Bayesian optimal reconstruction of the primordial power spectrum

The form of the primordial power spectrum has the potential to differentiate strongly between competing models of perturbation generation in the early universe and so is of considerable importance. The recent release of five years of Wilkinson Microwave Anisotropy Probe observations have confirmed the general picture of the primordial power spectrum as deviating slightly from scale invariance with a spectral tilt parameter of n(s) similar to 0.96. None the less, many attempts have been made to isolate further features such as breaks and cut-offs using a variety of methods, some employing more than similar to 10 varying parameters. In this paper, we apply the robust technique of the Bayesian model selection to reconstruct the optimal degree of structure in the spectrum. We model the spectrum simply and generically as piecewise linear in ln k between 'nodes' in k space whose amplitudes are allowed to vary. The number of nodes and their k-space positions are chosen by the Bayesian evidence so that we can identify both the complexity and location of any detected features. Our optimal reconstruction contains, perhaps, surprisingly few features, the data preferring just three nodes. This reconstruction allows for a degree of scale dependence of the tilt with the 'turn-over' scale occurring around k similar to 0.016 Mpc-1. More structure is penalized by the evidence as overfitting the data, so there is currently little point in attempting reconstructions that are more complex.