Emergence of cosmic space and horizon thermodynamics in the context of the quantum-deformed entropy

TL;DR

This paper explores how q-deformed entropy influences the emergence of cosmic space and horizon thermodynamics, deriving modified Friedmann equations and linking entropy maximization with the law of emergence in the universe.

Contribution

It introduces a q-deformed entropy framework into cosmology, deriving corresponding Friedmann equations and connecting entropy maximization with the universe's emergence law.

Findings

Derived Friedmann equations from q-deformed entropy and thermodynamics.

Established the connection between the law of emergence and entropy maximization.

Showed consistency of the q-deformed entropy approach with cosmic evolution.

Abstract

According to the quantum deformation approach to quantum gravity, the thermodynamical entropy of a quantum-deformed (q-deformed) black hole with horizon area $A$ established by Jalalzadeh is expressed as $S_q = \pi\sin \left( \frac{A}{8G\mathcal N} \right) /\sin\left(\frac{\pi}{2\mathcal N}\right)$, where $\mathcal N=L_q^2/L_{p}^2$ is the q-deformation parameter, $L_{p}$ denotes the Planck length, and $L_q$ denotes the quantum-deformed cosmic apparent horizon distance. In this paper, assuming that the q-deformed entropy is associated with the apparent horizon of the Friedmann-Robertson-Walker (FRW) universe, we derive the Friedmann equation from the unified first law of thermodynamics, ${dE = T_q dS_q + WdV}$. And this one obtained is in line with the Friedmann equation derived from the law of emergence proposed by Padmanabhan. It clearly shows the connection between the law of emergence and the unified first law of thermodynamics. Subsequently, in the context of the q-deformed horizon entropy, we investigate the constraints of entropy maximization, and result demonstrates the consistency of the law of emergence with the maximization of the q-deformed horizon entropy. Hence, the law of emergence can be understood as a tendency to maximize the q-deformed horizon entropy.