Quadratic effective energy--momentum tensor on uniform-density hypersurfaces during slow-roll inflation

Abstract

We investigate the quadratic-order effective energy--momentum tensor (2EMT) of scalar cosmological perturbations on uniform-density hypersurfaces during slow-roll inflation. The 2EMT is constructed from terms quadratic in the linear metric and inflaton perturbations, and is therefore a gauge-fixed effective source rather than a gauge-invariant observable. We impose the complete scalar gauge conditions $\delta\rho=0$ and $E=0$, express all perturbations in terms of the Bardeen potential $\Psi$, and evaluate the Fourier-space 2EMT in the long- and short-wavelength domains. We distinguish the ``strict'' infrared and ultraviolet limits from the ``intermediate'' regimes. The uniform-density and comoving results agree in the strict infrared limit. In the intermediate infrared regime, the dominant leading order remains the same, while explicit finite-gradient corrections distinguish the two gauges. In the ultraviolet, the 2EMT is enhanced by $1/\epsilon$ due to the slowly varying matter clock, $\rho_0'\propto\epsilon$, and the leading uniform-density 2EMT terms exhibit an additional enhancement by $1/\sigma_2^2$ $(\sigma_2\equiv \cal{H}/k)$ from the Laplacian term. The intermediate ultraviolet expansion makes the subleading gradient hierarchy explicit without changing the leading terms. We compare these results with newly recalculated longitudinal, spatially-flat, and comoving expressions, displayed in a more explicit form than in the earlier analysis. The comparison shows that the gauge dependence is structured: uniform-density and comoving slicings coincide for adiabatic super-Hubble modes, whereas the longitudinal and spatially-flat gauges are {\it slow-roll} suppressed in the strict infrared and become {\it gradient} dominated in the intermediate infrared.