The Large Number Limit of Multifield Inflation

We compute the tensor and scalar spectral index $n_t$, $n_s$, the tensor-to-scalar ratio $r$, the consistency relation $n_t/r$ in the general monomial multifield slow-roll inflation models with potentials $V \sim\sum_i\lambda_i \left|\phi_i\right|^{p_i}$. The general models give a novel relation that $n_t$, $n_s$ and $n_t/r$ are all proportional to the logarithm of the number of fields $N_f$ when $N_f$ is getting extremely large with the order of magnitude around $\mathcal{O}(10^{40})$. An upper bound $N_f\lesssim N_*e^{ZN_*}$ is given by requiring the slow variation parameter small enough where $N_*$ is the e-folding number and $Z$ is a function of distributions of $\lambda_i$ and $p_i$. Besides, $n_t/r$ differs from the single-field result $-1/8$ with substantial probability except for a few very special cases. Finally, we derive theoretical bounds $r>2/N_*$ ($r\gtrsim0.03$) and for $n_t$ which can be tested by observation in the near future.