Hyper Suprime-Cam Year 3 results: Cosmology from cosmic shear two-point correlation functions

We perform a blinded cosmology analysis with cosmic shear two-point correlation functions measured from more than 25 million galaxies in the Hyper Suprime-Cam three-year shear catalog in four tomographic redshift bins ranging from 0.3 to 1.5. After conservative masking and galaxy selection, the survey covers $416{\mathrm{deg}}^2$ of the northern sky with an effective galaxy number density of $15{\mathrm{arcmin}}^{-2}$ over the four redshift bins. The 2PCFs adopted for cosmology analysis are measured in the angular range; $7.1<\theta /\mathrm{arcmin}<56.6$ for ${\xi }_{+}$ and $31.2<\theta /\mathrm{arcmin}<248$ for ${\xi }_{-}$, with a total signal-to-noise ratio of 26.6. We apply a conservative, wide, flat prior on the photometric redshift errors on the last two tomographic bins, and the relative magnitudes of the cosmic shear amplitude across four redshift bins allow us to calibrate the photometric redshift errors. With this flat prior on redshift errors, we find ${\mathrm{\Omega }}_{\mathrm{m}}=0.256_{-0.044}^{+0.056}$ and $S_8\equiv {\sigma }_8\sqrt{{\mathrm{\Omega }}_{\mathrm{m}}/0.3}=0.769_{-0.034}^{+0.031}$ (both 68% C.I.) for a flat $\mathrm{\Lambda }$ cold dark matter cosmology. We find, after unblinding, that our constraint on $S_8$ is consistent with the Fourier space cosmic shear and the $3\times 2$ pt analyses on the same HSC dataset. We carefully study the potential systematics from astrophysical and systematic model uncertainties in our fiducial analysis using synthetic data, and report no biases (including projection bias in the posterior space) greater than $0.5\sigma$ in the estimation of $S_8$. Our analysis hints that the mean redshifts of the two highest tomographic bins are higher than initially estimated. In addition, a number of consistency tests are conducted to assess the robustness of our analysis. Comparing our result with Planck-2018 cosmic microwave background observations, we find a $\sim 2\sigma$ tension for the $\mathrm{\Lambda }\mathrm{CDM}$ model.

Figure 1

The map of effective number density $n_{\mathrm{eff}}$ of galaxies across four redshift bins. A rectangular region in GAMA09H ($132.5[\mathrm{deg}]<\mathrm{ra}<140[\mathrm{deg}]$, $1.6[\mathrm{deg}]<\mathrm{dec}<5[\mathrm{deg}]$ with very good seeing, a smaller number of input single exposures, and significant fourth-order PSF shape residual is removed from the original catalog.

Figure 2

The comparison between $n(z)$ distributions (solid line) estimated by the joint calibration with CAMIRA-LRG sample [24] and those estimated by stacking the DEmPz (dashed lines), dNNz (dot-dashed lines) and mizuki (dotted lines) photo-$z$ posteriors from individual galaxies. The shaded grey histogram is the number density as a function of redshift of CAMIRA-LRGs used to calibrate the $n(z)$ distributions of the solid lines. The median redshifts for the four redshift bins (solid black lines) are 0.44, 0.75, 1.03 and 1.31, and the median redshift for the overall sample is 0.80. The dashed black lines in the last two redshift bins are the $n(z)$ distributions after after the self-calibration in parameter inference (see text for details).

Figure 3

The ten 2PCFs including four autocorrelations and six cross-correlations between the four tomographic redshift bins (labeled with 1–4). This plot shows the 2PCFs on scales $5.3[\mathrm{arcmin}]<\theta <76[\mathrm{arcmin}]$ for ${\xi }_{+}$, and $23.2[\mathrm{arcmin}]<\theta <248$ [arcmin] for ${\xi }_{-}$. The unshaded region refers to the fiducial scale cut: $7.1[\mathrm{arcmin}]<\theta <56.6[\mathrm{arcmin}]$ for ${\xi }_{+}$, and $31.2[\mathrm{arcmin}]<\theta <248[\mathrm{arcmin}]$ for ${\xi }_{-}$. The errorbars are estimated with mock catalogs. The total SNR of the measured 2PCFs is 26.6. The solid lines are the best-fit model of our fiducial analysis, as discussed in Sec. 5a.

Figure 4

The normalized covariance matrix (correlation coefficients) estimated with mock catalogs. Note that this plot shows the coefficients on scales $5.3[\mathrm{arcmin}]<\theta <76[\mathrm{arcmin}]$ for ${\xi }_{+}$, and $23.2[\mathrm{arcmin}]<\theta <248[\mathrm{arcmin}]$ for ${\xi }_{-}$. The fiducial scale cut—$7.1<\theta <56.6[\mathrm{arcmin}]$ for ${\xi }_{+}$, and $31.2[\mathrm{arcmin}]<\theta <248[\mathrm{arcmin}]$ for ${\xi }_{-}$—is a subset of this scale range.

Figure 5

$B$-modes on 2PCFs measured from the HSC-Y3 catalog in four tomographic bins. The $p$-value of the measured $B$-modes relative to a model of exactly zero is 0.1143 for ${\xi }_{+}$ and 0.1237 for ${\xi }_{-}$, as shown in the legend. The errorbars are estimated with mock catalogs. The unshaded region refers to the fiducial scale cut.

Figure 6

The integrands in Eq. (13) which transform angular power spectra to correlation functions. We show the integrands for the smallest and largest angular bins for ${\xi }_{+}$ and ${\xi }_{-}$, respectively, for the correlation functions of the second redshift bin with itself.

Figure 7

Galaxy-star correlations, including cross-correlations between galaxy shape and star shape (left panel), and also between galaxy shape and star shape errors (right panel). We show the correlations to both second- ($e_{\mathrm{psf}}^{(2)}$) and fourth-order ($M_{\mathrm{psf}}^{(4)}$) star shapes and star shape errors. The points are the measurements from HSC data, and the solid lines are the best-fit model using star-star correlations [see Eqs. (25)–(28)].

Figure 8

The modeling errors (estimated parameters—true parameters) in the MAP (“$\times$”) and projected 1D mode (“$+$”) when applying our model to different synthetic mocks of 2PCFs including (from left to right in the legend) the baseline simulation, simulation with the camb linear power spectrum, owls-agn suppression, and the MiraTitan II nonlinear power spectrum. The gray lines are $0.3\sigma$ and $0.5\sigma$ contours of the 2D projected posterior in the parameter space of (${\mathrm{\Omega }}_{\mathrm{m}}$, $S_8$).

Figure 9

Tests applying our fiducial model to synthetic mocks of 2PCFs from different hydrodynamic simulations. The $y$-axis is the modeling errors in the projected 1D mode (estimation—truth) relative to the $1\sigma$ uncertainty in the projected mode estimated by analyzing real data. The $x$-axis is the small-scale cut that we applied. We simultaneously vary the cut on ${\xi }_{+}$ (denoted as ${\theta }_{\min }^{+}$) and the cut on ${\xi }_{-}$ (denoted as ${\theta }_{\min }^{-}$).

Figure 10

Posterior contours (68% and 95% C.I. [For all the 2D posteriors shown in this paper, we plot the 68% and 95% C.I.]) of the 2D projected posterior in the (${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$) plane (left panel) and (${\mathrm{\Omega }}_{\mathrm{m}}$, $S_8$) plane (right panel) for our fiducial analysis, where $S_8={\sigma }_8\sqrt{{\mathrm{\Omega }}_{\mathrm{m}}/0.3}$. In addition, we show the projected 1D mode (“$+$”) and the MAP estimated from PolyChord chains (“$\times$”).

Figure 11

The evaluation of goodness-of-fit with the ${\chi }^2$ value at the maximum a posteriori (MAP) obtained from the chain of the fiducial analysis (red vertical line). The reference distribution (blue histogram) is obtained by analyzing the 100 noisy mock-data vectors. The $p$-value is 0.18. The green cure is the best-fit ${\chi }^2$ distribution with effective degrees of freedom of 134.

Figure 12

The 68% C.I. of the 1D projected posterior on each of the parameters ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, $S_8$, $\mathrm{\Delta }{z}_3$, and $\mathrm{\Delta }{z}_4$ for different analysis setups. In total we have four groups divided by horizontal dashed lines: The first group is the fiducial setup; the second group is for different physical models; the third group is for different systematic models; the fourth group is for different samplers. The results with “$*$” (“$†$”) are sampled with PolyChord (emcee) and the others are sampled with MultiNest.

Figure 13

The marginalized 2D posteriors of analyses with the TATT model (fiducial), the NLA model (NLA) and without modeling of IA (no IA).

Figure 14

The contribution from IA (including both the GG and GI terms), based on resampling from our fiducial posterior, to our 2PCFs. The IA signal is shown in red, where the solid lines are the mean, and the shaded reg regions are the 95% confidence intervals. The blue points are the measured 2PCFs and the black lines are the model prediction with the MAP, which are the same as in Fig. 3.

Figure 15

The marginalized 2D posteriors of the flat $\mathrm{\Lambda }\mathrm{CDM}$ cosmology with fixed neutrino mass (fiducial; $m_{\nu }=0.06$) and free neutrino mass; $m_{\nu }\in [0.06,0.6]$.

Figure 16

The evaluation of statistical significance of the shifts in $\mathrm{\Delta }{z}_{3,4}$. Blue points show the $\mathrm{\Delta }{z}_{3,4}$ estimations by conducting our fiducial analysis on noisy mock 2PCFs. The red point is the estimation from our fiducial analysis on HSC-Y3 real data.

Figure 17

The 68% C.I.s of 1D projected modes on the parameters ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, $S_8$, $\mathrm{\Delta }{z}_3$ and $\mathrm{\Delta }{z}_4$ for different splits of the HSC-Y3 data. We have four groups divided by horizontal dashed lines. The first group is the fiducial setup; the second group examines different subfields, the third group is for removal of different redshift bins, and the fourth group is for different scale cuts. The fiducial results (marked with “$*$”) are sampled with PolyChord and the others are sampled with MultiNest. The $\mathrm{\Delta }{z}_3$ and $\mathrm{\Delta }{z}_4$ for “no $z_3$” and “no $z_4$” are missing, respectively, since the redshift bins are removed from the analysis.

Figure 18

The 2D posterior in the (${\mathrm{\Omega }}_{\mathrm{m}},{\sigma }_8$) plane for the fiducial analysis sampled with PolyChord, “no $z_3$” analysis sampled with both PolyChord (with ${}^{*}$) and MultiNest (without ${}^{*}$). The MultiNest sampler significantly underestimates the error on the parameters, particularly ${\mathrm{\Omega }}_{\mathrm{m}}$.

Figure 19

The evaluation of the statistical significance of the shift in $S_8$ for the test with large scale cut using 50 mock 2PCFs. The reference distribution (blue histogram) is obtained by analyzing the noisy mocks with the fiducial scale cut, and keeping large scales only. The red vertical line is the estimation of $\mathrm{\Delta }{S}_8$ from real data. The probability of finding $\mathrm{\Delta }{S}_8$ larger than the real analysis is 13%.

Figure 20

Similar to Figs. 12 and 17. The 68% C.I. of the 1D projected posterior on each of the parameters ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, $S_8$, $\mathrm{\Delta }{z}_3$ and $\mathrm{\Delta }{z}_4$ for the fiducial constraint (first row) and the constraint with $E$-mode-only 2PCFs (second row) after removing the $B$-mode signal in Fig. 5.

Figure 21

Comparison between our fiducial cosmology constraint and contemporary weak-lensing observations (i.e., KiDS-1000 and DES-Y3) and Planck-2018 CMB observation. The posterior data are plotted as published by each collaboration. These analyses have slightly different priors and astrophysical and systematic models.

Figure 22

Comparison between our fiducial constraint (blue contours) and the constraint on cosmology from the eBOSS BAO measurement (yellow contours), which is analyzed with the same priors in Table 2. The red contours are the joint estimation between the two observations.

Figure 23

This figure shows the normalized 1D marginalized posterior of our fiducial constraints on four parameters. The red lines are before the corrections for the boundary effect and multiplicative bias due to the smoothing in chainconsumer, and the blue lines are after the bias corrections. This illustrates how the posteriors for prior-dominated nuisance parameters are noticeably modified at the boundaries, whereas those for the cosmological parameters simply change by $\sim 10\%$ in width while remaining centered at the same parameter values.

Figure 24

Marginalized posteriors of the fiducial analysis for the various parameters in our analysis, including the cosmological (${\mathrm{\Omega }}_{\mathrm{m}}$, $S_8$, $A_s$, ${\sigma }_8$, $n_s$, $h_0$, ${\omega }_{\mathrm{b}}$), the parameter from hmcode 2016 encoding baryonic feedback ($A_{\mathrm{b}}$), and the TATT intrinsic alignment ($A_1$, $A_2$, ${\eta }_1$, ${\eta }_2$, $b_{\mathrm{ta}}$) parameters.

Figure 25

The marginalized 2D posteriors analyzed with different flat priors on $A_s$ (fiducial), $S_8$ and $\ln (A_s)$ sampled with MultiNest.

Figure 26

The marginalized 2D posteriors analyzed with different flat priors on $A_s$ (fiducial), $S_8$ and $\ln (A_s)$ sampled with PolyChord.

Figure 27

The marginalized 2D posteriors analyzed with different models for cold matter power spectrum.

Figure 28

The marginalized 2D posteriors analyzed with different systematic models.

Figure 29

The marginalized 2D posteriors in the (${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$) plane for six different subfields.

Figure 30

The marginalized 2D posteriors in the (${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$) plane with six different cuts on angular scales.

Figure 31

The marginalized 2D posteriors in the (${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$) plane when removing one of the redshift bins.

Figure 32

The marginalized 2D posteriors in the (${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$) plane for different samplers; MultiNest, PolyChord, and emcee.