Estimating CDM Particle Trajectories in the Mildly Non-Linear Regime of Structure Formation. Implications for the Density Field in Real and Redshift Space

TL;DR

This paper develops a trajectory-based framework to model CDM evolution in the mildly non-linear regime by perturbatively expanding around Lagrangian Perturbation Theory and calibrating displacement and velocity transfer functions with N-body simulations. The approach yields improved real- and redshift-space density and momentum predictions, achieving cross-correlations with fully non-linear fields exceeding 0.95 in real space down to k ~ 0.45 h/Mpc at z=0, and enabling large reductions in sample variance and substantial speed-ups for BAO-focused cosmological analyses. The method provides a path toward BAO peak reconstruction, mock catalog construction, and momentum-field reconstruction, with explicit treatment of redshift-space distortions and FoG effects through the tracked phase-space structure. The work demonstrates significant practical impact for accelerating cosmological parameter inference and improving the fidelity of synthetic surveys, with future extensions to biased tracers and higher-order perturbative corrections.

Abstract

We obtain approximations for the CDM particle trajectories starting from Lagrangian Perturbation Theory. These estimates for the CDM trajectories result in approximations for the density in real and redshift space, as well as for the momentum density that are better than what standard Eulerian and Lagrangian perturbation theory give. For the real space density, we find that our proposed approximation gives a good cross-correlation (>95%) with the non-linear density down to scales almost twice smaller than the non-linear scale, and six times smaller than the corresponding scale obtained using linear theory. This allows for a speed-up of an order of magnitude or more in the scanning of the cosmological parameter space with N-body simulations for the scales relevant for the baryon acoustic oscillations. Possible future applications of our method include baryon acoustic peak reconstruction, building mock galaxy catalogs, momentum field reconstruction.

Figures (11)

• Figure 1: Shown are various matter power spectra at $z=0$ for $\Lambda$CDM, similar to Fig. 1 of tassev (errorbars are suppressed for clarity). 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 solid curve shows the non-linear power spectrum (i.e. the "exact" power spectrum obtained from N-body simulations); the part of the NL power spectrum due to the "memory of the initial conditions" rpt is shown with the dashed line; the power due to the projection of the non-linear density field on our best approximation (cf. second line of eq. (\ref{['QE2']})) is given by the dotted curve. The residual mode-coupling power is 13.5% of the NL power at $k=0.5h/$Mpc for $z=0$. Here $R_\delta$ corresponds to the transfer function relating the NL density and the density given by our best approximation. A vertical line denotes the non-linear scale. The current Hubble expansion rate in units of 100 km/s/Mpc is given by $h$.
• Figure 2: Probability density, $p$, for the distribution of errors in determining the center-of-mass (CM) position/velocity of a given volume element; as well as the error in determining the rms velocity ($z=0$). We show the distributions after imposing the denoted cutoffs in the approximated density field, $\delta_\bigstar$. The approximate velocities, positions and density are calculated according to our best model (cf. second line of eq. (\ref{['QE2']})). CM velocities and positions for high-density regions are determined with worse precision than for voids, but still the error is within a couple of Mpc. The rms velocity responsible for the FoG effect is determined with worse precision but again within several Mpc. The rms linear displacement corresponding to $z=0$ is $\approx15\,$Mpc/h, or about an order of magnitude larger than the errors depicted in the plot. The probabilities are obtained after Gaussian smoothing on a scale of $1.3\,$Mpc$/h$.
• Figure 3: Cross-correlation and transfer functions for the CDM displacements and velocities. In each plot we show the denoted quantities for the four different approximations given in eq. (\ref{['QETZ']}, \ref{['QE2']}). The red curves show the corresponding cross-correlation function divided by $\delta^2(<k,\eta)$, to highlight the low-$k$ behavior. The non-linear scale (defined here as the linear power per logarithmic $k$-bin to be 1) is at $k_{NL}=0.25h/$Mpc for $z=0$ (denoted with vertical line) and $k_{NL}=0.74h/$Mpc for $z=1$.
• Figure 4: In this figure we show the different contributions to the power spectrum of the displacements (left panel) and velocities (right panel). The total quantities are denoted with $NL$, and the mode-coupling vector fields are split (in Lagrangian space) into irrotational and solenoidal parts. The plot is for $z=0$; and here $k$ is the wave-vector corresponding to Lagrangian space.
• Figure 5: Toy model for the ZA displacement and velocity transfer functions evaluated at $z=0$.