Fewer Mocks and Less Noise: Reducing the Dimensionality of Cosmological Observables with Subspace Projections

Creating accurate and low-noise covariance matrices represents a formidable challenge in modern-day cosmology. We present a formalism to compress arbitrary observables into a small number of bins by projection into a model-specific subspace that minimizes the prior-averaged log-likelihood error. The lower dimensionality leads to a dramatic reduction in covariance matrix noise, significantly reducing the number of mocks that need to be computed. Given a theory model, a set of priors, and a simple model of the covariance, our method works by using singular value decompositions to construct a basis for the observable that is close to Euclidean; by restricting to the first few basis vectors, we can capture almost all the constraining power in a lower-dimensional subspace. Unlike conventional approaches, the method can be tailored for specific analyses and captures nonlinearities that are not present in the Fisher matrix, ensuring that the full likelihood can be reproduced. The procedure is validated with full-shape analyses of power spectra from Baryon Oscillation Spectroscopic Survey (BOSS) DR12 mock catalogs, showing that the 96-bin power spectra can be replaced by 12 subspace coefficients without biasing the output cosmology; this allows for accurate parameter inference using only ∼100 mocks. Such decompositions facilitate accurate testing of power spectrum covariances; for the largest BOSS data chunk, we find the following: (a) analytic covariances provide accurate models (with or without trispectrum terms); and (b) using the sample covariance from the MultiDark-Patchy mocks incurs a ∼0.5σ shift in Ωm, unless the subspace projection is applied. The method is easily extended to higher order statistics; the ∼2000-bin bispectrum can be compressed into only ∼10 coefficients, allowing for accurate analyses using few mocks and without having to increase the bin sizes.