Inconsistencies of Tsallis Cosmology within Horizon Thermodynamics and Holographic Scenarios

TL;DR

This work probes Tsallis horizon entropy in two cosmological settings: Cai–Kim horizon thermodynamics and Tsallis holographic dark energy with both the Hubble and Granda–Oliveros cutoffs. It demonstrates that nonextensivity corrections scale as $H^{2(1-δ)}$, causing negative or complex dark-energy densities, diverging equations of state, or excessive early dark-energy contributions that spoil BBN and CMB constraints; consequently, viable cosmology exists only at the extensive limit $δ=1$, effectively recovering ΛCDM. The Granda–Oliveros cutoff requires delicate tuning ($α ≳ 2β$) to stabilize the radiation era, but this introduces inconsistencies in the matter era, while the Hubble cutoff drives sizable early dark energy for $δ$ near 1, undermining the standard radiation epoch. Overall, the study highlights nonperturbative instabilities of nonextensive horizon entropy in cosmology and argues for stringent dynamical viability tests when considering generalized entropy formalisms as alternatives to ΛCDM.

Abstract

We investigate the cosmological implications of Tsallis entropy in two widely discussed frameworks: the Cai-Kim thermodynamic derivation of the Friedman equations and the Tsallis holographic dark energy (HDE) scenario, considering both the Hubble scale and the Granda-Oliveros (GO) cutoff as infrared regulators. In both cases, the dynamics introduce a nonextensivity parameter $δ$, with the standard Bekenstein-Hawking entropy-area relation recovered for $δ= 1$. While previous studies have suggested that only small deviations from extensivity are observationally allowed, typically requiring $|1 - δ| \lesssim 10^{-3}$, here we go further and perform a systematic consistency analysis across the entire expansion history. We show that even mild departures from $δ= 1$ lead to pathological behavior in the effective dark energy sector: its density can become negative or complex, its equation of state may diverge, or it can contribute an unacceptably large early-time fraction that spoils radiation domination and violates BBN and CMB constraints. Our results sharpen and unify earlier hints of tension, providing a clear physical interpretation in terms of corrections that grow uncontrollably with the expansion rate toward the past. We conclude that within both the Cai-Kim and HDE formulations, a viable cosmology emerges only in the extensive limit, effectively reducing the models to $Λ$CDM. More broadly, our findings emphasize the importance of dynamical consistency and cosmological viability tests, when assessing nonextensive entropy formalisms as potential frameworks for describing the Universe's dynamics.

Figures (5)

• Figure 1: Density parameters $\Omega_{\text{DE}}$ and $\Omega_r$ at $z \approx 3200$ as functions of the nonextensivity parameter $\delta$. The expected condition $\Omega_r \approx 0.5$ is satisfied only near $\delta \approx 1$, while values $\delta < 1$ lead to an overshoot of $\Omega_r$ and negative $\Omega_\text{DE}$. For $\delta >1$, $\Omega_\text{DE}$ prematurely dominates the energy budget eliminating the radiation-to-matter transition epoch.
• Figure 2: Cosmological evolution of $\Omega_{\text{DE}}$, $\Omega_m$, and $\Omega_r$ for $\delta = 1.00037$, illustrating a standard sequence of radiation, matter, and dark energy domination. However, note that $\Omega_\text{DE}$ tends to grow at early times.
• Figure 3: Evolution of the dark energy equation of state $w_{\text{DE}}$ [Eq. \ref{['Eq: wDE']}] for different values of the non-extensivity parameter $\delta$. The black dashed line corresponds to the extensivity limit $\delta = 1$. For $\delta < 1$, $w_{\text{DE}}$ diverges at some redshift, while in all cases the solutions converge to $w_{\text{DE}} \to -1$ at late times.
• Figure 4: Deviation of the radiation density parameter from its standard $\Lambda$CDM evolution, as quantified by Eq. \ref{['Eq: Omega_r']}. For $\delta<1$ the correction enhances $\Omega_r$, forcing $\Omega_\text{DE}<0$ at early times. For $\delta>1$ the correction suppresses $\Omega_r$, leading to an excessive early dark energy component. Both effects are direct consequences of the $H^{2(2-\delta)}$ scaling of Tsallis corrections.
• Figure 5: Left: Evolution of $\Omega_r$ (red dot–dashed line) and the Tsallis holographic dark energy (HDE) density parameter $\Omega_\text{HDE}$ (black solid line) for the Hubble horizon cutoff. For $\delta = 1.01$, even at extremely high redshift ($z = 10^{14}$), dark energy contributes about $20\%$ of the total cosmic budget, severely disrupting the radiation-dominated era. Right: Evolution of $\Omega_\text{HDE}/\Omega_r$ for the Granda--Oliveros cutoff, considering $\alpha = 2\beta + 0.01$ and $\beta = 0.48$. When assuming radiation domination, this term should be near to zero. However, even for $\delta \approx 1$, this "correction" can be large enough to disrupt a proper radiation domination.