Hyper Suprime-Cam Year 3 results: Cosmology from galaxy clustering and weak lensing with HSC and SDSS using the minimal bias model

We present cosmological parameter constraints from a blind joint analysis of three two-point correlation functions measured from the Year 3 Hyper Suprime-Cam (HSC-Y3) imaging data, covering about $416{\mathrm{deg}}^2$, and the SDSS DR11 spectroscopic galaxies spanning the redshift range [0.15, 0.70]. We subdivide the SDSS galaxies into three luminosity-cut, and therefore nearly volume-limited samples separated in redshift, each of which acts as a large-scale structure tracer characterized by the measurement of the projected correlation function, $w_{\mathrm{p}}(R)$. We also use the measurements of the galaxy-galaxy weak-lensing signal $\mathrm{\Delta }\mathrm{\Sigma }(R)$ for each of these SDSS samples which act as lenses for a secure sample of source galaxies selected from the HSC-Y3 shape catalog based on their photometric redshifts. We combine these measurements with the cosmic shear correlation functions, ${\xi }_{\pm }(\vartheta )$ measured for our HSC source sample. We model these observables with the minimal bias model of the galaxy clustering observables in the context of a flat $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. We use conservative scale cuts, $R>12$ and $8h^{-1}\mathrm{Mpc}$ for $\mathrm{\Delta }\mathrm{\Sigma }$ and $w_{\mathrm{p}}$, respectively, where the minimal bias model is valid, in addition to conservative prior on the residual bias in the mean redshift of the HSC photometric source galaxies. We present various validation tests of our model as well as analysis methods. Our baseline analysis yields $S_8=0.775_{-0.038}^{+0.043}$ (68% C.I.) for the $\mathrm{\Lambda }\mathrm{CDM}$ model, after marginalizing over uncertainties in other parameters. Our value of $S_8$ is consistent with that from the Planck 2018 data, but the credible interval of our result is still relatively large. We show that various internal consistency tests based on different splits of the data are passed. Our results are statistically consistent with those of a companion paper, which extends this analysis to smaller scales with an emulator-based halo model, using $\mathrm{\Delta }\mathrm{\Sigma }(R)$ and $w_{\mathrm{p}}(R)$ down to $R>3$ and $2h^{-1}\mathrm{Mpc}$, respectively.

Figure 1

The cosmological constraint from the HSC Y3 data with the large-scale $3\times 2\mathrm{pt}$ analysis carried out in this work, along with the HSC Y3 small-scale $3\times 2\mathrm{pt}$ analysis in Miyatake et al. [16] and external experiments: Planck 2018 [9], DES Y3 $3\times 2\mathrm{pt}$ [70], and KiDS-1000 [8]. Here the marginal posterior distributions in one- or two-dimensional parameter space are shown for the main cosmological parameters constrained in this work, ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, and $S_8\equiv {\sigma }_8({\mathrm{\Omega }}_{\mathrm{m}}/0.3{)}^{0.5}$.

Figure 2

The cosmological parameter constraints for the baseline $3\times 2\mathrm{pt}$, $2\times 2\mathrm{pt}$, and cosmic shear analyses. Here, every analysis uses the informative prior on $\mathrm{\Delta }{z}_{\mathrm{ph}}$ from the HSC-Y3 small-scale analysis by Miyatake et al. [16].

Figure 3

Cosmological parameter constraints for HSC-Y3 $3\times 2\mathrm{pt}$ analyses with three different $\mathrm{\Delta }{z}_{\mathrm{ph}}$ prior choices; the informative prior taken from the fiducial small-scale analysis result, the uninformative prior $\mathcal{U}(-1,1)$, and the informative prior $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}})=\mathcal{N}(0,0.1)$.

Figure 4

Comparison between the measured signals and the best-fit model predictions for the baseline large-scale $3\times 2\mathrm{pt}$ analysis. From the top to the bottom panels, we show the comparison for $w_{\mathrm{p}}(R)$ and $\mathrm{\Delta }\mathrm{\Sigma }(R)$ for the three SDSS samples (LOWZ, CMASS1, and CMASS2), and the cosmic shear correlation functions, ${\xi }_{\pm }(\vartheta )$, respectively. In each panel, the black points with error bars denote the measured signals in each $R$ or $\vartheta$ bin, where the error bar is computed from the diagonal component of the covariance matrix. The solid line denotes the model prediction at the MAP (maximum a posteriori model of the chain), while the red-shaded regions show the 68% and 95% credible intervals of the model predictions in each bin. The blue shaded region in each panel indicates the range of $R$ or $\vartheta$ that is used for the cosmological parameter inference in this paper. See Sec. 3 for details of the model predictions. Note that the $x$-axes in the top and middle panels are the comoving distance in the reference cosmology (see the main text), $R^{\mathrm{ref}}$, but we omit the superscript “ref” for simplicity.

Figure 5

The evaluation of the goodness-of-fit with the ${\chi }^2({\theta }_{\mathrm{MAP}})$ value at the maximum a posteriori (MAP) model. The reference distribution (blue histogram) is obtained by analyzing 100 noisy mock data vectors. See the main text for how the noisy mock data vectors are generated. The vertical solid line denotes the observed ${\chi }^2$ value for the cosmology analysis of the actual HSC-Y3 and SDSS data. The probability of finding the ${\chi }^2$ value larger than the observed value ($p$ value) is about 43%.

Figure 6

Summary of internal consistency tests. The estimates of cosmological parameters, ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, and $S_8$, and the photo-$z$ parameter, $\mathrm{\Delta }{z}_{\mathrm{ph}}$, are summarized for each of the analysis setups in Table 2. Here, the central point is the mode and the error bar is the 68% highest density interval estimated from the one-dimensional posterior distribution of each parameter. For comparison, the shaded band is the constraint from the baseline analysis. The vertical black dotted line in the $\mathrm{\Delta }{z}_{\mathrm{ph}}$ panel denotes $\mathrm{\Delta }{z}_{\mathrm{ph}}=0$.

Figure 7

A statistical significance of the difference between the $S_8$ values from the two analyses, where we define $\mathrm{\Delta }{S}_8$ by the difference between the $S_8$ modes values in the 1D posteriors of the two analyses. Left panel: The result for $\mathrm{\Delta }{S}_8$ between the large-scale $2\times 2\mathrm{pt}$ and cosmic shear analyses. The blue histogram denotes the distribution of $\mathrm{\Delta }{S}_8$ that is obtained by carrying out these analyses on each of 100 realizations of the noisy mock data vector (see text for details). The red solid line is the observed $\mathrm{\Delta }{S}_8=0.10$ in the real data. The $p$-value is $p=0.2$. Right panel: The result for $\mathrm{\Delta }{S}_8$ between the large-scale $3\times 2\mathrm{pt}$ analysis in this paper and the small-scale $3\times 2\mathrm{pt}$ analysis in the companion paper by Miyatake et al. [16]. The observed difference is $\mathrm{\Delta }{S}_8=0.01$ and the probability to exceed this value by chance is $p=0.5$. The two arrows indicated by “assembly-$b$” and “assembly-$b$-ext” denote the expected difference values of $S_8$ obtained from the simulated synthetic data, where the assembly bias effects with different amplitudes are included.

Figure 8

Summary of the validation tests of the model and method. We apply the baseline analysis method to the synthetic data vectors in Table 3 to obtain constraints on the three cosmological parameters, ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, and $S_8$, and the photo-$z$ parameter $\mathrm{\Delta }{z}_{\mathrm{ph}}$. Similar types of validation tests are grouped by horizontal dotted lines. The superscript “$*$” denotes the analysis using the informative prior of $\mathrm{\Delta }{z}_{\mathrm{ph}}$ that is taken from the posterior distribution of the small-scale $3\times 2\mathrm{pt}$ analysis in Miyatake et al. [16] on the same synthetic data vector. The top section shows the results of the baseline analysis on the fiducial mock, i.e., the data vector uncontaminated by systematic effects. The second section shows the robustness of the minimum bias model against uncertainties in the galaxy bias or the galaxy-halo connection, where we use the different SDSS galaxy mock catalogs to simulate $\mathrm{\Delta }\mathrm{\Sigma }$ and $w_{\mathrm{p}}$ affected by the different galaxy-halo connections. The row “assembly-$b$-ext w $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}}=\mathcal{N}(0.1)$” shows the result obtained using the synthetic data for the “assembly-$b$-ext” mock, but using the informative Gaussian prior on $\mathrm{\Delta }{z}_{\mathrm{ph}}$ given by $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}})=\mathcal{N}(0,0.1)$. The fourth section shows the validation of the model of cosmic shear signal against baryonic effect contamination simulated by different versions of HMCode. The fifth section shows the results for the validation tests using the synthetic data that is affected by a systematic error in the mean source redshift by $|\mathrm{\Delta }{z}_{\mathrm{ph}}^{\mathrm{in}}|=0.2$ (see text for details). Here $|\mathrm{\Delta }{z}_{\mathrm{ph}}^{\mathrm{in}}|=0.2$ gives the worst-case scenario for the effect of the unknown systematic photo-$z$ error, because the small-scale $3\times 2\mathrm{pt}$ has the precision of $\sigma (\mathrm{\Delta }{z}_{\mathrm{ph}})\sim 0.1$ for the calibration of the photo-$z$ error parameter. The analysis with superscript “$*$” uses the informative prior of $\mathrm{\Delta }{z}_{\mathrm{ph}}$ taken from the small-scale $3\times 2\mathrm{pt}$ analysis on the same synthetic data (that is, this is our baseline analysis method). The analysis with the labels “$\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}}=\mathcal{N}(0,0.1)$” shows the result using the informative Gaussian prior around $\mathrm{\Delta }{z}_{\mathrm{ph}}=0$, and “$\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}}=\mathcal{U}(-1,1)$” shows the result using the uninformative flat prior. The last section is the validation of our PSF systematics modeling in which the synthetic cosmic shear signals include the PSF systematics up to the fourth moment of PSF.

Figure 9

Validation of the use of the $\mathrm{\Delta }{z}_{\mathrm{ph}}$ prior taken from the small-scale $3\times 2\mathrm{pt}$ analysis, which is our baseline analysis method. For this test, we use the synthetic data vector where we implemented the systematic error in the source redshift distribution modeled by $\mathrm{\Delta }{z}_{\mathrm{ph}}^{\mathrm{in}}=-0.2$ (see Sec. pp1-s1c). The blue contours show the results obtained when analyzing the synthetic data vector with a prior taken from small-scale $3\times 2\mathrm{pt}$ analysis method, while the orange contours show the results with an informative (but wrong!) prior $\mathcal{N}(0,0.1)$.

Figure 10

Marginalized posterior distribution for all the model parameters with derived parameters, ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, and $S_8$. Here the parameter constraints are obtained from the baseline $3\times 2\mathrm{pt}$ analysis (blue), the $2\times 2\mathrm{pt}$ -only analysis (orange), and the cosmic shear alone analysis (green).

Figure 11

Cosmological constraints with $3\times 2\mathrm{pt}$ but removing a single lens redshift bin from each analysis.

Figure 12

Cosmological constraints with $2\times 2\mathrm{pt}$ but removing a single lens redshift bin from each analysis.

Figure 13

Cosmological constraints with $3\times 2\mathrm{pt}$ but fixing one of the nuisance parameters to zero.

Figure 14

Cosmological constraints with $2\times 2\mathrm{pt}$ but fixing the cosmic shear related nuisance parameters to some fiducial values. In “no PSF error”, we fix ${\alpha }_{\mathrm{psf}}$ and ${\beta }_{\mathrm{psf}}$ to the center of the prior. In the “no IA” case, we fix $A_{\mathrm{IA}}=0$, while we set $A_{\mathrm{IA}}=5$ in the “extreme IA” case.

Figure 15

Cosmological constraints with $2\times 2\mathrm{pt}$ but fixing $n_{\mathrm{s}}$ and ${\omega }_{\mathrm{cdm}}\equiv {\mathrm{\Omega }}_{\mathrm{cdm}}{h}^2$ to the best-fit value of [9], and ${\omega }_{\mathrm{b}}\equiv {\mathrm{\Omega }}_{\mathrm{b}}{h}^2$ to the best fit of BBN [65, 66, 67].

Figure 16

The result of nestcheck [82] for the baseline chain, sampled in the real data analysis. The top panel shows the posterior volume as a function of the prior volume, $X$. The left panels show the uncertainty of the posterior distribution from an input nested sampling chain, where the uncertainty is estimated by bootstrapping the chain.

Figure 17

The comparison of posterior estimates from the nested sampling by MultiNest and the Markov Chain Monte Carlo sampling by the standard Metropolis algorithm.