Effective anisotropic stresses of the relic gravitons

Abstract

The effective anisotropic stresses induced by the scalar modes of the geometry depend on the coordinate system so that the comparison of the competing results is ultimately determined by the evolution of the pivotal variables in each particular gauge. After arguing that the only reasonable physical coordinate systems for this problem are the ones where the gauge freedom is completely fixed (like the longitudinal and the uniform curvature gauges), we propose a novel gauge-invariant strategy for the comparison of gauge-dependent results. Instead of employing the pivotal variables of a given coordinate system, the effective anisotropic stress is solely expressed in terms of the gravitating normal modes of the plasma and in terms of their conformal time derivatives. The new approach is explicitly gauge-invariant and when the wavelengths of the normal modes are either shorter or larger than the sound horizon, the physical limits of the anisotropic stresses are determined without relying on the specific details of the background evolution. The relevance of the proposed strategy is discussed in the general situation where the scalar anisotropic stress and the nonadiabatic pressure fluctuations are simultaneously present. We finally argue that the anisotropic stress can be most efficiently obtained from the second-order effective action of the curvature inhomogeneities.