Quantum modes in DBI inflation: Exact solutions and constraints from vacuum selection

We study a two-parameter family of exactly solvable inflation models with variable sound speed, and derive a corresponding exact expression for the spectrum of curvature perturbations. We generalize this expression to the slow-roll case, and derive an approximate expression for the scalar spectral index valid to second order in slow roll. We apply the result to the case of Dirac-Born-Infeld inflation, and show that for certain choices of slow-roll parameters, the Bunch-Davies limit (a) does not exist, or (b) is sensitive to stringy physics in the bulk, which in principle can have observable signatures in the primordial power spectrum.