Precise cosmological constraints from BOSS galaxy clustering with a simulation-based emulator of the wavelet scattering transform

We perform a reanalysis of the BOSS CMASS DR12 galaxy dataset using a simulation-based emulator for the wavelet scattering transform (WST) coefficients. Moving beyond our previous works, which laid the foundation for the first galaxy clustering application of this estimator, we construct a neural net-based emulator for the cosmological dependence of the WST coefficients and the 2-point correlation function multipoles, trained from the state-of-the-art suite of abacussummit simulations combined with a flexible halo occupation distribution (HOD) galaxy model. In order to confirm the accuracy of our pipeline, we subject it to a series of thorough internal and external mock parameter recovery tests, before applying it to reanalyze the CMASS observations in the redshift range $0.46<z<0.57$. We find that a joint $\mathrm{WST}+2$-point correlation function likelihood analysis allows us to obtain marginalized $1\sigma$ errors on the $\mathrm{\Lambda }\mathrm{CDM}$ parameters that are tighter by a factor of 2.5–6, compared to the 2-point correlation function, and by a factor of 1.4–2.5 compared to the WST-only results. This corresponds to a competitive 0.9%, 2.3% and 1% level of determination for parameters ${\omega }_c$, ${\sigma }_8\&n_s$, respectively, and also to a 0.7% and 2.5% constraint on derived parameters h and $f(z){\sigma }_8(z)$, in agreement with the Planck 2018 results. Our results reaffirm the constraining power of the WST and highlight the exciting prospect of employing higher-order statistics in order to fully exploit the power of upcoming stage-IV spectroscopic observations.

Figure 1

The galaxy number density of the original CMASS sample as a function of redshift $z$ (blue), shown with the final downsampled version of constant number density, ${\overline{n}}_g=2.9\times {10}^{-4}{h}^3/{\mathrm{Mpc}}^3$ (red), that we work with in order to match the constant density profile of the abacussummit mocks.

Figure 2

A histogram of the distribution of galaxy number densities of the galaxy mocks forming our original emulator parameter grid. Mocks with number densities lower than the cut-off ${\overline{n}}_g=2.9\times {10}^{-4}{h}^3/{\mathrm{Mpc}}^3$ (red vertical line) are discarded, while the rest are downsampled to exactly match this value, in order to ensure a robust modeling of the density-dependent WST estimator since this is the constant density value used in the abacussummit mocks. The outcome of this procedure forms the final emulator training set consisting of 151474 mocks, i.e. those lying on the right of the red vertical line in the histogram.

Figure 3

The median WST emulator error tested on the four leave-out cosmologies as a function of bin index of the WST coefficients vector. The $y$-axis denotes the emulator error normalized by the CMASS $1\sigma$ uncertainty.

Figure 4

The bias of the WST emulator tested on the four leave-out cosmologies for six randomly selected coefficients (bins) of the WST data vector. The legend shows the WST bin indices. The orange scatter points showcase the true and predicted values of the WST coefficients for each one of the 4000 leave-out tests, whereas the blue band corresponds to the $1\sigma$ uncertainty of the CMASS WST measurement.

Figure 5

Correlation matrix of the joint data vector consisting of the multipoles of the 2-point correlation function, $l=\{0,2\}$, and the 76 WST coefficients used in our analysis, evaluated from the 2048 realizations of the patchy mocks for the fiducial cosmology. The lower left and upper right subplots coincide with the individual correlation matrices of the two statistics, respectively, while the rest corresponds to the cross-correlations between them. The WST coefficients populate the data vector in order of increasing values of the $j_1$ and $l_1$ indices, with the $l_1$ index varied faster. The $2\times 2$ blocks on the lower left corner correspond to the auto- and cross- correlations of ${\xi }_0$ and ${\xi }_2$, from bottom to top and from left to right, respectively.

Figure 6

$\mathrm{\Lambda }\mathrm{CDM}$ recovery tests using our WST emulator to analyze 10 HOD configurations of the c001 (upper left), c003 (upper right) and c004 (lower left) hold-out cosmologies of our test set. We also show the marginalized $1\text{−}\sigma$ and $2\text{−}\sigma$ posteriors obtained by analyzing the mean data vector of the 24 additional realizations available for the fiducial c000 cosmology (lower right). The horizontal and vertical black dashed lines indicate the true values of the cosmological parameters in each case.

Figure 7

Recovery test using the Uchuu galaxy mock for the $\mathrm{\Lambda }\mathrm{CDM}$ cosmological parameters obtained using the monopole and quadrupole of the galaxy correlation function (red), the WST coefficients (blue) and their joint combination (black). The horizontal and vertical black dashed lines indicate the true values of the cosmological parameters.

Figure 8

Marginalized constraints on the $\mathrm{\Lambda }\mathrm{CDM}$ cosmological parameters obtained using the monopole and quadrupole of the galaxy correlation function (red), the WST coefficients (blue) and their joint combination (black) in order to analyze the BOSS CMASS observations. The results shown above were obtained after imposing a BBN Gaussian prior on the value of ${\omega }_b=0.02268\pm 0.00038$.

Figure 9

Marginalized constraints on the structure growth rate, $f(z){\sigma }_8(z)$, of our joint analysis in blue alongside other clustering constraints in the literature. We show the Planck 2018 [102] CMB constraints in black, with the corresponding 68% and 95% limits in shaded bands, together with the results from two other recent Abacus-based CMASS reanalyses using the small-scale 2-point correlation function [103] and the density split clustering statistic [51, 52]. Additionally, we show clustering constraints from BOSS LOWZ small-scale RSD [97], BOSS full-shape power spectrum [135], BOSS large-scale $\mathrm{RSD}+\mathrm{BAO}$ [156], BOSS small-scale RSD [157] eBOSS small-scale RSD [158], eBOSS large-scale $\mathrm{RSD}+\mathrm{BAO}$ [159, 160], BOSS DR12 large-scale power spectrum [161] and the 6dF Galaxy Survey [162].

Figure 10

All 76 WST coefficients evaluated from the BOSS CMASS dataset (black circles) plotted together with the best-fit prediction obtained from our likelihood analysis (solid blue line). The WST coefficients populate the data vector in order of increasing values of the $j_1$ and $l_1$ indices, with the $l_1$ index varied faster.

Figure 11

Marginalized constraints on extensions to $\mathrm{\Lambda }\mathrm{CDM}$ obtained using the joint $\mathrm{WST}+\text{correlation}$ function combination in order to analyze the BOSS CMASS observations. The results shown above were obtained after imposing a BBN Gaussian prior on the value of ${\omega }_b=0.02268\pm 0.00038$.

Figure 12

Probability density function of the ${\chi }^2$ distribution of the WST coefficients as measured from the 2048 realizations of the patchy mocks (blue) plotted together with a theoretical ${\chi }^2$ distribution with $N_{\mathbf{d}}=76$ degrees of freedom (black line) and a Gaussian distribution with the same mean and covariance (orange). The WST estimator does not exhibit any significant deviations from a Gaussian distribution.

Figure 13

The same ${\chi }^2$ analysis as in Fig. 12 is repeated for the multipoles of the 2-point correlation function that we use as the benchmark in our analysis.

Figure 14

Marginalized constraints on the full set of $\mathrm{cosmological}+\mathrm{HOD}$ parameters obtained using the WST coefficients (blue) and the joint combination of $\mathrm{WST}+\mathrm{correlation}$ function multipoles (black) in order to analyze the BOSS CMASS observations. The results shown above were obtained after imposing a BBN Gaussian prior on the value of ${\omega }_b=0.02268\pm 0.00038$.

Figure 15

Marginalized constraints on the $\mathrm{\Lambda }\mathrm{CDM}$ cosmological parameters obtained using the WST coefficients without the inclusion of the emulator error, $C_{\mathrm{emu}}$, in Eq. (20), shown in the blue contours. The result of the main WST analysis using the full covariance (originally shown in Fig. 8) is also plotted in red, for comparison.

Figure 16

Marginalized constraints on the $\mathrm{\Lambda }\mathrm{CDM}$ cosmological parameters obtained from the joint $\mathrm{WST}+\text{correlation}$ function analysis using a flat prior on the value of ${\omega }_b$ (red), as opposed to the main analysis that used a Gaussian prior (24), in blue (and originally presented in Fig. 8).

Figure 17

Fractional changes to the WST data vector when a sharp top-hat filter with various $k_{\max }$ cutoff values is applied to the galaxy field before the evaluation in Eq. (6), with respect to the original evaluation of our main analysis. This example evaluation corresponds to the NGC part of the BOSS dataset.