An upper bound on the number of e-foldings

TL;DR

The paper investigates whether inflation can entail an arbitrarily large number of e-foldings if the universe ultimately approaches a small positive cosmological constant and becomes asymptotically de Sitter (dS). By invoking Cosmological Complementarity and a finite dS Hilbert space, it derives entropy-based bounds on $N_e$ tied to the inflationary energy density $\Lambda_I$ and the asymptotic cosmological constant $\lambda$, with a primary bound $N_e<\frac{1+3\kappa}{6(1+\kappa)}\log\frac{\Lambda_I}{\lambda}$ and a maximum at $\kappa=1$; a refined holographic gauge yields $N_e\le\frac{1}{4}\log\left(\frac{\Lambda_I}{\lambda}\right)+\frac{1}{12}\log\left(\frac{M_P^4}{\lambda}\right)$, which gives numerically about $N_e\sim88$. These results imply that truly asymptotic dS futures constrain inflationary histories, and that models with many e-foldings that seem to approach a fixed dS vacuum may in fact correspond to metastable or non-dS scenarios, with potential observational or theoretical discriminants forthcoming from holographic or SUSY-breaking considerations.

Abstract

If the present acceleration of the universe is due to an asymptotically de Sitter universe with small cosmological constant, and the principle of Cosmological Complementarity is valid, then the number of e-foldings during inflation is bounded.