Constraining the primordial power spectrum using a differentiable likelihood

Abstract The simplest inflationary models predict the primordial power spectrum (PPS) of curvature perturbations to be nearly scale-invariant. However, various other models of inflation predict deviations from this behaviour, motivating a data-driven approach to reconstruct the PPS and constrain its shape. In this work, we present a novel method that employs a fully differentiable pipeline to reconstruct the PPS using Gaussian processes and uses neural network emulators for fast and differentiable theoretical predictions. By leveraging gradient-based sampling techniques, such as Hamiltonian Monte Carlo, our approach efficiently samples the high-dimensional parameter space of cosmological parameters and the free-form PPS, enabling joint constraints on both. Applying this framework to Planck 2018 Cosmic Microwave Background (CMB) temperature anisotropy data we find our reconstructed PPS to be consistent with near scale-invariance on small scales, while exhibiting large uncertainties at large scales, driven mostly by cosmic variance. Our results show an overestimation of the PPS amplitude compared to ΛCDM predictions from the Planck 2018 analysis, which we attribute to our choice of a wider prior on the optical depth τ based on Planck 2015 measurements. Adopting a prior consistent with Planck 2018 measurements brings our results into full agreement with previous work. To ensure robustness of our results, we validate our differentiable pipeline against a non-differentiable framework, and also demonstrate that our results are insensitive to the choice of Gaussian process hyperparameters. These promising results and the flexibility of our pipeline make it ideally suited for application to additional data sets such as CMB polarisation as well as Large-Scale Structure probes, thus moving towards multi-probe primordial power spectrum reconstruction.