Cosmological constraints from the nonlinear galaxy bispectrum

We present the cosmological constraints from analyzing higher-order galaxy clustering on small nonlinear scales. We use SimBIG, a forward modeling framework for galaxy clustering analyses that employs simulation-based inference to perform highly efficient cosmological inference using normalizing flows. It leverages the predictive power of high-fidelity simulations and robustly extracts cosmological information from regimes inaccessible with current standard analyses. In this work, we apply SimBIG to a subset of the BOSS galaxy sample and analyze the redshift-space bispectrum monopole, $B_0(k_1,k_2,k_3)$, to $k_{\max }=0.5h/\mathrm{Mpc}$. We achieve $1\sigma$ constraints of ${\mathrm{\Omega }}_m=0.293_{-0.027}^{+0.027}$ and ${\sigma }_8=0.783_{-0.038}^{+0.040}$, which are more than 1.2 and $2.4\times$ tighter than constraints from standard power spectrum analyses of the same dataset. We also derive 1.4, 1.4, $1.7\times$ tighter constraints on ${\mathrm{\Omega }}_b$, $h$, $n_s$. This improvement comes from additional cosmological information in higher-order clustering on nonlinear scales and, for ${\sigma }_8$, is equivalent to the gain expected from a standard analysis on a $\sim 4\times$ larger galaxy sample. Even with our BOSS subsample, which only spans 10% of the full BOSS volume, we derive competitive constraints on the growth of structure: $S_8=0.774_{-0.053}^{+0.056}$. Our constraint is consistent with results from both cosmic microwave background and weak lensing. Combined with a ${\omega }_b$ prior from big bang nucleosynthesis, we also derive a constraint on $H_0={67.6}_{-1.8}^{+2.2}\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ that is consistent with early Universe constraints.

Figure 1

The bispectrum monopole ($B_0$; top panel) and reduced bispectrum monopole ($Q_0$; bottom panel) of a subset of simulated galaxy catalogs in our training set. The catalogs are constructed using the SimBIG forward model from the Quijote $N$-body simulations and include BOSS survey realism. We randomly select 200 out of the 20,000 catalogs. We present a subset of 1,354 triangle configurations with $k_1,k_2,k_3<k_{\max }=0.25h/\mathrm{Mpc}$,for clarity. The configurations are ordered by looping through $k_3$ in the inner most loop and $k_1$ in the outer most loop with $k_1\le k_2\le k_3$. For reference, we include $B_0$ measured from the observed BOSS CMASS sample (black) with error bars estimated from the TEST0 simulations. The observed $B_0$ is well within our training dataset.

Figure 2

The NDE accuracy test that shows the SBC validation of the SimBIG $B_0(k_{123}<0.5h/\mathrm{Mpc})$ posterior estimate. We present the distribution of the rank statistics, which are derived by comparing the true parameter values to the inferred marginalized 1D posteriors. The rank statistics are calculated using 2,000 validation simulations that were excluded from training the posterior estimate. For an accurate estimate of the true posterior, the rank statistic would be uniformly distributed (black dashed). Overall, we estimate unbiased posteriors of all of the $\mathrm{\Lambda }\mathrm{CDM}$ cosmological parameters.

Figure 3

Posteriors of $({\mathrm{\Omega }}_m,{\sigma }_8)$ inferred using the SimBIG bispectrum analysis for a random subset of the TESTO0 (top), TEST1 (center), and TEST2 (bottom) simulations. We mark the 68 and 84 percentiles of the posteriors with the contours. We also include the true $({\mathrm{\Omega }}_m,{\sigma }_8)$ of the test simulations in each panel (black $\times$). The comparison between the posteriors and the true parameter values qualitatively show good agreement for each test simulations.

Figure 4

Comparison of the compressed bispectrum likelihood, $p(B_0^{(c)}|{\theta }_{\mathrm{fid}})$, computed on the three sets of test simulations: TEST0 (blue), TEST1 (orange), and TEST2 (green). $B_0^{(c)}$ is derived by taking the mean of the marginalized 1D SimBIG $B_0(k_{123}<0.5h/\mathrm{Mpc})$ posterior for the $\mathrm{\Lambda }\mathrm{CDM}$ parameters, an optimal compression of the cosmological information in $B_0$. In each panel, we mark the corresponding $\mathrm{\Lambda }\mathrm{CDM}$ parameters. The likelihoods are at the fixed fiducial cosmologies and parameter values of the test sets. We present the distribution of $B_0^{(c)}-\overline{B_0^{(c)}}$ because TEST2 simulations are constructed using different fiducial parameter values than the TEST0 and TEST1 simulations. Overall, we find excellent agreement among the likelihoods of the different test simulations and conclude that our $B_0$ analysis is robust to modeling choices in our forward model.

Figure 5

Posterior distribution of all parameters inferred using the SimBIG $B_0$ analysis to $k_{\max }<0.5h/\mathrm{Mpc}$ from BOSS CMASS SGC. In the top set of panels, we present the cosmological parameters. In the bottom, we present the halo occupation parameters. The axis ranges of the panels represent the prior range. We place significant constraints on all $\mathrm{\Lambda }\mathrm{CDM}$ parameters and a number of the halo occupation parameters (e.g. $\log M_{\min }$, $\log M_0$, and ${\eta }_{\mathrm{sat}}$).

Figure 6

Left: Posterior of cosmological parameters inferred from $B_0$ using SimBIG. In the diagonal panels we present the marginalized 1D posterior of each parameter. The other panels present the 2D posteriors that illustrate the degeneracies between two parameters. The contours mark the 68 and 95 percentiles. By robustly analyzing $B_0$ down to nonlinear regimes, $k_{\max }=0.5h/\mathrm{Mpc}$, we place significant constraints on all $\mathrm{\Lambda }\mathrm{CDM}$ parameters without any priors from BBN of CMB experiments. Right: We focus on the posteriors of ${\mathrm{\Omega }}_m$ and ${\sigma }_8$, the parameters that can be most significantly constrained by galaxy clustering alone. We derive ${\mathrm{\Omega }}_m=0.293_{-0.027}^{+0.027}$ and ${\sigma }_8=0.783_{-0.038}^{+0.040}$. Our ${\mathrm{\Omega }}_m$ and ${\sigma }_8$ constraints are >10 and 50% tighter than the $P_{\ell }(k<k_{\max }=0.25h/\mathrm{Mpc})$ constraints from a PT approach [11] (orange) and an emulator approach [13] (green). This improvement comes from simultaneously exploiting higher-order and nonlinear cosmological information.

Figure 7

(${\mathrm{\Omega }}_m,{\sigma }_8$) posterior from theSim BIG $B_0$ analysis to $k_{\max }=0.3h/\mathrm{Mpc}$ (red dashed). For comparison, we include posteriors from $P_{\ell }$ analyses (Ivanov et al. [11], orange; Kobayashi et al. [13], green) and the SimBIG $B_0(k_{123}<0.5h/\mathrm{Mpc})$ analysis (black). The contours represent the 68 and 95 percentiles. We find overall good agreement among the posteriors. Furthermore, the improvement we find from $B_0(k_{123}<0.3h/\mathrm{Mpc})$ over $P_{\ell }$ is consistent with the improvement from $B_0$ found in the literature (e.g. [24]). Our $B_0(k_{123}<0.3h/\mathrm{Mpc})$ posterior is significantly broader than our $B_0(k_{123}<0.5h/\mathrm{Mpc})$. This demonstrates that there is additional higher-order cosmological information in the nonlinear regime, $0.3<k<0.5h/\mathrm{Mpc}$, that we can robustly analyze using simbig.