Scalar$-$Tensor Gravity as a Probe of Generalized Black Hole Entropy

TL;DR

This paper links generalized black hole entropy functionals with scalar-tensor gravity, deriving explicit potentials that influence cosmological models and are testable by observations.

Contribution

It provides a geometric framework connecting entropy proposals with scalar-tensor theories, deriving specific scalar potentials for various entropy models.

Findings

Derived explicit scalar potentials for different entropy models.

Showed the potentials lead to distinctive cosmological behaviors.

Framework compatible with current observational constraints.

Abstract

We develop a geometric realization of a broad class of generalized black hole entropy functionals by establishing their direct correspondence with the Misner$-$Sharp quasilocal mass and the Wald Noether$-$charge entropy in scalar$-$tensor theories of gravity. The resulting models feature a scale-dependent effective gravitational coupling, whose functional dependence is determined by the underlying entropy parameters. Within this framework, we derive explicit Einstein-frame scalar potentials: for Barrow entropy, a steep exponential potential; for Tsallis$-$Cirto entropy, an exponential potential governed by the nonextensivity parameter; and for quantum-gravity and entanglement$-$induced corrections, an approximately linear potential. These distinct potentials generate characteristic cosmological phenomenology, with implications for inflationary dynamics, late-time dark-energy behavior, and non-singular bouncing cosmologies. The framework is compatible with current constraints from solar-system tests, big-bang nucleosynthesis, and pulsar-timing observations, and it yields predictions that can be probed by forthcoming observational surveys. In this way, the analysis establishes a unified and geometrically grounded connection between information$-$theoretic entropy proposals and gravitational field theory.