Slow roll in simple non-canonical inflation

We consider inflation using a class of non-canonical Lagrangians for which the modification to the kinetic term depends on the field, but not its derivatives. We generalize the standard Hubble slow roll expansion to the non-canonical case and derive expressions for observables in terms of the generalized slow roll parameters. We apply the general results to the illustrative case of 'slinky' inflation, which has a simple, exactly solvable, non-canonical representation. However, when transformed into a canonical basis, slinky inflation consists of a field oscillating on a multi-valued potential. We calculate the power spectrum of curvature perturbations for slinky inflation directly in the non-canonical basis, and show that the spectrum is approximately a power law on large scales, with a 'blue' power spectrum. On small scales, the power spectrum exhibits strong oscillatory behaviour. This is an example of a model in which the widely used solution of Garriga and Mukhanov gives the wrong answer for the power spectrum.