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On Decoupling the Integrals of Cosmological Perturbation Theory
Abstract
Perturbation theory (PT) is often used to model statistical observables capturing the translation and rotationinvariant information in cosmological density fields. PT produces higherorder corrections by integration over linear statistics of the density fields weighted by kernels resulting from recursive solution of the fluid equations. These integrals quickly become highdimensional and naively require increasing computational resources the higher the order of the corrections. Here we show how to decouple the integrands that often produce this issue, enabling PT corrections to be computed as a sum of products of independent 1D integrals. Our approach is related to a commonly used method for calculating multiloop Feynman integrals in Quantum Field Theory, the Gegenbauer Polynomial $x$Space Technique (GPxT). We explicitly reduce the three terms entering the 2loop power spectrum, formally requiring 9D integrations, to sums over successive 1D radial integrals. These 1D integrals can further be performed as convolutions, rendering the scaling of this method $N_{\rm g} \log N_{\rm g}$ with $N_{\rm g}$ the number of grid points used for each Fast Fourier Transform. This method should be highly enabling for upcoming largescale structure redshift surveys where model predictions at an enormous number of cosmological parameter combinations will be required by Monte Carlo Markov Chain searches for the bestfit values.
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