The Quantum Corrected Mode Function and Power Spectrum for a Scalar Field during Inflation

Abstract

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 ${\Delta }_{\phi }^2$. 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 $n_{\phi }$ and the running of the spectral index ${\alpha }_{\phi }$. They imply that the spectrum is slightly red-tilted; hence, the amplitudes of fluctuations grow slightly towards the larger scales.

33 pages

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Cited by in corpus (16)

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• Characters of different secular effects in various patches of de Sitter space
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• Imprint of entanglement entropy in the power spectrum of inflationary fluctuations
• Primordial cosmic fluctuations for variable gravity
• Cosmic fluctuations from quantum effective action
• Fermionic influence (action) on inflationary fluctuations
• A quantum heating as an alternative of reheating
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• Radiative Corrections from Heavy Fast-Roll Fields during Inflation
• More on loops in reheating: Non-gaussianities and tensor power spectrum
• BRST Quantization of Cosmological Perturbations
• Entropy mode loops and cosmological correlations during perturbative reheating