Perturbation Theory Reloaded: Analytical Calculation of Non-linearity in Baryonic Oscillations in the Real Space Matter Power Spectrum

TL;DR

The paper addresses predicting the non-linear real-space matter power spectrum with percent-level accuracy for future high-redshift galaxy surveys. It uses 3rd-order perturbation theory to compute the next-to-leading order corrections, expressing P(k,z) via $P(k,z) = D^2(z) P_{11}(k) + D^4(z) [2 P_{13}(k) + P_{22}(k)]$ and validating against extensive N-body simulations. The key findings are that PT agrees with simulations to better than 1% for Delta^2(k,z) ≤ 0.4, that non-linear distortions of baryon acoustic oscillations can be corrected nearly exactly at z>1, and that PT outperforms empirical halo-model fits. These results imply that PT can be a powerful, parameter-free tool for precision cosmology, with planned extensions to redshift-space distortions, halo biasing, and higher-order statistics such as the bispectrum.

Abstract

We compare the non-linear matter power spectrum in real space calculated analytically from 3rd-order perturbation theory with N-body simulations at 1<z<6. We find that the perturbation theory prediction agrees with the simulations to better than 1% accuracy in the weakly non-linear regime where the dimensionless power spectrum, Delta^2(k)=k^3P(k)/2pi^2, which approximately gives variance of matter density field at a given k, is less than 0.4. While the baryonic acoustic oscillation features are preserved in the weakly non-linear regime at z>1, the shape of oscillations is distorted from the linear theory prediction. Nevertheless, our results suggest that one can correct the distortion caused by non-linearity almost exactly. We also find that perturbation theory, which does not contain any free parameters, provides a significantly better fit to the simulations than the conventional approaches based on empirical fitting functions to simulations. The future work would include perturbation theory calculations of non-linearity in redshift space distortion and halo biasing in the weakly non-linear regime.

Figures (5)

• Figure 1: Power spectrum at $z=1$, 2, 3, 4, 5 and 6 (from top to bottom), derived from $N$-body simulations (dashed lines), perturbation theory (solid lines), and linear theory (dot-dashed lines). We plot the simulation data from 512, 256, 128, and 64 $h^{-1}$ Mpc simulations at $k\leq 0.24~h~{\rm Mpc}^{-1}$, $0.24<k\leq 0.5~h~{\rm Mpc}^{-1}$, $0.5<k\leq 1.4~h~{\rm Mpc}^{-1}$, and $1.4<k\leq 5~h~{\rm Mpc}^{-1}$, respectively. Note that we did not run 64 $h^{-1}$ Mpc simulations at $z=1$ or 2.
• Figure 2: ( Top) Dimensionless power spectrum, $\Delta^2(k)$. The solid and dashed lines show perturbation theory calculations and $N$-body simulations, respectively. The dotted lines show the predictions from halo approach smith/etal:2003. The dot-dashed lines show the linear power spectrum. ( Bottom) Residuals. The errorbars show the $N$-body data divided by the perturbation theory predictions minus one, while the solid curves show the halo model calculations given in smith/etal:2003 divided by the perturbation theory predictions minus one. The perturbation theory predictions agree with simulations to better than 1% accuracy for $\Delta^2(k)\lesssim 0.4$.
• Figure 3: Non-linearity in baryonic acoustic oscillations. All of the power spectra have been divided by a smooth power spectrum without baryonic oscillations from equation (29) of eisenstein/hu:1998. The errorbars show $N$-body simulations, while the solid lines show perturbation theory calculations. The dot-dashed lines show the linear theory predictions. Perturbation theory describes non-linear distortion on baryonic oscillations very accurately at $z>1$. Note that different redshift bins are not independent, as they have grown from the same initial conditions. The $N$-body data at $k<0.24$ and $k>0.24~h~{\rm Mpc}^{-1}$ are from 512 and 256 $h^{-1}$ Mpc box simulations, respectively.
• Figure 4: Non-linearity and the amplitude of matter fluctuations, $\sigma_8$. In each panel the lines show the linear spectrum and non-linear spectrum with $\sigma_8=0.7$, 0.8, 0.9 and 1.0 from bottom to top.
• Figure 5: Convergence test. ( Left) Fractional differences between the power spectra from $N$-body simulations in $L_{\rm box}=512$, 256, and 128 $h^{-1}$ Mpc box (from bottom to top lines) and the perturbation theory predictions in $k<1.5~h~{\rm Mpc}^{-1}$. ( Right) The same as left panel, but for simulations in $L_{\rm box}=512$, 256, 128, and 64 $h^{-1}$ Mpc box (from bottom to top lines) in the expanded range in wavenumber, $k<5~h~{\rm Mpc}^{-1}$.