The Mildly Non-Linear Regime of Structure Formation

TL;DR

This work develops a physically motivated framework for the mildly non-linear regime of structure formation, centering on the Zel'dovich approximation to capture bulk flows that contaminate equal-time statistics. It introduces a transfer-function approach, P_{NL} = ilde{R}^2 P_Z + ilde{P}_{MC}, enabling accurate reconstruction of the non-linear power spectrum from cheaply computed ZA/2LPT realizations and substantial reduction of sample variance in N-body simulations. The authors provide practical estimators for P_{NL} and demonstrate order-of-magnitude improvements in variance and large-scale accuracy, including BAO-scale robustness, with a simple analytic model P_{NL} ≈ P_Z [1 + (2950/1323) δ^2(<k/2)]. These results accelerate cosmological parameter estimation that relies on mildly non-linear scales and offer a reliable pathway to sub-percent BAO measurements.

Abstract

We present a simple physically motivated picture for the mildly non-linear regime of structure formation, which captures the effects of the bulk flows. We apply this picture to develop a method to significantly reduce the sample variance in cosmological N-body simulations at the scales relevant to the Baryon Acoustic Oscillations (BAO). The results presented in this paper will allow for a speed-up of an order of magnitude (or more) in the scanning of the cosmological parameter space using N-body simulations for studies which require a good handle of the mildly non-linear regime, such as those targeting the BAO. Using this physical picture we develop a simple formula, which allows for the rapid calculation of the mildly non-linear matter power spectrum to percent level accuracy, and for robust estimation of the BAO scale.

Figures (13)

• Figure 1: Shown schematically are various matter power spectra at $z=0$ for $\Lambda$CDM. The power spectra are divided by a smooth BBKS BBKS power spectrum with shape parameter $\Gamma=0.15$ in order to highlight the wiggles due to the BAO. The non-linear power spectrum (i.e. the "exact" power spectrum obtained from N-body simulations) is given by $P_{NL}$; the linear power spectrum by $P_L$; the part due to the "memory of the initial conditions" by $R^2P_L$; the power spectrum calculated in the Zel'dovich approximation by $P_Z$; the power due to the projection of the non-linear density field on the Zel'dovich density field by $\tilde{R}^2P_Z$. The green (long dashed) arrow represents the power generated from mode-coupling due to bulk flows, free-streaming and structure formation; while the blue (dot-dashed) arrow represents the power generated by structure formation alone. The current Hubble expansion rate in units of 100 km/s/Mpc is given by $h$.
• Figure 2: The cross-correlation coefficient between the non-linear overdensity and the overdensity in linear theory and the ZA as a function of scale at $z=0$. The cross-correlation coefficient, $\rho_{NL,L}$, is very close to the non-linear propagator in RPT, despite their different definitions. Note that the Zel'dovich density field is well-correlated with the non-linear density field well into the mildly non-linear regime.
• Figure 3: We show the overdensity $\delta(<k)$ and rms particle displacements $\sigma_v(<k)$ integrated up to $k$, calculated in linear theory at $z=0$. The quantity $k\sigma_v(<k)$ is due to the bulk flows and it regulates the destruction of the cross-correlation between the linear and non-linear density fields. Note that $k\sigma_v(<k)$ is larger than $\delta(<k)$ in the mildly non-linear regime. The peak of the velocity power spectrum is at $k_*$ which is well below the non-linear scale $k_{NL}$ when the linear density power per logarithmic $k$-interval reaches 1. Therefore $\sigma_v$ is roughly constant in the mildly non-linear regime. In the plot we marked with vertical lines the positions of $k_*$ and $k_{NL}$ for reference. See the text for further discussion.
• Figure 4: We show the raw non-linear matter power spectrum at $z=0$ (with 2-sigma errorbars for the average $P_{NL}$) obtained from an ensemble of 10 N-body simulations. We also show the linear and Zel'dovich power spectra for comparison. All power spectra are divided by a smooth BBKS power spectrum to highlight the BAO wiggles.
• Figure 5: The same as in Figure \ref{['fig:RAW']} with the non-linear power spectrum calculated from the same 10 simulations but using the 2LPT estimator, eq. (\ref{['restore2']}).