New Gravity Model Delivers Exact Solutions for Bouncing Universes

What if the Big Bang was never actually a bang at all, but rather the moment a shrinking universe stopped, rebounded, and began expanding again? A new theoretical study published in The European Physical Journal C suggests that this cosmic bounce scenario, long considered mathematically elusive, can be realized exactly within an extended framework of gravity known as f(T, T_G) gravity. The work, carried out by Abdul Malik Sultan and Ali Ahmad Sabir of the University of Okara in Pakistan together with Jackson Levi Said of the University of Malta, demonstrates that a wide family of singularity-free cosmological models emerges naturally from a theory in which gravity is mediated not by spacetime curvature, as in Einstein’s general relativity, but by torsion, a geometric property associated with the twisting of spacetime itself.

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