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

We present cosmology results from a blinded joint analysis of cosmic shear, ${\xi }_{\pm }(\vartheta )$, galaxy-galaxy weak lensing, $\mathrm{\Delta }\mathrm{\Sigma }(R)$, and projected galaxy clustering, $w_{\mathrm{p}}(R)$, measured from the Hyper Suprime-Cam three-year (HSC-Y3) shape catalog and the Sloan Digital Sky Survey (SDSS) DR11 spectroscopic galaxy catalog—a $3\times 2\mathrm{pt}$ cosmology analysis. We define luminosity-cut, and therefore nearly volume-limited, samples of SDSS galaxies to serve as the tracers of $w_{\mathrm{p}}$ and as the lens samples for $\mathrm{\Delta }\mathrm{\Sigma }$ in three spectroscopic redshift bins spanning the range $0.15<z<0.7$. For the ${\xi }_{\pm }$ and $\mathrm{\Delta }\mathrm{\Sigma }$ measurements, we use a single sample of about seven million source galaxies over $416{\mathrm{deg}}^2$, selected from HSC-Y3 based on having photometric redshifts (photo-$z$) greater than 0.75. The deep, high-quality HSC-Y3 data enable significant detections of the $\mathrm{\Delta }\mathrm{\Sigma }$ signals, with integrated signal-to-noise ratio $S/N\sim 24$ in the range $3\le R/[h^{-1}\mathrm{Mpc}]\le 30$ over the three lens samples. ${\xi }_{\pm }$ has $S/N\sim 19$ in the range $8^{\prime }\le \vartheta \le 50^{\prime }$ and ${30}^{\prime }\le \vartheta \le 150^{\prime }$ for ${\xi }_{+}$ and ${\xi }_{-}$, respectively. For cosmological parameter inference, we use the dark emulator package, combined with a halo occupation distribution prescription for the relation between galaxies and halos, to model $w_{\mathrm{p}}$ and $\mathrm{\Delta }\mathrm{\Sigma }$ down to quasinonlinear scales, and we estimate cosmological parameters after marginalizing over nuisance parameters. In our baseline analysis we employ an uninformative flat prior of the residual photo-$z$ error, given by $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}})=\mathcal{U}(-1,1)$, to model a residual bias in the mean redshift of HSC source galaxies. Comparing the relative lensing amplitudes for $\mathrm{\Delta }\mathrm{\Sigma }$ in the three redshift bins and for ${\xi }_{\pm }$ with the single HSC source galaxy sample allows us to calibrate the photo-$z$ parameter $\mathrm{\Delta }{z}_{\mathrm{ph}}$ to the precision of $\sigma (\mathrm{\Delta }{z}_{\mathrm{ph}})\simeq 0.09$. With these methods, we obtain a robust constraint on the cosmological parameters for the flat $\mathrm{\Lambda }\mathrm{CDM}$ model: $S_8={\sigma }_8({\mathrm{\Omega }}_{\mathrm{m}}/0.3{)}^{0.5}={0.763}_{-0.036}^{+0.040}$, or the best-constrained parameter given by $S_8^{\prime }={\sigma }_8({\mathrm{\Omega }}_{\mathrm{m}}/0.3{)}^{0.22}=0.721\pm 0.028$, determined with about 4% fractional precision. Based on multidimensional tension metrics, HSC-Y3 data exhibits about $2.5\sigma$ tension with the cosmological constraint inferred by Planck for the $\mathrm{\Lambda }\mathrm{CDM}$ model, and hints at a nonzero residual photo-$z$ bias implying that the true mean redshift of the HSC galaxies at $z\gtrsim 0.75$ is higher than that implied by the original photo-$z$ estimates.

Figure 1

The 1D and 2D posterior distributions in the sub-space of $S_8$, ${\sigma }_8$ and ${\mathrm{\Omega }}_{\mathrm{m}}$ for the flat $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. The blue dark (light) shaded regions denote the 68% (95%) credible interval, respectively, for our HSC-Y3 $3\times 2\mathrm{pt}$ baseline analysis in Table 3. For comparison we show the results for other recent cosmological analyses. The red contours are from the DES-Y3 $3\times 2\mathrm{pt}$ analysis [8]. The blue contours are from the KiDS-1000 analysis [9] with cosmic shear (“CS”) and galaxy-galaxy weak lensing (“GGL”) (see text for details). The green contours are the Planck 2018 results using the primary CMB anisotropy information (“TT, TE, $\mathrm{EE}+\mathrm{lowE}$”) [11]. Note that the degeneracy direction of the HSC-Y3 result in each 2D subspace of the parameters are different from those of DES-Y3 and KiDS-1000, since the relative constraining powers of the cosmological parameters for different observables are different.

Figure 2

The green solid line in each panel denotes the model prediction at the maximum a posteriori (MAP) for the baseline analysis in Fig. 1, while the data points with error bars are the measured signals. The upper-row three panels are for the projected correlation functions of galaxies, $w_{\mathrm{p}}(R)$, for the LOWZ, CMASS1 and CMASS2 samples in the redshift ranges $z=[0.15,0.35]$, [0.43, 0.55] and [0.55, 0.70], respectively. The middle-row three panels are for the galaxy-galaxy weak lensing using the HSC galaxies as source sample, $\mathrm{\Delta }\mathrm{\Sigma }(R)$, for the same LOWZ, CMASS1, and CMASS2 samples as lens samples, respectively. The bottom-row two panels are for the cosmic shear correlation functions, ${\xi }_{\pm }(\vartheta )$. For illustration purpose, we show $R\times w_{\mathrm{p}}(R)$, $R\times \mathrm{\Delta }\mathrm{\Sigma }(R)$ and $\vartheta \times {\xi }_{\pm }(\vartheta )$. The red shaded regions around the green line denote the 68% and 95% credible intervals of the model predictions in each separation bin, which are computed from the posterior distributions in the Bayesian cosmology inference. Note that the errors are computed from the diagonal components of the covariance matrix. The blue-color shaded region in each panel denotes the range of projected or angular separation bins that is used for the cosmology analysis.

Figure 3

An evaluation of the goodness-of-fit of the best-fit (MAP) model in Fig. 1. The histogram shows the distribution of the ${\chi }^2$ values at the MAP model, obtained by applying the same baseline analysis to 100 noisy mock datasets (see text for details). The vertical black line denotes the measured ${\chi }^2$-value (${\chi }^2=85.1$) at MAP for the actual analysis of HSC-Y3 and SDSS data. The probability of finding the ${\chi }^2$ value larger than the observed value ($p$ value) is about 41%.

Figure 4

A summary of the cosmological parameters and the residual photo-$z$ error parameter $\mathrm{\Delta }{z}_{\mathrm{ph}}$, estimated from each of the different analysis setups in Table 3. The vertical dashed line in the panel of $\mathrm{\Delta }{z}_{\mathrm{ph}}$ denotes $\mathrm{\Delta }{z}_{\mathrm{ph}}=0$, i.e., the case of no residual photo-$z$ error or equivalently the case that the mean redshift estimate of HSC source galaxies inferred from the photo-$z$ estimates is perfect. As our default choice, we employ the uninformative flat prior on the residual photo-$z$ error, $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}})=\mathcal{U}(-1,1)$. As explained in Table 3, the analysis with superscript “${}^{*}$” denotes an analysis using the informative Gaussian prior on the photo-$z$ error, given by $\mathcal{N}=(-0.06,0.08)$, which is inferred from the posterior of the baseline $3\times 2\mathrm{pt}$ analysis. The analysis with superscript “${}^{†}$” denotes an analysis using the Gaussian prior on $\mathrm{\Delta }{z}_{\mathrm{ph}}$ with mean around $\mathrm{\Delta }{z}_{\mathrm{ph}}=0$, given by $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}})=\mathcal{N}(0,0.1)$. The analysis with label “$\sigma (\mathrm{\Delta }{z}_{\mathrm{ph}})=0.2$ prior” denotes the result using the Gaussian prior, $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}})=\mathcal{N}(0,0.2)$.

Figure 5

Comparison of the cosmological constraints from the different cosmological analyses of the HSC-Y3 data. The result labeled as “$3\times 2\mathrm{pt}$ large scale” uses the same data vector as that in this paper, but uses the perturbation theory based model template when comparing the predictions to the measurements on large scales (Sugiyama et al. [24]). “Cosmic shear tomography: Real” is the result from the cosmic shear tomography analysis using the real-space cosmic shear correlation functions with 4 tomographic redshift bins (Li et al. [25]). “Cosmic shear tomography: Fourier” is from the cosmic shear tomography using the cosmic shear power spectra (Dalal et al. [26]). The two cosmic shear results use the different scale cuts.

Figure 6

The importance of the uniformative prior on the residual photo-$z$ error parameter ($\mathrm{\Delta }{z}_{\mathrm{ph}}$) in the cosmology analysis. Left panel: the 1D and 2D posterior distributions obtained using the different observables: the baseline ($3\times 2\mathrm{pt}$), the $2\times 2\mathrm{pt}$ ($\mathrm{\Delta }\mathrm{\Sigma }\times w_{\mathrm{p}}$), and the cosmic shear correlations. For all the analyses, we employ the flat prior of $\mathrm{\Delta }{z}_{\mathrm{ph}}$: $\mathcal{U}(-1,1)$ as our baseline analysis. Right: the posterior distributions for the $3\times 2\mathrm{pt}$ analyses when using the different priors of $\mathrm{\Delta }{z}_{\mathrm{ph}}$: the baseline analysis (the flat prior), the Gaussian prior of $\mathcal{N}(0,0.1)$ and the case fixing $\mathrm{\Delta }{z}_{\mathrm{ph}}=0$, respectively. The result of $\mathrm{\Delta }{z}_{\mathrm{p}}=0$ approximately corresponds to the case where the redshift distribution of HSC source galaxies is as inferred from the photo-$z$ estimate, because the prior is as informative as $\sigma (\mathrm{\Delta }{z}_{\mathrm{ph}})\sim {10}^{-2}$. The different treatments of $\mathrm{\Delta }{z}_{\mathrm{ph}}$ affect the mode value and the size of the credible interval of the cosmological parameters.

Figure 7

The posterior distributions of the cosmological parameters when using different scale cuts of $R=[4,6]$ or $[8,12]h^{-1}\mathrm{Mpc}$ for the $w_{\mathrm{p}}$ and $\mathrm{\Delta }\mathrm{\Sigma }$ signals, respectively, in the cosmology analysis. The gray contours are the same as those in Fig. 1. Note that we used the Gaussian prior of the residual photo-$z$ error parameter, $\mathcal{N}(-0.06,0.08)$, that is inferred from the baseline $3\times 2\mathrm{pt}$ analysis and the results include the cosmic shear information too, where we used the same range of the angular separations as in the baseline analysis.

Figure 8

The shaded histogram shows the expected distribution of the differences between the $S_8$ values obtained from the $3\times 2\mathrm{pt}$ analyses using the different scale cuts of (2, 3) and $(8,12)h^{-1}\mathrm{Mpc}$ for $w_{\mathrm{p}}$ and $\mathrm{\Delta }\mathrm{\Sigma }$, respectively, assuming that $w_{\mathrm{p}}$ and $\mathrm{\Delta }\mathrm{\Sigma }$ are not contaminated by the assembly bias effect. To obtain the distribution, we perform the same cosmology analysis on each of the 100 realizations of the noisy mock data vector. Note that in this inference simulation, we employ the prior on the residual photo-$z$ error, $\mathrm{\Delta }{z}_{\mathrm{ph}}$ obtained from the fiducial analysis of $(2,3)h^{-1}\mathrm{Mpc}$ scale cut to each realization for the (8, 12)-analyses, as we did for the actual analysis. When there is no assembly bias effect, the $S_8$ values from the (2, 3)- and (8, 12)-scale cuts should be consistent with each other, and the actual observed difference of $S_8$, as denoted by the vertical solid line, is consistent with the distribution from the synthetic data vector. The probability of finding $\mathrm{\Delta }{S}_8$ larger than the observed value ($p$-value) is about 50%. 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 assembly bias effects with different amplitudes are included.

Figure 9

The posterior distribution of the parameters when the Gaussian prior of $\mathrm{\Pi }({\mathrm{\Omega }}_{\mathrm{m}})=\mathcal{N}(0.3,0.01)$ as motivated by BAO constraints is added. This analysis was done as a part of the post-unblinding analysis.

Figure 10

Summary of model validation tests for the $3\times 2\mathrm{pt}$ analysis. Constraints on cosmological parameters ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, and $S_8$ and the photo-$z$ shift parameter $\mathrm{\Delta }{z}_{\mathrm{ph}}$ are shown in each panel from left to right. The input cosmological parameters used for making synthetic data vectors are indicated by the vertical dashed lines in the panels. The shaded regions are the 68% confidence intervals from the constraints in the fiducial “$3\times 2\mathrm{pt}$” case.

Figure 11

The posterior distributions for baseline “$3\times 2\mathrm{pt}$,” “$2\times 2{\mathrm{pt}}^{*}$,” and “cosmic ${\mathrm{shear}}^{*}$” analysis setups in Table 3. The contours show the 68% and 96% credible intervals. The constraints from “$2\times 2{\mathrm{pt}}^{*}$” and “cosmic ${\mathrm{shear}}^{*}$” are consistent with the “$3\times 2\mathrm{pt}$” analysis. Note that since we use the prior on the residual photo-$z$ errors ($\mathrm{\Delta }{z}_{\mathrm{ph}}$) for “$2\times 2{\mathrm{pt}}^{*}$” and “cosmic ${\mathrm{shear}}^{*}$” derived from the “$3\times 2\mathrm{pt}$” analysis, the constraining power of “$2\times 2{\mathrm{pt}}^{*}$” is similar to “$3\times 2\mathrm{pt}$,” but that of “cosmic ${\mathrm{shear}}^{*}$” is still weaker than “$3\times 2\mathrm{pt}$.” For the constraints from the $2\times 2\mathrm{pt}$ and cosmic shear analysis with the uniform prior on $\mathrm{\Delta }{z}_{\mathrm{ph}}$, see the left panel in Fig. 6.

Figure 12

Similar to Fig. 11, but the results from “$3\times 2\mathrm{pt}$, w/o LOWZ,” “$3\times 2\mathrm{pt}$, w/o CMASS1,” and “$3\times 2\mathrm{pt}$, w/o CMASS2” analysis setups in Table 3. We do not see any significant changes in the cosmological constraints from the baseline “$3\times 2\mathrm{pt}$” analysis when excluding one of the lens samples in our analysis.

Figure 13

Similar to Fig. 11, but the results from “$2\times 2\mathrm{pt}$, w/o ${\mathrm{LOWZ}}^{*}$,” “$2\times 2\mathrm{pt}$, w/o CMASS1*,” and “$2\times 2\mathrm{pt}$, w/o CMASS2*” analysis setups in Table 3. We do not see any significant changes in the cosmological constraints from the baseline “$3\times 2\mathrm{pt}$” analysis when excluding one of the lens samples in our analysis.

Figure 14

Similar to Fig. 11, but the results from “no photo-$z$ error,” “no shear error,” and “fix mag. bias” analysis setups in Table 3. When the source redshift distribution is fixed, corresponding to the case “no photo error,” the credible interval of the cosmological parameters are significantly tightened, but the central values are shifted. In particular, fixing the source redshift distribution as indicated from the photo-$z$ estimates rather than using the flat prior $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}})=\mathcal{U}(-1,1)$ in our baseline analysis leads to a higher value of $S_8$ than that of the baseline analysis. We do not see any significant changes in the cosmological constraints for the other cases.

Figure 15

Similar to Fig. 11, but the results from “no PSF error,” “no IA error,” and “extreme IA” analysis setups in Table 3. For these setups, we do not see any significant changes in the cosmological constraints from the baseline analysis.

Figure 16

Similar to Fig. 11, but the results from “$3\times 2{\mathrm{pt}}^{†}$,” “$2\times 2{\mathrm{pt}}^{†}$,” and “cosmic ${\mathrm{shear}}^{†}$” analysis setups in Table 3. Here the analyses with superscript ${}^{†}$ uses the Gaussian prior on the photo-$z$ error prior parameter, $\mathrm{\Pi }(\mathrm{\Delta }{z}_{\mathrm{ph}})=\mathcal{N}(0,0.1)$. The use of the Gaussian prior still affects the $S_8$ constraints for the “$3\times 2{\mathrm{pt}}^{†}$,” “$2\times 2{\mathrm{pt}}^{†}$” analyses.

Figure 17

Similar to Fig. 11, but the results from “XMM $(\sim 33{\mathrm{deg}}^2{)}^{*}$,” “GAMA15H $(\sim 41{\mathrm{deg}}^2{)}^{*}$,” “HECTOMAP $(\sim 43{\mathrm{deg}}^2{)}^{*}$,” “GAMA09H $(\sim 78{\mathrm{deg}}^2{)}^{*}$,” “VVDS $(\sim 96{\mathrm{deg}}^2{)}^{*}$,” and “WIDE12H $(\sim 121{\mathrm{deg}}^2{)}^{*}$” analysis setups in Table 3. Given that the analyses of each field are almost uncorrelated, the cosmological constraints are consistent with each other.

Figure 18

Similar to Fig. 11, but the results from “dempz&wx,” “mizuki,” and “dnnz” analysis setups in Table 3. “The dnnz” analysis exhibits the largest deviation in $S_8$, but it is still $\sim 0.5\sigma$.

Figure 19

Similar to Fig. 11, but the results from “w/o star weight” analysis setup in Table 3. We do not see any significant changes in the cosmological constraints in the “w/o star weight” analysis.

Figure 20

Similar to Fig. 11, but the results from “off-centering” and “incompleteness” analysis setups in Table 3. We do not see any significant changes in the cosmological constraints in these analysis.

Figure 21

Similar to Fig. 11, but the results from “$\sigma (\mathrm{\Delta }m)=0.1$ prior” and “$\sigma (\mathrm{\Delta }{z}_{\mathrm{ph}})=0.2$ prior” analysis setups in Table 3. Note that these are the postunblinding analyses. We do not see a significant change in the “$\sigma (\mathrm{\Delta }m)=0.1$ prior” analysis, meaning that the degradation of constraining power is mostly due to the use of flat prior on the residual photo-$z$ errors. We see only a slight change in the “$\sigma (\mathrm{\Delta }{z}_{\mathrm{ph}})=0.2$” analysis without improvement in the constraining power. This means that with this prior width, we are reaching to the limit of the flat prior.

Figure 22

Similar to Fig. 11, but the results from “2 cosmo paras” analysis setup in Table 3. Note that this is the post-unblinding analyses. When fixing cosmological parameters other than ${\mathrm{\Omega }}_{\mathrm{de}}$ and $\ln ({10}^{10}{A}_{\mathrm{s}})$ to the Planck CMB constraints [11], we obtain better cosmological constraints, but the shift in $\mathrm{\Delta }{z}_{\mathrm{ph}}$ is unchanged.

Figure 23

The posterior distributions of all parameters sampled in the baseline $3\times 2\mathrm{pt}$ analysis.

Figure 24

The result of nestcheck [132] for the baseline $3\times 2\mathrm{pt}$ analysis, 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 25

The comparison of posterior estimates of our baseline analysis from the nested sampling by MultiNest and the Markov Chain Monte Carlo sampling by the standard Metropolis algorithm. We do not see significant difference in the cosmological constraints from MultiNest and the Metropolis algorithm, meaning MultiNest provides sufficiently accurate estimate of credible levels.