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

## TL;DR

This paper introduces a fast, accurate, and IR-safe one-loop nonlinear matter power spectrum computation method using Unified Lagrangian Perturbation Theory, validated against emulators and suitable for large-scale structure analysis.

## Contribution

The authors develop a novel ULPT-based framework that improves convergence and computational efficiency for nonlinear power spectrum modeling, with open-source implementation.

## Key findings

- ULPT achieves 2-3% accuracy up to k ≈ 0.4 h/Mpc for z ≥ 0.5
- The method is computationally efficient, taking about 2 seconds per evaluation
- ULPT accurately reproduces BAO features and is validated against emulators.

## 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).