Inflation and de Sitter Thermodynamics

TL;DR

This work shows that in quasi-de Sitter inflation, the energy flow of the slowly rolling inflaton through the apparent horizon, via the thermodynamic relation $dE=TdS$ with $S=A/(4G)$ and $T=H/(2π)$, reproduces the background Einstein equation in the slow-roll limit, $\dot H=-4πG\dot φ^2$. Extending to perturbations, the authors derive a linearized relation that matches the Einstein equations for inflaton fluctuations and Φ, and find that horizon entropy is perturbed by metric fluctuations, leading to horizon-area wiggles whose dispersion grows with time, in line with the stochastic inflation picture. They also connect the black hole accretion of a rolling scalar to the same TdS law, illustrating a close thermodynamic correspondence across horizons. Collectively, the results provide a thermodynamic and holographic lens on inflationary dynamics and cosmological fluctuations, clarifying when holography constrains or does not constrain the quantum field theory during inflation and highlighting the stochastic nature of horizon area during this era.

Abstract

We consider the quasi-de Sitter geometry of the inflationary universe. We calculate the energy flux of the slowly rolling background scalar field through the quasi-de Sitter apparent horizon and set it equal to the change of the entropy (1/4 of the area) multiplied by the temperature, dE=TdS. Remarkably, this thermodynamic law reproduces the Friedmann equation for the rolling scalar field. The flux of the slowly rolling field through the horizon of the quasi-de Sitter geometry is similar to the accretion of a rolling scalar field onto a black hole, which we also analyze. Next we add inflaton fluctuations which generate scalar metric perturbations. Metric perturbations result in a variation of the area entropy. Again, the equation dE=TdS with fluctuations reproduces the linearized Einstein equations. In this picture as long as the Einstein equations hold, holography does not put limits on the quantum field theory during inflation. Due to the accumulating metric perturbations, the horizon area during inflation randomly wiggles with dispersion increasing with time. We discuss this in connection with the stochastic decsription of inflation. We also address the issue of the instability of inflaton fluctuations in the ``hot tin can'' picture of de Sitter horizon.

Figures (3)

• Figure 1: Penrose diagram of de Sitter spacetime in the flat FRW coordinates (left) and the static coordinates (right). Each point represent a sphere $S^2$. Its radius at the horizon (dashed line on the left, edge of diamond on the right) is equal to $\frac{1}{H}$.
• Figure 2: Field profiles $\phi(t, r)$ outside a black hole for the runaway potential $V(\phi)=1/\phi^2$ at two subsequent time moments; horizontal axis is a tortoise coordinate $r_*$. Red (solid) curves are the numerical solutions, and green (pale) curves are obtained with the delayed field approximation. The horizontal line corresponds to the homogeneous initial conditions.
• Figure 5: A sketch of the apparent horizon of radius $\rho$. Left: the horizon in quasi-de Sitter geometry where $\rho=\frac{1}{H}$ very slowly increases with time. Right: the same but with small metric perturbations where $\rho=\frac{1}{H} (1+\Phi)$ is wiggling with time due to the randomly varying $\Phi$.