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}: $S_{vN} = - \sum_{k} \Big[ \ln(1-\xi_k) + \frac{\xi_k\,\ln(\xi_k)}{1-\xi_k} \Big]$. For super-Hubble fluctuations we find $\xi_k = -\frac{Y^2}{48\pi^2}\Big\{2\,N_T\,\ln(k/k_f) + \ln^2(k/k_f)\Big\}$ 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.