Adiabatic modes in cosmology

We show that the field equations for cosmological perturbations in the Newtonian gauge always have an adiabatic solution, for which a quantity R is nonzero and constant in all eras in the limit of large wavelength, so that it can be used to connect observed cosmological fluctuations in this mode with those at very early times. There is also a second adiabatic mode, for which R vanishes for a large wavelength, and in general there may be nonadiabatic modes as well. These conclusions apply in all eras and whatever the constituents of the universe, under only a mild technical assumption about the wavelength dependence of the field equations for a large wavelength. In the absence of anisotropic inertia, the perturbations in the adiabatic modes are given for a large wavelength by universal formulas in terms of the Robertson-Walker scale factor. We discuss an apparent discrepancy between these results and what appears to be a conservation law in all modes found for a large wavelength in the synchronous gauge: it turns out that, although equivalent, the synchronous and Newtonian gauges suggest inequivalent assumptions about the behavior of the perturbations for a large wavelength.