Effective field theory during inflation: Reduced density matrix and its quantum master equation

We study the power spectrum of super-Hubble fluctuations of an inflatonlike scalar field, the “system,” coupled to another scalar field, the “environment” during de Sitter inflation. We obtain the reduced density matrix for the inflaton fluctuations by integrating out the environmental degrees of freedom. These are considered to be massless and conformally coupled to gravity as a proxy to describe degrees of freedom that remain sub-Hubble all throughout inflation. The time evolution of the density matrix is described by a quantum master equation, which describes the decay of the vacuum state, the production of particles and correlated pairs and quantum entanglement between super and sub-Hubble degrees of freedom. The quantum master equation provides a nonperturbative resummation of secular terms from self-energy (loop) corrections to the inflaton fluctuations. In the case studied here these are Sudakov-type double logarithms which result in the decay of the power spectrum of inflaton fluctuations upon horizon crossing with a concomitant violation of scale invariance. The reduced density matrix and its quantum master equation furnish a powerful nonperturbative framework to study the effective field theory of long wavelength fluctuations by tracing short wavelength degrees of freedom.

Figure 1

The correlation functions $G^{>}(\overrightarrow{x}-\overrightarrow{y},\eta ,{\eta }^{\prime }),G^{<}(\overrightarrow{x}-\overrightarrow{y},\eta ,{\eta }^{\prime })$.

Figure 2

The state $|{\mathrm{\Psi }}^{(1)}(\eta )⟩$: production of correlated $\chi$ and $\psi$ particles from the vacuum state.

Figure 3

The states in $|{\mathrm{\Psi }}^{(2)}(\eta )⟩$ that contribute to ${\rho }_r(\eta )$: $(a)=|{\mathrm{\Psi }}^{(2)}(\eta ){⟩}_{II};(b)=|{\mathrm{\Psi }}^{(2)}(\eta ){⟩}_I$.