Abstract
A perturbative description of Large Scale Structure is a cornerstone of our understanding of the observed distribution of matter in the universe. Renormalization is an essential and defining step to make this description physical and predictive. Here we introduce a systematic renormalization procedure, which neatly associates counterterms to the UV-sensitive diagrams order by order, as it is commonly done in quantum field theory. As a concrete example, we renormalize the one-loop power spectrum and bispectrum of both density and velocity. In addition, we present a series of results that are valid to all orders in perturbation theory. First, we show that while systematic renormalization requires temporally non-local counterterms, in practice one can use an equivalent basis made of local operators. We give an explicit prescription to generate all counterterms allowed by the symmetries. Second, we present a formal proof of the well-known general argument that the contribution of short distance perturbations to large scale density contrast δ and momentum density π(k) scale as k2 and k, respectively. Third, we demonstrate that the common practice of introducing counterterms only in the Euler equation when one is interested in correlators of δ is indeed valid to all orders.
Highlights
A robust and accurate understanding of the gravitational clustering of Dark Matter is one of the main goals of cosmology
We introduce a renormalization procedure that works systematically to all orders in perturbation theory and give an explicit prescription to construct all effective counterterms allowed by the symmetries
We provide an explicit prescription to generate the counterterms allowed by symmetries to all orders
Summary
A robust and accurate understanding of the gravitational clustering of Dark Matter is one of the main goals of cosmology. Our systematic renormalization makes explicit the implications of the long memory of the short-scale modes, of the equivalence principle and neatly organizes the generation of new counterterms To renormalize the one and two-loop power spectrum and one-loop bispectrum, it is sufficient to include counterterms to the Euler equation [2, 18, 20,21,22] It is well-known that this approach works to all orders and for all statistics of δ, and is equivalent to a particular definition of velocity in terms of momentum current.. In terms of the more general velocity used in [20] and in the systematic renormalization, double softness implies a highly non-trivial relation among counterterms to all orders.
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