On the Soft Limit of the Large Scale Structure Power Spectrum: UV Dependence

Garny, Mathias; Porto, Rafael A.; Sagunski, Laura; Konstandin, Thomas

doi:10.1088/1475-7516/2015/11/032

Article
Report number	arXiv:1508.06306 ; DESY-15-148 ; CERN-PH-TH-2015-198 ; ICTP-SAIFER-15-127 ; CERN-PH-TH-2015-198
Title	On the Soft Limit of the Large Scale Structure Power Spectrum: UV Dependence
Author(s)	Garny, Mathias (CERN) ; Konstandin, Thomas (DESY) ; Porto, Rafael A. (ICTP-SAIFR, Sao Paulo) ; Sagunski, Laura (DESY)
Publication	2015-11-18
Imprint	25 Aug 2015
Number of pages	31
Note	25+5 pages, 10 figures, comments added, matches published version
In:	JCAP 1511 (2015) 032
DOI	10.1088/1475-7516/2015/11/032
Subject category	Astrophysics and Astronomy
Abstract	We derive a non-perturbative equation for the large scale structure power spectrum of long-wavelength modes. Thereby, we use an operator product expansion together with relations between the three-point function and power spectrum in the soft limit. The resulting equation encodes the coupling to ultraviolet (UV) modes in two time-dependent coefficients, which may be obtained from response functions to (anisotropic) parameters, such as spatial curvature, in a modified cosmology. We argue that both depend weakly on fluctuations deep in the UV. As a byproduct, this implies that the renormalized leading order coefficient(s) in the effective field theory (EFT) of large scale structures receive most of their contribution from modes close to the non-linear scale. Consequently, the UV dependence found in explicit computations within standard perturbation theory stems mostly from counter-term(s). We confront a simplified version of our non-perturbative equation against existent numerical simulations, and find good agreement within the expected uncertainties. Our approach can in principle be used to precisely infer the relevance of the leading order EFT coefficient(s) using small volume simulations in an `anisotropic separate universe' framework. Our results suggest that the importance of these coefficient(s) is a $\sim 10 \%$ effect, and plausibly smaller.
Copyright/License	arXiv nonexclusive-distrib. 1.0 Preprint: © 2015-2024 CERN (License: CC-BY-4.0)

$Moments in \eqref{eq:sig3} as a function of $k_{max}$, at zero redshift. Results based on $N$-body simulations presented in \cite{Hahn:2014lca}. The dependence on short-distance scales is less relevant at earlier times. Note that $\sigma_{\delta\delta}\equiv \sigma_{11}, \sigma_{\delta\theta}\equiv \sigma_{12}, \sigma_{\theta\theta}\equiv \sigma_{22}$.$ $Numerical results for for the coefficients $c_{11}=c_{\delta\delta}, c_{12}=c_{\delta\theta}, c_{22}=c_{\theta\theta}$ as a function of redshift, obtained from the (truncated) non-perturbative equation \eqref{eq:flow_final}, using velocity correlation spectra from $N$-body simulations performed in \cite{Hahn:2014lca} as input. The black squares in the plot correspond to numerical results for the $c_{\delta\delta}$ coefficient for the density power spectrum in the soft limit \cite{Foreman:2015, Baldauf:2015aha}. The error bands are a rough estimate --at the $\sim 10 \%$ level-- of the errors in the non-perturbative equation, induced by ignoring $\alpha$-terms and using the approximation in \eqref{disc1}. The agreement is remarkably good, especially at higher redshifts.$ $On the upper panels, solid lines reproduce the results from Fig.~\ref{fig:sig3n}. Dashed lines show the one-loop, dotted lines the two-loop, and dot-dashed lines the three-loop standard perturbation theory results. The two figures are given for two different cutoff scales $k_{max}=1(10)\,h/$Mpc, respectively. The solid lines are affected at the $\simeq 2\%$ level by the change in cutoff. For the lower panels, on the left we plot $\sigma_{ab}^2(k_{\rm max})$ (in dashed lines) computed at one-loop as a function of cutoff. On the right, we plot (in dotted lines) the two-loop calculation. We notice, once again, the relatively strong dependence on $k_{\rm max}$ beyond the non-linear scale. This is the case also for $\sigma^2_{12}(k_{\rm max})$, especially at two-loops, and also for $\sigma^2_{22}(k_{\rm max})$ to a lesser degree. On the other hand, the non-perturbative results remain essentially unaltered.$ Show more plots

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Element opprettet 2015-08-27, sist endret 2023-03-15

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