The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: RSD measurement from the power spectrum and bispectrum of the DR12 BOSS galaxies

TL;DR

This work advances the use of the galaxy bispectrum from the DR12 BOSS data to jointly constrain growth and geometry by combining isotropic bispectrum measurements with power-spectrum moments, while accounting for redshift-space distortions, bias, and Alcock–Paczynski distortions. It employs a full covariance from 2048 MD-Patchy mocks, expands the triangle configurations analyzed, and includes AP effects, yielding improved constraints on fσ8, Hr_s, and D_A/r_s for LOWZ and CMASS. The results are largely consistent with earlier BOSS analyses but reveal a mild tension with General Relativity when Planck priors are included, quantified by γ ≈ 0.73. The methodology demonstrates how integrating bispectrum information breaks the f–σ8 degeneracy and enhances cosmological inferences, setting a framework for future large-scale structure surveys.

Abstract

We measure and analyse the bispectrum of the final, Data Release 12, galaxy sample provided by the Baryon Oscillation Spectroscopic Survey, splitting by selection algorithm into LOWZ and CMASS galaxies. The LOWZ sample contains 361\,762 galaxies with an effective redshift of $z_{\rm LOWZ}=0.32$, and the CMASS sample 777\,202 galaxies with an effective redshift of $z_{\rm CMASS}=0.57$. Combining the power spectrum, measured relative to the line-of-sight, with the spherically averaged bispectrum, we are able to constrain the product of the growth of structure parameter, $f$, and the amplitude of dark matter density fluctuations, $σ_8$, along with the geometric Alcock-Paczynski parameters, the product of the Hubble constant and the comoving sound horizon at the baryon drag epoch, $H(z)r_s(z_d)$, and the angular distance parameter divided by the sound horizon, $D_A(z)/r_s(z_d)$. After combining pre-reconstruction RSD analyses of the power spectrum monopole, quadrupole and bispectrum monopole; with post-reconstruction analysis of the BAO power spectrum monopole and quadrupole, we find $f(z_{\rm LOWZ})σ_8(z_{\rm LOWZ})=0.427\pm 0.056$, $D_A(z_{\rm LOWZ})/r_s(z_d)=6.60 \pm 0.13$, $H(z_{\rm LOWZ})r_s(z_d)=(11.55\pm 0.38)10^3\,{\rm kms}^{-1}$ for the LOWZ sample, and $f(z_{\rm CMASS})σ_8(z_{\rm CMASS})=0.426\pm 0.029$, $D_A(z_{\rm CMASS})/r_s(z_d)=9.39 \pm 0.10$, $H(z_{\rm CMASS})r_s(z_d)=(14.02\pm 0.22)10^3\,{\rm kms}^{-1}$ for the CMASS sample. We find general agreement with previous BOSS DR11 and DR12 measurements. Combining our dataset with {\it Planck15} we perform a null test of General Relativity (GR) through the $γ$-parametrisation finding $γ=0.733^{+0.068}_{-0.069}$, which is $\sim2.7σ$ away from the GR predictions.

Figures (22)

• Figure 1: Power spectrum data: the top sub-panels display the measured LOWZ- (left panel) and CMASS-DR12 (right panel), monopole (blue symbols) and quadrupole (red circles) power spectra. For both cases, the measurements correspond to a combination of the northern and southern galaxy caps according to their effective areas as presented in Eq. \ref{['eq:NGCSGC']}. The error-bars correspond to the dispersion among 2048 realisations of the MD-Patchy mocks. The black solid lines correspond to the best-fitting model, calculated from a full fit to the power spectrum and bispectrum moments, with parameters as listed in Table \ref{['table:results']}. The middle sub-panel shows the ratio between the power spectrum multipole measurements and the best-fitting models. In the bottom sub-panel the difference between the data and the model, $\Delta P\equiv P^{\rm data}-P^{\rm model}$, relative to the statistical error of the data, $\sigma_P$, is presented. The black dashed lines represent a $2\sigma$ deviation (95.4%) confidence level. In the middle and bottom sub-panel the quadrupole measurements have been horizontally displaced for clarity.
• Figure 2: Bispectrum data: the top sub-panels display the measured LOWZ- (top panel) and CMASS-DR12 (bottom panel) bispectrum monopole for different triangular shapes: equilateral triangles (red squares), isosceles triangles (blue circles) and scalene triangles (green triangles), ordered sequentially in $k_1$, $k_2$ and $k_3$ (see text for details of the ordering), and covering $0.03\leq k_i\,[h{\rm Mpc}^{-1}]\leq 0.18$ for the LOWZ sample and $0.03\leq k_i\,[h{\rm Mpc}^{-1}]\leq 0.22$ for the CMASS sample. As for the power spectrum, the measurements correspond to a combination of the northern and southern galactic caps, described by Eq. \ref{['eq:NGCSGC']}. The displayed error-bars correspond to the dispersion among 2048 realisations of the MD-Patchy mocks. The black solid line represent the best-fitting model using the parameters of Table \ref{['table:results']}. The middle and the bottom sub-panel show the deviation of the model respect to the data, as it is shown in Fig. \ref{['fig:powerspectrum_data']} for the power spectrum.
• Figure 3: Correlation coefficients for the power spectrum monopole - power spectrum quadrupole - bispectrum monopole from the LOWZ-DR12 sample (left panels) and from the CMASS-DR12 sample (right panels), extracted from the 2048 realisations of the MD-Patchy galaxy mocks. The dashed black lines marks the limit between the auto-correlation and the cross-correlation parts of the covariance. For the LOWZ sample the covariance contains the elements with $0.02\leq k\, [h{\rm Mpc}^{-1}]\leq 0.18$ for the power spectrum monopole, $0.04\leq k\, [h{\rm Mpc}^{-1}] \leq 0.18$ for the power spectrum quadrupole and $0.03\leq k_i\, [h{\rm Mpc}^{-1}] \leq 0.18$ for the bispectrum monopole. For the CMASS sample the covariance contains the elements with $0.02\leq k\, [h{\rm Mpc}^{-1}]\leq 0.22$ for the power spectrum monopole, $0.04\leq k\, [h{\rm Mpc}^{-1}] \leq 0.22$ for the power spectrum quadrupole and $0.03\leq k_i\, [h{\rm Mpc}^{-1}] \leq 0.22$ for the bispectrum monopole. Red crosses mark the position of equilateral bispectra. The ordering of the triangles follows that presented in Fig. \ref{['fig:bispectrum_data']}, and therefore, the bispectra corresponding to the indices between crosses share the same value of $k_1$.
• Figure 4: Statistical diagonal errors for the measured LOWZ and CMASS bispectrum, as labeled, as a function of the same triangle index presented in Fig. \ref{['fig:bispectrum_data']}. The top sub-panels show in solid black line the diagonal statistical errors computed from the rms of the 2048 realisations of MD-Patchy mocks, divided by the measured bispectrum from the data. The different coloured symbols on the top of the black line represent the different triangle shapes with the same colour and symbol notation than in Fig. \ref{['fig:bispectrum_data']}. In the bottom sub-panels the ratio between the diagonal errors computed from the 1000 realisations of the qpm mocks and the 2048 realisations of the MD-Patchy mocks is displayed.
• Figure 5: The top sub-panels display the performance of the bispectrum of the MD-Patchy mocks (black solid line for the mean of the mocks and grey lines for each of the first 100 mock realisation), the qpm mocks (black dashed line), compared to the data with the $1\sigma$ error-bars extracted from the $\it rms$ of the MD-patchy mocks (red symbols) as a function of the triangle index (same definition that in Fig. \ref{['fig:bispectrum_data']}). The bottom sub-panels display the difference between the data and the mean of the MD-Patchy mocks relative to the error of the data: $(B^{\rm data}-B^{\rm mocks})/\sigma_B$ The left panel display the results for LOWZ and the right panel for CMASS, as labeled. From the bottom sub-panels we see that, both for LOWZ and CMASS samples, the mocks reproduce well the data at large scales, but as we explore smaller scales (higher triangle index) there is a systematic offset of order of $\sim2\sigma$ between the data and the mocks, having the data a higher amplitude.