmological dynamics. The authors note that while the perturbative scheme has limitations (e.g., regarding the precise nature of the horizon's trapping properties and foliation invariance), the results provide strong evidence for the survival of such objects through the bounce, potentially offering candidates for dark matter or seeds for supermassive black holes in the post-bounce universe. The work extends previous analyses by removing arbitrary assumptions about the scale factor and utilizing a consistent perturbative expansion within a scalar-tensor framework.