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

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 $[{\hbar G}/{2\pi}] H^2,$ and $2/N_{sur}$ for the first and second approaches respectively. Similarly, using the Kodama-Hayward temperature, the asymptotic fluctuations are $[{5\hbar G}/{2\pi}] H^2,$ 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 $\sqrt{3/\Lambda},$ 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.