The Coyote Universe I: Precision Determination of the Nonlinear Matter Power Spectrum

TL;DR

This paper demonstrates that gravity-only N-body simulations can calibrate the nonlinear matter power spectrum to ~1% accuracy out to k ~ 1 h/Mpc for 0 ≤ z ≤ 1, given large volumes (~1 Gpc^3), ~10^9 particles, early initialization (z_in ~ 200), and careful control of force resolution and time stepping. It compares PM and tree-PM codes, develops robust power-spectrum estimation, and performs extensive convergence tests across box size, mass resolution, aliasing, and stepping schemes. A key result is that matching low- and high-resolution spectra yields a reliable spectrum that deviates by only ~1% up to k ≈ 0.5 h/Mpc, while HaloFit underestimates power by ~5% in the same range. The work lays the groundwork for a 3-paper series that will build an emulator and publicly release precise predictions for a range of cosmologies, advancing the use of simulations in precision weak-lensing analyses.

Abstract

Near-future cosmological observations targeted at investigations of dark energy pose stringent requirements on the accuracy of theoretical predictions for the clustering of matter. Currently, N-body simulations comprise the only viable approach to this problem. In this paper we demonstrate that N-body simulations can indeed be sufficiently controlled to fulfill these requirements for the needs of ongoing and near-future weak lensing surveys. By performing a large suite of cosmological simulation comparison and convergence tests we show that results for the nonlinear matter power spectrum can be obtained at 1% accuracy out to k~1 h/Mpc. The key components of these high accuracy simulations are: precise initial conditions, very large simulation volumes, sufficient mass resolution, and accurate time stepping. This paper is the first in a series of three, with the final aim to provide a high-accuracy prediction scheme for the nonlinear matter power spectrum.

Figures (21)

• Figure 1: Ratio of the E-mode correlation function with and without an assumed suppression of the power spectrum mimicking a possible systematic error in the matter power spectrum. This figure demonstrates that a gradual decrease in the accuracy of the matter power spectrum on small scales will not lead to a catastrophic error in the weak lensing prediction. The green line with $k_F=10\,h\,{\rm Mpc}^{-1}$ corresponds to error properties which will be close to the degradation we expect for the matter power spectrum presented in this paper (see text).
• Figure 2: Upper panel: Synthetic data from a HaloFit run. Lower panel: Synthetic data from a combination of several $N$-body runs. In both cases the black line shows the underlying power spectrum from which the data was drawn and the red points show 34 data points with error bars. At small spatial scales, the assumed error is 1%, rising to 10% at large scales due to increased sample variance.
• Figure 3: Posterior distributions for the five parameters under consideration. Upper panel: Results for the analysis of the HaloFit synthetic data set analyzed with a set of HaloFit power spectra. The red dots indicate the true values. As is to be expected, the constraints on the parameters are very good. Lower panel: Results for the HaloFit-based analysis of the $N$-body synthetic data set. Note that the constraints for $_m$ and $w$ are now incorrect at $\sim 20$%.
• Figure 4: Upper panel: Comparison of dimensionless power spectra from a handful of $N$-body codes, taken from the data of CodeCompare for the "LCDMb" box: a $\Lambda$CDM model with $\Omega_m=0.314$, $h=0.71$, $n_s=0.99$, and $L_{\rm box}=256\,h^{-1}$Mpc with 256$^3$ particles. The two PM codes, MC$^2$ and PMM, were run on a 1024$^3$ grid (with a grid-to-particle ratio of 4:1, a factor of two higher than used for the PM runs in this paper). The FLASH run had a base grid of 256$^3$ and a refinement level of two. Therefore, the force resolution of the purely grid-based codes is roughly a factor of ten lower than for the other codes (the different force kernels make a precise comparison difficult). The dotted lines show the 1% agreement limit. The high force-resolution codes agree to ${\mathcal{O}}(1\%)$ up to $k\sim 1\,h\,{\rm Mpc}^{-1}$ despite different choices for the force softening and other numerical parameters. Lower panel: Comparison of GADGET-2 and ART for a simulation with 1024$^3$ particles and $L_{\rm box}=1~h^{-1}$Gpc. The cosmological parameters are very close to those for our major runs, the main difference being the starting redshift of $z_{\rm in}=65.66$. The agreement of the two codes is better than 1% over all scales. In addition, we compare one of the PM runs used in this paper with respect to GADGET-2. The agreement is also at ${\mathcal{O}}(1\%)$. We re-emphasize that our goal is to derive simulation requirements for percent level accuracy and finding a good balance between efficient computing and accuracy. By tuning code parameters, the agreement between different codes may be improved, but this would defeat the purpose of testing for robustness.
• Figure 5: Top two panels: comparison of outputs at $z=10$ using different starting redshifts. The particles are colored with respect to their velocities. The simulation box is 8$\,h^{-1}$Mpc on a side. The simulation shown in the left panel was started at $z_{\rm in}=250$, while for the other, $z_{\rm in}=50$. In the simulation started at $z_{\rm in}=50$, structures formed by $z=10$ are not as concentrated as in simulations with a high-$z$ start, leading to the possible lowering of halo masses. The lower panel shows differences along a filament. In this case a line was drawn between each particle position in the two different data sets. The longer the line, the larger the difference due to the two different initial redshifts. For more details see Har08HarHei08Luk07.