Unified galaxy power spectrum measurements from 6dFGS, BOSS, and eBOSS

TL;DR

The paper develops a matrix-based framework to analyze galaxy power spectrum multipoles, explicitly incorporating wide-angle effects and the survey window function. By recasting window convolution and wide-angle corrections as linear matrix operations, it enables both convolved-model and deconvolved-data pipelines and validates that BAO analyses yield consistent likelihoods across methods. The approach is applied to 6dFGS DR3, BOSS DR12, and eBOSS DR16 QSO, with publicly released window/wide-angle matrices, covariance data, and a Python toolkit (pk_tools) to enable broader adoption and reproducibility. The work advances accessibility and consistency in cosmological analyses of large-scale structure data, while noting limitations related to binning and the need for square window matrices for deconvolution. Overall, it provides a practical, extensible framework for leveraging current and future galaxy surveys in RSD/BAO studies.

Abstract

We make use of recent developments in the analysis of galaxy redshift surveys to present an easy to use matrix-based analysis framework for the galaxy power spectrum multipoles, including wide-angle effects and the survey window function. We employ this framework to derive the deconvolved power spectrum multipoles of 6dFGS DR3, BOSS DR12 and the eBOSS DR16 quasar sample. As an alternative to the standard analysis, the deconvolved power spectrum multipoles can be used to perform a data analysis agnostic of survey specific aspects, like the window function. We show that in the case of the BOSS dataset, the Baryon Acoustic Oscillation (BAO) analysis using the deconvolved power spectra results in the same likelihood as the standard analysis. To facilitate the analysis based on both the convolved and deconvolved power spectrum measurements, we provide the window function matrices, wide-angle matrices, covariance matrices and the power spectrum multipole measurements for the datasets mentioned above. Together with this paper we publish a \code{Python}-based toolbox to calculate the different analysis components. The appendix contains a detailed user guide with examples for how a cosmological analysis of these datasets could be implemented. We hope that our work makes the analysis of galaxy survey datasets more accessible to the wider cosmology community.

Figures (16)

• Figure 1: The window function multipoles of the low redshift bin $0.2 < z < 0.5$ of BOSS DR12, NGC, calculated following eq. (\ref{['eq:2Dwindow']}) at zero order in the wide-angle expansion ($n=0$). The corresponding window function multipoles at first order in the wide-angle expansion ($n=1$) are shown in figure \ref{['fig:wll1_NGC_z1']}. The three columns correspond to the contributions of the three even multipoles to the five non-zero multipoles, including the dipole and octopole. Each window function is plotted multiple times as a function of $k'$ with fixed values of $k$ (vertical dashed lines). This window function does not account for any bin averaging, but is evaluated at specific values $k$ with $16\,384$ values of $k'$ (here we focus on $0 < k' < 0.12\, h \, {\rm Mpc}^{-1}$, while in practice we calculate the full range of $0 < k' < 0.4\, h \, {\rm Mpc}^{-1}$). For visual purposes, each of the sub-panels is re-normalized by the factor written in the right hand corner. The final window function matrix used for the actual analysis requires bin averaging in both $k_{\rm o}$ and $k_{\rm th}$ (see eq. \ref{['eq:intW']} for more details).
• Figure 2: The window function multipoles of the low redshift bin $0.2 < z < 0.5$ of BOSS DR12, NGC, calculated following eq. (\ref{['eq:2Dwindow']}) at first order in the wide-angle expansion ($n=1$). The corresponding window function multipoles at zero order in the wide-angle expansion ($n=0$) are shown in figure \ref{['fig:wll0_NGC_z1']}. The two columns correspond to the contributions of the two odd multipoles to the five non-zero multipoles. Each window function is plotted multiple times as a function of $k'$ with fixed values of $k$ (vertical dashed lines). For visual purposes, each of the sub-panels is re-normalized by the factor written in the right hand corner.
• Figure 3: The window function matrix ($W_{\ell\ell'}$, see section \ref{['sec:window_matrix']}) and wide-angle matrix ($M_{\ell\ell'}$, section \ref{['sec:wamatrix']}) of BOSS DR12 NGC in the low redshift bin. The wide-angle transformation matrix maps the theoretical power spectrum multipoles with three even multipoles to a vector with five multipoles (only accounting for wide-angle effects at order $n=1$). The window function maps the five (theoretical) power spectrum multipoles (in bins of $\Delta k_{\rm th}$) to the observed/convolved power spectrum multipoles (in bins of $\Delta k_{\rm o}$), which can be compared to the data. For plotting purposes here we use $\Delta k_{\rm th} = \Delta k_{\rm o} = 0.01\, h \, {\rm Mpc}^{-1}$, while the analysis in this paper uses $\Delta k_{\rm o} = 0.01\, h \, {\rm Mpc}^{-1}$ and $\Delta k_{\rm th} = 0.001\, h \, {\rm Mpc}^{-1}$. The window function matrix is not symmetric, since the contribution of the monopole to the dipole is different to the contribution of the dipole to the monopole. The diagonal elements are not $1$, since they correspond to the bin integral of the window function within the bin $\Delta k_{\rm th}$ (see eq. \ref{['eq:intW']}), which is only $1$ for $\Delta k \gg k_f$ as shown in figure \ref{['fig:Ik']}. The z-axis is logarithmic for better contrast.
• Figure 4: Comparison of the window function terms $W_{\ell0}$ for different binning schemes at $k_{\rm o}=0.055\, h \, {\rm Mpc}^{-1}$ for BOSS DR12 NGC in the low redshift bin (z1). The yellow line shows the window with bins of $\Delta k_{\rm o} = \Delta k_{\rm th} = 0.001\, h \, {\rm Mpc}^{-1}$. The red line shows the window with larger observational bins of $\Delta k_{\rm o} = 0.01\, h \, {\rm Mpc}^{-1}$, by averaging $10$ bins of the yellow line. The green line is averaged over both $k_{\rm o}$ and $k_{\rm th}$. The black dashed lines show the edges of the bin at $k_{\rm min} = 0.05\, h \, {\rm Mpc}^{-1}$ and $k_{\rm max} = 0.06\, h \, {\rm Mpc}^{-1}$. The relative heights of the three windows are scaled to make them comparable. The red line corresponds to our default choice.
• Figure 5: Comparison of convolved power spectrum multipole models based on different window function binning choices of $\Delta k_{\rm th}$ using BOSS DR12 NGC in the low redshift bin (z1). The window function transforms the theoretical binning ($\Delta k_{\rm th}$) into the binning used for the data ($\Delta k_{\rm o}$). The reference model is based on the current standard method Wilson2015:1511.07799v2Beutler2016:1607.03149v1, which Hankel transforms the model into configuration-space, multiplies with the real-space window and Hankel transforms back into Fourier space. The window function uses $\Delta k_{\rm o}=0.01\, h \, {\rm Mpc}^{-1}$ in all cases. The window functions provided with this paper uses a default binning of $\Delta k_{\rm th} = 0.001\, h \, {\rm Mpc}^{-1}$, except of the case of deconvolution, where we use $\Delta k_{\rm th} = 0.01\, h \, {\rm Mpc}^{-1}$.