[Paper Review] Corrections to Friedmann equations inspired by Kaniadakis entropy

Ahmad Sheykhi|arXiv (Cornell University)|Feb 25, 2023

Cosmology and Gravitation Theories50 references4 citations

TL;DR

This paper derives modified Friedmann equations by assuming the apparent horizon entropy of a Friedmann-Robertson-Walker universe follows the generalized Kaniadakis entropy, a relativistic statistical entropy formalism. Using the first law of thermodynamics with a work term, the study shows that the geometry of spacetime is corrected, introducing new terms resembling dark energy, while the generalized second law of thermodynamics remains valid, confirming entropy non-decrease over time.

ABSTRACT

Adopting the thermodynamics-gravity conjecture, and assuming the entropy associated with the apparent horizon of the Friedmann-Robertson-Walker (FRW) universe has the form of the generalized Kaniadakis entropy, we extract the modified Friedmann equations describing the evolution of the universe using the first law of thermodynamics on the apparent horizon. We then investigate the validity of the generalized second law of thermodynamics for the universe enclosed by the apparent horizon.

Motivation & Objective

• To investigate the cosmological implications of modifying the entropy of the apparent horizon using the generalized Kaniadakis entropy formalism.
• To derive corrected Friedmann equations by applying the first law of thermodynamics on the apparent horizon, treating entropy as a geometric quantity.
• To ensure the energy content of the universe remains unchanged, focusing instead on geometric corrections due to modified entropy.
• To verify the validity of the generalized second law of thermodynamics in the presence of Kaniadakis entropy corrections.
• To establish a foundation for exploring early-universe phenomena such as baryogenesis and primordial nucleosynthesis in this modified cosmological framework.

Proposed method

• Assumes the entropy of the apparent horizon follows the generalized Kaniadakis entropy, parameterized by a dimensionless Kaniadakis parameter $ K \in (-1, 1) $.
• Applies the first law of thermodynamics on the apparent horizon in the form $ dE = T_h dS_h + W dV $, including a work term due to volume change.
• Derives the modified apparent horizon radius $ \tilde{r}_A $ by solving the energy flux equation under the modified entropy, leading to a corrected Hubble parameter.
• Uses the Gibbs equation to model matter field entropy evolution inside the horizon, assuming thermal equilibrium with the horizon temperature.
• Combines horizon and matter entropy time derivatives to compute the total entropy rate, ensuring $ T_h(\dot{S}_h + \dot{S}_m) \geq 0 $.
• Derives modified Friedmann equations (25) and (26) that include a $ \alpha \tilde{r}_A^4 $-dependent correction term, where $ \alpha = \frac{K^2}{6} $, modifying the geometry without altering energy content.

Experimental results

Research questions

• RQ1 How do the Friedmann equations of the FRW universe change when the apparent horizon entropy is replaced by the generalized Kaniadakis entropy?
• RQ2 What are the cosmological implications of the modified Friedmann equations, particularly in terms of dark energy-like behavior and late-time acceleration?
• RQ3 Does the generalized second law of thermodynamics remain valid when the horizon entropy is described by Kaniadakis statistics?
• RQ4 How does the inclusion of a work term in the first law affect the derivation of the modified field equations?
• RQ5 Can the Kaniadakis parameter $ K $ induce a phantom-like equation of state without violating thermodynamic consistency?

Key findings

• The modified Friedmann equations (25) and (26) include a new correction term proportional to $ \alpha \tilde{r}_A^4 $, where $ \alpha = \frac{K^2}{6} $, modifying the geometry of spacetime.
• The cosmological constant emerges as a constant of integration in the derived equations, indicating a geometric origin in this framework.
• The total entropy $ S = S_h + S_m $, combining horizon and matter field entropy, is always non-decreasing over time, confirming the generalized second law of thermodynamics.
• The time derivative of the total entropy satisfies $ T_h(\dot{S}_h + \dot{S}_m) = 8\pi^2 G H \tilde{r}_A^5 (\rho + p)^2 (1 + \alpha \tilde{r}_A^4)^{-1} \geq 0 $, ensuring thermodynamic consistency.
• The correction terms induced by Kaniadakis entropy lead to a dark energy-like effective fluid, with the equation of state potentially remaining in the phantom regime.
• The approach preserves the energy content of the universe while modifying the geometric part of the field equations, which is physically more consistent than modifying matter content.