⚛️ general relativity

Kruglov-entropy cosmology from apparent-horizon thermodynamics: background dynamics and observational constraints

This paper investigates two cosmological models derived from Kruglov's nonadditive entropy at the apparent horizon, finding that while the second model remains phenomenologically similar to $\Lambda$ CDM, the first is strongly disfavored by observational data, demonstrating that the specific thermodynamic implementation of generalized entropy critically determines the viability of such cosmologies.

Original authors: Candrasyah Muhammad, Burin Gumjudpai, Nandan Roy, Pornrad Srisawad

Published 2026-09-28

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CC0 1.0

Original authors: Candrasyah Muhammad, Burin Gumjudpai, Nandan Roy, Pornrad Srisawad

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 is expanding, and not just expanding, but speeding up. Since astronomers first noticed this acceleration decades ago, the leading explanation has been a mysterious force called dark energy, often modeled as a simple, unchanging cosmological constant. This standard model works remarkably well with most observations, yet it leaves physicists with nagging questions about why this force exists and why it has the specific strength it does. To find answers, researchers look at the very edge of what we can see: the cosmic horizon. In the language of modern physics, the boundary of our observable universe acts like a surface that holds information, much like a black hole does. By treating this boundary as a thermodynamic object with entropy, or disorder, scientists can derive the rules that govern how the universe expands. This approach suggests that the expansion we see might not be driven by a mysterious substance at all, but rather by the fundamental way information is stored on the cosmic horizon.

A recent study by a team of researchers in Thailand explores two different ways to apply a specific, non-standard formula for this horizon entropy to the history of the universe. They focused on a mathematical proposal known as Kruglov entropy, which modifies the traditional rules of how information scales with area. The researchers did not just assume this formula works; they built two distinct cosmological models based on it and then tested them against the most precise measurements of the universe's expansion available today. These measurements include the ripples in the distribution of galaxies, known as baryon acoustic oscillations, and the afterglow of the Big Bang. The goal was to see if this new entropy formula could explain the universe's behavior better than the standard model, or if it leads to contradictions with what we observe.

The team constructed two separate scenarios, both starting from the same entropy formula but using different physical logic to translate that formula into the equations that describe cosmic expansion. The first model, which followed a direct thermodynamic approach, resulted in a universe that behaves quite differently from our own. When the researchers ran this model against the data, it predicted a universe that is expanding much more slowly than we measure it to be today. Specifically, the model suggested a value for the current expansion rate, known as the Hubble constant, of approximately 62.4 kilometers per second per megaparsec. This is significantly lower than the value of about 67 to 73 derived from the standard model and other local measurements. Furthermore, this version of the universe would require a different mix of matter and energy, and it failed to match the observed distances to galaxies and the cosmic microwave background. The data strongly rejected this version, indicating that this direct application of the entropy formula does not describe our reality.

The second model took a different path, deriving the expansion rules by integrating the entropy correction into the energy density of the vacuum. This approach yielded a result that looked almost identical to the standard cosmological model. In this scenario, the universe expands at the same rate as we observe, with a Hubble constant consistent with current measurements, and the amount of matter and dark energy matches current measurements almost perfectly. The model even allowed for a tiny deviation in the behavior of dark energy, suggesting it might be slightly stronger than a simple constant, but this difference was so small it was barely distinguishable from the standard model. However, the researchers found that this model required two extra parameters to describe the entropy effects, and the data did not provide enough evidence to say these extra numbers were necessary. The model fit the data just as well as the standard one, but it did not offer a clear improvement.

The study concludes that while the idea of using generalized entropy to explain cosmic acceleration is mathematically interesting, its success depends entirely on how the math is translated into physical laws. One way of doing the translation leads to a universe that is clearly wrong according to current observations, while the other way leads to a universe that is indistinguishable from the standard model. This means that simply having a new entropy formula is not enough to solve the mysteries of dark energy; the specific mechanism used to connect that formula to the expansion of space is the deciding factor. The researchers emphasize that their work was limited to the smooth, large-scale background of the universe and did not include the clumping of matter or the formation of structures. Future work will need to test these ideas against the complex details of how galaxies form, but for now, the study shows that the path from abstract entropy to a working cosmology is narrow and highly sensitive to the details of the theory.

Technical Summary: Kruglov-entropy cosmology from apparent-horizon thermodynamics

Problem Statement
The standard $\Lambda$ CDM model, while phenomenologically successful, faces theoretical challenges regarding the cosmological constant's value and the coincidence problem, alongside observational tensions such as the Hubble tension. Generalized entropy-area relations offer a pathway to explore deviations from standard Bekenstein–Hawking entropy, potentially generating effective dark energy contributions through modified horizon physics. Specifically, Kruglov proposed a nonadditive entropy ( $S_K$ ) that deforms the standard entropy law via a dimensionless parameter $\gamma$ . While previous work has constrained this entropy using Big Bang nucleosynthesis, its implications for late-time background cosmological dynamics remain to be fully explored. This paper investigates two distinct thermodynamic realizations of Kruglov's entropy to determine their observational viability against current cosmological data.

Methodology
The authors formulate two cosmological models based on Kruglov's nonadditive entropy applied to the apparent horizon of a spatially flat FLRW universe, retaining radiation and pressureless matter.

Model I (Direct Thermodynamic Construction): Following Kruglov's original proposal and the Cai–Kim thermodynamic construction, the Bekenstein–Hawking entropy in the Raychaudhuri equation is replaced by the Kruglov entropy. This leads to an implicit, modified Friedmann equation. The model contains no additional cosmological constant term; the effective dark energy arises entirely from the entropy correction. The entropy scale parameter $B$ is not sampled independently but is fixed by the exact present-day closure condition ( $z=0$ ).
Model II (Entropy-Integral Construction): Proposed in this work, this model utilizes an entropy-integral prescription derived from the unified first law at the apparent horizon. The effective dark energy density is defined as the contribution arising specifically from the non-Bekenstein–Hawking part of the entropy, with the standard BH part subtracted to ensure the limit recovers a cosmological constant. This model introduces two dimensionless entropy-sector parameters, $\beta$ and $\lambda$ .

Both models are expressed in terms of the dimensionless Hubble parameter $E^2(z)$ . The authors solve the resulting implicit differential equations analytically for derivatives to ensure numerical stability. The models are constrained using:

DESI DR2 Baryon Acoustic Oscillations (BAO): 13 measurements.
Compressed CMB Likelihood: A Gaussian constraint on $\theta_s$ , $\omega_b$ , and $\omega_{bc}$ .
DES-Y5 Type-Ia Supernovae (SNe): 1820 SNe (used in the D2 combination).

The analysis is restricted to homogeneous background observables; no perturbation or growth data are included. Parameter estimation is performed using the Cobaya framework with Metropolis-Hastings sampling.

Key Contributions

Dual Realization Comparison: The paper demonstrates that two different thermodynamic constructions based on the same nonadditive entropy functional yield sharply different phenomenological consequences at the background level.
Numerical Implementation: The authors provide a stable numerical framework for solving implicit Friedmann equations derived from generalized entropies, ensuring exact closure at $z=0$ and handling the radiation component essential for BAO and CMB distance calculations.
Observational Constraints: The study provides the first comprehensive constraints on Kruglov-entropy cosmology using high-precision late-time data (DESI DR2 and DES-Y5), moving beyond early-Universe constraints.

Results

Model I: This model is strongly disfavoured by the data. The exact closure condition forces the model to a significantly lower Hubble constant ( $H_0 \simeq 62.4$ km s $^{-1}$ Mpc $^{-1}$ ) and a higher matter density ( $\Omega_{m0} \simeq 0.36$ ) compared to $\Lambda$ CDM. The effective dark energy equation of state is non-phantom ( $w_{DE,0} \simeq -0.81$ ). The inclusion of supernova data does not alleviate this tension. Information criteria (AIC/BIC) show a large degradation in fit quality ( $\Delta \chi^2 \approx 39-62$ ) relative to $\Lambda$ CDM.
Model II: This model remains very close to $\Lambda$ CDM in standard cosmological parameters ( $H_0 \simeq 66.8$ , $\Omega_{m0} \simeq 0.326$ ). The effective equation of state is slightly phantom ( $w_{DE,0} \simeq -1.003$ ). While the best-fit likelihood is comparable to $\Lambda$ CDM, the two additional entropy parameters ( $\beta, \lambda$ ) are strongly degenerate and prior-sensitive. Consequently, information criteria penalize the model for the extra degrees of freedom, providing no evidence that the additional freedom is required by the data.
Kinematic Diagnostics: Model I predicts a significantly less negative deceleration parameter today compared to $\Lambda$ CDM, while Model II tracks $\Lambda$ CDM closely at low redshift.

Significance and Claims
The paper concludes that the observational plausibility of entropy-based cosmology hinges critically on how the generalized horizon entropy is incorporated into the Friedmann dynamics. The direct apparent-horizon modification (Model I) is strongly constrained and disfavoured by current distance measurements, whereas the entropy-integral realization (Model II) can mimic $\Lambda$ CDM closely but offers no statistical preference over the standard model given current data.

The authors emphasize that this analysis is limited to the homogeneous background. They state that a definitive test of Kruglov-entropy cosmology requires a consistent perturbation theory, a full recombination history, and a complete Boltzmann calculation, which are left for future work. The current results serve as a necessary preliminary filter, establishing that the thermodynamic realization method fundamentally dictates the model's viability before more complex perturbation treatments are developed.

Technical Summary: Kruglov-entropy cosmology from apparent-horizon thermodynamics

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