Bayesian inference methodology for Primordial Power Spectrum reconstructions from Large Scale Structure

We use Bayesian inference and nested sampling to develop a non-parametric method to reconstruct the primordial power spectrum $P_{\mathcal{R}}(k)$ from Large Scale Structure (LSS) data. The performance of the method is studied by applying it to simulations of the clustering of two different object catalogues, low-$z$ (ELGs) and high-$z$ (QSOs), and considering two different photometric errors. These object clusterings are derived from different templates of the primordial power spectrum motivated by models of inflation: the Standard Model power law characterized by the two parameters $A_s$ and $n_s$; a local feature template; and a global oscillatory template. Our reconstruction method involves sampling $N$ knots in the log $\{k,P_{\mathcal{R}}(k)\}$ plane. We use two statistical tests to examine the reconstructions for signs of primordial features: a global test comparing the evidences and a novel local test quantifying the power of the hypothesis test between the power law model and the marginalized probability over $N$ model. The method shows good performance in all scenarios considered. In particular, the tests show no feature detection for the SM. The method is able to detect power spectrum deviations at a level of $\approx 2\%$ for all considered features, combining either the low-$z$ or the high-$z$ redshift bins. Other scenarios with different redshift bins, photometric errors, feature amplitudes and detection levels are also discussed. In addition, we include a first application to real data from the Sloan Digital Sky Survey Luminous Red Galaxy Data Release 4 (SDSS LRG 04), finding no preference for deviations from the primordial power law. The method is flexible, model independent, and suitable for its application to existing and future LSS catalogues.