Evolution of fluctuations in horizon energy and its dependence on the degrees of freedom

TL;DR

The paper investigates how thermal fluctuations of the Hubble horizon energy evolve in an expanding universe, using two complementary frameworks: a canonical ensemble treatment with both Gibbons-Hawking and Kodama-Hayward horizon temperatures, and a holographic equipartition approach based on horizon degrees of freedom $N_{sur}$. It finds that, for Gibbons-Hawking temperature, the asymptotic relative fluctuations scale as $|\sigma_E^2/\langle E_H\rangle^2| \to (\hbar G)/(2\pi)\,H^2$, while Kodama-Hayward temperatures yield a larger asymptotic magnitude (five times larger at end-de Sitter), and deviations during matter domination depend on the dynamical term $\dot H$. The DOF-based method shows $|\sigma_H^2/\langle E_H\rangle^2| \propto 2/N_{sur}$, with the same $H^2$-scaling and a clear inverse dependence on horizon degrees of freedom, connecting fluctuations to horizon entropy. Importantly, the authors propose a relation $L_p^2 \Lambda \propto |\sigma_H^2/\langle E_H\rangle^2|$ that, in principle, could constrain the cosmological constant if horizon fluctuations could be measured, yielding rough bounds compatible with the observed small value of $\Lambda$ and offering a thermodynamic perspective on the cosmological constant problem.

Abstract

Taking account of the thermal nature of the Hubble horizon of the expanding universe, we analysed the evolution of relative fluctuations of horizon energy. For this analysis, we used two approaches: (i) by treating the Hubble horizon as a system in canonical ensemble, and (ii) by considering the microscopic degrees of freedom on the horizon. In both approaches, we obtained the relative fluctuations by using two different definitions of the horizon temperature; first, the Gibbons-Hawking temperature, and second, the Kodama-Hayward temperature. For a given temperature, both approaches yield the same general evolution for the fluctuations. In the asymptotic limit, the relative energy fluctuations corresponding to the Gibbons-Hawking temperature, is $[{\hbar G}/{2π}] H^2,$ and $2/N_{sur}$ for the first and second approaches respectively. Similarly, using the Kodama-Hayward temperature, the asymptotic fluctuations are $[{5\hbar G}/{2π}] H^2,$ and $10/N_{sur}.$ This implies that, the magnitude of the relative fluctuations of the horizon energy is higher in the case of Kodama-Hayward temperature. The inverse dependence of the fluctuation on $N_{sur},$ the number of degrees of freedom on the horizon, reflects a familiar behaviour in ordinary thermal systems: fluctuations decrease as the number of degrees of freedom increases. Notably, we also found that the relative energy fluctuations establish a connection between the Planck length scale $L_p,$ characteristic length scale of the very early epoch of the universe, and $\sqrt{3/Λ},$ the length scale associated with the late-time accelerated phase. This relationship can offer valuable insights that could help in addressing the cosmological constant problem.