Bigger uncertainties and the Big Bang

We use Bayesian hierarchical models and recent results from the theory of minimax confidence interval estimation to study the effect of prior information in a cosmological inverse problem. We consider the effect of prior information on uncertainty estimates of a linear functional of an infinite-dimensional model x, given noisy observations . The model represents the cosmic microwave background (CMB), which is the radiation left over from the Big Bang. The linear functional is related to the important cosmological question of whether the CMB temperature varies with direction in the sky; such variation is required by cosmological theories to account for the observed large-scale heterogeneity of matter and energy in the Universe. Evidence of this heterogeneity is the non-zero quadrupole term in the CMB detected by the COBE satellite in 1992. Estimation of the quadrupole is an interesting ill-posed problem that requires more information than cosmologists expected. Previously published quadrupole estimates relied on constraints such as artificially truncating the spherical harmonic expansion of the CMB fluctuations, or on modelling the effect of unestimated high-frequency terms, without accounting for model uncertainty. If these implicit constraints were relaxed, the uncertainty would be several to dozens of times larger than reported in the astrophysical literature. We study the dependence of quadrupole estimates to a series of increasingly stringent constraints. We show that no useful estimates can be obtained from COBE data without assuming a particular class of prior cosmological models. Even restricting the spectrum to lie in a two-parameter family of models commonly used in cosmology does not suffice without positing a prior probability distribution on those two parameters.