⚛️ general relativity

Modified Cosmology from Mass-to-Horizon Relation: Background Evolution

This paper demonstrates that thermodynamically consistent generalized horizon-entropy models are severely constrained by background cosmological evolution, allowing only minor deviations from the standard Bekenstein-Hawking area law to preserve the observed radiation-matter-dark energy sequence and a $\Lambda$ CDM-like universe.

Original authors: Pranav Prasanthan (Institute of Physics, University of Szczecin, Szczecin, Poland), Hussain Gohar (Institute of Physics, University of Szczecin, Szczecin, Poland), Vincenzo Salzano (Institute of Physi

Published 2026-07-02

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Original authors: Pranav Prasanthan (Institute of Physics, University of Szczecin, Szczecin, Poland), Hussain Gohar (Institute of Physics, University of Szczecin, Szczecin, Poland), Vincenzo Salzano (Institute of Physics, University of Szczecin, Szczecin, Poland)

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

Imagine the universe as a giant, expanding balloon. For decades, physicists have tried to understand what's inside that balloon and why it's stretching faster and faster. The standard explanation is a mysterious "dark energy" pushing it out, but this theory has some annoying glitches, like needing to be tuned very precisely to work.

This paper asks a different question: What if the rules of the game aren't quite right? Specifically, what if the way we count the "information" or "disorder" (entropy) on the edge of our observable universe (the horizon) is slightly different from the textbook version?

The authors, Pranav Prasanthan, Hussain Gohar, and Vincenzo Salzano, decided to test a new, more flexible set of rules for this "edge of the universe." Here is what they found, explained simply:

The Setup: A New Thermodynamic Recipe

Think of the universe's edge (the horizon) like the surface of a soap bubble. In standard physics, the "heat" and "energy" of this bubble are linked to its surface area in a very strict, linear way (like a straight line on a graph).

The authors tried a "generalized" recipe. Imagine the surface area of the bubble doesn't just grow in a straight line, but maybe curves slightly, or has a tiny extra bump on it. They used a specific thermodynamic rule (the "Cai–Kim formulation") to see what happens to the universe's expansion if they tweak this surface rule.

The Experiment: Tweaking the Knobs

They turned two main "knobs" on their new formula:

The Shape Knob ( $m$ ): This changes how the surface area scales. It's like asking, "What if the bubble's surface grows a tiny bit faster or slower than a perfect sphere?"
The Volume Knob ( $\gamma$ ): This changes the overall "size" or weight of the energy associated with that surface.

They also looked at two other potential "glitches" in the system:

Quantum Entanglement: Tiny, invisible threads connecting particles across the horizon.
Quantum Gravity: Effects from the very smallest scales of the universe (Planck scale).

The Results: The Universe is Picky

The authors ran computer simulations to see how the universe would evolve from the Big Bang to today with these new rules. The results were surprisingly strict.

1. The "Goldilocks" Zone is Tiny
The universe is incredibly sensitive to these changes.

If you turn the Shape Knob ( $m$ ) even a tiny bit away from the standard setting: The universe breaks.
- If you turn it one way, the "dark energy" becomes negative (like trying to push a balloon with a vacuum instead of air), causing the math to blow up.
- If you turn it the other way, you get too much dark energy too early. It's like putting a giant rocket booster on a model airplane; it flies off the table before it can even get to the runway. The standard sequence of "Radiation $\to$ Matter $\to$ Dark Energy" gets scrambled.
The Verdict: To keep the universe looking like the one we see today, the Shape Knob must be set to the standard value within a margin of error so small it's almost zero (like $0.9999$ vs $1.0001$).

2. The Volume Knob ( $\gamma$ ) Helps, But Only a Little
They found that if they tweaked the Shape Knob, they could sometimes "fix" the broken universe by adjusting the Volume Knob at the same time. It's like if you add too much sugar to a cake, you can sometimes fix the taste by adding a tiny bit of salt.

However, this "fix" only works for very small adjustments. If you try to make a big change to the rules, no amount of tweaking the other knob can save the model. The universe still ends up looking weird or breaking down.

3. The "Quantum" Effects are Invisible

Entanglement: They found that if the "threads" connecting particles are just right (a specific range of strength), they can create a small, temporary burst of dark energy in the early universe. This is interesting, but it has to be very weak, or it ruins the early universe's history.
Quantum Gravity: The effects from the smallest scales of reality (Planck scale) are so incredibly tiny that they are completely drowned out. It's like trying to hear a whisper in a hurricane. These corrections are effectively zero for the evolution of the universe.

The Big Conclusion

After testing all these fancy new theories, the authors found that the universe forces us back to the standard model.

The "New" Physics: The universe allows for some deviation from the standard rules, but only in a very narrow, safe neighborhood.
The Result: Any version of this theory that actually works looks almost exactly like the standard $\Lambda$ CDM model (the current best model of the universe with a cosmological constant).
The Takeaway: You can't just invent a new entropy rule and expect a totally different universe. The background evolution of the cosmos is a strict judge. If your new rule doesn't produce a universe that looks like ours (with the right mix of radiation, matter, and dark energy), it's rejected.

In short: The paper shows that while we can imagine many different ways the universe's "edge" could behave, nature is very conservative. It only accepts the standard rules, or perhaps a version so close to the standard rules that we can barely tell the difference. The "new" physics doesn't change the story of the universe; it just confirms that the old story is very hard to beat.

Technical Summary: Modified Cosmology from Mass-to-Horizon Relation

Problem Statement
While the standard $\Lambda$ CDM model successfully describes a wide range of cosmological observations, it faces theoretical challenges regarding fine-tuning, the coincidence problem, and observational tensions such as the $H_0$ and $S_8$ discrepancies. Alternative approaches often explore modified gravity or dynamical dark energy. A specific route involves the gravity-thermodynamics correspondence, where Einstein's equations are derived from the first law of thermodynamics applied to the apparent horizon. However, recent studies indicate that applying generalized horizon entropies (e.g., Tsallis, Rényi, Barrow, or quantum-gravity corrected entropies) directly with standard Hawking or Cai-Kim temperatures leads to thermodynamic and holographic inconsistencies. Specifically, these combinations often fail to satisfy the Clausius relation ($dE = T dS$). To restore consistency, a "mass-to-horizon relation" (MHR) was previously introduced to link the horizon mass to the entropy functional. This paper investigates the cosmological implications of this generalized MHR framework, specifically examining whether such thermodynamically consistent entropy modifications can yield viable cosmological backgrounds distinct from $\Lambda$ CDM.

Methodology
The authors employ the Cai-Kim formulation of the first law of thermodynamics applied to the apparent horizon of a spatially homogeneous and isotropic FLRW universe.

Thermodynamic Framework: They utilize the Cai-Kim temperature ( $T_{CK}$ ), which remains positive throughout cosmic evolution, and the generalized mass-to-horizon entropy ( $S_G$ ) derived from a unified MHR. This entropy functional encompasses non-extensive corrections (via parameters $m, \gamma$ ) and quantum-gravity/entanglement corrections (via parameters $\beta, \alpha$ ).
Derivation of Field Equations: By substituting the generalized entropy and the Cai-Kim temperature into the first law ( $\delta Q = Th dSh$ ), the authors derive modified Friedmann equations. These equations are reinterpreted as standard Friedmann equations with an effective dark energy density ( $\varepsilon_{de}$ ) arising from the entropy corrections.
Numerical Analysis: The resulting system of differential equations is solved numerically for the scale factor range $a \in [10^{-10}, 1]$ . The analysis employs arbitrary-precision arithmetic to handle the extreme smallness of the Planck length relative to the Hubble scale.
Viability Constraints: The models are tested against strict observational bounds on the effective dark energy density parameter ( $\Omega_{de}$ ) at key epochs: Big Bang Nucleosynthesis (BBN), matter-radiation equality, and the Cosmic Microwave Background (CMB) era. Specifically, $\Omega_{de}$ is constrained to be $< 0.045$ at BBN and $< 0.02$ at CMB decoupling to preserve the standard radiation-matter-dark energy sequence.

Key Contributions and Results
The study systematically analyzes three classes of entropy modifications:

Standard Area-Law Extensions ( $\beta = 0$ ):
- This class includes Tsallis-Cirto and Barrow entropies, governed by the exponent $m$ and coupling $\gamma$ .
- Sensitivity: The background evolution is extremely sensitive to deviations from the standard area law ( $m=1$ ). For a fixed $\gamma=1$ , the viable range for $m$ is restricted to a narrow interval $0.99986 \leq m \leq 1$ .
- Pathologies: Deviations where $m > 1$ lead to negative effective dark energy densities at early times, while $m < 1$ generates excessive early dark energy, both disrupting the standard cosmological sequence.
- Compensation: Viable models with $m \neq 1$ can be recovered only by simultaneously adjusting the coupling parameter $\gamma$ . However, even with this adjustment, deviations larger than $|m-1| \sim 10^{-3}$ are generally excluded.
Quantum-Entanglement Corrections ( $\beta = 1$ ):
- These corrections introduce a power-law term governed by exponent $\alpha$ .
- Early Dark Energy: These corrections can generate a moderate early dark energy component. However, strong entanglement (low $\alpha$ ) leads to $\Omega_{de}$ values that violate BBN and CMB bounds.
- Viability: Phenomenologically acceptable scenarios exist only for a narrow range of parameters (e.g., $\alpha \approx 2.04$ ) where the correction decays sufficiently fast to allow late-time $\Lambda$ CDM-like behavior.
Quantum-Gravity Corrections ( $\alpha = 4$ ):
- These corrections (logarithmic or power-law) are governed by $\beta$ .
- Suppression: The correction terms are suppressed by the Planck scale factor $(\ell_p H_0 / c)^{3-m} \sim 10^{-122}$ . Consequently, these corrections are observationally irrelevant on cosmological scales. The dynamics are dominated by the leading area-law terms rather than the quantum-gravity corrections.

Significance and Conclusions
The paper concludes that background cosmology alone imposes stringent constraints on thermodynamically consistent generalized entropy constructions.

Narrow Viability: All viable models predict a $\Lambda$ CDM-like cosmological background at the present epoch. The entropy exponent $m$ and the coupling parameter $\gamma$ are confined to a narrow neighborhood of the standard Bekenstein-Hawking values ( $m \approx 1, \gamma \approx 1$ ).
Constraints on Generalizations: Non-extensive generalizations (Tsallis-Cirto, Barrow) are tightly constrained, with viable parameters restricted to values extremely close to the standard limit ( $\delta \approx 1, \Delta \approx 0$ ).
Irrelevance of QG Corrections: Quantum-gravity corrections, while theoretically motivated, remain observationally irrelevant due to Planck-scale suppression.
Implication: The study demonstrates that maintaining thermodynamic consistency via the mass-to-horizon relation does not easily allow for significant deviations from standard cosmology; rather, it reinforces the standard area law as the only phenomenologically robust choice for the current epoch.

The authors note that these conclusions are based exclusively on background evolution and early-universe constraints, with a comprehensive statistical analysis using full cosmological datasets reserved for future work.

The Setup: A New Thermodynamic Recipe

The Experiment: Tweaking the Knobs

The Results: The Universe is Picky

The Big Conclusion

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