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We study a two-parameter family of exactly solvable inflation models withvariable sound speed, and derive a corresponding exact expression for thespectrum of curvature perturbations. We generalize this expression to the slowroll case, and derive an approximate expression for the scalar spectral indexvalid to second order in slow roll. We apply the result to the case of DBIinflation, and show that for certain choices of slow roll parameters, theBunch-Davies limit (a) does not exist, or (b) is sensitive to stringy physicsin the bulk, which in principle can have observable signatures in theprimordial power spectrum.

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