PRISM: Sparse recovery of the primordial power spectrum

Laboratoire AIM, UMR CEA-CNRS-Paris 7, Irfu, SAp, CEA Saclay, 91191 Gif sur Yvette Cedex, France
e-mail: paniez.paykari@cea.fr; francois.lanusse@cea.fr

Received: 19 July 2013
Accepted: 2 March 2014

Abstract

Aims. The primordial power spectrum describes the initial perturbations in the Universe which eventually grew into the large-scale structure we observe today, and thereby provides an indirect probe of inflation or other structure-formation mechanisms. Here, we introduce a new method to estimate this spectrum from the empirical power spectrum of cosmic microwave background maps.

Methods. A sparsity-based linear inversion method, named PRISM, is presented. This technique leverages a sparsity prior on features in the primordial power spectrum in a wavelet basis to regularise the inverse problem. This non-parametric approach does not assume a strong prior on the shape of the primordial power spectrum, yet is able to correctly reconstruct its global shape as well as localised features. These advantages make this method robust for detecting deviations from the currently favoured scale-invariant spectrum.

Results. We investigate the strength of this method on a set of WMAP nine-year simulated data for three types of primordial power spectra: a near scale-invariant spectrum, a spectrum with a small running of the spectral index, and a spectrum with a localised feature. This technique proves that it can easily detect deviations from a pure scale-invariant power spectrum and is suitable for distinguishing between simple models of the inflation. We process the WMAP nine-year data and find no significant departure from a near scale-invariant power spectrum with the spectral index n_s = 0.972.

Conclusions. A high-resolution primordial power spectrum can be reconstructed with this technique, where any strong local deviations or small global deviations from a pure scale-invariant spectrum can easily be detected.