Cosmological constraints from the redshift-space galaxy skew spectra

Extracting the non-Gaussian information of the cosmic large-scale structure (LSS) is vital in unlocking the full potential of the rich datasets from the upcoming stage-IV galaxy surveys. Galaxy skew spectra serve as efficient beyond-two-point statistics, encapsulating essential bispectrum information with computational efficiency akin to power spectrum analysis. This paper presents the first cosmological constraints from analyzing the full set of redshift-space galaxy skew spectra of the data from the SDSS-III BOSS, accessing cosmological information down to nonlinear scales. Employing the simbig forward modeling framework and simulation-based inference via normalizing flows, we analyze the CMASS-SGC subsample, which constitute approximately 10% of the full BOSS data. Analyzing the scales up to $k_{\max }=0.5h^{-1}\mathrm{Mpc}$, we find that the skew spectra improve the constraints on ${\mathrm{\Omega }}_{\mathrm{m}},{\mathrm{\Omega }}_{\mathrm{b}},h$, and $n_s$ by 34%, 35%, 18%, 10%, respectively, compared to constraints from previous simbig power spectrum multipoles analysis, yielding ${\mathrm{\Omega }}_{\mathrm{m}}=0.288_{-0.034}^{+0.024}$, ${\mathrm{\Omega }}_{\mathrm{b}}=0.043_{-0.007}^{+0.005}$, $h=0.759_{-0.050}^{+0.104}$, $n_{\mathrm{s}}=0.918_{-0.090}^{+0.041}$ (at 68% confidence limit). On the other hand, the constraints on ${\sigma }_8$ are weaker than from the power spectrum. Including the big bang nucleosynthesis (BBN) prior on baryon density reduces the uncertainty on the Hubble parameter further, achieving $h=0.750_{-0.032}^{+0.034}$, which is a 38% improvement over the constraint from the power spectrum with the same prior. Compared to the simbig bispectrum (monopole) analysis, skew spectra offer comparable constraints on larger scales ($k_{\max }<0.3h^{-1}\mathrm{Mpc}$) for most parameters except for ${\sigma }_8$.

Figure 1

Schematic view of the simbig pipeline; to generate synthetic data that is statistically indistinguishable from BOSS observations, we utilize 2,000 $N$-body simulations from the quijote suite run at different cosmologies, apply Rockstar halo finder to identify halos, populate halos with galaxies using an extended (9-parameter) HOD model, and finally apply the survey realism. The resulting 20,000 synthetic mocks are then used to train normalizing flows as neural density estimators to obtain approximate posterior distributions, $q_{\varphi }$, for cosmological and HOD parameters from given summary statistics $\mathbf{x}$, which in this paper refer to the skew spectra).

Figure 2

Measurements of the 14 skew spectra from the subsample of the BOSS DR12-SGC galaxies before the shot noise subtraction. Different colors correspond to two smoothing scales of $R=5h^{-1}\mathrm{Mpc}$ (in red) and $R=10h^{-1}\mathrm{Mpc}$ (in black). Due to smoothing, skew spectra approach zero at small scales.

Figure 3

The three contributions to the Poisson shot noise of individual galaxy skew spectra given in Eq. (3). The smoothing scale is set to $R=5h^{-1}\mathrm{Mpc}$. Grey curves denote the total shot noise contribution.

Figure 4

Accuracy test: shown are the rank statistics of the validation dataset for the configuration $R=5h^{-1}\mathrm{Mpc}$ and $k_{\max }=0.5{\mathrm{Mpc}}^{-1}h$. For most of the parameters, the rank statistics are uniformly distributed. The slight $\cap$-shape in the ${\mathrm{\Omega }}_{\mathrm{m}}$ plot is an indication of an under-confident (conservative) error bar.

Figure 5

Robustness test: the distribution of the difference between the inferred posterior mean ${\overline{\theta }}_i$ and the fiducial value ${{\theta }_i}^{\mathrm{fid}}$ normalized by the standard deviation of the posterior ${\sigma }_{{\theta }_i}$ for each cosmological parameter on the three sets of test simulations.

Figure 6

Posterior distribution of cosmological parameters from the power spectrum multipoles (gray) and the skew spectra (blue) setting $k_{\max }=0.5h^{-1}\mathrm{Mpc}$ for both. The smoothing scale for the skew spectra is set to $R=5h^{-1}\mathrm{Mpc}$.

Figure 7

Posterior distribution of cosmological parameters from the power spectrum (gray), the skew spectra (blue), and the bispectrum monopole (red) imposing the BBN prior, and setting $k_{\max }=0.25h^{-1}\mathrm{Mpc}$ for the first two and $k_{\max }=0.3{\mathrm{Mpc}}^{-1}h$ for the third. The smoothing scale is set to $R=10h^{-1}\mathrm{Mpc}$ for the skew spectra.

Figure 8

Posterior distribution of cosmological parameters inferred the skew spectra (blue) and power spectrum (gray) with a cutoff scale $k_{\max }=0.5h^{-1}\mathrm{Mpc}$ including the BBN prior and smoothing scale $R=5h^{-1}\mathrm{Mpc}$.

Figure 9

Rank statistics of the validation data for analysis of the skew spectra with $R=10h^{-1}\mathrm{Mpc}$ and $k_{\max }=0.25{\mathrm{Mpc}}^{-1}h$.

Figure 10

Distribution of the difference between the inferred posterior mean $\overline{{\theta }_i}$ and the fiducial value ${{\theta }_i}^{\mathrm{fid}}$ normalized by the standard deviation of the posterior ${\sigma }_{{\theta }_i}$ for each cosmological parameter on the three sets of test simulations.

Figure 11

Left: posterior distribution of cosmological parameters with (purple) and without (green) subtracting the Poisson shot noise in skew spectra estimators. Right: posterior distribution of cosmological parameters for three choices of small-scale cutoff: $k_{\max }=0.15{\mathrm{Mpc}}^{-1}h$ in red, $k_{\max }=0.25{\mathrm{Mpc}}^{-1}h$ in orange, and $k_{\max }=0.5{\mathrm{Mpc}}^{-1}h$ in green.

Figure 12

Posterior distribution of cosmological parameters for two choices of the smoothing scale: $R=5h^{-1}\mathrm{Mpc}$ in green and $R=10h^{-1}\mathrm{Mpc}$ in orange. The small-scale cutoff is set to $k_{\max }=0.5{\mathrm{Mpc}}^{-1}h$ in both cases.