Systematic effects in the sound horizon scale measurements

TL;DR

This work assesses three potential systematics that could bias BAO-based distance measurements when using the correlation-function peak as a minimal, model-independent ruler: nonlinear structure growth, scale-dependent galaxy bias, and survey window-function errors. By deriving the statistical error on the peak location, evaluating nonlinear and bias-induced shifts with halo-model and toy-model approaches, and analyzing window-function effects, the authors find that for redshifts $z\gtrsim 1$ the biases are typically smaller than or comparable to statistical errors in ambitious surveys, with nonlinear evolution causing $<0.3\%$ shifts and scale-dependent bias usually at $\lesssim 1\%$ (often less), while window-function uncertainties must be carefully controlled (to a few percent in the relevant correlation-function scale) to keep biases below $1\%$. They also demonstrate that RMS photometric zero-point errors of $\lesssim 0.14$ mag at $z\sim1$ (red galaxies) and $\lesssim 0.01$ mag at $z\sim3$ (LBGs) are required to maintain accuracy, and that a simple peak-of-the-correlation-function estimator yields errors comparable to full power-spectrum ML approaches. Overall, the study supports the viability of using the BAO sound horizon as a standard ruler while highlighting practical targets for controlling systematic biases in future surveys.

Abstract

We investigate three potential sources of bias in distance estimations made assuming that a very simple estimator of the baryon acoustic oscillation (BAO) scale provides a standard ruler. These are the effects of the non-linear evolution of structure, scale-dependent bias and errors in the survey window function estimation. The simple estimator used is the peak of the smoothed correlation function, which provides a variance in the BAO scale that is close to optimal, if appropriate low-pass filtering is applied to the density field. While maximum-likelihood estimators can eliminate biases if the form of the systematic error is fully modeled, we estimate the potential effects of un- or mis-modelled systematic errors. Non-linear structure growth using the Smith et al. (2003) prescription biases the acoustic scale by <0.3% at z>1 under the correlation-function estimator. The biases due to representative but simplistic models of scale-dependent galaxy bias are below 1% at z>1 for bias behaviour in the realms suggested by halo model calculations, which is expected to be below statistical errors for a 1000 sq.degs. spectroscopic survey. The distance bias due to a survey window function errors is given in a simple closed form and it is shown it has to be kept below 2% not to bias acoustic scale more than 1% at z=1, although the actual tolerance can be larger depending upon galaxy bias. These biases are comparable to statistical errors for ambitious surveys if no correction is made for them. We show that RMS photometric zero-point errors (at limiting magnitude 25 mag) below 0.14 mag and 0.01 mag for redshift z=1 (red galaxies) and z=3 (Lyman-break galaxies), respectively, are required in order to keep the distance estimator bias below 1%.

Figures (6)

• Figure 1: The nonlinear dark matter power spectrum (upper panel) and respective correlation function (lower panel) for our fiducial cosmological model at the redshift $z=1$. Baryon acoustic oscillations are seen as wiggles in the power spectrum and the 'hump' at scale $102.1 \; \, h^{-1} \mathrm{Mpc}$ in the correlation function. For scales larger than the baryon acoustic peak the correlation function becomes negative (dashed line).
• Figure 2: Relative statistical error of the characteristic scale $r_c$ in the real space correlation function vs. filtering scale $R$. Note the competing effects of the shot noise subtraction (small $R$) and baryon wiggles removal (large $R$), both due to the smoothing. Shown are predictions for the modest 1000 sq. degs. survey covering volume of $1.4 \, h^{-3} \mathrm{Gpc}^3$ around $z=1$ ($\Delta z = 0.5$) and spatial galaxy density (solid lines, from top to bottom) $2.5\times 10^{-3} \, h^{3} \mathrm{Mpc}^{-3}$, $12.5\times 10^{-3} \, h^{3} \mathrm{Mpc}^{-3}$, $25\times 10^{-3} \, h^{3} \mathrm{Mpc}^{-3}$. The dashed line is the sample-variance-limited case. For the full-hemisphere survey, all values should scaled with survey volume as $V_s^{-1/2}$, lowering errors by a factor of 4.5. Nonlinear evolution has been modelled with the 2003MNRAS.341.1311S prescription.
• Figure 3: The ratio of the nonlinear matter power spectrum for a model with baryon fraction $f_b = 17\%$, total matter density $\Omega_m = 0.27$ and $\sigma_8=0.9$ to the CMB normalised linear power spectrum $P^0_{\mathrm{lin}}(k)$ with no baryons for redshifts $z=3$ (long dashed), $z=1$ (short dashed), $z=0.3$ (dotted), $z=0$ (solid). The normalised linear power spectrum is also shown (dot dashed). Nonlinear evolution modelling in based on the 2003MNRAS.341.1311S fitting formula and can be compared to the numerical results of 2005ApJ...633..575S in their Fig. 1. Note that the cosmological model assumed here differs from the one used throughout the paper.
• Figure 4: Effect of the nonlinear matter evolution on the measured sound horizon scale from the real space correlation function. We plot the characteristic scale position $r_c$ as a function of redshift for our fiducial cosmological model (solid line). The horizontal line (dashed) shows the characteristic scale for the linear matter evolution, $r_c = 102.4 \, h^{-1} \mathrm{Mpc}$. The error bars represent statistical errors in the baryon-peak position from the hemisphere-scale survey, assuming that we use galaxies lying in bins of width $\Delta z =0.5$.
• Figure 5: The galaxy bias from the halo model for two schemes of populating haloes with galaxies: when number of galaxies occupying a halo is proportional to the halo mass (solid lines), and when there are central and satellites galaxies with $M_1 = 20 M_{\mathrm{min}}$ (dotted lines). The minimum mass of the haloes inhabited by galaxies for each occupation scheme are $0.1 M_{\star}$, $M_{\star}$ and $10 M_{\star}$ (from bottom to top for each family of curves). Redshift $z=1$ is assumed for which $M_{\star} = 8 \times 10^{11} h^{-1} M_{\odot}$. The bias relation as given by the generic function described in Sec. \ref{['toymodel']} is overplotted (dashed lines).