Holographic beta function in de Sitter space

TL;DR

This paper explores the holographic beta function in de Sitter space, linking de Sitter entropy with the distribution of conformal zero modes, and proposes a duality between inflationary and stochastic descriptions via gravitational Fokker-Planck equations.

Contribution

It introduces a holographic framework connecting de Sitter entropy with conformal zero mode distributions and derives a gravitational Fokker-Planck equation governing their evolution.

Findings

De Sitter entropy is identified with the distribution entropy of conformal zero modes.

The gravitational Fokker-Planck equation describes the evolution of the distribution function.

Two solution types: UV complete spacetime and inflationary spacetime with power potentials.

Abstract

The scale invariance of the universe is slightly broken by slow roll parameters. It is likely the slow roll is dual to the random walk. We investigate the distribution function of the conformal zeromode. We identify de Sitter entropy $S_{dS}$ with the distribution entropy of the conformal zeromode $\rho(\omega)$. We have collected convincing support on our postulate. The semiclassical evidence is that the both are given by the gravitational coupling $1/g=\log N/2$ where $g=G_NH^2/\pi$ and $N$ is the e-folding number. We show the renormalized distribution function obeys gravitational Fokker-Planck equation (GFP) and Langevin equations . Under the Gaussian approximation, they boil down to a simple first order partial differential equation. The identical equation is derived by the thermodynamic arguments in the inflationary space-time. GFP determines the evolution of de Sitter entropy of the universe. It coincides with $\beta$ function of $g$. We find two types of the solutions of GFP:(1) UV complete spacetime and (2) inflationary spacetime with power potentials. The maximum entropy principle favors the scenario: (a) born small $\epsilon$ and (b) grow large by inflation. We like to convey the emerging notion of de Sitter duality. The inflationary universe: (bulk/geometrical) is dual to the stochastic space-time on the boundary (cosmological horizon ) as the both are the solutions of GFP.