On the Renormalization of the Effective Field Theory of Large Scale Structures

TL;DR

The paper tackles the UV-divergence problem of standard perturbation theory by employing the Effective Field Theory of Large Scale Structures (EFToLSS), which integrates out short-scale modes and adds consistent effective terms to the large-scale fluid equations. Focusing on an Einstein–de Sitter background with no-scale initial power $P_{in}(k)\propto k^{n}$, it demonstrates that the induced short-scale terms have the right scale and time dependence to cancel UV divergences at one loop, with the result expected to hold at higher loops. By leveraging self-similarity, the authors derive a simple, renormalized one-loop power spectrum valid for any $n>-3$, and show how the relative importance of corrections depends on $n$ (notably for $n oughly -1.5$, relevant to our Universe, where speed-of-sound and dissipative corrections can dominate over two-loop terms). The work compares EFToLSS predictions with self-similar EdS simulations, finding good agreement and advocating EFToLSS as the consistent framework for perturbation theory in cosmology, while outlining future directions including simulations, realization-by-realization tests, and potential Lagrangian EFT implementations.

Abstract

Standard perturbation theory (SPT) for large-scale matter inhomogeneities is unsatisfactory for at least three reasons: there is no clear expansion parameter since the density contrast is not small on all scales; it does not fully account for deviations at large scales from a perfect pressureless fluid induced by short-scale non-linearities; for generic initial conditions, loop corrections are UV-divergent, making predictions cutoff dependent and hence unphysical. The Effective Field Theory of Large Scale Structures successfully addresses all three issues. Here we focus on the third one and show explicitly that the terms induced by integrating out short scales, neglected in SPT, have exactly the right scale dependence to cancel all UV-divergences at one loop, and this should hold at all loops. A particularly clear example is an Einstein deSitter universe with no-scale initial conditions P_in=A k^n. After renormalizing the theory, we use self-similarity to derive a very simple result for the final power spectrum for any n, excluding two-loop corrections and higher. We show how the relative importance of different corrections depend on n. For n=-1.5, relevant for our universe, pressure and dissipative corrections are more important than the two-loop corrections.

Figures (2)

• Figure 1: We show the exponent of $k/k_{NL}$ for various contributions to the power spectrum as function of the power in the initial conditions $P_{lin}\propto k^{n}$. The full orange lines are zero, one and two loop contributions from SPT (from bottom to top); the black dot-dashed line is the term $\Delta_{c_{s}^{2}}$ while the dotted red line refers to $\Delta_{J}$. The three background regions labeled 1-loop, $c_{s}^{2}$ and $J$ indicate which is the most important correction to $\Delta_{lin}$ at that value of $n$. For our universe, near the non-linear scale $n\sim-1.5$ (eg. Smith:2002dz).
• Figure 2: The plots show the fractional difference of EFToLSS (red, thick line) and, when available, SPT (blue, dashed line) predictions from the simulations of OW for $n=-1.5,-1$ and of W for $n=-1$. We show the same plot both with a linear and with a log abscissa. For $n=-1$ when both simulations are available the top line refers to W. The $k$-range shown and the ambiguity $\mathcal{O} (1)$ in the definition of $k_{NL}$ are discussed in the middle of section 4.