The effective field theory of large scale structure at three loops

Abstract

We study the power spectrum of dark matter density fluctuations in the framework of the Effective Field Theory of Large Scale Structures (EFTofLSS) up to three loop orders. In principle, several counter-terms may be needed to handle the short-distance sensitivity in perturbation theory. However, we show that a small number of extra coefficients are sufficient to match numerical simulations with percent accuracy when a generic renormalization prescription is implemented (allowing for running of the individual counter-terms). We show that the level of accuracy increases with respect to the two loop results, up to k 0.4 h Mpc −1 at redshift z = 0, although the overall improvement is somewhat marginal. At the same time, we argue there is evidence that the behavior of the loop expansion in the EFTofLSS is typical of an asymptotic series, already on the brink of its maximum predictive power (at z = 0). Hence, the inclusion of higher orders will likely deteriorate the matching to data, even at moderate values of k. Part of the reason for this behavior is due to large contributions to the (renormalized) power spectrum at three loop order from mildly non-linear scales, even after the UV counter-terms are included. In conclusion, the EFTofLSS to three loop orders provides the best approximation to the (deterministic part of the) power spectrum in the weakly non-linear regime at z = 0, and higher loops are not expected to improve our level of accuracy.

The effective field theory and perturbative analysis for log-density fields

“…Comparing it to N-body simulations, we give strong evidence that the series constructed by that manner does not converge to the expected simulation result at low redshift even on large scales 3 . It is a consequence of the lack of convergence of the PT series for δ on intermediate scales, already pointed out by [4,28,29]. We then propose an alternative perturbative approach for A directly through the equations of motion, which we label as EoM-APT.…”

“…The total one and two-loop contributions for the Taylor-APT method can be seen by dashed lines on the top panel of Fig 4. The three-loop contribution is shown in orange and it was calculated only for the first modes measured in the simulation suite 4 . We compare the full PT contribution to the simulation measurements on the bottom left panel of Fig 4. As pointed out by [15] and [16], the measured power spectrum for A differs from the power spectrum for δ already on the large scales.…”

“…This motivates us to build in the following section a perturbative approach for A that is not constructed on top of δ R . 4 Calculating each point of the three-loop spectrum of A for both the Taylor-APT and EoM-APT schemes is much more time consuming than for δ due to the symmetrizations needed by the new kernels. It is beyond this project's scope to analyse the full-shape spectrum at three loops, such that we only calculated it for the first modes.…”

“…Notice that one can determine a convergence radius for δ[4], while for the A case it already fails on the large scales. As will be shown later in the text, the k modes no longer decouple for A even in the limit k → 0.…”

The Schrödinger-Poisson method for Large-Scale Structure

“…Nevertheless, reaching percent accuracy with acceptable computational effort is not a trivial task [13]. The major alternative are well-known perturbation theory techniques and variants thereof [14][15][16][17][18][19][20][21][22][23], as well as effective descriptions such as the halo model [24].…”