MPTbreeze: A fast renormalized perturbative scheme

TL;DR

This work introduces MPTbreeze, a fast renormalized perturbative scheme that uses multi-point propagators (MP) as an expansion basis to predict the nonlinear matter power spectrum $P(k)$. By matching low-$k$ perturbative kernels to high-$k$ resummed damping, the authors construct practical expressions for two-, three-, and four-point propagators and assemble them into a three-term expansion for $P(k)$ that remains accurate at BAO scales across redshifts. The approach is validated against extensive N-body simulations with a fiducial cosmology and a cosmological suite, achieving ~2% accuracy up to $k$ near the damping scale $\sigma_d^{-1}$, with evaluation times of only a few seconds per run. The MP expansion is shown to be robust across cosmologies, and the authors publicly release the MPTbreeze code, highlighting its potential for efficient likelihood analyses in large-scale structure surveys. Limitations include a focus on mildly nonlinear, BAO-scale regimes, with suggested future work to combine MP resummation with halo-model or high-$k$ approaches to extend applicability to smaller scales.

Abstract

We put forward and test a simple description of multi-point propagators (MP), which serve as building-blocks to calculate the nonlinear matter power spectrum. On large scales these propagators reduce to the well-known kernels in standard perturbation theory, while at smaller scales they are suppresed due to nonlinear couplings. Through extensive testing with numerical simulations we find that this decay is characterized by the same damping scale for both two and three-point propagators. In turn this transition can be well modeled with resummation results that exponentiate one-loop computations. For the first time, we measure the four components of the non-linear (two-point) propagator using dedicated simulations started from two independent random Gaussian fields for positions and velocities, verifying in detail the fundamentals of propagator resummation. We use these results to develop an implementation of the MP-expansion for the nonlinear power spectrum that only requires seconds to evaluate at BAO scales. To test it we construct six suites of large numerical simulations with different cosmologies. From these and LasDamas runs we show that the nonlinear power spectrum can be described at the ~ 2% level at BAO scales for redshifts in the range [0-2.5]. We make a public release of the MPTbreeze code with the hope that it can be useful to the community.

Figures (10)

• Figure 1: Two-point (nonlinear) propagator for the density field: model vs. measurements in N-body simulations at $z=0,0.5,1$. The model in Eq. (\ref{['eq:prop']}) performs remarkably well at all redshifts shown. The dashed line shows the corresponding high-$k$ limit (which is only reached at very high-$k$, not shown here). Lower panels show the ratio of the measurements to the two different analytic descriptions and stress that the accuracy of Eq. (\ref{['eq:prop']}) is at the percent level.
• Figure 2: Components of the Nonlinear propagator. We show for the first time the four individual components of the nonlinear propagator (normalized to the linear growth factor) measured in dedicated simulations with independent $\delta$ and $\theta$ initial conditions. In our model the decay of the density propagators $G_{\delta\delta}$ and $G_{\delta\theta}$ is given by $\exp[(13/25)f(k) D^2(z)]$ while the velocity components $G_{\theta\delta}$ and $G_{\theta\theta}$ are given by $\exp[(13/25)g(k) D^2(z)]$, see Eqs. (\ref{['eq:Gfull']},\ref{['eq:mixmodeIC2']}). Dashed lines show for reference the decay obtained in the high-$k$ limit $\exp(-k^2 (13/25) \sigma_{\rm d}^2 /2)$ (same for all).
• Figure 3: Nonlinear three-point propagator: analytic predictions vs. measurements. Different panels show measurements of $\Gamma^{(2)}(q_1,q_2,q_3)$ in our fiducial ensemble of simulations for different triangular configurations, as indicated in the y-axis label (see text for details). Solid line corresponds to the interpolation scheme proposed in this paper, see Eq. (\ref{['eq:g2']}). Dashed line to the one introduced in Bernardeau et al. (2011) (RegPT), see Eq. (\ref{['eq:g2regPT']}). They are mostly indistinguishable for almost all configurations and agree with the measurements remarkably well. In particular for right panels where we show the configuration that contribute the most to the one-loop power spectrum at (fixed) wavenumber $k = 0.06\, h^{-1} \, {\rm Mpc}$ ($0.1\, h^{-1} \, {\rm Mpc}$). Error bars correspond to the variance over the ensemble and results (in left and middle panels) are plotted against $k_3=k$.
• Figure 4: The multi-point propagator expansion presented in this paper (solid blue line) against measurements of $P(k)$ in our FID ensemble of N-body simulations (top entry in Table \ref{['cosmologies']}) at $z=0$, $0.5$ and $1$. The dotted red line is linear theory and solid black is halofit. The evaluation time of the multi-point expansion shown in each panel is at most five seconds.
• Figure 5: Nonlinear two-point propagator at z = 0 for different cosmological models. In each panel symbols with error bars correspond to the measurements of the propagator over four simulations of the given cosmology. In solid red lines we show the prescription used throughout this paper corresponding to the exponentiation of the most-growing one-loop contribution: $D_+ \exp{\left[D_+(z)^2f(k)\right]}$. It agrees with the measurements at the sub-percent level for all the scales of interest in all cases studied.