Improved forecasts for the baryon acoustic oscillations and cosmological distance scale

TL;DR

<3-5 sentence high-level summary> The paper develops a refined forecasting framework for cosmological distance measurements from baryon acoustic oscillations (BAO) that accounts for nonlinear degradation via Lagrangian displacements. It derives a compact 2-D (and 1-D) Fisher-matrix-based fitting formula for D_A(z) and H(z) errors, isolating BAO information from broadband shape and redshift-space distortions, and validates the approach against N-body simulations. The method accommodates redshift distortions and photometric redshift errors, and demonstrates strong agreement with full Fisher calculations and χ^2 analyses, offering a practical tool for planning future surveys and assessing dark energy constraints. It also quantifies the impact of reconstruction on improving BAO precision and provides guidance for applying the formula across spectroscopic and photometric datasets.

Abstract

We present the cosmological distance errors achievable using the baryon acoustic oscillations as a standard ruler. We begin from a Fisher matrix formalism that is upgraded from Seo & Eisenstein (2003). We isolate the information from the baryonic peaks by excluding distance information from other less robust sources. Meanwhile we accommodate the Lagrangian displacement distribution into the Fisher matrix calculation to reflect the gradual loss of information in scale and in time due to nonlinear growth, nonlinear bias, and nonlinear redshift distortions. We then show that we can contract the multi-dimensional Fisher matrix calculations into a 2-dimensional or even 1-dimensional formalism with physically motivated approximations. We present the resulting fitting formula for the cosmological distance errors from galaxy redshift surveys as a function of survey parameters and nonlinearity, which saves us going through the 12-dimensional Fisher matrix calculations. Finally, we show excellent agreement between the distance error estimates from the revised Fisher matrix and the precision on the distance scale recovered from N-body simulations.

Figures (3)

• Figure 1: 2-D errors from equation (\ref{['eq:F2D']}) (black lines) and errors from the full Fisher matrix calculations (blue lines) for WMAP3. Upper panels: $c=\Sigma_\parallel/\Sigma_\perp=1$. Lower panels : $c=2$. Left: distance errors on $D_A$. Right: distance errors on $H$. Off-diagonal terms in the middle field of each panel are defined as $r=C_{12}/\sqrt{C_{11}C_{22}}$. The bottom field of each panel shows the discrepancy between the 2-D errors and the full-D errors as a ratio of the two. The shaded region corresponds to 2% of discrepancy. We find that the 2-D model gives excellent fits to the errors from the full Fisher matrix calculations. Solid lines : $nP_{0.2}=76.1$, short-dashed : $nP_{0.2}=7.61$, dotted : $nP_{0.2}=3.81$, long-dashed : $nP_{0.2}=0.761$, dot-short-dashed : $nP_{0.2}=0.381$, and dot-long-dashed : $nP_{0.2}=0.076$. These values are chosen because $nP_{0.2}\sim 0.76$ is appropriate for the Luminous Red Galaxy sample from SDSS in real space.
• Figure 2: Errors on $D_A$ for photometric redshift surveys. We assume a redshift error of $\Sigma_z=34h^{-1}{\rm\;Mpc}$. We compare the 2-D errors from equation (\ref{['eq:F2D']}) using $R(k,\mu)$ in equation (\ref{['eq:Rsig']}) (black lines) and the errors from the full Fisher matrix calculations (blue lines) for WMAP3. The lower field of the panel shows the discrepancy between the 2-D errors and the full-D errors as a ratio of the two. The 2-D errors from our fitting formula are in good agreement with the full-D errors: the discrepancy is at most $8\%$ but smaller for $nP_{0.2}=3-10$. Solid lines : $nP_{0.2}=76.1$, short-dashed : $nP_{0.2}=7.61$, dotted : $nP_{0.2}=3.81$, long-dashed : $nP_{0.2}=0.761$, dot-short-dashed : $nP_{0.2}=0.381$, and dot-long-dashed : $nP_{0.2}=0.076$.
• Figure 3: The fractional errors on $D_A/s$ ( left) and $sH$ ( right) available as a function of redshift assuming redshift bins $\Delta z=0.1$ and a $3\pi$ sr survey area. The bottom line in each case is the cosmic variance limit, assuming perfect linearity and no shot noise. The top line assumes unreconstructed level of non-linearity and a shot noise level of $nP_{0.2}=3$. The middle line uses the same shot noise and assumes that reconstruction can halve the values of $\Sigma_\perp$ and $\Sigma_\parallel$.