The long freeze: an asymptotically static universe from holographic dark energy

TL;DR

This paper investigates whether holographic dark energy (HDE) can yield an asymptotically static universe, termed the long freeze. Using a generalized Nojiri-Odintsov cutoff for the HDE density $\rho_{HDE}=3 c^2 L^{-2}$, the authors derive explicit solutions where the Hubble rate $H$ decays to zero, the energy density and pressure vanish, and the scale factor $a$ approaches a finite value $a_f$. They show that for a broad class of models with $L=(\beta \dot H+f(H))^{-1/2}$, long freeze occurs when $f(H)\sim H^n$ with $1\le n<2$, and they analyze how adding nonrelativistic matter typically leads to a finite-time recollapse unless the cutoff is discontinuous. The work clarifies the conditions under which an asymptotically static cosmic fate is viable within HDE and highlights the dependence on the cutoff choice, with implications for late-time cosmology and model-building.

Abstract

We show that some holographic dark energy models can lead to a future evolution of the universe in which the scale factor $a$ is asymptotically constant, while $\dot a \rightarrow 0$ and the corresponding energy and pressure densities also vanish. We provide specific examples of such models and general conditions that can lead to an asymptotically static universe, which we have called the ``long freeze." In some cases, such evolution can follow an arbitrarily long exponential expansion essentially identical to the asymptotic evolution of $Λ$CDM. When nonrelativistic matter is added to the holographic dark energy, it tends to destroy the long freeze behavior, driving the universe to recollapse. We show that a long freeze evolution is still possible, but only for a more limited set of HDE models.

Figures (1)

• Figure 1: The scale factor $a$ as a function of the time $t$, both in arbitrary units, for the HDE model given by Eq. (\ref{['genmodel']}) with the indicated functional forms for $f(H)$.