⚛️ general relativity

Modified Cosmology from Mass-to-Horizon Relation: Observational Bounds

This study uses Markov chain Monte Carlo analyses of multiple cosmological datasets to constrain a class of modified cosmologies derived from a generalized mass-to-horizon relation, finding that while these models can partially alleviate the Hubble tension by adjusting the Hubble constant, they are statistically disfavored compared to the standard $\Lambda$ CDM model and fail to fully resolve the discrepancy.

Original authors: Pranav Prasanthan, Hussain Gohar, Vincenzo Salzano

Published 2026-07-22

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Pranav Prasanthan, Hussain Gohar, Vincenzo Salzano

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, scientists have been trying to figure out exactly how fast this balloon is inflating. They have two very different rulers to measure it. One ruler, based on the "baby pictures" of the universe (the Cosmic Microwave Background), says the balloon is growing at a steady, moderate pace. The other ruler, based on "grown-up" stars and exploding supernovae nearby, says the balloon is puffing up much faster. This disagreement is a massive headache for cosmologists, known as the "Hubble tension," and it suggests that our current rulebook for how the universe works might be missing a few pages.

The rulebook in question is called the "Standard Model" of cosmology, and it relies heavily on a concept called entropy. Think of entropy as a measure of how much information or "disorder" is stored on the surface of a cosmic horizon—the edge of the observable universe. For a long time, scientists assumed this information storage followed a simple, straight-line rule: double the size of the horizon, and you double the information. This idea, born from the work of giants like Bekenstein and Hawking, links gravity and thermodynamics (heat and energy) in a beautiful way. But what if that rule isn't a straight line? What if the universe's "information hard drive" has a fractal, bumpy, or quantum-entangled texture that changes how it stores data? That's the question this paper tackles.

The authors of this study decided to test a new, more flexible version of the universe's rulebook. They proposed a "Mass-to-Horizon Relation" where the connection between the size of the universe's edge and the energy inside it isn't just a simple straight line. Instead, they introduced a few "knobs" or parameters that could twist and turn the rules of thermodynamics. One knob, called the "entropy exponent" ( $m$ ), controls how the information scales with size. Another, the "coupling parameter" ( $\gamma$ ), acts like a volume dial for gravity's strength. A third, the "entanglement amplitude" ( $f_B$ ), accounts for weird quantum connections that might exist across the horizon.

The team took these modified rules and ran them through a massive simulation engine, feeding them the best data we have: the light from thousands of exploding stars (supernovae), the rhythmic ripples in the distribution of galaxies (baryon acoustic oscillations), the ticking clocks of aging stars (cosmic chronometers), and the ancient glow of the Big Bang (CMB). They wanted to see if tweaking these knobs could fix the "Hubble tension" and make the two different rulers agree on the universe's expansion speed.

Here is what they found, and it's a bit of a plot twist. First, the "entropy exponent" knob ( $m$ ) is stuck almost exactly where the old rulebook said it should be. The data shows that $m$ is incredibly close to 1 (specifically, the difference is less than 0.0001 in many cases). This means the universe's information storage is likely a smooth, straight line, not a bumpy fractal. The universe is stubbornly sticking to the standard rules.

Second, when they allowed the "volume dial" ( $\gamma$ ) or the "quantum entanglement" knob ( $f_B$ ) to move, the universe's expansion speed ( $H_0$ ) did indeed shift upward. It went from about 0.688 to roughly 0.70–0.71. This helped narrow the gap between the "baby picture" and "grown-up" measurements, reducing the tension from a screaming 4-sigma disagreement to a more manageable 1.2 to 2.6 sigma. It's like finding a slightly better pair of glasses that makes the two blurry rulers look more similar.

However, the paper delivers a final, decisive verdict. Even though these new models made the fit to the data slightly better, a statistical test called "Bayesian evidence" strongly rejected them. The authors explain that the models didn't actually solve the problem; they just absorbed the tension by adding extra complexity. It's like trying to balance a wobbly table by adding a fourth leg that you have to constantly adjust; the table stops wobbling, but the extra leg isn't a real solution—it's just a crutch. The math shows that the standard model (with no extra knobs) is still the most likely description of reality. The universe, it seems, prefers its simple, straight-line rules over these fancy, modified thermodynamics. The tension remains, and the mystery of the universe's expansion speed is still unsolved.

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

Problem Statement
The thermodynamic nature of gravitational horizons, established by the Bekenstein–Hawking entropy and Hawking temperature, provides a foundation for deriving cosmological dynamics via the gravity–thermodynamics correspondence. However, recent generalizations of horizon entropy motivated by non-extensive statistics, quantum gravity, and entanglement often lead to thermodynamic inconsistencies when paired with the standard linear mass-to-horizon relation (MHR). To resolve this, a unified framework was proposed (Paper I [1]) that generalizes the MHR to ensure consistency with a broad class of generalized entropies. This paper addresses the observational viability of this framework, specifically testing whether deviations from standard horizon thermodynamics can be constrained by current cosmological data and whether such deviations can alleviate the Hubble tension ( $H_0$ discrepancy) between early-universe (CMB) and late-universe (local distance ladder) measurements.

Methodology
The authors perform a comprehensive observational test of modified Friedmann equations derived from a generalized MHR. The framework introduces three primary parameters:

Entropy exponent ( $m$ ): Governs the scaling of the horizon entropy.
MHR coupling parameter ( $\gamma$ ): A dimensionless scaling factor for the mass-horizon relation.
Entanglement-correction amplitude ( $f_B$ ): Represents the magnitude of subleading quantum/entanglement corrections (derived from parameter $\beta$ and exponent $\alpha$ ).

The study utilizes a Markov Chain Monte Carlo (MCMC) analysis using the emcee sampler to constrain eight physically motivated sub-cases of the model. These cases range from fixing parameters to recover standard $\Lambda$ CDM (where $m=\gamma=1, \beta=0$ ) to freeing combinations of $m$ , $\gamma$ , and the entanglement amplitude.

Datasets
The analysis combines four distinct geometric datasets:

Pantheon+ (with SH0ES calibration): 1701 Type Ia supernovae, including 77 in Cepheid-host galaxies to break the $H_0$ – $M$ degeneracy.
Cosmic Chronometers (CC): 33 measurements of $H(z)$ from passively evolving galaxies.
DESI DR2: Baryon Acoustic Oscillation (BAO) measurements covering $0.1 \lesssim z \lesssim 4.2$ .
Planck 2018: Compressed CMB distance priors (shift parameters $R$ and $l_a$ ) and density constraints.

Key Contributions and Technical Results

Tight Constraints on the Entropy Exponent ( $m$ ):
When the MHR coupling is fixed ( $\gamma=1$ ), the entropy exponent is tightly constrained to $|m-1| \lesssim \mathcal{O}(10^{-4})$ . This precision is not due to direct sensitivity to horizon geometry but arises from a "scale-hierarchy amplification." Because the Planck-to-Hubble ratio $X \approx 10^{-61}$ , a fractional deviation in $m$ is amplified by a factor of $\sim 140$ in the effective dark energy density ( $\Omega_{\Lambda}$ ), which is tightly constrained by CMB and BAO data. When $\gamma$ is freed, the constraint relaxes to $\mathcal{O}(10^{-3})$ due to the $m$ – $\gamma$ degeneracy.
Impact on the Hubble Tension:
Freeing the MHR coupling ( $\gamma$ ) or the entanglement amplitude ( $f_B$ ) allows the models to absorb the Hubble tension.
- Coupling ( $\gamma$ ): Models with free $\gamma$ (e.g., BTC-II) shift the inferred Hubble constant to $h \simeq 0.70\text{--}0.71$ , reducing the tension between Planck and SH0ES from $\sim 4\sigma$ to $\sim 1.2\text{--}2.6\sigma$ .
- Entanglement ( $f_B$ ): Including the entanglement correction (Case II) yields a $\sim 2\sigma$ preference for a non-zero amplitude ( $f_B \approx 0.009$ ) only when SH0ES data is included, also raising $h$ to $\sim 0.70$ .
- Resolution Status: No scenario fully resolves the tension within a flat universe; the maximum $h$ achieved is $0.7148 \pm 0.0076$ (BTC-II), which remains $\sim 1.2\sigma$ below the SH0ES central value.
Bayesian Model Comparison:
Despite the improved $\chi^2$ fits when SH0ES data is included, the Bayesian log-evidence ( $\ln Z$ ) disfavors all extended models relative to the standard $\Lambda$ CDM baseline across all dataset combinations. The evidence differences range from $\Delta \ln Z \approx -1$ (weak disfavor) to $\approx -16$ (strong disfavor). The authors attribute the apparent preference for modified entropy in SH0ES-anchored chains to the "absorption" of the Hubble tension by the additional parameters, rather than genuine evidence for modified horizon thermodynamics. The Occam penalty for the extra prior volume outweighs the gain in fit quality.
Reparametrization and Degeneracies:
The study highlights severe degeneracies between $m$ and $\gamma$ (slope $\approx -140$ in the parameter plane) and between the entanglement exponent $\alpha$ and the likelihood. To handle these, the authors employ physically motivated reparametrizations (sampling $A$ and $f_B$ instead of raw parameters) and apply Jacobian corrections to the log-posterior, ensuring efficient sampler convergence and accurate posterior reconstruction.

Significance and Claims
The paper concludes that while the generalized MHR framework is mathematically consistent and capable of shifting the Hubble constant to alleviate the tension, current observational data does not provide positive evidence for deviations from standard Bekenstein–Hawking horizon entropy. The tight bounds on $m$ and the lack of Bayesian evidence suggest that the standard $\Lambda$ CDM model remains the most robust description of the data. The observed shifts in $H_0$ are interpreted as a statistical accommodation of the existing CMB–SH0ES discrepancy rather than a detection of new physics in horizon thermodynamics. The authors note that a complete test requires future work on cosmological perturbations and full CMB power spectrum analyses, which are beyond the scope of this background-only study.

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

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