Causal compensated perturbations in cosmology

We develop a framework within which to calculate cosmological perturbations generated causally by "stiff"-matter sources, namely, sources whose self-gravity is unimportant in determining their dynamics. Examples of "stiff" matter which may be important in cosmology are cosmic strings, domain walls, and cosmic texture. Our method is manifestly causal and is applicable even when the sources are highly relativistic, are coherent over scales which are large in comparison with the horizon length, or are highly localized in space. We show that in order to satisfy energy and momentum conservation, the gravitational fields of the source must be compensated by perturbations in the matter and gravitational fields, and we discuss the role of such compensation in obtaining the initial inhomogeneities and in their subsequent evolution. We derive exact Green function solutions for the matter density and gravitational stresses in the matter-dominated era, and obtain analytic and numerical approximations to the Green functions at late times in a universe containing both radiation and matter. In order to clarify the nature of energy and momentum conservation in the expanding universe, we define a cosmological energy-momentum pseudotensor, and use it to demonstrate that the power spectrum of the source and perturbations must be bounded on large scales.