The 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 $\Delta^2_\varphi$. The corrections
are secular, and have 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 $n_\varphi$ and the running of the
spectral index $\alpha_\varphi$. They imply that the spectrum is slightly
red-tilted; hence, the amplitudes of fluctuations grow slightly towards the
larger scales.
