Non-linear structure formation and the acoustic scale

TL;DR

BAO serve as a precise standard ruler for cosmological distances, but nonlinear growth and redshift distortions can induce sub-percent shifts in the measured acoustic scale. Using a large-volume PM simulation suite and flexible template fitting, the study shows real-space shifts up to $\sim0.45\%$ and redshift-space shifts about $\sim0.25$–$0.54\%$, all manageable with reconstruction and robust against reasonable template variations. The Fisher matrix forecasts broadly agree with the simulation results, giving confidence in distance inferences such as $D_A(z)/r_s$ and $H(z)r_s$ from BAO. With reconstruction, the shifts can be reduced to $<0.1\%$ across the redshift range, solidifying BAO as a practical, high-precision standard ruler for probing dark energy.

Abstract

We present high signal-to-noise measurements of the acoustic scale in the presence of nonlinear growth and redshift distortions using 320(Gpc/h)^3 of cosmological PM simulations. Using simple fitting methods, we obtain robust measurements of the acoustic scale with scatter close to that predicted by the Fisher matrix. We detect and quantify the shift in the acoustic scale by analyzing the power spectrum: we detect at greater than 5 sigma a decrease in the acoustic scale in the real-space matter power spectrum of 0.2% at z=1.5, growing to 0.45% at z=0.3. In redshift space, the shifts are about 25% larger: we detect a decrease of 0.25% of at z=1.5 and 0.54% at z=0.3. Despite the nonzero amounts, these shifts are highly predictable numerically, and hence removable within the standard ruler analysis of clustering data. Moreover, we show that a simple density-field reconstruction method substantially reduces the scatter and nonlinear shifts of the acoustic scale measurements: the shifts are reduced to less than 0.1% at z=0.3-1.5, even in the presence of non-negligible shot noise. Finally, we show that the ratio of the cosmological distance to the sound horizon that would be inferred from these fits is robust to variations in the parameterization of the fitting method and reasonable differences in the template cosmology.

Figures (8)

• Figure 1: Real-space (top) and redshift-space (bottom) power spectrum $P(k)$ from our $N$-body simulations for z=0.3, 0.7, 1.0, and 1.5. The power spectra are divided by $P_{\rm smooth}$, the nowiggle form from EH98, scaled with $D^2$. The dashed line is the linear power spectrum. The gray lines indicate the large-scale amplitude expected from linear theory.
• Figure 2: Real-space power spectra at $z=0.3$ and 1.5 divided by the initial linear power spectrum: we divide $P_{\rm res} = (P_{\rm nl}-A(k))/B(k)$ by $P_{\rm lin}$. The oscillatory feature represents the amount of degradation of the BAO. Red error bars are Gaussian errors centered at $P_{\rm res}/P_{\rm lin}$. Black solid lines are for the template power spectra $P_m (k/\alpha)/P_{\rm lin} (k)$ constructed by using $\Sigma_{\rm m}=7.6h^{-1}{\rm\;Mpc}$ and $4.5h^{-1}{\rm\;Mpc}$ at $z=0.3$ and 1.5, respectively. Blue solid lines represent the nowiggle form, i.e., $P_{\rm smooth}(k/\alpha)/P_{\rm lin}(k)$.
• Figure 3: Left and Middle panels: the growth of nonlinear shift in $\alpha-1$ with redshift. The data points show the mean of $\alpha$ as a function of redshift; the errors show the scatter. The points are correlated. In the left panel, we overplot curves proportional to $D^1$; in the middle panel, we overplot curves proportional to $D^2$. Right panel: the $\alpha-1$ at $z=0.3$ and $z=1.5$ for 40 simulations. One can see that a given simulation produces $\alpha$ that are highly correlated between redshifts.
• Figure 4: Redshift-space power spectra at $z=0.3$ and 1.5 divided by the initial linear power spectrum: we divide $P_{\rm res} = (P_{\rm nl}-A(k))/[B(k)F_{\rm fog}]$ by $P_{\rm lin}$. The oscillatory feature represents the amount of degradation of the BAO. Red error bars are Gaussian errors centered at $P_{\rm res}/P_{\rm lin}$. Black solid lines: $P_m (k/\alpha)/P_{\rm lin} (k)$ constructed by using $\Sigma_{\rm m}=9.0h^{-1}{\rm\;Mpc}$ and $6.0h^{-1}{\rm\;Mpc}$ at $z=0.3$ and 1.5, respectively. Blue solid lines represent the nowiggle form, i.e., $P_{\rm smooth}(k/\alpha)/P_{\rm lin}(k)$.
• Figure 5: Real-space $P_{\rm nl}(k)$ of the 1%-sample after reconstruction (red lines), divided by the nowiggle form from EH98, in comparison to the nonlinear power spectra before reconstruction (blue lines). The gray lines are for the large-scale amplitude expected from linear theory. The dashed lines are for linear power spectrum. The effect of reconstruction is most apparent at $z=0.3$ and is smallest at $z=1.5$.