Dark matter clustering: a simple renormalization group approach

I compute a renormalization group (RG) improvement to the standard beyond-linear-order Eulerian perturbation theory (PT) calculation of the power spectrum of large-scale density fluctuations in the Universe. At $z=0$, for a power spectrum matching current observations, lowest order RGPT appears to be as accurate as one can test using existing numerical simulation-calibrated fitting formulas out to at least $k\ensuremath{\simeq}0.3h\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$; although inaccuracy is guaranteed at some level by approximations in the calculation (which can be improved in the future). In contrast, standard PT breaks down virtually as soon as beyond-linear corrections become non-negligible, on scales even larger than $k=0.1h\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$. This extension in range of validity could substantially enhance the usefulness of PT for interpreting baryonic acoustic oscillation surveys aimed at probing dark energy, for example. I show that the predicted power spectrum converges at high $k$ to a power law with index given by the fixed-point solution of the RG equation. I discuss many possible future directions for this line of work. The basic calculation of this paper should be easily understandable without any prior knowledge of RG methods, while a rich background of mathematical physics literature exists for the interested reader.