Black Hole Persistence in New General Relativity

Abstract: We investigate whether black holes can persist through the bounce with a minimal scale factor in a non-singular cosmology, whereby black holes from a previous contracting phase survive into the current expanding one. We do so by studying a generalized McVittie spacetime which embeds a spherically symmetric black hole in a positive spatial curvature bouncing FLRW cosmological background within the modified theory of teleparallel new general relativity. There are no further assumptions on the spacetime (e.g., on the form of the scale factor) initially, and the local evolution is derived from the field equations of the theory, utilizing a perturbative scheme which is valid ``near the bounce". To leading order we obtain a simple bounce solution similar to that in general relativity for a closed FLRW model with a positive cosmological constant, but in which the curvature term in the Friedmann equation is re-normalized within new general relativity. Qualitatively the minimum of the bounce at $t=0$ changes, but near the bounce the evolution remains symmetric. The central inhomogeneity evolves at higher perturbative orders, where the details depend on the arbitrary constants of the perturbative solution. Hence the evolution of the local horizon during the bounce changes qualitatively, where the effects depend on the signs of the perturbation, and the symmetry across the bounce is disrupted due to a linear term.

• The paper demonstrates that black holes can persist through a non-singular bounce in a closed FLRW universe by leveraging teleparallel New General Relativity.
• The methodology uses a perturbative expansion of the McVittie solution around the bounce, ensuring regularity and decay of inhomogeneities.
• The findings highlight that positive spatial curvature and re-normalized curvature parameters are crucial for maintaining a distinct black hole horizon across cosmic cycles.

Black Hole Persistence in Non-Singular Bouncing Cosmologies within New General Relativity

Introduction and Motivation

The persistence of black holes through non-singular bounces in cosmological models is a critical question for both theoretical and observational cosmology. Bouncing cosmologies—where the universe transitions from a contracting to an expanding phase through a finite, nonzero minimal scale factor—arise as alternatives to $\Lambda$CDM to avert the initial singularity problem inherent in standard cosmology. Models featuring such bounces have been realized through a variety of mechanisms, including modifications of gravity (e.g., Horndeski, $F(R)$, teleparallel theories), violation of the null energy condition (NEC), or quantum gravitational effects.

The fate of black holes during bounces has notable implications for the origin of large-scale structures, dark matter (via primordial or persistent black holes), and the formation of supermassive black holes (SMBHs) at high redshift, as increasingly evidenced by JWST observations. The scenario wherein black holes persist ("pre-big-bang black holes," PBBBHs) across cosmic cycles could influence structure formation and offer signatures distinguishing them from black holes formed post-bounce.

This work addresses the dynamics of black holes embedded in a closed positively curved FLRW bouncing cosmology within the framework of teleparallel New General Relativity (NGR), focusing particularly on the evolution and persistence of local black hole horizons near the bounce.

Theoretical Framework: McVittie Solutions and NGR

The McVittie solution, and especially its generalizations with time-dependent central mass, serves as a paradigm for embedding a black hole in an (expanding or contracting) cosmological background. While previous studies have considered spatially flat ($k=0$) backgrounds and employed ad hoc scale factors or mass functions, the present work instead pursues an exact, locally consistent construction utilizing the field equations of NGR with positive spatial curvature.

NGR is a teleparallel theory where the dynamical variable is the tetrad, and gravity is sourced by torsion rather than curvature. The theory includes new parameters (e.g., $b_3$ in the action), allowing for modifications of the effective curvature terms and further flexibility to match observations. The challenge is to treat a central inhomogeneity (the black hole) as a perturbation of the homogeneous isotropic bouncing background and study its effect on the evolution of local horizons near the bounce.

Methodological Contribution: Perturbative Construction Around the Bounce

The analysis applies a local perturbative scheme, treating the general McVittie metric as the seed and expanding both in the inhomogeneity (via a parameter $\epsilon$) and near the bounce (time $t=0$). To leading order, the background is described by a closed ($k=+1$) FLRW solution exhibiting a non-singular bounce at $t=0$, with the minimal scale factor $a_+$ controlled not only by usual cosmological constants but also by the NGR parameter $b_3$.

The perturbative solution involves expanding the metric functions and matter variables to first order in the central inhomogeneity. The field equations at this order (after imposing a local perfect-fluid equation of state) yield a unique, closed system for the time-dependent profiles of the perturbations. These equations are solved analytically up to quadratic order in $F(R)$0, with all integration constants fixed by requirements of regularity (notably, a well-behaved GR limit as $F(R)$1), decay of the perturbation at large "radius," and the correct asymptotics for the black hole mass function and central inhomogeneity.

A crucial result is that only the $F(R)$2 case (closed universe) naturally allows for an inhomogeneous bounce in this framework; for $F(R)$3 or $F(R)$4, no regular central inhomogeneity persists through the bounce within the perturbative scheme.

Key Results

1. NGR Bounce Solution and Re-Normalized Curvature:

The scale factor for the closed FLRW bounce in NGR is analogous to the GR solution but with the curvature term re-normalized: $F(R)$5. This allows tuning of the minimal scale factor $F(R)$6 and potentially alleviates tensions with observations.

2. Evolution of the Central Inhomogeneity:

The central inhomogeneity evolves through the bounce, with its effects given by perturbative corrections parameterized by an amplitude $F(R)$7 (the effective "smallness" of the black hole mass relative to the background), equation-of-state parameter $F(R)$8, and the NGR parameter $F(R)$9.

3. Dynamics of the Local Horizon:

To leading order, the local horizon radius $k=0$0 follows the (symmetric) dynamics of the background FLRW solution:

$k=0$1

where the $k=0$2 coefficient ensures a contraction-expansion profile through the bounce. The black hole perturbation alters the dynamics at higher orders (via $k=0$3), introducing an asymmetric component across the bounce, the sign and magnitude of which depend on $k=0$4 and $k=0$5.

4. Regularity and Boundary Conditions:

The construction ensures that the perturbations are regular and decaying towards the largest spatial radius $k=0$6 (in isotropic coordinates for $k=0$7), thus preserving the FLRW asymptotics. The solution also smoothly recovers the GR case ($k=0$8), demonstrating the consistency of the perturbative method.

5. Theoretical Consistency:

Conservation of the energy-momentum tensor is preserved at the perturbative level, and the field content matches that of an imperfect fluid transitioning to a perfect fluid at the bounce. The appearance of linear-in-$k=0$9 terms in the corrected horizon location corresponds to disrupted time symmetry induced by the inhomogeneity.

Contradictory Claims and Comparison to Previous Work

The analysis demonstrates the possibility of black hole persistence through a non-singular bounce, with the key claim that for a closed, positively curved universe in NGR, black holes can cross the bounce with modifications to their horizon radius and symmetry properties. This result challenges the prior notion—applicable to some GR and Galileon settings—that the homogeneous bounce precludes black hole survival or that the horizon merges with the cosmological horizon, resulting in a naked singularity.

Past works (e.g., Perez & Romero; Corman et al.) examined either numerically imposed bounces or constructed solutions with prescribed (and sometimes ad hoc) metrics and scale factors, typically for $b_3$0. This paper achieves local analytic control near the bounce and clarifies that spatial curvature ($b_3$1) is essential for black hole survival, at least in the context of the NGR-based bounce.

Implications and Future Perspectives

The results extend the catalogue of settings in which black holes persist across bounces and provide a model for investigating the role of PBBBHs in early universe cosmology—pertinent to the formation of SMBHs, seeding of large-scale structures, and potentially dark matter via leftover black hole populations.

From a theoretical standpoint, the perturbative machinery developed herein is generalizable to other modified gravity scenarios (such as scalar-tensor or $b_3$2 gravity) and lays groundwork for fully non-linear or numerical investigations of black hole evolution through cosmological bounces.

Future research directions include:

• Analysis of nonlinear black hole perturbations and merger dynamics during the bounce.
• Application of the method to alternative bouncing mechanisms (e.g., in Horndeski, Loop Quantum Cosmology).
• Quantitative confrontation with observational constraints (e.g., matching to high redshift SMBH observations).
• Further study of gravitational wave signatures from black holes surviving from previous cycles.

Conclusion

This work rigorously constructs a locally valid model of a black hole embedded in a closed, bouncing FLRW universe within NGR, demonstrating black hole persistence through the bounce without recourse to ad hoc assumptions on the background or metric functions. The mechanism requires positive spatial curvature and is shown to be consistent, regular, and adaptable to observational criteria. The evolution of the local horizon is modified in detail by the perturbations, with potential asymmetric features across the bounce. The methodology and results provide both a technical advance in cosmological inhomogeneous solutions in modified gravity and open a pathway towards resolving outstanding questions on the cosmological role of black holes in bouncing or cyclic models.