Inverting the angular correlation function

The two point angular correlation function is an excellent measure of structure in the Universe. To extract from it the three-dimensional power spectrum, one must invert Limber's equation. Here we perform this inversion using a Bayesian prior constraining the smoothness of the power spectrum. Among other virtues, this technique allows for the possibility that the estimates of the angular correlation function are correlated from bin to bin. The outputs of this technique are estimators for the binned power spectrum and a full covariance matrix. Angular correlations mix small and large scales but after the inversion, small-scale data can be trivially eliminated, thereby allowing for realistic constraints on theories of large-scale structure. We analyse the automated plate measurement (APM) catalogue as an example, comparing our results with previous results. As a by-product of these tests, we find - in rough agreement with previous work - that APM places stringent constraints on cold dark matter inspired models, with the shape parameter constrained to be 0.25 +/- 0.04 (using data with wavenumber k less than or equal to 0.1 h Mpc(-1)). This range of allowed values uses the full power spectrum covariance matrix, but assumes negligible covariance in the off-diagonal angular correlation error matrix, which is estimated with a large angular resolution of 0.5 degrees (in the range 0.5 degrees and 20 degrees).