A 2% Distance to z=0.35 by Reconstructing Baryon Acoustic Oscillations - II: Fitting Techniques

TL;DR

This work develops and validates a robust BAO analysis pipeline that integrates reconstruction with a new, smooth covariance modeling approach to measure the acoustic scale from SDSS DR7 LRGs. Using 160 LasDamas mocks, the authors demonstrate that reconstruction sharpens the BAO feature, reduces the non-linear damping scale by a factor ~1.8, and yields a precise measurement of $D_V(z)/r_s$ with $D_V(z=0.35)/r_s = 8.88 \\pm 0.17$ after reconstruction. The modified Gaussian covariance matrix (MGCM) is shown to closely approximate mock covariances and to be robust to changes in template cosmology and fitting choices. Applying the method to DR7 data, they obtain $D_V(z=0.35)/r_s = 8.88 \\pm 0.17$ post-reconstruction, with BAO detected at $>4\\sigma$, illustrating the practical impact of reconstruction for current and future large-scale structure surveys.

Abstract

We present results from fitting the baryon acoustic oscillation (BAO) signal in the correlation function obtained from the first application of reconstruction to a galaxy redshift survey, namely, the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) luminous red galaxy (LRG) catalogue. We also introduce more careful approaches for deriving a suitable covariance matrix and fitting model for galaxy correlation functions. These all aid in obtaining a more accurate measurement of the acoustic scale and its error. We validate our reconstruction, covariance matrix and fitting techniques on 160 mock catalogues derived from the LasDamas simulations in real and redshift space. We then apply these techniques to the DR7 LRG sample and find that the error on the acoustic scale decreases from ~3.5% before reconstruction to ~1.9% after reconstruction. This factor of 1.8 reduction in the error is equivalent to the effect of tripling the survey volume. We also see an increase in our BAO detection confidence from ~3-sigma to ~4-sigma after reconstruction with our confidence level in measuring the correct acoustic scale increasing from ~3-sigma to ~5-sigma. Using the mean of the acoustic scale probability distributions produced from our fits, we find D_v/r_s = 8.89 +/- 0.31 before reconstruction and 8.88 +/- 0.17 after reconstruction.

Figures (20)

• Figure 1: The diagonal (black) and 6th off-diagonal (red) of the mock (circles) and modified Gaussian (crosses) covariance matrices in redshift space before reconstruction (top) and after reconstruction (bottom). The mock covariance matrix shows clear signs of noise. The modified Gaussian covariance matrices take on the form given in Equations (\ref{['eqn:modc']} & \ref{['eqn:noise']}) with $\sigma_s=4h^{-1}{\rm\;Mpc}$. Before reconstruction, $c_0=0.89$, $c_1=0.46$, $c_2=1.34$, $c_3=2.32 \times 10^{-7}$ and after reconstruction $c_0=0.89$, $c_1=0.30$, $c_2=1.45$, $c_3=1.87 \times 10^{-7}$. One can see that the modified Gaussian covariance matrices are good smoothed approximations to the mock covariance values.
• Figure 2: The ratio of $\mathfrak{P}(k;c_0,c_1,c_2)$ terms (see Equation (\ref{['eqn:modc']}) and surrounding text) found in the definition of the modified Gaussian covariance matrix (MGCM). These MGCMs were all fit to the covariances calculated from the LasDamas mocks in redshift space after reconstruction. The numerator corresponds to MGCMs constructed using 3 non-LasDamas cosmologies. The denominator corresponds to the MGCM in the LasDamas cosmology. The 3 non-LasDamas cosmologies are WMAP7+BAO+$H_0$ (solid line) and the 1$\sigma$ limits of this cosmology (+1$\sigma$ is shown as the dotted line and -1$\sigma$ is shown as the dashed line). It is seen that the 3 lines are all $\sim1$ to within $\sim5\%$. This indicates that if we input a power spectrum with cosmology different to LasDamas, our modification parameters can balance this input and the noise terms to recover a covariance matrix that matches the expected LasDamas covariances fairly well.
• Figure 3: The values of $B^2$ versus $\alpha$ fit from the mocks in redshift space before reconstruction. To ensure that $B^2$ is non-negative, these values were obtained through fitting the 160 mock redshift-space correlation functions using the non-linear parameter $\log(B^2)$ instead of $B^2$. The solid red line indicates the median $B^2$ value and the solid black line indicates the mean. The dashed red lines indicate the 16th and 84th percentiles of $B^2$ (quoted with the median $\widetilde{B^2}$). The dashed black lines correspond to the 1$\sigma$ deviations from the mean (quoted with the mean $\overline{B^2}$). One can see that $B^2$ can reach values as high as $\sim2.1$ and as low as $\sim0.3$. This substantial variation is possible because the $A(r)$ term can compensate, and is therefore not physically motivated. Hence to disfavour these extreme values, we place a weak Gaussian prior on $\log(B^2)$ that has mean equal to 0 and standard deviation equal to 0.4.
• Figure 4: (left) Fits to the average redshift-space correlation function of the mocks (black crosses) using Equation (\ref{['eqn:fform']}) with $A(r)$ being $poly0$ (dotted green line), $poly2$ (dash-dotted blue line), fiducial form (Equation (\ref{['eqn:aform']})) (solid black line) and $poly4$ (dashed red line). (right) The corresponding residuals of the fits (note that the fitting range is $30<r<2000h^{-1}{\rm\;Mpc}$). One can see that the fit using the fiducial form matches the data better than the fits with $poly0$ and $poly2$. However, the improvement between the fiducial form and $poly4$ is negligible as reflected by the similar shapes of the solid and dashed curves. These results motivate our choice of $A(r)$ given in Equation (\ref{['eqn:aform']}). We have also allowed $\Sigma_{nl}$ to vary in these fits. Using the fiducial form, we find $\Sigma_{nl}=8.1h^{-1}{\rm\;Mpc}$, which is close to the value we assumed in deriving the covariance matrix.
• Figure 5: Fits to the average of the mock redshift-space correlation functions before and after reconstruction. The black crosses are the mock data before reconstruction and the black line is its best-fit model. The red crosses are the mock data after reconstruction and the red line is its best-fit model. We have allowed $\Sigma_{\rm nl}$ to vary in these fits, the results are summarized on the plot. We find that before reconstruction, the shift in the acoustic peak is already very small ($\alpha\sim1$), so we do not expect reconstruction to shift the peak much closer to its predicted linear theory position. However, we find that $\Sigma_{\rm nl}$ was reduced by a factor of 1.8 after reconstruction, indicating that reconstruction was able to mitigate the acoustic peak smearing due to non-linear structure growth.