Imprint of entanglement entropy in the power spectrum of inflationary fluctuations

Abstract

If the inflaton couples to other degrees of freedom that populate the post-inflationary stage, such coupling modifies the dynamics of the inflaton \emph{during} inflation. We consider light fermions Yukawa coupled to the inflaton as "unobserved" degrees of freedom integrated out of the total density matrix. Tracing out these degrees of freedom yields a \emph{mixed} density matrix whose time evolution is described by an effective field theory. We show that the coupling leads to profuse fermion pair production for super-Hubble inflaton fluctuations which lead to the \emph{growth of entanglement entropy during inflation}. The power spectrum of inflaton fluctuations features scale invariance violations $\mathcal{P}(k)={\mathcal{P}}_0(k)\exp \{8{\xi }_k\}$ with corrections to the \emph{index and its running directly correlated with the entanglement entropy}: $S_{vN}=-{\sum }_k[\ln (1-{\xi }_k)+\frac{{\xi }_k\ln ({\xi }_k)}{1-{\xi }_k}]$. For super-Hubble fluctuations we find ${\xi }_k=-\frac{Y^2}{48{\pi }^2}\{2N_T\ln (k/k_f)+{\ln }^2(k/k_f)\}$ with $Y$ the Yukawa coupling, $N_T$ the total number of e-folds during inflation, and $k_f$ a "pivot" scale corresponding to the mode that crosses the Hubble radius at the end of inflation.

Comments: published version