Quantum corrected mode function and power spectrum for a scalar field during inflation

We compute the one- and two-loop corrected mode function of a massless minimally coupled scalar endowed with a quartic self-interaction in the locally de Sitter background of an inflating universe for a state which is released in Bunch-Davies vacuum at time t=0. We then employ it to correct the scalar's tree-order scale invariant power spectrum Δφ2. The corrections are secular, and have a scale-dependent part that can be expanded in even powers of k/(Ha), where k is the comoving wave number, H is the expansion rate and, a is the cosmic scale factor. At one-loop, the scale invariant shift in the power spectrum grows as (Ht)2 in leading order. The k-dependent shifts, however, are constants for each mode, in the late time limit. At two-loop order, on the other hand, the scale invariant shift grows as (Ht)4 whereas the k-dependent shifts grow as (Ht)2, in leading order. We finally calculate the scalar's spectral index nφ and the running of the spectral index αφ. They imply that the spectrum is slightly red-tilted; hence, the amplitudes of fluctuations grow slightly toward the larger scales.