We study the corrections to the power spectra of curvature and tensor perturbations and the tensor-to-scalar ratio r in single field slow roll inflation with standard kinetic term due to initial conditions imprinted by a “ fast-roll ” stage prior to slow roll .
For a wide range of initial inflaton kinetic energy , this stage lasts only a few e-folds and merges smoothly with slow-roll thereby leading to non-Bunch-Davies initial conditions for modes that exit the Hubble radius during slow roll .
We describe a program that yields the dynamics in the fast-roll stage while matching to the slow roll stage in a manner that is independent of the inflationary potentials .
Corrections to the power spectra are encoded in a “ transfer function ” for initial conditions \mathcal { T } _ { \alpha } ( k ) , \mathcal { P } _ { \alpha } ( k ) = P ^ { BD } _ { \alpha } ( k ) \mathcal { T } _ { \alpha } ( k ) , implying a modification of the “ consistency condition ” for the tensor to scalar ratio at a pivot scale k _ { 0 } : r ( k _ { 0 } ) = -8 n _ { T } ( k _ { 0 } ) \big [ { \mathcal { T } _ { T } ( k _ { 0 } ) } / { \mathcal { T } _ { \mathcal { % R } } ( k _ { 0 } ) } \big ] .
We obtain \mathcal { T } _ { \alpha } ( k ) to leading order in a Born approximation valid for modes of observational relevance today .
A fit yields \mathcal { T } _ { \alpha } ( k ) = 1 + A _ { \alpha } k ^ { - p } \cos [ 2 \pi \omega k / H _ { sr } + \varphi _ { % \alpha } ] , with 1.5 \lesssim p \lesssim 2 , \omega \simeq 1 and H _ { sr } the Hubble scale during slow roll inflation , where curvature and tensor perturbations feature the same p, \omega for a wide range of initial conditions .
These corrections lead to both a suppression of the quadrupole and oscillatory features in both P _ { R } ( k ) and r ( k _ { 0 } ) with a period of the order of the Hubble scale during slow roll inflation .
The results are quite general and independent of the specific inflationary potentials , depending solely on the ratio of kinetic to potential energy \kappa and the slow roll parameters \epsilon _ { V } , \eta _ { V } to leading order in slow roll .
For a wide range of \kappa and the values of \epsilon _ { V } ; \eta _ { V } corresponding to the upper bounds from Planck , we find that the low quadrupole is consistent with the results from Planck , and the oscillations in r ( k _ { 0 } ) as a function of k _ { 0 } could be observable if the modes corresponding to the quadrupole and the pivot scale crossed the Hubble radius very few ( 2 - 3 ) e-folds after the onset of slow roll .
We comment on possible impact on the recent BICEP2 results .