A self consistent treatment of observations in a realistically inhomogeneous universe involves relativistic cosmological calculation of both the influence of the inhomogeneity on the metric and the metric on the light. Often both are treated only within the late time, short wavelength Newtonian approximation where the density determines a spatial metric fluctuation by the Poisson equation and the metric influences the propagation through an essentially local impulse. Here we use recent improvements beyond "Newtonian" order to investigate relativistically light propagation, especially from the last scattering surface of the cosmic microwave background radiation (CMB).
In Sect. 2 we use a relativistically rigorous metric (Futamase 1989) describing a universe deviating by gravitational potential perturbations from FRW, accurate to a well defined order, to obtain the Einstein field equations. The Green function solution of Jacobs, Linder, & Wagoner (1992=JLW1; 1993=JLW2) relates the gravitational potential to the density inhomogeneity for arbitrary density contrasts, i.e. without restrictions to the linear regime. We identify those regions in parameter space where post-Newtonian effects are appreciable, as well as deriving analytic expressions for the derivatives of the potential, useful in light propagation calculations. In Sect. 3 we concentrate on the Sachs-Wolfe effect, where density inhomogeneities generate temperature anisotropies in the CMB, evaluating the deviation of the Green function results from the standard Newtonian ones. The results are given both numerically and by analytic order of magnitude arguments to reveal where the deviations arise. In addition we find an effective correction to the density power spectrum which alters the intrinsic scaling of its amplitude from small to large scales.
While the redshift of light propagating through inhomogeneities is well studied (see, e.g., Ma & Bertschinger 1995 for the Sachs-Wolfe effect in linear perturbation theory and Seljak 1996 for the Rees-Sciama effect in numerical simulations), it has been considered within linearized general relativity and not the post-Newtonian formalism of Futamase 1989, JLW1, and JLW2. That possibly significant differences may arise can be seen from the diffusion equation analogy of JLW2 as well as the relativistic approaches of Kodama & Sasaki 1984 and Futamase & Schutz 1983, which agree with the results here that there exist corrections to the linearized general relativistic solution - the Poisson-Newton equation between the gravitational potential perturbation appearing in the metric, , and the density fluctuation . That is just the late time, short wavelength "Newtonian" approximation.
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998