On the reach of perturbative methods for dark matter density fields

TL;DR

The paper probes the reach of perturbative methods for dark matter density fields by applying Lagrangian Perturbation Theory within an EFT framework to map from Lagrangian to Eulerian space and by incorporating IR-resummation via density transfer functions. It demonstrates that a full, non-perturbative density transformation built on LPT, together with EFT counterterms and stochastic contributions, can match N-body results with percent-level accuracy up to k ≈ 0.25 h Mpc^{-1} at z=0, with stochastic effects limiting further reach. A beyond-one-loop approach using transfer functions captures higher-order effects and bulk flows, revealing that the stochastic term remains the dominant irreducible source of error on the scales of interest. The study clarifies the relation between Lagrangian and Eulerian stochastic terms, highlights the challenges in modeling k^4-scaling for the stochastic contribution, and provides a framework for testing IR-resummed EFT predictions at the field level. Overall, IR-resummed LPT-based methods offer a practical route to benchmark and extend the validity of EFT predictions for large-scale structure.

Abstract

We study the mapping from Lagrangian to Eulerian space in the context of the Effective Field Theory (EFT) of Large Scale Structure. We compute Lagrangian displacements with Lagrangian Perturbation Theory (LPT) and perform the full non-perturbative transformation from displacement to density. When expanded up to a given order, this transformation reproduces the standard Eulerian Perturbation Theory (SPT) at the same order. However, the full transformation from displacement to density also includes higher order terms. These terms explicitly resum long wavelength motions, thus making the resulting density field better correlated with the true non-linear density field. As a result, the regime of validity of this approach is expected to extend that of the Eulerian EFT, and match that of the IR-resummed Eulerian EFT. This approach thus effectively enables a test of the IR-resummed EFT at the field level. We estimate the size of stochastic, non-perturbative contributions to the matter density power spectrum. We find that in our highest order calculation, at redshift z=0 the power spectrum of the density field is reproduced with an accuracy of 1 % (10 %) up to k=0.25 h/Mpc (k=0.46 h/Mpc). We believe that the dominant source of the remaining error is the stochastic contribution. Unfortunately, on these scales the stochastic term does not yet scale as $k^4$ as it does in the very low-k regime. Thus, modeling this contribution might be challenging.

Figures (10)

• Figure 1: Power spectrum of the displacement divergence $-\text{i} \bm k \cdot \bm s$ (left panel) and ratio of this power spectrum to linear theory (right panel). We show the non-linear displacement divergence from the simulations as well as the LPT and EFT terms. The 3LPT displacement overpredicts the true non-linear displacement. This agreement can be fixed by adding the leading order counterterm (green dotted) until $k=0.2 \ h\text{Mpc}^{-1}$, but for larger wavenumbers the counterterm clearly makes the agreement between simulations and theory worse.
• Figure 2: Performance of LPT density power spectra for 1LPT/Zel'dovich and 3LPT evaluated for the same initial conditions as the simulation for redshifts $z=0$ (left panel) and $z=1$ (right panel). We clearly see the missing power in both the 1LPT and 3LPT power spectra. This failure can be reduced by setting the maximum wavenumber or LPT cutoff to $k_\text{max}=0.4\ h\text{Mpc}^{-1}$. The same effect can be achieved by adding a $k^2 P_{11}$ counter term as predicted in the EFT framework. The jaggedness of the 1LPT lines compared to the higher orders of LPT arises is due to the missing higher order terms that also cancel cosmic variance.
• Figure 3: Scale dependence of the coefficient $\alpha$ of the leading order counterterm in the L simulation (left panel) and the M simulation (right panel). We consider the one loop LPT model (green, lower) and the Zel'dovich model (red, upper). The solid lines show constraints from the cross spectrum with the Zel'dovich field, whereas dashed lines show constraints from the auto spectrum. The horizontal gray lines indicate the value of the EFT coefficient employed in Fig. \ref{['fig:leftperformance']}. The difference between the green and red lines on large scales arises from the low-$k$ limit of $P_{13,\text{L}}$ ($48/63 \sigma_d^2 k^2 P$) as well as $\sigma_{d,13}^2$ and $\sigma_{d,22}^2$ entering in Eq. \ref{['eq:dterm']}. The difference between the green lines in the two panels on large scales is given by the cutoff dependence of $\sigma_{d,13}^2$ and $\sigma_{d,22}^2$.
• Figure 4: Cumulative contributions to the displacement dispersion up to a certain maximum wavenumber $k_\text{max}$ from the various contributions to one loop LPT. We show both the bare contributions of LPT (solid) as well as the ones from the displacements regularized by the transfer functions (dashed). The non-linear displacement dispersion coincides with the linear one before a transfer function is employed. The transfer function has a mild influence on the $\sigma_{d,22}^2$ and $\sigma_{d,11}^2$ part, but significantly reduces the contribution from $\sigma_{d,13}^2$.
• Figure 5: Error power spectra at redshift $z=0$ for the L simulation and various orders of LPT and tLPT. Left panel: Total power. The gray lines show the expectation for the stochastic term for $k_\text{nl}=0.3\ h\text{Mpc}^{-1}$ and $k_\text{nl}=0.5\ h\text{Mpc}^{-1}$, respectively. We see a decrease in mode coupling as we go to higher orders and implement transfer functions on the density. Right panel: Ratio of the error power spectrum and the full non-linear power spectrum.