A Measure of de Sitter Entropy and Eternal Inflation

TL;DR

The paper derives a macroscopic bound on de Sitter space in NEC-respecting, non-eternal inflation: the horizon area must grow by at least one Planck unit per e-fold, making the observable inflationary spectrum effectively bounded by $e^{S_{\rm dS}}$ modes and linking this limit to holography and black-hole thermodynamics. It uses an EFT of adiabatic perturbations with the Goldstone mode $\pi$ to show that $\frac{dS}{dN} \gg 1$ generally holds, implying $N_{\rm tot} \ll S_{\rm end}$, whereas eternal inflation corresponds to $dS/dN$ near unity, violating the bound. The null energy condition is crucial; NEC-violating theories like ghost inflation evade the bound and disrupt black-hole thermodynamics, suggesting they lie in the swammland of incompatible gravitational theories. The work thus provides a thermodynamic viewpoint on de Sitter entropy and locality in quantum gravity, with clear implications for the landscape of inflationary models and future research on covariant holography.

Abstract

We show that in any model of non-eternal inflation satisfying the null energy condition, the area of the de Sitter horizon increases by at least one Planck unit in each inflationary e-folding. This observation gives an operational meaning to the finiteness of the entropy S_dS of an inflationary de Sitter space eventually exiting into an asymptotically flat region: the asymptotic observer is never able to measure more than e^(S_dS) independent inflationary modes. This suggests a limitation on the amount of de Sitter space outside the horizon that can be consistently described at the semiclassical level, fitting well with other examples of the breakdown of locality in quantum gravity, such as in black hole evaporation. The bound does not hold in models of inflation that violate the null energy condition, such as ghost inflation. This strengthens the case for the thermodynamical interpretation of the bound as conventional black hole thermodynamics also fails in these models, strongly suggesting that these theories are incompatible with basic gravitational principles.

Figures (8)

• Figure 1: Nice slices in Kruskal coordinates (left) and in the Penrose diagram (right). The singularity is at $T^2-X^2=1$.
• Figure 2: The entanglement entropy for an evaporating black hole as a function of time. After a time of order of the evaporation time the EFT prediction (blue line) starts violating the holographic bound (dashed line). The correct behavior (red line) must reduce to the former at early times and approach the latter at late times. At the final stages, $t \simeq t_{\rm Planckian}$, curvatures are large and EFT breaks down.
• Figure 3: Love in an inflationary Universe.
• Figure 4: A given cosmological history is a classical trajectory in field space (red line), parameterized by time. The Goldstone field $\pi$ describes small local fluctuations along the classical solution. In general other light oscillation modes, transverse to the trajectory will also be present, and $\pi$ can be mixed with them. In the picture $\varphi_1$ and $\varphi_2$ are the modes that locally diagonalize the quadratic Lagrangian of perturbations. The blue ellipsoid gives the typical size of quantum fluctuations.
• Figure 5: The null energy condition is violated whenever $F'(\delta g^{00})$ enters the shaded region, $F' + M^2_{Pl} \dot H > 0$. Since $F'$ starts with a strictly positive slope at the origin, to avoid this one needs that higher derivatives of $F$ bend $F'$ away from the NEC-violating region. The smaller $|\dot H|$, the stronger the needed 'bending'. This can make $\pi$ fluctuations strongly coupled at $H$.