The research, carried out by B. Yildirim and A. A. Coley of the Department of Mathematics and Statistics at Dalhousie University in Halifax, Canada, approaches the problem through the mathematics of scalar-tensor gravity. This class of theories extends Einstein’s general relativity by adding a scalar field that dynamically couples to the curvature of spacetime. Scalar-tensor gravity has long attracted attention in cosmology because it provides a natural mathematical setting in which nonsingular bouncing universes can be realized. In such models, the scalar field’s evolution can drive a contraction to a halt and trigger a rebound, replacing the dreaded big bang singularity with a smooth, finite transition. But while bouncing cosmologies have been studied extensively in homogeneous settings, the inclusion of localized objects such as black holes makes the field equations dramatically harder to solve.

To make progress, the authors adopted a perturbative strategy, treating the black hole as a small deviation from an otherwise perfectly uniform cosmos. At leading order in their perturbative scheme, controlled by a small parameter epsilon, the solution is a spatially flat Friedmann-Lemaître-Robertson-Walker, or FLRW, spacetime undergoing a bounce, sourced by a perfect fluid of radiation. This background captures the essence of a bouncing cosmology in the simplest possible form: the universe contracts, reaches a minimum size at a moment identified with a conformal time coordinate eta equal to zero, and then re-expands. This leading-order solution respects what the authors call the parabolic structure of the bounce, a smoothness condition on how the scale factor behaves at the turning point.

At the next order in the expansion, the team embedded a central inhomogeneity into the bouncing background using a generalized McVittie geometry. The original McVittie solution, constructed in 1933, is a celebrated exact solution of general relativity that describes a mass concentrated at the center of an expanding universe, providing a mathematically tractable bridge between black hole physics and cosmology. By generalizing this construction to the scalar-tensor setting and treating it perturbatively, Yildirim and Coley encoded the gravitational imprint of a localized compact object within the contracting and rebounding cosmos. The perturbations appear as first-order corrections to the metric and to the scalar field, and the coupled field equations were solved as a series expansion carried up to fourth order in the parameter eta near the bounce.

A central technical challenge arose from the nature of the matter content near the inhomogeneity. In the vicinity of a concentrated mass, the stress-energy generically becomes anisotropic, meaning that pressure differs along the radial direction compared with the tangential directions. The authors therefore first allowed an anisotropic fluid with separate radial and tangential pressures, whose diagonal components suffice to solve the diagonal components of the field equations. They then imposed the physically motivated condition that the stress-energy reduce to a perfect fluid, one with a single isotropic pressure, far from the center. The resulting perfect fluid solution contains three arbitrary functions, which are constrained by demanding that the spacetime smoothly asymptote to the homogeneous FLRW background as the radial coordinate tends to infinity, ensuring that the black hole’s influence fades with distance as it must.

With suitable initial conditions chosen to preserve the parabolic structure of the bounce, a remarkable simplification emerged. The solution’s integration constants consolidate into a single quantity, denoted d0, which the authors identify as the true perturbative parameter of the problem. When d0 is set to zero, every perturbation vanishes and the spacetime reverts exactly to the homogeneous bouncing FLRW universe. When d0 is small but nonzero, a localized inhomogeneity, and with it a small evolving horizon, appears in the geometry. This clean parametrization means the entire structure of the black hole embedding is controlled by one number, allowing the authors to track precisely how the compact object’s gravitational field responds to the cosmic contraction and rebound.

The key result of the analysis concerns the fate of that horizon. Yildirim and Coley find a small evolving horizon whose radius scales linearly with the perturbative parameter, roughly as d0, and which they interpret as the horizon of the central inhomogeneity. Crucially, this horizon persists through the bounce at eta equals zero, supporting the interpretation that a black hole present before the cosmological transition survives it and continues to exist in the expanding universe on the other side. Intriguingly, the evolution is not symmetric about the bounce: the horizon’s behavior on the contracting side differs from its behavior on the expanding side, suggesting that the cosmic reversal leaves a subtle imprint on the object even as it survives. In a cosmological context, such findings resonate with long-standing speculations about black holes from a previous cosmic epoch, sometimes discussed in connection with ideas about the origin of supermassive black holes and the possible relics of a pre-bounce universe.

The authors are careful about what their construction does and does not establish. The small horizon they track is a future outer trapping horizon, a locally defined surface characterized by the convergence properties of outgoing and ingoing light rays, and it behaves as such in a two-sided neighborhood of the bounce. Because the entire construction is local and perturbative, it does not demonstrate the existence of a global event horizon, the teleological boundary beyond which nothing can escape to infinity. The distinction matters in a dynamical spacetime, where global horizons are notoriously difficult to define and can depend on the entire future evolution of the cosmos. Still, the persistence of a local trapping horizon through a nonsingular bounce is a nontrivial and suggestive result, indicating that the mechanisms of black hole formation and survival may be more robust under extreme cosmological conditions than simpler arguments had implied.

The work also fits into a broader effort to understand how inhomogeneities behave in bouncing scenarios. Previous studies have explored whether structure formation is possible through a bounce, whether primordial black holes can survive ekpyrotic contractions, and how numerical simulations of nonsingular bouncing spacetimes handle black holes and their horizons. By providing an analytic, perturbative solution in scalar-tensor gravity, the Dalhousie team adds a complementary tool to this mostly numerical and heuristic literature. The mathematical framework, combining generalized McVittie geometries with a controlled expansion around a radiation-dominated bounce, offers a concrete laboratory in which questions about horizons, matter anisotropies, and scalar-field dynamics can be addressed with precision. While the analysis is idealized, assuming spherical symmetry, a radiation fluid, and a small perturbation strength, it demonstrates that black hole persistence through a bounce is a mathematically consistent possibility in a well-motivated class of gravitational theories, bringing the speculative picture of black holes bridging cosmic epochs one step closer to rigorous footing.

Subject of Research: Perturbative scalar-tensor cosmology modeling black hole survival through a nonsingular bouncing universe

Article Title: Black hole persistence in scalar-tensor theories

Article References: Yildirim, B., & Coley, A. A. (2026). Black hole persistence in scalar-tensor theories. General Relativity and Gravitation, 58(9), Article 111. https://doi.org/10.1007/s10714-026-03610-6

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03610-6

Keywords: black holes, scalar-tensor gravity, bouncing cosmology, general relativity, McVittie metric, trapping horizon, FLRW spacetime, cosmic bounce, perturbation theory, theoretical cosmology, nonsingular cosmology, gravitational physics

The foundation of the new analysis lies in teleparallel gravity, an alternative formulation of Einstein’s theory in which the central dynamical object is the tetrad, a field that attaches a local frame to every point in spacetime. In this picture, the torsion scalar T replaces the Ricci scalar R of general relativity, and the simplest version of the theory, known as the teleparallel equivalent of general relativity, reproduces all the standard predictions of Einstein’s framework. The advantage of the teleparallel route is technical: its field equations remain second order, avoiding the mathematical pathologies that plague many higher-derivative extensions. Extending this theory by adding an arbitrary function of the torsion scalar yields f(T) gravity, which has been widely studied as a way to explain the accelerated expansion of the late universe without invoking a cosmological constant or dark energy.

But f(T) gravity alone, the authors argue, cannot capture all of the higher-order torsional corrections that matter in the extreme gravity regimes of the very early universe. To fill this gap, theorists introduced the teleparallel analogue of the Gauss-Bonnet invariant, denoted T_G, a four-dimensional topological quantity built from combinations of the contortion tensor that acquires dynamical significance when coupled to nontrivial functions. The resulting theory, f(T, T_G) gravity, keeps the field equations at second order while enriching the gravitational dynamics with genuinely new terms that become important precisely where classical descriptions break down, namely near the hypothetical moment of the Big Bang singularity.

The central achievement of the new paper is an exact analytical reconstruction. Rather than solving the modified Friedmann equations numerically, the team began with six carefully chosen bouncing scale factors, the mathematical functions that describe how the size of the universe evolves in time. These included the symmetric bounce, in which contraction mirrors expansion around the bounce point; the superbounce, in which a sharply contracting cosmos reverses without a singularity; the oscillatory bounce, which produces repeated cycles of contraction and expansion; the matter bounce, rooted in loop quantum cosmology and capable of generating a scale-invariant spectrum of primordial perturbations; the Little Rip class of phantom-like evolutions; and the ekpyrotic bounce, in which slow contraction driven by a steep negative potential smooths the cosmos before a controlled transition to expansion.

For each scenario, the researchers expressed the scale factor as a function of the torsion scalar and examined three distinct classes of gravitational Lagrangian. The first, an additive separable form written as g(T) plus h(T_G), lets the torsion and Gauss-Bonnet sectors contribute independently and splits the Friedmann equation into two manageable ordinary differential equations. The second, a multiplicative coupling of the form T_G multiplied by g(T), captures direct interactions between the two geometric invariants. The third, written as T plus T_G g(T), preserves the standard teleparallel theory at low energies while allowing higher-order corrections to dominate near the bounce. In several cases the reconstructed solutions involve sophisticated special functions, including the Gauss hypergeometric function, the error function, and the incomplete Gamma function, reflecting the rich structure hidden inside these torsion-based models.

The analysis also tracked the key physical quantities through the bounce. Plotting the scale factor, the Hubble parameter, the energy density, and the effective pressure for every model, the authors verified that each scenario remains finite at the transition: the Hubble parameter passes smoothly from negative values during contraction, through zero at the bounce, to positive values during expansion, while the energy density and pressure stay bounded. This regular behavior stands in sharp contrast to standard cosmology, where curvature invariants and densities diverge at the initial singularity, signaling the breakdown of general relativity itself at the moment when the theory is needed most.

A crucial test of physical viability comes from the vacuum condition, the requirement that the reconstructed gravitational Lagrangian reduce correctly in the absence of matter, so that flat Minkowski spacetime is recovered when the torsional invariants vanish. The team found that this criterion discriminates sharply between the model classes. The multiplicative coupling and its minimally extended variant generally satisfy the constraint for the symmetric bounce and the superbounce, while the separable additive model fails, producing a trivial, physically unacceptable Lagrangian. For the matter bounce, only the additive model survives the vacuum test, while the coupled forms require setting the matter density parameter to zero, a condition that undermines a scenario fundamentally driven by dust-like matter. For the ekpyrotic case, the vacuum condition reduces one model to a purely torsion-dominated contribution, revealing a tension between mathematical consistency and the matter content the scenario requires.

The broader significance of the work lies in its unifying scope. Previous studies of bouncing cosmologies in modified gravity, including landmark analyses of superbounce and loop-quantum ekpyrosis in curvature-based F(R), F(G), and F(T) theories, typically treated single backgrounds with numerical or approximate reconstruction methods. By contrast, the new analysis derives exact reconstruction schemes for six distinct cosmological histories within one consistent teleparallel Gauss-Bonnet framework, allowing a direct, side-by-side comparison of their gravitational structures. The authors emphasize that the resulting models can describe a smooth transition from contraction to expansion without exotic matter violating energy conditions, positioning f(T, T_G) gravity as a serious competitor to inflation as a description of the earliest moments of cosmic history.

Of course, mathematical elegance is not the same as observational truth. The authors note that the reconstructed Lagrangians are reliable primarily near the bounce and must ultimately be tested against data. A natural next step, they suggest, is to compare the predicted Hubble expansion profiles with observations from cosmic chronometers, type Ia supernovae, and baryon acoustic oscillation measurements, which would prune the wide family of solutions down to a smaller set of observationally viable models. Extensions of the analysis should also address the stability of the solutions under perturbations and the shape of the primordial perturbation spectrum they generate. If those tests succeed, the picture they paint is striking: a universe that never began with a singularity, but with a quiet, torsion-driven rebound from a previous contracting epoch, described exactly by equations that Einstein’s own theory could never provide.

Subject of Research: Exact cosmological bouncing solutions in modified teleparallel f(T, T_G) gravity

Article Title: Exact cosmological bouncing solutions in (f(T,T_G)) gravity

Article References: Sultan, A. M., Sabir, A. A., & Said, J. L. (2026). Exact cosmological bouncing solutions in $$f(T,T_G)$$ gravity. The European Physical Journal C, 86(9), Article 1061. https://doi.org/10.1140/epjc/s10052-026-16154-5

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16154-5

Keywords: f(T, T_G) gravity, teleparallel gravity, bouncing cosmology, initial singularity, Gauss-Bonnet term, torsion scalar, matter bounce, ekpyrotic universe, oscillatory bounce, superbounce, modified gravity, early universe