The mass-richness relation of optically-selected clusters from weak
  gravitational lensing and abundance with Subaru HSC first-year data
  Constraining the relation between the richness $N$ and the halo mass $M$ over
a wide redshift range for optically-selected clusters is a key ingredient for
cluster-related science in optical surveys, including the Subaru Hyper
Suprime-Cam (HSC) survey. We measure stacked weak lensing profiles around 1747
HSC CAMIRA clusters over a redshift range of $0.1\leq z_{\rm cl}\leq 1.0$ with
$N\geq 15$ using the HSC first-year shear catalog covering $\sim$$140$ ${\rm
deg^2}$. The exquisite depth and image quality of the HSC survey allow us to
measure lensing signals around the high-redshift clusters at $0.7\leq z_{\rm
cl}\leq 1.0$ with a signal-to-noise ratio of 19 in the comoving radius range
$0.5\lesssim R\lesssim 15 h^{-1}{\rm Mpc}$. We constrain richness-mass
relations $P(\ln N|M,z)$ of the HSC CAMIRA clusters assuming a log-normal
distribution without informative priors on model parameters, by jointly fitting
to the lensing profiles and abundance measurements under both Planck and WMAP
cosmological models. We show that our model gives acceptable $p$-values when we
add redshift dependent terms which are proportional to $\ln (1+z)$ and $[\ln
(1+z)]^{2}$ into the mean and scatter relations of $P(\ln N|M,z)$. Such terms
presumably originate from the variation of photometric redshift errors as a
function of the redshift. We show that the constraints on the mean relation
$\langle M|N \rangle$ are consistent between the Planck and WMAP models,
whereas the scatter values $\sigma_{\ln M|N}$ for the Planck model are
systematically larger than those for the WMAP model. We also show that the
scatter values for the Planck model increase toward lower richness values,
whereas those for the WMAP model are consistent with constant values as a
function of richness. This result highlights the importance of the scatter in
the mass-richness relation for cluster cosmology.
