Time delay by primordial density fluctuations: its biasing effect on the observed mean curvature of the Universe

Abstract

In this paper we specifically studied one aspect of foreground primordial matter density perturbations: the relative gravitational time delay between a pair of light paths converging towards an observer and originating from two points on the last scattering surface separated by the physical scale of an acoustic oscillation. It is found that time delay biases the size of acoustic oscillations {\it systematically} towards smaller angles, or larger harmonic numbers $\ell$, i.e. the mean geometry as revealed by CMB light becomes that of an open Universe if $\Omega=1$. Since the effect is second order, its standard deviation $\delta\ell/\ell \sim (\delta\Phi)^2$ where $(\delta\Phi)^2 \sim 10^{-9}$ is the normalization of the primordial matter spectrum $P(k)$, the consequence is too numerically feeble to warrant a re-interpretion of WMAP data. If, however, this normalization were increased to $\delta\Phi \gtrsim 0.01$ which is still well within the perturbation limit, the shift in the positions of the acoustic peaks would have been substantial enough to implicate inflationary $\Lambda$CDM cosmology. Thus $\Omega$ is not the only parameter (and by deduction inflation cannot be the only mechanism) of relevance to the understanding of {\it observed} large scale geometry. The physics that explains why $\delta\Phi$ is so small also plays a crucial role, but since this is a separate issue independent of inflation, might it be less artificial to look for an alternative solution to the flatness problem altogether?