In a paper published in The European Physical Journal C, Shulan Li, Jian-Pin Wu and Xian-Hui Ge construct a new non-singular cosmological model by matching a Minkowski-core regular black hole to a modified version of the Oppenheimer–Snyder collapse framework. Its interior is described by a spatially flat Friedmann–Robertson–Walker metric, exactly the geometry of an expanding dust-filled universe, while its exterior is a static, spherically symmetric vacuum spacetime. The two regions are joined smoothly at the boundary surface using the Darmois–Israel junction conditions, which require the induced metric and the extrinsic curvature to be equal on either side of the interface. This makes the collapse model a natural bridge between black hole spacetimes and cosmological spacetimes.

The key innovation of the new work lies in the choice of exterior geometry. In the classical Oppenheimer–Snyder scenario, the exterior is the Schwarzschild metric, and the interior cosmology is the standard big bang cosmology of general relativity. In recent years, however, researchers have explored modified versions of the collapse model in which the exterior is described by a regular, that is non-singular, black hole metric. A previous study used the Hayward black hole, a regular black hole whose core asymptotically approaches de Sitter spacetime, and found that the matching interior cosmology is itself non-singular, replacing the big bang singularity with an asymptotically de Sitter epoch. The new paper asks the parallel question: what cosmology arises when the exterior is a regular black hole with a Minkowski core, a spacetime that becomes flat near its center?

The answer comes from substituting the metric function of the Minkowski-core black hole into the junction condition relating the interior Hubble parameter to the exterior geometry. This yields a modified Friedmann equation in which the square of the Hubble parameter equals the classical expression, eight pi rho over three, multiplied by an exponential suppression factor that depends on the energy density. The exponential factor has a dramatic effect at high densities: it forces the Hubble parameter toward zero as the density grows without bound, rather than allowing it to diverge as in classical general relativity. Notably, the authors require the black hole parameters to satisfy a specific condition so that the mass of the collapsing dust ball drops out of the cosmological evolution equation, a requirement that turns out to coincide with the condition that the maximum curvature of the black hole is independent of its mass.

Comparing the new model against three benchmarks, the classical standard cosmology, loop quantum cosmology, and the Hayward cosmology, the authors find that all three non-singular models share identical late-time behavior, recovering the standard expansion of the universe at low densities. But at very early times, in the high-density regime, they differ drastically. In loop quantum cosmology, the scale factor reaches a minimum value at a finite time and then bounces, with the rate of change of the scale factor vanishing and changing sign, and the energy density bounded above by a critical density. In the Hayward cosmology, the scale factor decreases toward zero asymptotically, never reaching it in finite time, while the square of the Hubble parameter approaches a constant, producing an asymptotically de Sitter epoch. In the new cosmology, the scale factor undergoes a very slow but persistent decrease into the infinite past, and the Hubble parameter approaches zero, meaning the universe asymptotically approaches flat Minkowski spacetime rather than a bounce or a de Sitter plateau.

The authors also computed the Ricci and Kretschmann curvature scalars for both the cosmological and black hole spacetimes. In the classical case, these scalars diverge as the scale approaches zero, reflecting the singularity. In the new model, by contrast, the curvature scalars first increase, then rapidly decrease, and finally vanish as the scale tends to zero, mirroring the behavior of the Minkowski-core black hole near its center. Importantly, in all three non-singular cosmologies the affine parameter along timelike geodesics spans the full range from minus infinity to plus infinity, meaning these spacetimes are timelike geodesically complete. The big bang singularity is not merely softened; it is replaced by an infinite past in which the universe exists smoothly, with vanishing curvature.

Extending the analysis to a universe filled with a massive scalar field rather than dust, the authors show that inflation is a generic feature in all three non-singular frameworks. Phase trajectories in the scalar field’s phase space, starting from widely different initial conditions, are all drawn toward an inflationary phase characterized by slow-roll behavior, and then undergo damped oscillations before settling at a stable fixed point corresponding to zero energy density. In the new cosmology, the scalar field exhibits a striking symmetry between its behavior in the asymptotic past and future: trajectories show continuous damped oscillations at both ends of cosmic time, becoming denser in the phase plane both far from the origin and near it. This mirrors the geometric symmetry of the model, in which the universe approaches Minkowski spacetime in both the remote past and the remote future.

The authors also examined more realistic scenarios involving multiple coexisting fluids, including radiation, dust, and a positive cosmological constant. Without a cosmological constant, the new cosmology is asymptotically Minkowski in both limits. When a positive cosmological constant is included, the exponential suppression term at high densities kills the dynamical matter contribution in the infinite-past limit, leaving the square of the Hubble parameter approaching the constant Lambda over three. The early-time asymptotics therefore shift from Minkowski to de Sitter, and the symmetry between past and future is preserved but in the de Sitter form. The authors contrast this with the Hayward cosmology, whose intrinsic de Sitter core adds to the externally supplied cosmological constant, producing a larger effective cosmological constant in the infinite past. In neither case does the cosmological constant reintroduce a primordial singularity.

A crucial question is whether the asymptotic Minkowski past is a robust feature or an artifact requiring fine-tuned initial conditions. Using a dynamical systems analysis with an inverse-density variable, the authors show that the Minkowski limit corresponds to a fixed point of the homogeneous dynamics. Because the exponential factor makes all derivatives of the evolution function vanish at the fixed point, standard linear stability analysis is inconclusive, but a direct sign argument suffices: the system evolves away from the fixed point in the forward time direction, making it a repeller, and therefore, by time reversal, an attractor in the backward direction. All trajectories converge to the Minkowski state as time tends to minus infinity, regardless of initial data. The same conclusion holds for the cosmological-constant-extended version, though the convergence rate changes from logarithmic to exponential and the asymptotic Hubble parameter becomes nonzero. The asymptotic-past state is thus a robust attractor requiring no extreme fine-tuning.

The authors note that the qualitatively distinct very-early-time behaviors of the three non-singular models could, in principle, leave observable imprints in the cosmic microwave background and in primordial gravitational waves, with the quasi-Minkowski pre-inflationary phase of the new model potentially suppressing large-scale perturbation amplitudes. They caution, however, that quantitative predictions require a full treatment of linear perturbations, which lies beyond the scope of the present homogeneous analysis. The work remains entirely theoretical and numerical, with no experimental data involved. Nevertheless, it offers a compelling new entry in the family of singularity-free cosmologies, one in which the universe neither begins at a finite-time singularity nor ends in one, but instead unfolds smoothly from an infinite, flat, and quiet past toward an equally tranquil future.

Subject of Research: A theoretical non-singular cosmological model derived from matching a Minkowski-core regular black hole in a modified Oppenheimer–Snyder collapse framework

Article Title: Non-singular cosmologies matching regular black holes

Article References: Li, S., Wu, J.-P., & Ge, X.-H. (2026). Non-singular cosmologies matching regular black holes. The European Physical Journal C, 86(10), 1236. https://doi.org/10.1140/epjc/s10052-026-16398-1

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16398-1

Keywords: cosmology, regular black holes, Oppenheimer-Snyder collapse, big bang singularity, loop quantum cosmology, Friedmann equation, inflation, dynamical systems, Minkowski spacetime, de Sitter, theoretical physics, spacetime singularities