Sub-Planckian Two-Field Inflation Consistent with the Lyth Bound

The BICEP2 observation of a large tensor-to-scalar ratio, $r = 0.20^{+0.07}_{-0.05}$, implies that the inflaton $\phi$ in single-field inflation models must satisfy $\phi \sim 10M_{Pl}$ in order to produce sufficient inflation. This is a problem if interaction terms suppressed by the Planck scale impose a bound $\phi \; ^{<}_{\sim} \; M_{Pl}$. Here we consider whether it is possible to have successful sub-Planckian inflation in the case of two-field inflation. The trajectory in field space cannot be radial if the effective single-field inflaton is to satisfy the Lyth bound. By considering a complex field $\Phi$, we show that a near circular but aperiodic modulation of a $|\Phi|^{4}$ potential can reproduce the results of $\phi^2$ chaotic inflation for $n_{s}$ and $r$ while satisfying $|\Phi|\; ^{<}_{\sim} \; 0.01 M_{Pl}$ throughout. More generally, for models based on a $|\Phi|^{4}$ potential, the simplest sub-Planckian models are equivalent to $\phi^{2}$ and $\phi^{4/3}$ chaotic inflation.