On the reach of perturbative descriptions for dark matter displacement fields

TL;DR

This work probes how far perturbative descriptions can reliably predict dark matter displacement fields by performing field-level comparisons between LPT/EFT predictions and N-body simulations with matched initial phases to suppress cosmic variance. The authors demonstrate a nonzero leading EFT coefficient α and identify a stochastic displacement term that uncorrelated with LPT terms, imposing a fundamental 1% error floor in the non-linear power spectrum at k ≈ 0.2 h/Mpc (z=0). They show that 1-loop EFT substantially extends the regime of accuracy (to k ≈ 0.15 h/Mpc) beyond 1-loop LPT, and that transfer-function based tLPT is close to EFT in performance, while higher-order terms are limited by the stochastic floor. The results underscore the importance of explicitly modeling stochastic displacements within EFT and establish a practical, field-level testing framework that minimizes cosmic variance and yields robust constraints on perturbative approaches for upcoming large-scale structure surveys.

Abstract

We study Lagrangian Perturbation Theory (LPT) and its regularization in the Effective Field Theory (EFT) approach. We evaluate the LPT displacement with the same phases as a corresponding $N$-body simulation, which allows us to compare perturbation theory to the non-linear simulation with significantly reduced cosmic variance, and provides a more stringent test than simply comparing power spectra. We reliably detect a non-vanishing leading order EFT coefficient and a stochastic displacement term, uncorrelated with the LPT terms. This stochastic term is expected in the EFT framework, and, to the best of our understanding, is not an artifact of numerical errors or transients in our simulations. This term constitutes a limit to the accuracy of perturbative descriptions of the displacement field and its phases, corresponding to a $1\%$ error on the non-linear power spectrum at $k=0.2 h$/Mpc at $z=0$. Predicting the displacement power spectrum to higher accuracy or larger wavenumbers thus requires a model for the stochastic displacement.

Figures (25)

• Figure 1: Diagrams for the one-loop (first row) and two-loop (second and third row) contributions to the power spectrum of the scalar displacement $\phi$. The empty squares symbolize the LPT kernels $L_n$, while the linear matter power spectrum is represented by the black dots. The explicit formulae for the expressions are given in App. \ref{['sec:appendix_lpt']}.
• Figure 2: Power spectra of the LPT displacements up to 5th order, for one particular realization of the linear density field $\delta_0$ with cutoff $k_\text{max}=0.6 h\text{Mpc}^{-1}$.
• Figure 3: Left panel: Power index for each contribution to the power spectrum of $\phi$, as a function of the power index of the linear density power spectrum $P_\text{lin}$. Right panel: Power index of the no-wiggle power spectrum from Eisenstein & Hu Eisenstein:1997ik.
• Figure 4: Comparison between the scalings from EdS (shaded areas) and the actual loop contributions to the power spectrum (solid lines). The scalings correspond to $n = -1.4$, $k_\text{nl} = 1.05$ h/Mpc and $b$ floating between $0.1$ and $10$. These scalings approximately match the 1-loop terms, but fail at reproducing the higher order terms.
• Figure 5: Left panel: contributions to $P_{13}$ from the large-scale density modes (red line) and small-scale displacement modes (blue line). Right panel: contributions to $P_{22}$ from the large-scale density modes (blue line) and the $\eta$ term. The dominant contribution comes from the small-scale displacement modes, which only enter the expression of $P_{13}$ and not $P_{22}$.