Semi-classical Einstein equations:descend to the ground state

TL;DR

This paper investigates the behavior of the energy-momentum tensor in various quantum states during inflation, demonstrating how the cosmological constant evolves and decreases exponentially in certain conditions, with implications for the ground state dynamics.

Contribution

It introduces a detailed analysis of the expectation value of the energy-momentum tensor in different states during inflation, linking it to the evolution of the cosmological constant and ground state behavior.

Findings

Expectation value decreases exponentially at the local maximum of the potential.

Confirmation of the descent of the expectation value in stochastic inflation models.

Cosmological constant at large times is proportional to the thermal dissipation rate raised to the 4/3 power.

Abstract

The time-dependent cosmological term arises from the energy-momentum tensor calculated in a state different from the ground state. We discuss the expectation value of the energy-momentum tensor on the rhs of Einstein equations in various (approximate)pure as well as mixed states. We apply the classical slow-roll field evolution as well as the Starobinsky and warm inflation stochastic equations in order to calculate the expectation value. We show that in a state concentrated at the local maximum of the double-well potential the expectation value is decreasing exponentially. We confirm the descend of the expectation value in the stochastic inflation model. We calculate the cosmological constant $\Lambda$ at large time as the expectation value of the energy density with respect to the stationary probability distribution. We show that $\Lambda\simeq \gamma^{\frac{4}{3}} where $\gamma$ is the thermal dissipation rate.