Non-linear Evolution of Baryon Acoustic Oscillations from Improved Perturbation Theory in Real and Redshift Spaces

TL;DR

Non-linear BAO evolution is challenging to model with standard perturbation theory due to non-perturbative effects and redshift-space distortions. The authors apply a closure approximation within renormalized perturbation theory and use the Born approximation to derive analytic expressions for the non-linear power spectrum and two-point statistics in real and redshift space. In real space, the improved PT achieves percent-level agreement with N-body results within a defined $k$-range, and the BAO peak in the correlation function is robust to high-$k$ propagator details; in redshift space, monopole predictions improve when the velocity dispersion $\sigma_{\rm v}$ is fitted to simulations, but higher multipoles reveal deficiencies that require more sophisticated distortion modeling. Overall, the framework provides a fast, non-perturbative route to precise BAO templates for current and future galaxy surveys and can be extended to more complex cosmologies and forward-modeling approaches.

Abstract

We study the non-linear evolution of baryon acoustic oscillations in the matter power spectrum and correlation function from the improved perturbation theory (PT). Based on the framework of renormalized PT, we apply the {\it closure approximation} that truncates the infinite series of loop contributions at one-loop order, and obtain a closed set of integral equations for power spectrum and non-linear propagator. The resultant integral expressions keep important non-perturbative properties which can dramatically improve the prediction of non-linear power spectrum. Employing the Born approximation, we then derive the analytic expressions for non-linear power spectrum and the predictions are made for non-linear evolution of baryon acoustic oscillations in power spectrum and correlation function. A detailed comparison between improved PT results and N-body simulations shows that a percent-level agreement is achieved in a certain range in power spectrum and in a rather wider range in correlation function. Combining a model of non-linear redshift-space distortion, we also evaluate the power spectrum and correlation function in correlation function. In contrast to the results in real space, the agreement between N-body simulations and improved PT predictions tends to be worse, and a more elaborate modeling for redshift-space distortion needs to be developed. Nevertheless, with currently existing model, we find that the prediction of correlation function has a sufficient accuracy compared with the cosmic-variance errors for future galaxy surveys with volume of a few (Gpc/h)^3 at z>=0.5.

Figures (14)

• Figure 1: Diagrammatic notion of the initial power spectrum (left), linear propagator (middle), and tree vertex (right). The linear propagator satisfies the equation (\ref{['eq:linear_prop']}) with boundary condition $g_{ab}(\eta,\eta)=\delta_{ab}$. The explicit expression of vertex function $\gamma_{abc}$ is given by Eq. (\ref{['eq:def_Gamma']}).
• Figure 2: Diagrammatic representation of the power spectrum and non-linear propagator in closure approximation. The thick lines represent the full-order quantities, while the thin line indicates the linear-order one. The second terms at right-hand side indicate the irreducible one-loop diagrams of the mode-coupling terms, $P_{ab}^{\rm(MC,1\hbox{-}loop)}$ and $G_{ab}^{\rm(MC,1\hbox{-}loop)}$. In the renormalized PT, the mode-coupling term is expressed as an infinite sum of the irreducible loop corrections. Truncating the infinite sum at one-loop order and adopting the tree-level approximation of the full vertex function, we obtain the closed system of power spectrum and propagator, as shown in the figure.
• Figure 3: Diagrammatic representation for the perturbative treatment of the power spectrum with the Born approximation, i.e., Eq. (\ref{['eq:Pk_Born']}).
• Figure 4: Convergence properties of standard PT (left) and improved PT (right) expansions in the matter power spectrum. In each panel, the higher-order contributions to the total power spectrum labeled as $P_{\rm nl}$ is separately plotted. In left panel, one-loop and two-loop corrections in the standard PT, $P_{11}^{\rm 1\hbox{-}loop}$ and $P_{11}^{\rm 2\hbox{-}loop}$, are plotted, while in right panel, the mode-coupling corrections $P_{11}^{\rm(MC1)}$ and $P_{11}^{\rm(MC2)}$ in the improved PT given at Eqs. (\ref{['eq:CLA_MC1']}) and (\ref{['eq:CLA_MC2']}) respectively shown (labeled as MC1 and MC2), together with the first term in Eq. (\ref{['eq:Pk_Born']}) (labeled as G$^2$P$_0$). Note that dashed lines indicate the negative values.
• Figure 5: Ratios of power spectrum to smoothed reference spectrum, $P(k)/P_{\rm no\hbox{-}wiggle}(k)$, given at redshifts $z=3$(top), $1$(middle) and $0$(bottom). Cosmological parameters used in the wmap3 simulations are adopted to compute the power spectrum from standard PT and improved PT, and the results are compared with N-body simulations (symbols with error-bars). The reference spectrum $P_{\rm no\hbox{-}wiggle}(k)$ is calculated from the no-wiggle formula of the linear transfer function in Ref. Eisenstein:1997ik. In each panel, dotted, dashed and solid lines represent the linear, standard PT and improved PT results, respectively. In left panel, leading-order results of standard PT and improved PT are shown, while in right panel, the results including the higher-order corrections are plotted.