RegPT: Direct and fast calculation of regularized cosmological power spectrum at two-loop order

TL;DR

RegPT develops a regularized Γ-expansion framework for cosmological perturbation theory, yielding accurate two-loop predictions of the matter power spectrum and correlation function in the weakly nonlinear regime and validating them against large N-body simulations. The key advance is the regularized multi-point propagator, which interpolates between standard PT at low k and resummed high-k behavior, ensuring robust convergence. To enable efficient exploration of cosmological models, the paper introduces RegPT-fast, a method that reconstructs target spectra from precomputed fiducial-kernel sets, reducing multi-dimensional integrals to one-dimensional ones and delivering speedups to a few seconds. Comprehensive validation against N-body simulations and a cosmic emulator across 38 cosmologies demonstrates the method’s accuracy and practicality, and the authors release an accompanying Fortran code for public use.

Abstract

We present a specific prescription for the calculation of cosmological power spectra, exploited here at two-loop order in perturbation theory (PT), based on the multi-point propagator expansion. In this approach power spectra are constructed from the regularized expressions of the propagators that reproduce both the resummed behavior in the high-k limit and the standard PT results at low-k. With the help of N-body simulations, we show that such a construction gives robust and accurate predictions for both the density power spectrum and the correlation function at percent-level in the weakly non-linear regime. We then present an algorithm that allows accelerated evaluations of all the required diagrams by reducing the computational tasks to one-dimensional integrals. This is achieved by means of pre-computed kernel sets defined for appropriately chosen fiducial models. The computational time for two-loop results is then reduced from a few minutes, with the direct method, to a few seconds with the fast one. The robustness and applicability of this method are tested against the power spectrum cosmic emulator from which a wide variety of cosmological models can be explored. The fortran program with which direct and fast calculations of power spectra can be done, RegPT, is publicly released as part of this paper.

Figures (13)

• Figure 1: Diagrammatic representation of the standard PT expansion.
• Figure 2: Example of the multi-point propagator, $\Gamma_a^{(4)}$. A large filled circle symbolically represents all possible contributions that enter into the fully non-linear propagator. A part of those contributions can be seen graphically using PT expansion (see Figs. \ref{['fig:diagram_Gamma2spt']} and \ref{['fig:diagram_Gamma2reg']} for three-point propagator $\Gamma_a^{(2)}$).
• Figure 4: Diagrammatic representation of the standard PT expansion for three-point propagator, $\Gamma_a^{(2)}$. For fastest growing-mode contribution, the standard PT kernels, $F_{a,{\rm sym}}^{(n)}$, form the basic pieces of PT expansion, depicted as incoming lines connected to a single outgoing line at the shaded circle. In the case of $\Gamma_a^{(2)}$, the leading-order contribution is $F_{a,{\rm sym}}^{(2)}$, and successively the kernels $F_{a,{\rm sym}}^{(4)}$ and $F_{a,{\rm sym}}^{(6)}$ appear as higher-order contributions, for which pairs of the incoming lines are glued at the crossed circle, which indicates the initial power spectrum $P_0$, forming closed loops.
• Figure 6: Contribution of multi-point propagators to the power spectrum, $P(k)=P_{11}(k)$ at $z=1$. Magenta, green, and blue curves represent the power spectrum contributions from the first, second, and third terms at the right-hand-side of Eq. (\ref{['eq:pk_Gamma_reg_2loop']}), respectively, each of which just corresponds to the diagram in Fig. \ref{['fig:diagram_pk']}, involving $\Gamma_{\rm reg}^{(1)}$, $\Gamma_{\rm reg}^{(2)}$, and $\Gamma_{\rm reg}^{(3)}$. Summing up these contributions, total power spectrum is shown in black solid line. For reference, linear power spectrum is also plotted as dotted line.
• Figure 7: Sensitivity of the power spectrum prediction at $z=1$ to the UV cutoff in the estimation of $\sigma_{\rm d}$. Top panel shows each contribution of the power spectrum corrections involving $\Gamma^{(1)}$ (magenta), $\Gamma^{(2)}$ (green), and $\Gamma^{(3)}$ (blue), respectively. Bottom panel shows the total sum of power spectrum divided by the smooth reference power spectrum, $P_{\rm no\hbox{-}wiggle}(k)$, which is calculated from the no-wiggle formula of the linear transfer function in Ref. Eisenstein:1997ik. In each case, top lines represent the results obtained by setting $\sigma_{\rm d}=0$, while undermost lines show the cases adopting the value of $\sigma_{\rm d}$ without UV cutoff. The middle six lines represent the cases adopting the running UV cutoff in estimating $\sigma_{\rm d}$, with cutoff $k_\Lambda(k)=k$, $k/2$, $k/3$, $k/5$, $k/10$, and $k/20$ (from bottom to top). As a reference, linear theory prediction is also plotted in both panels (dotted).