Cosmological dynamics and structure formation in a generalized mass-to-horizon entropy-inspired modified gravity

Abstract: In this article, our goal is to investigate the cosmological dynamics and structure formation in a modified cosmological framework inspired by a generalized mass-to-horizon entropy relation and consistent with the Clausius relation. Invoking the gravity-thermodynamics conjecture leads to alterations in the Friedmann equations as well as Hubble parameter evolution. The effects of the generalized entropy on various cosmographic parameters and on the growth of the linear matter perturbations by constructing perturbed field equations via employing spherical collapse formalism in a flat Friedmann-Lemaître-Robertson-Walker background have been explored. We discuss a novel and well-known diagnostic approach to differentiate various cosmological models vis-à-vis flat and non-flat $Λ$CDM frameworks, and find that the generalized mass-to-horizon entropy-inspired modified cosmology ($n\ne 1$) successfully passes all the litmus tests by falsifying both the flat and non-flat $Λ$CDM paradigms. It is shown that this model also satisfies the requirements for the Universe to achieve thermodynamic equilibrium in the distant future. We also study the halo mass function and cluster number counts in this modified gravity scenario. All the results are compared with the fiducial $Λ$CDM profile, showing that the additional entropic correction influences the expansion history, the growth rate of structures, and the abundance of collapsed halos. We observe that the more massive collapsed structures are less abundant and form at later epochs, which is expected from the hierarchical model of large-scale structure formation.}

• The paper introduces a GMHE-inspired cosmological model that modifies the Friedmann equations and challenges ΛCDM tensions through a tunable entropic parameter n.
• It demonstrates that variations in n alter the cosmic acceleration transition and higher-order cosmographic parameters, affecting the linear growth of density perturbations.
• The study applies the Sheth-Mo-Tormen formalism to reveal n-dependent shifts in halo abundance and mass function, providing observable tests for modified gravity.

Cosmological Dynamics and Structure Formation in a Generalized Mass-to-Horizon Entropy-Inspired Modified Gravity

Introduction and Theoretical Framework

This work develops and analyzes a cosmological scenario derived from a generalized mass-to-horizon entropy (GMHE) relation, rooted in the thermodynamic-gravity correspondence. The approach generalizes the standard Bekenstein-Hawking area law by considering a nonlinear relationship between the mass enclosed by the apparent horizon and the horizon radius. The resultant gravity theory modifies the Friedmann equations, affecting both background cosmological evolution and the linear growth of structure. The model is parameterized by a nonlinearity index $n$, with $n=1$ returning the ΛCDM case.

A primary motivation is to address theoretical and observational tensions in ΛCDM, such as the $H_0$ and $\sigma_8$ discrepancies, while providing a systematic framework to explore entropic modifications compatible with the Clausius relation. The entropy functional reduces to various nonextensive and quantum-corrected forms (e.g., Tsallis-Cirto and Barrow entropies) for specific choices of $n$. The theory is constructed to ensure the generalized entropy is consistent with gravitational dynamics and cosmological thermodynamic equilibrium.

Modified Friedmann Dynamics and Cosmography

The background cosmological expansion is fundamentally altered by the entropic parameter $n$, resulting in power-law corrections to the Friedmann equations. The evolution of the Hubble parameter $E(z)$ is strongly $n$-dependent—deviations from $n=1$ cause nontrivial modifications in the expansion rate at all redshifts.

Figure 1: Normalized Hubble parameter $E(z)$ as a function of redshift for different $n=1$0, highlighting the expansion rate modifications relative to ΛCDM.

Despite the modified expansion, the effective equation of state and $n=1$1 parameters maintain the same functional dependence as ΛCDM due to the conservation structure, but the true physical impact emerges in the cosmographic sector: the deceleration, jerk, snap, lerk, and $n=1$2-parameters all exhibit explicit model-dependence for $n=1$3.

Figure 2: The evolution of the effective equation of state $n=1$4 with redshift; the behavior converges to the de Sitter value at $n=1$5 irrespective of $n=1$6.

Figure 3: Deceleration parameter $n=1$7 shows a shift in the transition redshift between deceleration and acceleration as $n=1$8 varies.

Significantly, the model generically predicts:

• A shift in the acceleration transition redshift $n=1$9 with $H_0$0: higher $H_0$1 delays the onset of cosmic acceleration.
• The jerk and higher-order cosmographic parameters deviate from standard values, becoming nontrivial functions of $H_0$2 and $H_0$3. For example, at $H_0$4 the jerk is $H_0$5 (ΛCDM), but for $H_0$6 this degeneracy is lifted, providing a robust test against cosmological data.

  Figure 4: Jerk parameter $H_0$7 demonstrates the sensitivity of higher-order expansion kinematics to entropic modifications; deviations from $H_0$8 exist at $H_0$9.

The model comprehensively passes geometric diagnostic "null tests" designed to rule out both flat and non-flat ΛCDM via the $\sigma_8$0, $\sigma_8$1, $\sigma_8$2, and $\sigma_8$3 functions—an important demonstration of theoretical distinguishability:

Figure 5: The $\sigma_8$4 diagnostic identifying explicit deviations for $\sigma_8$5; this falsifies ΛCDM in both flat and non-flat scenarios.

Linear Perturbation Theory and Structure Formation

The analysis extends beyond background dynamics, focusing on the linear growth of density perturbations using the top-hat spherical collapse (SC) model, cast within the modified Friedmann context. The key result is an explicit, exact growth equation for the matter density contrast $\sigma_8$6, incorporating the nonlinearity parameter $\sigma_8$7.

Figure 6: Redshift evolution of the linear matter density contrast $\sigma_8$8 for varying $\sigma_8$9. Larger $n$0 yields faster growth and earlier nonlinearity, diverging from the ΛCDM growth law.

Summary of structural implications:

• For $n$1, perturbations grow more efficiently, with early structure formation and enhanced clustering.
• For $n$2, growth is slower, potentially addressing the $n$3 problem by suppressing small-scale structure.
• The logarithmic growth rate $n$4 and $n$5 both display nontrivial dependence on $n$6 at low $n$7 (late times), where differences from ΛCDM are maximized.

  Figure 7: Logarithmic growth rate $n$8 as a function of redshift, quantifying the suppression or enhancement of structure formation due to $n$9.

  

  Figure 8: Product $n$0 demonstrates how the entropic parameter $n$1 shifts the epoch of maximal growth, potentially impacting interpretations of redshift-space distortion data.

Nonlinear Collapse: Halo Mass Function and Number Counts

The Sheth-Mo-Tormen (SMT) formalism is deployed to analyze the halo mass function and number counts, explicitly incorporating the entropic modifications into the multiplicity and mass function calculations. The model predicts:

• A shift in the formation epoch of massive halos for varying $n$2, with more massive halos being less abundant and forming later as $n$3 increases.
• The overall number of collapsed halos decreases with increasing $n$4, consistent with reduced growth at late times.

  Figure 9: SMT halo multiplicity function $n$5 for $n$6, illustrating sensitivity to $n$7 and the resulting shift in the mass function's peak.

  

  Figure 10: Differential halo mass function $n$8 versus redshift for a fixed mass, indicating changes in structure abundance.

Full-sky number counts per redshift and mass bin, calculated over comoving volume, exhibit an $n$9-dependent suppression in number counts for larger structures, and align with the hierarchical formation paradigm: smaller halos are more numerous and form earlier.

Figure 11: Number counts $E(z)$0 as a function of redshift for two mass bins, highlighting how the structure abundance and formation epoch respond to the entropy exponent $E(z)$1.

Implications, Observational Prospects, and Future Directions

The GMHE-inspired modification realizes a family of cosmological models with clear, falsifiable deviation from standard ΛCDM at both the background and perturbative levels. Theoretical implications include:

• The model reproduces the essential features of cosmic acceleration, but predicts observable alterations in the growth rate, structure formation, and cosmographic diagnostics.
• The presence of a tunable entropic parameter $E(z)$2 allows the possibility to ameliorate or exacerbate current cosmological tensions ($E(z)$3, $E(z)$4) by shifting the expansion and growth histories.
• The unambiguous falsification of ΛCDM by diagnostic functions and the modification of the matter power spectrum place these models squarely within reach of next-generation cosmological and large-scale structure surveys.

On the practical front, the framework is amenable to confrontation with data from CMB, SNe Ia, BAO, cosmic chronometers, and direct probes of growth (e.g., redshift-space distortions and weak lensing number counts). Future developments should:

• Extend the analysis to nonlinear structure formation, which is dominant in the late universe and sensitive to physics beyond ΛCDM.
• Perform Markov Chain Monte Carlo constraints using multi-probe datasets to quantify the permitted range of $E(z)$5.
• Study the impact on high-redshift observables and the implications for early structure and reionization.

Conclusion

The GMHE-inspired cosmology provides a technically consistent and phenomenologically rich generalization of standard gravity and cosmology, introducing testable deviations from ΛCDM in both the background and the evolution of structure. The $E(z)$6 parameter smoothly deforms all observable sectors, and its effects are manifest in expansions, growth rates, and halo statistics. This establishes the GMHE framework as a principal candidate for probing the intersection of gravitational thermodynamics and observational cosmology and motivates comprehensive future observational tests using both linear and nonlinear structure probes.