Entropy Perturbations in N-flation

In this paper we study the entropy perturbations in N-flation by using the $\d\ma{N}$ formalism. We calculate the entropy corrections to the power spectrum of the overall curvature perturbation $P_{\z}$. We obtain an analytic form of the transfer coefficient $T^2_{\ma{R}\ma{S}}$, which describes the correlation between the curvature and entropy perturbations, and investigate its behavior numerically. It turns out that the entropy perturbations cannot be neglected in N-flation, because the amplitude of entropy components is approximately in the same order as the adiabatic one at the end of inflation $T^2_{\ma{R}\ma{S}}\sim\ma{O}(1)$. The spectral index $n_S$ is calculated and it becomes smaller after the entropy modes are taken into account, i.e., the spectrum becomes redder, compared to the pure adiabatic case. Finally we study the modified consistency relation of N-flation, and find that the tensor-to-scalar ratio ($r\simeq0.006$) is greatly suppressed by the entropy modes, compared to the pure adiabatic one ($r\simeq0.017$) at the end of inflation.