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 WMAP observat ions have confirmed the general picture of the primordial power spectrum as deviating s lightly from scale invariance with a spectral tilt parameter of ns � 0:96. Nonetheless, many attempts have been made to isolate further features such as breaks and cutoffs using a variety o f methods, some employing more than� 10 varying parameters. In this paper we apply the robust techni que of 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, p erhaps, surprisingly few features, the data preferring just three nodes. This reconstruction allo ws for a degree of scale dependence of the tilt with the ‘turn-over’ scale occuring around k � 0:016 Mpc 1 . More structure is penalised by the evidence as over-fitting the data, so there is c urrently little point in attempting reconstructions that are more complex.