The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: combining correlated Gaussian posterior distributions

TL;DR

The paper introduces a formal method to combine multiple Gaussian posterior constraints from correlated cosmological measurements into a single consensus set that preserves all information via the full cross-covariance. It provides exact expressions for the consensus covariance and mean, and shows how to handle cases where analyses constrain different parameter spaces. By applying the method to BAO and full-shape (BAO+RSD) analyses of MD-Patchy BOSS mocks, the authors demonstrate tighter consensus constraints and quantify information gain across redshift bins. The approach, demonstrated on mock data and discussed for potential systematic considerations, offers a practical pathway to maximize cosmological information from large galaxy surveys.

Abstract

The cosmological information contained in anisotropic galaxy clustering measurements can often be compressed into a small number of parameters whose posterior distribution is well described by a Gaussian. We present a general methodology to combine these estimates into a single set of consensus constraints that encode the total information of the individual measurements, taking into account the full covariance between the different methods. We illustrate this technique by applying it to combine the results obtained from different clustering analyses, including measurements of the signature of baryon acoustic oscillations (BAO) and redshift-space distortions (RSD), based on a set of mock catalogues of the final SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). Our results show that the region of the parameter space allowed by the consensus constraints is smaller than that of the individual methods, highlighting the importance of performing multiple analyses on galaxy surveys even when the measurements are highly correlated. This paper is part of a set that analyses the final galaxy clustering dataset from BOSS. The methodology presented here is used in Alam et al. (2016) to produce the final cosmological constraints from BOSS.

Figures (8)

• Figure 1: The mean 68% and 95% two-dimensional constraints on the parameters $D_{\rm M}(z)(r_{\rm d}^{\rm fid}/r_{\rm d})$ and $H(z)(r_{\rm d}/r_{\rm d}^{\rm fid})$ obtained by applying the BAO-only analyses of Ross2016 and Beutler2016a to 996 MD-Patchy BOSS mock catalogues for the redshift bins indicated in the legend. The results are in excellent agreement with the true underlying values of these parameters, indicated by the dotted lines. The full information from these measurements can be combined into a set of consensus constraints (black solid lines) as described in Section \ref{['sec:combination']}. The dashed lines correspond to the combination of the results obtained by averaging the logarithm of the two posterior distributions.
• Figure 2: Scatter plots of the BAO-only constraints on $D_{\rm M}(z)(r_{\rm d}^{\rm fid}/r_{\rm d})$ and $H(z)(r_{\rm d}/r_{\rm d}^{\rm fid})$ obtained from the configuration and Fourier space BAO-only analyses of 996 MD-Patchy mock catalogues for $0.4 < z < 0.6$. Although the results obtained from these methods are highly correlated, their correlation coefficients, $r$, are not exactly one, indicating that additional information can be obtained from their combination.
• Figure 3: Correlation matrix corresponding to the full covariance $\mathbfss{C}_{\rm tot}$ of the BAO-only constraints on $D_{\rm M}(z)/r_{\rm d}$, and $H(z)\times r_{\rm d}$, constructed from the individual MD-Patchy mock catalogues in configuration and Fourier space. The blocks $\mathbfss{C}_{ij}$ indicated by the dashed lines correspond to the auto and cross-covaraince matrices of the two methods.
• Figure 4: The mean 68% and 95% two-dimensional constraints on the parameters $D_{\rm M}(z)(r_{\rm d}^{\rm fid}/r_{\rm d})$, $H(z)(r_{\rm d}/r_{\rm d}^{\rm fid})$ and $f\sigma_8(z)$ inferred from our mock BOSS catalogues for $0.4 < z < 0.6$. The filled contours correspond to the results obtained by means of full-shape fits of the Legendre multipoles, $\xi_{\ell}(s)$ (magenta) and $P_{\ell}(k)$ (blue) and clustering wedges $\xi_{3{\rm w}}(s)$ (orange) and $P_{3{\rm w}}(k)$ (green), using the methodology of our companion papers Satpathy2016Beutler2016bSanchez2016Grieb2016. The obtained constraints are good agreement with the true underlying values of these parameters, indicated by the dotted lines. The black solid contours correspond to the combination of these measurements into a set of consensus constraints, computed as described in Section \ref{['sec:combination']}. The dashed lines correspond to the combination of the results obtained by averaging the logarithms of the four posterior distributions.
• Figure 5: Correlation matrix corresponding to the total covariance $\mathbfss{C}_{\rm tot}$ of the full-shape fits of the Legendre multipoles and clustering wedges in configuration and Fourier space constructed from the individual MD-Patchy mock catalogues. The blocks $\mathbfss{C}_{ij}$ indicated by the dashed lines correspond to the auto and cross-covaraince matrices of the different methods.