A Closure Theory for Non-linear Evolution of Cosmological Power Spectra

TL;DR

The paper tackles the challenge of predicting non-linear cosmological matter power spectra beyond linear theory. It introduces a non-linear statistical closure, specifically Direct-Interaction Approximation (DIA), to derive a closed set of evolution equations for $P_{ab}(k;\eta)$, $R_{ab}(k;\eta,\eta')$, and the nonlinear propagator $G_{ab}(k;\eta|\eta')$, enabling analytic integral solutions. The authors show that these closed equations reproduce standard one-loop perturbation theory and relate to renormalized perturbation theory results, while also providing exact integral representations that encode nonlinear mode coupling. Using the Born approximation and a matched propagator, they obtain analytic predictions for the non-linear power spectrum and BAO features, which qualitatively agree with N-body trends but indicate the need for higher-order corrections and full numerical closure for precision at small scales.

Abstract

We apply a non-linear statistical method in turbulence to the cosmological perturbation theory and derive a closed set of evolution equations for matter power spectra. The resultant closure equations consistently recover the one-loop results of standard perturbation theory and beyond that, it is still capable of treating the non-linear evolution of matter power spectra. We find the exact integral expressions for the solutions of closure equations. These analytic expressions coincide with the renormalized one-loop results presented by Crocce & Scoccimarro (2006,2007). By constructing the non-linear propagator, we analytically evaluate the non-linear matter power spectra based on the first-order Born approximation of the integral expressions and compare it with those of the renormalized perturbation theory.

Figures (4)

• Figure 1: Approximate solutions for non-linear propagators $\widetilde{G}_{1}(k|z,\,z_{\rm init})=G_{11}+G_{12}$ ( left) and $\widetilde{G}_{2}(k|z,\,z_{\rm init})=G_{21}+G_{22}$ ( right) as function of wave number. In each panel, solid lines show the results from closure theory, while the dashed lines are the propagators based on the renormalized perturbation theory (RPT) CS2006b. For comparison, we also plot the results from the perturbation theory ($1$-loop). From left to right, the lines indicate $z=0,~2$ and $5$ with initial redshift $z_{\rm init}=35$.
• Figure 2: Power spectrum of density fluctuations $P_{11}(k;z)$ at $z=1$, obtained from the first-order Born approximation to the integral solution (see Eq.[\ref{['eq:Pk_Born']}]). The contributions to the total power spectrum are separately plotted as indicated by $P^{\rm(I)}(k)$ and $P^{\rm(II)}(k)$ in the panel and the total power spectrum, $P^{\rm(I)}(k)+P^{\rm(II)}(k)$, is depicted as the dashed lines. Note that in evaluating the power spectrum, the approximate solutions for the non-linear propagators $G_{ab}^{\rm approx}$ were used. Thick and thin lines indicate the results using the approximate solutions $G_{ab}^{\rm approx}$ from the closure theory and RPT, respectively.
• Figure 3: Ratio of non-linear power spectrum to smoothed linear spectrum, $P(k)/P_{\rm no\hbox{-}wiggle}(k)$, given at specific redshifts, $z=3$, $2$, $1$ and $0.5$. The error bar represents the N-body results taken from JK2006, in which different color indicates the results with different box size (see their paper in detail). Here, smoothed linear spectra $P_{\rm no\hbox{-}wiggle}(k)$ were calculated from the linear transfer function without baryon acoustic oscillation according to the fitting formula of EH1998 (Eq.[29] of their paper). The non-linear power spectra are obtained from the first-order Born approximation to the integral solution (Eq.[\ref{['eq:Pk_Born']}]), with approximate solutions of the non-linear propagator given by closure theory (thick) and RPT (thin). For comparison, one-loop predictions from the standard perturbation theory are plotted in dashed lines. Also, in panels with $z=1$ and $0.5$, maximum wave number for limitation of one-loop perturbation is indicated by vertical arrows, according to the criterion, $\Delta^2(k)\equiv k^3P(k)/(2\pi^2)\hbox{$\; \buildrel < \over \sim \;$}0.4$JK2006.
• Figure 4: Logarithmic derivative of non-linear power spectrum, $d\ln P(k)/d\ln k$, at redshifts $z=3$, $2$, $1$ and $0.5$. Line types and labels are the same as in Fig.\ref{['fig:ratio_Pk']}.