One-Loop Nonlinear Matter Power Spectrum from Unified Lagrangian Perturbation Theory: Fast Computation and Comparison with Emulators

TL;DR

This paper introduces ULPT, a unified, IR-safe perturbative framework to compute the one-loop nonlinear matter power spectrum by decomposing the density field into a Jacobian deviation and a displacement-mapping factor. The authors develop a fast numerical pipeline using FFTLog and FAST-PT to evaluate the displacement-mapping integrals and validate the approach against Dark Emulator and Euclid Emulator 2 across 100 cosmologies, achieving 2–3% accuracy up to $k \,\simeq\,0.4\,h\,\mathrm{Mpc}^{-1}$ for $z \ge 0.5$, with consistent configuration-space results down to $r \simeq 10\,h^{-1}\mathrm{Mpc}$. A key contribution is an IR-resummed interpretation of BAO damping, where exponential suppression by displacement and mild nonlinear sharpening from the source term reproduce BAO features seen in simulations. The work also provides a fast, open-source Python package (ulptkit) and discusses future extensions to biased tracers, redshift-space distortions, reconstruction, and higher-loop corrections, highlighting ULPT as a robust tool for modeling LSS in galaxy surveys.

Abstract

We present a fast and accurate formulation for computing the nonlinear matter power spectrum at one-loop order based on Unified Lagrangian Perturbation Theory (ULPT). ULPT decomposes the density field into the Jacobian deviation, capturing intrinsic nonlinear growth, and the displacement-mapping factor, accounting for large-scale distortions due to bulk flows. This structural separation leads to a natural division of the power spectrum into a source term and a displacement-mapping factor, ensuring infrared (IR) safety by construction. We implement an efficient numerical algorithm using FFTLog and FAST-PT, achieving approximately 2-second evaluations on a standard laptop. The results are validated against simulation-based emulators, including the Dark Emulator and Euclid Emulator 2. Across 100 sampled cosmologies, ULPT agrees with emulator predictions at the 2--3\% level up to \( k \simeq 0.4\,h\,\mathrm{Mpc}^{-1} \) for \( z \geq 0.5 \), without any nuisance parameters. Similar agreement is found in configuration space, where the two-point correlation function remains accurate down to \( r \simeq 10\,h^{-1}\mathrm{Mpc} \). Compared to standard perturbation theory, which fails at small scales due to series expansion of the displacement factor, ULPT maintains convergence by preserving its full exponential form. We also clarify the mechanism of BAO damping: exponential suppression by displacement and peak sharpening by nonlinear growth. The combination accurately reproduces BAO features seen in simulations. ULPT thus offers a robust, IR-safe, and computationally efficient framework for modeling large-scale structure in galaxy surveys. The numerical implementation developed in this work is publicly released as the open-source Python package \texttt{ulptkit} (https://github.com/naonori/ulptkit).

Figures (11)

• Figure 1: Displacement correlation functions $\sigma_0^2(r)$ and $\sigma_2^2(r)$, computed at redshift $z = 0$ from the linear power spectrum using FFTLog. This figure focuses on the small-scale regime, $r \leq 2\,h^{-1}\mathrm{Mpc}$. The original FFTLog results, which exhibit spurious oscillations at small separations due to the finite $k_{\rm max}$ cutoff, are shown as blue curves. To suppress these instabilities, we apply a linear interpolation between $r = 0.75\,h^{-1}\mathrm{Mpc}$ and $r = 0$, connecting the analytic values at the origin to the numerical ones at the cutoff scale.
• Figure 2: Full-range displacement correlation functions at $z=0$. The magenta solid line shows the monopole component $\sigma_0^{2}(r)$, and the blue dashed line shows the quadrupole component $\sigma_2^{2}(r)$. The small-scale behavior for $r < 0.75\,h^{-1}\mathrm{Mpc}$ is smoothly interpolated as described in Fig. \ref{['fig:sigma_small']}.
• Figure 3: One-loop ULPT power spectrum at $z = 0.0$. Upper panel: The one-loop ULPT power spectrum is decomposed into the convolution-free component, $P_{\rm CF}/P_{\rm nw}$, and the cumulative convolution-containing contributions, $\sum_{n=0}^{N_{\rm C}} P_{\rm CC}^{[n]}/P_{\rm nw}$, for $N_{\rm C} = 0, 1, 2, 3, 4$. Here, $P_{\rm nw}$ denotes the no-wiggle linear power spectrum. The total ULPT power spectrum, which is sufficiently converged and can be regarded as the full result, is defined as $P_{\rm ULPT} = P_{\rm CF} + \sum_{n=0}^{4} P_{\rm CC}^{[n]}$ and is also shown. Lower panel: Convergence test of the truncated expansion in $P_{\rm CC}$, shown as the fractional deviation from the fully summed result $P_{\rm ULPT}$. The color coding is consistent with the upper panel: $N_{\rm C} = 0$, 1, 2, and 3 correspond to blue, orange, green, and red, respectively. The results demonstrate that the series converges rapidly, achieving sub-percent accuracy for $N_{\rm C} \geq 2$.
• Figure 4: Comparison of the one-loop power spectra at redshifts $z = 0$, $0.5$, and $1.0$. Upper panels: Ratios of the one-loop source power spectrum $P_{\rm J}$, the full ULPT power spectrum $P_{\rm ULPT}$, and the standard SPT power spectrum $P_{\rm SPT}$ to the no-wiggle linear spectrum $P_{\rm nw}$. All spectra include one-loop corrections. The SPT prediction is computed by expanding the displacement-mapping factor to $\mathcal{O}(P_{\rm lin})$, whereas ULPT retains its full exponential form. At $z = 0$ (left panel), $P_{\rm ULPT}$ and $P_{\rm SPT}$ begin to diverge around $k \simeq 0.1\,h\,{\rm Mpc}^{-1}$, with the latter exhibiting excessive nonlinear growth due to the breakdown of the perturbative expansion. Lower panels: Ratios $P_{\rm ULPT}/P_{\rm J}$ at each redshift, illustrating the modulation induced by the displacement-mapping factor. At $z = 0$, the modulation is purely suppressive, reaching 3--5% up to $k \simeq 0.4\,h\,{\rm Mpc}^{-1}$. At higher redshifts ($z = 0.5$ and $z = 1.0$), the modulation becomes scale-dependent: while a suppression of 2--3% persists at large scales ($k \lesssim 0.2\,h\,{\rm Mpc}^{-1}$), an enhancement up to 5% emerges at smaller scales ($k \gtrsim 0.2\,h\,{\rm Mpc}^{-1}$), with the effect being most pronounced at $z = 1.0$. These ratios also exhibit oscillatory features around the BAO scale, reflecting the difference in how $P_{\rm ULPT}$ and $P_{\rm J}$ encode the BAO signal. A more detailed analysis of BAO damping is presented in Fig. \ref{['fig:xi_BAO']}.
• Figure 5: Fractional difference in the nonlinear matter power spectrum between Euclid Emulator 2 and Dark Emulator, evaluated at redshift $z=0$ for 100 randomly sampled cosmological models within the overlapping parameter space of the two emulators. The quantity plotted is $(P_{\mathrm{Euclid}} - P_{\mathrm{Dark}})/P_{\mathrm{Dark}}$. While the deviation exceeds 5% in some individual models, the mean deviation (magenta line) remains within $\pm 1\%$ for $k \gtrsim 0.05\,h\,\mathrm{Mpc}^{-1}$, confirming the mutual consistency of the two emulators in the nonlinear regime. At larger scales ($k \lesssim 0.05\,h\,\mathrm{Mpc}^{-1}$), the mean deviation can reach up to 2%, primarily due to sample variance in Dark Emulator, as discussed in Sec. \ref{['sec:various']}.