Generalized Mass-to-Horizon Entropic Cosmology

Updated 2 January 2026

Generalized mass-to-horizon entropic cosmology is a framework that extends gravity–thermodynamics using a power-law mass–horizon relation to derive generalized horizon entropy.
It modifies the Friedmann equations to produce an emergent dark energy sector that can mimic ΛCDM while interpolating between phantom and quintessence behaviors.
Observational tests from SNe Ia, BAO, and structure surveys constrain its parameters, linking entropy generalizations to precision cosmology and cosmic acceleration.

Generalized mass-to-horizon entropic cosmology is a theoretical framework that extends the gravity–thermodynamics paradigm by introducing a non-linear mass–horizon relation to derive generalized horizon entropies, yielding a modified cosmological dynamics. This approach is motivated by the gravity–thermodynamics conjecture, which posits that Einstein gravity and its extensions emerge from thermodynamic considerations of spacetime, with the cosmological horizon serving as the thermodynamic boundary. The resulting models encapsulate Bekenstein–Hawking area law and several of its quantum/phenomenological generalizations as specific cases, and naturally accommodate an emergent, effective dark energy sector whose properties are controlled by the underlying entropic parameters. Precision comparisons with cosmological datasets show that these models can reproduce ΛCDM in a specific parameter limit and provide insights into the microphysical origin of late-time cosmic acceleration (Basilakos et al., 31 Mar 2025, Luciano et al., 18 Aug 2025, Denkiewicz et al., 26 Dec 2025).

1. Mass–Horizon Relation and Generalized Horizon Entropy

At the foundation of the framework is the generalization of the mass–horizon relation (MHR), which replaces the linear Misner–Sharp or Schwarzschild scaling by a power law,

$M = \gamma\,\frac{c^2}{G} L^n,$

where $M$ is the mass enclosed by the cosmological horizon of radius $L$ , $n$ is the entropic exponent ( $n\geq0$ ), and $\gamma$ is a normalization with units $[\text{length}]^{1-n}$ . For $n=1$ , $\gamma=1$ , one recovers the standard area law, but for general $n$ , the mass and thus the associated entropy and thermodynamical relations are deformed.

Applying the Clausius relation,

$dE = T_H\, dS,$

where $T_H = (\hbar c)/(2\pi k_B L)$ is the Hawking temperature, and $E = Mc^2$ , yields the generalized entropy

$S_n(L) = \gamma\,\frac{2n}{n+1}\,L^{n-1}\,S_{BH},$

where $S_{BH} = A/(4G)$ with $A=4\pi L^2$ . Thus, the entropy scales as $S_n \propto L^{n+1}$ . Various forms of black hole and horizon entropy, such as Barrow ( $n=1+\Delta$ ), Tsallis–Cirto (volume, $n=3$ in 3+1 D), and quantum-corrected forms, are recovered as special cases of this construction (Gohar et al., 2023, Gohar, 8 Oct 2025).

2. Thermodynamical Derivation of the Modified Friedmann Equations

The generalized mass–to–horizon entropy enters the cosmological dynamics through the application of the unified first law of thermodynamics on the apparent horizon,

$dU = T_h\, dS - W\, dV,$

where $U = \rho V$ , $W = (\rho - p)/2$ , $V = (4\pi/3) L^3$ . Substituting the appropriate expressions for $T_h$ and $S_n$ leads to a set of modified Friedmann equations,

$\dot H = -4\pi \left[\rho + p + \rho_{DE} + p_{DE}\right],$

$H^2 = \frac{8\pi}{3}\left(\rho + \rho_{DE}\right),$

in which all corrections from the generalized entropy are encapsulated in the emergent 'dark energy': $\rho_{DE} = \frac{3}{8\pi}\left[\frac{\Lambda}{3} + H^2 - \frac{2\gamma n}{3-n} H^{3-n}\right],$

$p_{DE} = -\frac{1}{8\pi}\left[\Lambda + 2\dot H + 3H^2 - 2\gamma n H^{1-n}\left(\dot H + \frac{3}{3-n}H^2\right)\right].$

For $n=\gamma=1$ , these reduce to the standard ΛCDM energy and pressure (Basilakos et al., 31 Mar 2025, Luciano et al., 18 Aug 2025).

3. Phenomenology: Effective Dark Energy and Observational Constraints

The emergent dark energy sector arising from generalized mass–to–horizon entropy displays an effective equation-of-state (EoS) $w_{DE}(z)$ that can interpolate between phantom-like and quintessence-like behaviors, and generically asymptotes to $w_{DE}\to-1$ in the far future. Analytic solutions for the dark energy density parameter $\Omega_{DE}(z)$ , its EoS, and the deceleration parameter $q(z)$ can be derived, illuminating the dynamical role of entropic parameters:

Sub-extensive scaling ( $n<1$ ) yields an EoS that may be phantom ( $w<-1$ ) at high $z$ , becoming quintessence-like ( $w>-1$ ) at low $z$ .
Super-extensive scaling ( $n>1$ ) reverses this time sequence. Observational constraints from Type Ia supernovae, cosmic chronometers, DESI DR2 BAO, and SH0ES $H_0$ priors place the best-fit entropic parameters at $n \lesssim 1,\, \gamma \gtrsim 1$ , with $n=\gamma=1$ (ΛCDM) within $\sim1\sigma$ in all fits. Statistical model selection using AIC shows only a weak preference for ΛCDM due to the additional parameters in the entropic model, but the entropic framework remains fully consistent with extant cosmological data (Luciano et al., 18 Aug 2025).

Dataset Combination	$H_0$ [km/s/Mpc]	$n$	$\gamma$
SN+BAO	$69.8^{+4.3}_{-4.3}$	$0.923^{+0.075}_{-0.066}$	$1.59^{+0.65}_{-0.65}$
SN+CC+BAO	$69.2^{+1.6}_{-1.6}$	$0.928^{+0.068}_{-0.068}$	$1.60^{+0.66}_{-0.66}$
SN+BAO+SH0ES	$73.6^{+1.0}_{-1.0}$	$0.923^{+0.069}_{-0.069}$	$1.57^{+0.67}_{-0.67}$
SN+CC+BAO+SH0ES	$72.28^{+0.90}_{-0.90}$	$0.945^{+0.070}_{-0.070}$	$1.70^{+0.86}_{-0.67}$

(Luciano et al., 18 Aug 2025)

4. Structure Formation, Gravitational Waves, and Cosmic Evolution

The modified entropic background induces changes in linear structure growth. The linear density contrast $\delta(a)$ grows as $\delta(a)\propto a^{2n/(3-n)}$ , so the growth rate $f(a)=2n/(3-n)$ recovers the standard Einstein–de Sitter value for $n=1$ and deviates for $n\neq1$ —suppressed for sub-extensive ( $n<1$ ), enhanced for super-extensive ( $n>1$ ) scaling (Luciano, 1 Oct 2025). For $|n-1| \sim 0.1$ , changes in $f\sigma_8(z)$ at the 5–10% level are possible, making redshift-space distortion surveys (DESI, Euclid) a sensitive probe.

The evolution of the primordial gravitational wave spectrum is also sensitive to $n$ , through its impact on $H(a)$ during horizon re-entry. The relic PGW background can be suppressed or enhanced by $\sim 20\%$ at LISA/ET frequencies for $n$ deviations at the 10% level. Pulsar timing arrays already constrain $n\gtrsim0.88$ at $1\sigma$ to avoid PGW overproduction (Luciano, 1 Oct 2025).

5. Thermodynamic Consistency, Emergence, and Holography

The construction ensures thermodynamic consistency between horizon entropy, temperature, and the first law. By deriving the entropy from the MHR with the Hawking temperature, the Clausius relation is always satisfied. The model also satisfies the generalized second law for $n\gamma>0$ , as the total (horizon plus bulk) entropy always increases during cosmic expansion (Sheykhi, 19 Dec 2025). Furthermore, this framework provides a microphysical basis for the emergence of cosmic space in the sense of Padmanabhan, relating the difference between surface and bulk degrees of freedom to horizon entropy change. For $n=1$ , the Bekenstein area law is recovered, with $G_\text{eff}=G$ ; for $n\neq1$ , the effective gravitational coupling becomes $G_\text{eff} = (3-n)/(2 n \gamma) G$ (Sheykhi, 19 Dec 2025).

6. Extensions, Observational Viability, and Future Directions

The mass-to-horizon entropic cosmology accommodates a broad class of entropy functionals including all known generalizations (Tsallis, Barrow, quantum-corrected, etc.), unified by the generalized MHR (Gohar, 8 Oct 2025). Observational comparisons using Pantheon+ SNe Ia, BAO, Planck CMB distance priors, SH0ES, and structure growth probes show that, for $\log_{10}\gamma < -3$ , these models are statistically undistinguishable from ΛCDM, while in specific regimes (e.g., when structure growth data is included), there can be slight to moderate statistical preference for the entropic scenario (Denkiewicz et al., 26 Dec 2025, Gohar et al., 2023).

The baryogenesis mechanism provides further constraints on the parameter space: successful gravitational baryogenesis in this context requires $0 < 1-n \lesssim 10^{-2}$ , independently linking horizon entropy microphysics with observed cosmological asymmetries (Luciano et al., 3 Nov 2025). The framework thus predicts that any deviation from $n=1$ is tightly constrained by early and late-universe observations.

Further tests will require full CMB and large-scale structure likelihoods, precise measurements of the cosmic expansion rate and structure growth, and possibly signatures in the stochastic gravitational-wave background. Extensions of the MHR to multiparameter forms allow construction of even more general entropy types compatible with both thermodynamic and holographic principles (Gohar, 8 Oct 2025).

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