Galaxy clustering analysis with SimBIG and the wavelet scattering transform

The non-Gaussian spatial distribution of galaxies traces the large-scale structure of the Universe and therefore constitutes a prime observable to constrain cosmological parameters. We conduct Bayesian inference of the $\mathrm{\Lambda }\mathrm{CDM}$ parameters ${\mathrm{\Omega }}_m$, ${\mathrm{\Omega }}_b$, $h$, $n_s$, and ${\sigma }_8$ from the Baryon Oscillation Spectroscopic Survey CMASS galaxy sample by combining the wavelet scattering transform (WST) with a simulation-based inference approach enabled by the SimBIG forward model. We design a set of reduced WST statistics that leverage symmetries of redshift-space data. Posterior distributions are estimated with a conditional normalizing flow trained on 20,000 simulated SimBIG galaxy catalogs with survey realism. We assess the accuracy of the posterior estimates using simulation-based calibration and quantify generalization and robustness to the change of forward model using a suite of 2000 test simulations. When probing scales down to $k_{\max }=0.5h/\mathrm{Mpc}$, we are able to derive accurate posterior estimates that are robust to the change of forward model for all parameters, except ${\sigma }_8$. We mitigate the robustness issues with ${\sigma }_8$ by removing the WST coefficients that probe scales smaller than $k\sim 0.3h/\mathrm{Mpc}$. Applied to the Baryon Oscillation Spectroscopic Survey CMASS sample, our WST analysis yields seemingly improved constraints obtained from a standard perturbation-theory-based power spectrum analysis with $k_{\max }=0.25h/\mathrm{Mpc}$ for all parameters except $h$. However, we still raise concerns on these results. The observational predictions significantly vary across different normalizing flow architectures, which we interpret as a form of model misspecification. This highlights a key challenge for forward modeling approaches when using summary statistics that are sensitive to detailed model-specific or observational imprints on galaxy clustering.

Figure 1

Diagram representing the different components of the SimBIG forward model.

Figure 2

Visualization of the 3D oriented wavelets designed for this work. (a) Radial (left and center) and angular (right) profiles for a subset of our 3D wavelets with fixed orientation. We show the radial profiles in both physical (left) and Fourier (center) space. The angular part is shown in Fourier space only. Our wavelets cover Fourier space with their passbands. (b) Planar cuts of the real (top row) and imaginary (bottom row) parts of $2^{3j}\psi \lambda$ for $\lambda =(j,\vartheta ,\phi )=(3,0,0)$ (the wavelet is oriented along the $z$ axis). The planar cuts are shown for $x,y,z=0$ from left to right, respectively.

Figure 3

Definition of the spherical coordinates with respect to the survey geometry. The redshift axis is $z$. The elevation angle $\vartheta$ measures deviation from $z$. The azimuthal angle $\phi$c measures rotation around $z$. The rounded rectangle represents a cut through the survey geometry.

Figure 4

Posterior accuracy test using SBC. We present the distributions of the rank statistics (Sec. 3a) for each of the cosmological parameters. These are derived from the validation set used during posterior estimation. The rank statistics should be uniformly distributed for an ideal posterior estimate (black dashed). Our results indicate accurate estimates for ${\mathrm{\Omega }}_b$, $h$, and $n_s$ but underconfidence for ${\mathrm{\Omega }}_m$ and ${\sigma }_8$.

Figure 5

Posterior distributions for $({\mathrm{\Omega }}_m,{\sigma }_8)$ inferred from WST measurements of 10 randomly selected test simulations from: TEST0 (top), TEST2 (middle), and TEST2 (bottom). The true parameters are marked in orange, and the contours represent the 68 and 95 percentiles. For TEST0 and TEST1, the true parameters lie expectedly within the percentiles of the posteriors. However, for TEST2, the true ${\sigma }_8$ tend to lie systematically higher than the inferred posteriors.

Figure 6

Distributions of the differences between the posterior mean $\mu$ and the true parameter ${\theta }^{\mathrm{fid}}$ normalized by the posterior standard deviation $\sigma$, for each cosmological parameter and test set. The distributions agree across the test sets for ${\mathrm{\Omega }}_m$, ${\mathrm{\Omega }}_b$, $h$, and $n_s$, but not for ${\sigma }_8$. This indicates that our predictions for ${\sigma }_8$ are not robust to variations of the forward model.

Figure 7

Same as Fig. 6 but using summary statistics where WST coefficients with $j<j_{\min }=2$ are discarded. This removes galaxy clustering information for scales smaller than $k_{\max }\approx 0.3h/\mathrm{Mpc}$. We focus on ${\sigma }_8$. By imposing a stringent scale cut we improve the robustness of our WST analysis.

Figure 8

Posterior distributions for the $\mathrm{\Lambda }\mathrm{CDM}$ cosmological parameters inferred from the BOSS CMASS SGC sample using WST with SimBIG. On the left, we present constraints using the full WST statistics, which probe scales down to $k_{\max }=0.5h/\mathrm{Mpc}$. On the right we present constraints using the $j_{\min }=2$ WST statistics, where modes with $k\gtrsim 0.3h/\mathrm{Mpc}$ are discarded. The contours represent the 68 and 95 percentiles. We also show for comparison posteriors from the SimBIG $P_{\ell }$ analysis (gray, [26, 27]) and the [15] PT based $P_{\ell }$ analysis (black dashed).

Figure 9

Observational constraints on ${\sigma }_8$ and ${\mathrm{\Omega }}_m$ obtained across the 10 best models included in the ensemble model with $j_{\min }=2$. This important variability suggests a form of model misspecification.

Figure 10

Same as Fig. 8 (left) but also including predictions of the HOD parameters.

Figure 11

Same as Fig. 8 (right) but also including predictions of the HOD parameters.