Bias-Limited Extraction of Cosmological Parameters

It is known that modeling uncertainties and astrophysical foregrounds can potentially introduce appreciable bias in the deduced values of cosmological parameters. While it is commonly assumed that these uncertainties will be accounted for to a sufficient level of precision, the level of bias has not been properly quantified in most cases of interest. We show that the requirement that the bias in derived values of cosmological parameters does not surpass nominal statistical error, translates into a maximal level of overall error $O(N^{-1/2})$ on $|\Delta P(k)|/P(k)$ and $|\Delta C_{l}|/C_{l}$, where $P(k)$, $C_{l}$, and $N$ are the matter power spectrum, angular power spectrum, and number of (independent Fourier) modes at a given scale $l$ or $k$ probed by the cosmological survey, respectively. This required level has important consequences on the precision with which cosmological parameters are hoped to be determined by future surveys: In virtually all ongoing and near future surveys $N$ typically falls in the range $10^{6}-10^{9}$, implying that the required overall theoretical modeling and numerical precision is already very high. Future redshifted-21-cm observations, projected to sample $\sim 10^{14}$ modes, will require knowledge of the matter power spectrum to a fantastic $10^{-7}$ precision level. We conclude that realizing the expected potential of future cosmological surveys, which aim at detecting $10^{6}-10^{14}$ modes, sets the formidable challenge of reducing the overall level of uncertainty to $10^{-3}-10^{-7}$.