⚛️ general relativity

Regular Cosmologies: Three Distinct Routes

This paper investigates three distinct prescriptions for extending regular black hole metrics to cosmology, finding that while all models successfully replace the initial singularity with a de Sitter phase and avoid phantom energy behavior, they diverge at late times and cannot simultaneously account for both inflation and dark energy using a single regularization length.

Original authors: Sergio Bravo Medina, Marek Nowakowski

Published 2026-09-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sergio Bravo Medina, Marek Nowakowski

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe began in a state of unimaginable density and heat, a moment where the known laws of physics break down and space-time itself seems to tear apart. For decades, physicists have been haunted by this "singularity," a point of infinite density that marks the beginning of time in the standard Big Bang model. Just as a collapsing star is predicted to crush itself into a similar point of infinite density at the center of a black hole, the early universe is predicted to have started from one. To fix this, many researchers have explored the idea of "regular" black holes. These are theoretical objects where the crushing center is replaced by a smooth, finite core, much like a dense ball of energy rather than a mathematical point of nothingness. This core behaves like a patch of space that is pushing outward, a state known as de Sitter space, which prevents the collapse from ever becoming truly infinite.

The question that drives this new research is whether the same trick used to save the center of a black hole can also save the beginning of the universe. If the early universe was indeed a regular, non-singular state, what would its history look like? Would it still expand and cool in the way we observe today, or would the rules change? A team of researchers set out to test this idea by taking six different mathematical models of these regular black holes and asking a simple question: if we treat the entire universe as if it were made of the "stuff" inside one of these cores, how does it evolve? They did not just look at the math; they followed three distinct ways of translating the static geometry of a black hole into the expanding story of a cosmos, effectively running the universe forward in time to see if it matches what we actually see in the sky.

The researchers found that when they applied the most direct translation of these models, the universe behaves in a very specific way. It starts in a phase of rapid, smooth expansion, driven entirely by the finite energy of the core. This looks exactly like the period of "inflation" that cosmologists believe happened fractions of a second after the Big Bang, a time when the universe grew exponentially fast. However, as the universe expands, this driving force fades away, and the universe slows down, eventually settling into a state dominated by dust-like matter. Crucially, this model does not produce the late-time acceleration we see today, where the expansion of the universe is speeding up again due to "dark energy." Instead, the model suggests that the regular core is a one-time event: it provides the initial push for inflation but then disappears, leaving the universe to coast. The researchers showed that for five of the six models they tested, this early inflationary phase is mathematically clean and free of singularities, meaning the universe could have existed forever in the past without ever hitting a point of infinite density, though it would still have a limit to how far back you can trace a light beam.

The study also tackled a major puzzle in modern cosmology: recent observations from the Dark Energy Spectroscopic Instrument (DESI) have hinted that the "dark energy" driving the current acceleration might be changing over time, potentially crossing a critical threshold where the universe's expansion accelerates faster than a simple constant would allow. The researchers proved that the smooth, regular cores they were studying cannot do this. Because the density of these cores simply decreases as the universe grows, they can never cross that threshold. To get the behavior suggested by the new data, the universe would need a "shell" of energy—a bump in the density profile that goes up and then down—which a standard regular core does not have. This means that if the new data is correct, the dark energy we see today cannot come from the same smooth, regular mechanism that might have fixed the Big Bang.

When the team tried two other ways of connecting the black hole models to the expanding universe, the results were slightly different but told a consistent story. These alternative methods treated the core's energy as a "running vacuum," a type of energy that changes its strength as the universe grows. In these scenarios, the models could produce a form of dark energy that fits the current data reasonably well, matching the observed rate of acceleration. However, even with these adjustments, the models still could not cross the critical threshold into the "phantom" regime suggested by the latest measurements. The researchers found that while the different methods agreed on the early, inflationary phase, they disagreed on the details of the late-time expansion. This disagreement is significant because it shows that our predictions for the future of the universe depend heavily on exactly how we choose to translate the geometry of a black hole into the language of cosmology.

Ultimately, the paper paints a picture of a universe that is regular and safe from the infinite densities of the past, but one that requires a specific kind of structure to explain the present. The smooth, regular cores act as a perfect engine for the beginning, providing a clean start to the cosmos, but they are too simple to explain the complex, changing nature of dark energy we see today. If the universe is indeed behaving in the way the latest data suggests, it implies that the dark energy filling space is not a simple, smooth vacuum but something more complex, perhaps containing a shell of energy that a standard regular black hole does not possess. The work does not solve the mystery of dark energy, but it draws a sharp line in the sand: the smooth, regular solutions that fix the Big Bang cannot be the same thing that drives the current acceleration of the universe.

Technical Summary: Regular Cosmologies: Three Distinct Routes

Problem Statement
The paper addresses the problem of extending the concept of "regular black holes"—spacetimes that replace the central curvature singularity with a non-singular de Sitter core—to early universe cosmology. While regular black holes are well-studied as static solutions where the mass function $m(r) \sim r^3$ near the origin, translating this into a dynamical Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology is not unique. The central challenge is determining how to map the position-dependent effective cosmological term, $\Lambda_{\text{eff}}(r) = 8\pi G \rho_{\text{MS}}(r)$ (where $\rho_{\text{MS}}$ is the Misner-Sharp energy density), onto the expanding universe. The authors investigate whether this regularisation mechanism can simultaneously resolve the Big Bang singularity and account for late-time cosmic acceleration (dark energy), or if these phenomena require distinct physical scales.

Methodology
The authors analyze six distinct regular black hole metrics: Hayward, Bardeen, Dymnikova, Nicolini–Smailagic–Spallucci (NSS), Jusufi–Singleton (JS), and Bonanno–Reuter. They apply three distinct "routes" (prescriptions) to map these static metrics to cosmological dynamics:

Route N (Newtonian/McCrea–Milne): A heuristic construction treating the universe as a homogeneous sphere of the regular black hole "fluid." The comoving radius is identified with the scale factor ( $R \propto a$ ). The Friedmann equation is derived from Newtonian energy balance using the potential defined by the metric function $f(r)$ . This route treats the regular black hole as the sole content of the universe, with no separate matter or radiation components.
Route A/B (Running Vacuum): The effective cosmological term $\Lambda_{\text{eff}}(r)$ is promoted to a function of the scale factor by identifying the metric radius with the Hubble radius ( $r \to r_H = H^{-1}$ ). This term is added to a standard background of matter and radiation, treating the core as a "running vacuum" (similar to Shapiro-Solà models). Route A assumes a constant equation of state $w=-1$ with energy transfer, while Route B treats the vacuum as a self-conserved fluid with a variable equation of state.
Route C (Quasi-local/Faraoni): The source of expansion is defined by the Misner-Sharp-Hernandez mass enclosed within the cosmological apparent horizon ( $R_{AH} = H^{-1}$ ). This defines a quasi-local mean density that sources the Friedmann equation.

The study involves solving the resulting Friedmann equations analytically or numerically, calculating cosmographic parameters (deceleration $q$ , jerk $j$ , Om diagnostic), and comparing the predictions against DESI DR2 data and the $\Lambda$ CDM model.

Key Contributions and Results

Route N: Inflationary Core, No Late-Time Dark Energy
- For metrics with a de Sitter core ( $m(r) \sim r^3$ ), Route N yields a universe that begins in a non-singular de Sitter phase ( $H \to \text{const}$ , $a \propto e^{Ht}$ ) and transitions smoothly to a dust-dominated era ( $a \propto t^{2/3}$ ).
- The Hayward metric admits a closed-form analytic solution for the scale factor. Other metrics require special functions or numerical integration.
- Crucial Finding: Route N describes a primordial inflationary model with a "graceful exit" to dust. It does not produce late-time acceleration. The single regularisation length $\ell$ sets the inflationary scale; attempting to use it for late-time dark energy fails due to scale hierarchy conflicts (the term redshifts too rapidly or requires unphysical mass scales).
- The model is curvature-regular but past null incomplete, consistent with the Borde-Guth-Vilenkin theorem for inflationary phases.
The Crossing Criterion: Monotone Cores Cannot Cross $w = -1$
- The authors establish a structural theorem: for any metric with a monotonically decreasing Misner-Sharp density (a standard feature of regular cores satisfying the weak energy condition), the reconstructed effective equation of state $w_{\text{eff}}$ can never cross the phantom divide ( $w = -1$ ).
- A crossing $w < -1$ (phantom behavior) requires a local maximum in the density profile—a "density shell." Among the six metrics, only the Jusufi–Singleton metric possesses such a shell, but only in a self-energy-dominated regime ( $M/\ell \gtrsim 15$ ) outside the standard regular core branch.
- Consequently, the dynamical dark energy suggested by recent DESI data (which hints at a phantom crossing) cannot be generated by a standard monotone regular core.
Routes A/B and C: Late-Time Behavior and Data Comparison
- Route A/B: When applied to late times with matter/radiation, these routes yield "thawing quintessence" models. They reach the DESI $w_0$ band but fail to match the magnitude of $|w_a|$ (the time variation) required by the data. The Gaussian profiles (Dymnikova, NSS) sit closer to $\Lambda$ CDM than the rational profiles (Hayward, Bardeen).
- Route C: This quasi-local approach yields results systematically closer to $\Lambda$ CDM than Route A/B. The source term decays as $H^3$ at late times (due to mass saturation), whereas Route A/B decays faster ( $H^4$ to $H^6$ ).
- Prescription Dependence: While all three routes agree on the early-time de Sitter core, they diverge significantly at late times. The spread in predicted parameters ( $\Delta w_0 \approx 0.17$ for rational cores) indicates that late-time predictions are highly sensitive to the choice of mapping (prescription) rather than just the underlying metric.
Validation with Quantum Gravity
- The authors validate Route N by applying it to the quantum Oppenheimer-Snyder metric (derived from Loop Quantum Cosmology). Route N correctly reproduces the known LQC bounce equation, confirming the method's consistency with established quantum gravity results in that specific limit.

Significance and Claims
The paper claims that regular black hole cosmology is primarily a theory of the early universe. The same mechanism that removes the central singularity (the de Sitter core) naturally provides a primordial inflationary phase with a finite, non-singular beginning.

However, the authors modestly conclude that this framework does not unify inflation and dark energy into a single scale. The regularisation length $\ell$ that cures the singularity sets the inflationary scale, while the late-time acceleration requires a separate mechanism or scale. Furthermore, the framework imposes a strict barrier: standard regular cores (monotone density) cannot explain the phantom crossing ( $w < -1$ ) hinted at by DESI. If such a crossing is confirmed, it would imply that the dark sector is not a regular core but a "hairy" black hole with a density shell, potentially violating energy conditions.

The work serves as a classification of how different regularisation mechanisms (rational vs. Gaussian vs. self-energy) behave under different cosmological mappings, highlighting that while the early universe physics is robust across routes, late-time cosmology is deeply prescription-dependent.

Technical Summary: Regular Cosmologies: Three Distinct Routes

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