Imprint of entanglement entropy in the power spectrum of inflationary fluctuations

If the inflaton couples to other degrees of freedom (d.o.f.) that populate the postinflationary stage, such coupling modifies the dynamics of the inflaton during inflation. We consider light fermions Yukawa coupled to the inflaton as “unobserved” d.o.f. integrated out of the total density matrix. Tracing out these d.o.f. yields a 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 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 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.

Figure 1

The first-order state ${\mathrm{\Psi }}^{(1)}$ is a multiparticle state with an inflaton (dashed line) kinematically entangled with a fermion-antifermion pair (solid lines) with $\overrightarrow{k}=-\overrightarrow{p}-\overrightarrow{q}$.

Figure 2

The second-order state ${\mathrm{\Psi }}^{(2b)}$ (solid line) is a fermion-antifermion self-energy loop. The dashed lines are inflaton states.