Perturbation Theory Reloaded II: Non-linear Bias, Baryon Acoustic Oscillations and Millennium Simulation In Real Space

TL;DR

We address how to extract cosmological information from the real-space galaxy power spectrum using third-order perturbation theory with a local, stochastic bias model. The authors validate the approach against Millennium Simulation data across redshifts 1–6, showing accurate modeling in the weakly non-linear regime. By marginalizing over three bias parameters, they recover an unbiased distance scale D, related to $D_A(z)$ and $H(z)$, with about 3% accuracy, and discuss degeneracies and the potential of the bispectrum to break them. The study highlights the practical viability of PT for high-redshift galaxy surveys and outlines future work to include redshift-space distortions and larger simulations for tighter cosmological constraints.

Abstract

We calculate the non-linear galaxy power spectrum in real space, including non-linear distortion of the Baryon Acoustic Oscillations, using the standard 3rd-order perturbation theory (PT). The calculation is based upon the assumption that the number density of galaxies is a local function of the underlying, non-linear density field. The galaxy bias is allowed to be both non-linear and stochastic. We show that the PT calculation agrees with the galaxy power spectrum estimated from the Millennium Simulation, in the weakly non-linear regime (defined by the matter power spectrum) at high redshifts, $1\le z\le6$. We also show that, once 3 free parameters characterizing galaxy bias are marginalized over, the PT power spectrum fit to the Millennium Simulation data yields unbiased estimates of the distance scale, $D$, to within the statistical error. This distance scale corresponds to the angular diameter distance, $D_A(z)$, and the expansion rate, $H(z)$, in real galaxy surveys. Our results presented in this paper are still restricted to real space. The future work should include the effects of non-linear redshift space distortion. Nevertheless, our results indicate that non-linear galaxy bias in the weakly non-linear regime at high redshifts is reasonably under control.

Figures (23)

• Figure 1: Matter power spectrum at $z=0$, 1, 2, 3, 4, 5 and 6 (from top to bottom) derived from the Millennium Simulation (dashed lines), the 3rd-order PT (solid lines), and the linear PT (dot-dashed lines).
• Figure 2: Dimensionless matter power spectrum, $\Delta^2(k)$, at $z=1$, 2, 3, 4, 5, and 6. The dashed and solid lines show the Millennium Simulation data and the 3rd-order PT calculation, respectively. The dot-dashed lines show the linear power spectrum.
• Figure 3: Fractional difference between the matter power spectra from the 3rd-order PT and that from the Millennium Simulation, $P_m^{sim}(k)/P_m^{PT}-1$ (dots with errorbars). The solid lines show the perfect match, while the dashed lines show $\pm 2\%$ accuracy. We also show $k_{max}(z)$, below which we trust the prediction from the 3rd-order PT, as a vertical dotted line.
• Figure 4: Distortion of BAOs due to non-linear matter clustering. All of the power spectra have been divided by a smooth power spectrum without baryonic oscillations from eq. (29) of eisenstein/hu:1998. The error bars show the simulation data, while the solid lines show the PT calculations. The dot-dashed lines show the linear theory calculations. The power spectrum data shown here have been taken from Figure 6 of springel/etal:2005.
• Figure 5: Halo power spectra from the Millennium Simulation at $z=1$, 2, 3, 4, 5, and 6. Also shown in smaller panels are the residual of fits. The points with errorbars show the measured halo power spectra, while the solid, dashed, and dot-dashed lines show the best-fitting non-linear bias model (Eq. (\ref{['eq:3rd_PT_Pk']})), the best-fitting linear bias with the non-linear matter power spectrum, and the best-fitting linear bias with the linear matter power spectrum, respectively. Both linear models have been fit for $k_{max,linear}=0.15~[h~\mathrm{Mpc}^{-1}]$, whereas $k_{max}(z)$ given in Table \ref{['table:kmax']} (also marked in each panel) have been used for the non-linear bias model.