Peaks in weak lensing mass maps for cluster astrophysics and cosmology

Abstract

Clusters of galaxies can be identified from the peaks in weak lensing aperture mass maps constructed from weak lensing shear catalogs. Such purely gravitational cluster selection differs considerably from traditional cluster selection based on the baryonic properties of clusters. In this review, we present the basics and applications of weak lensing shear-selected cluster samples. Detailed studies of the baryonic properties of shear-selected clusters shed new light on cluster astrophysics. The purely gravitational selection indicates that the selection function can be quantified more easily and robustly, which is crucial for deriving accurate cosmological constraints from the abundance of shear-selected clusters. Recent advances in shear-selected cluster studies are driven by the Subaru Hyper Suprime-Cam survey, in which more than 300 shear-selected clusters with a signal-to-noise ratio ＞ 5 were identified. It is argued that various systematic effects in cosmological analysis can be mitigated by carefully selecting the setup of the analysis, including the selection of kernel functions and the source galaxy sample.

1. Introduction

The standard Λ-dominated Cold Dark Matter (ΛCDM) cosmological model requires the presence of two unknown components: dark matter and dark energy. The nature of dark matter and dark energy are central problems in modern cosmology. It is expected that a clue to their nature can be obtained by detailed studies of the structure of the Universe because information on both the expansion history of the Universe and the nature of dark matter is imprinted in the matter distribution of the Universe.¹⁾ Because the distribution of matter is dominated by dark matter that cannot be directly observed, we must resort to indirect methods to study its distribution.

Weak gravitational lensing takes advantage of small distortions (“shears”) in distant galaxies due to the deflection of light rays by intervening gravitational fields and provides a powerful means of studying the distribution of dark matter.²⁾ The gravitational lensing effect can be robustly predicted by general relativity and allows us to map the total mass distribution directly, including dark matter.

Clusters of galaxies provide other means of studying the matter distribution in the Universe.³⁾ Clusters are the most massive gravitationally bound objects in the Universe, and their abundance and internal structure are determined mainly by the gravitational dynamics of dark matter. Clusters of galaxies can be easily and securely identified in several different observations, including optical imaging and spectroscopy to identify member galaxies and observations of X-rays and the Sunyaev-Zel’dovich effect originating from hot gas in clusters of galaxies.⁴⁾

Additionally, clusters of galaxies can be identified directly from weak gravitational lensing data.^5),6) The small distortions of the galaxies can be inverted to the projected mass distribution (“convergence”), which is referred to as a mass map, as convergence and shear are calculated by second derivatives of the lens potential. Clusters of galaxies are identified as high signal-to-noise ratio peaks in weak lensing mass maps. Such cluster identification represents a purely gravitational selection that is insensitive to complicated baryon physics. This is a clear advantage over cluster selection in optical, X-ray, and the Sunyaev-Zel’dovich effect.

In this paper, we review clusters of galaxies selected from mass map peaks, which are sometimes referred to as shear-selected clusters. As detailed below, constructing a large sample of shear-selected clusters require deep and wide imaging surveys, which have been enabled only recently. We describe how these shear-selected clusters advance our understanding of clusters of galaxies, and how they can be used to place tight constraints on cosmological parameters. We note that, for constraining cosmological parameters, mass map peaks without associations with clusters of galaxies are sometimes sufficient observables that can be predicted from cosmological models. In this review, we focus on mass map peaks as proxies for galaxy clusters.

2. Basics of gravitational lensing

2.1. Lens equation.

Under the geometric optics approximation, the propagation of light is determined by the geodesic equation

  

\[\frac{d^2x^\mu}{d\lambda^2} +\Gamma^{\mu}{}_{\alpha\beta}\frac{dx^\alpha}{d\lambda}\frac{dx^\beta}{d\lambda} =0, \label{eq:geodesic_def_geoeq}\][1]

where λ is an affine parameter and Γ^μ_αβ are the Christoffel symbols. The geodesic equation can often be rewritten as

  

\[\frac{d}{d\lambda}\left(g_{\mu\nu}\frac{dx^\nu}{d\lambda}\right) -\frac{1}{2}g_{\alpha\beta,\mu}\frac{dx^\alpha}{d\lambda}\frac{dx^\beta}{d\lambda}=0, \label{eq:geodesic_def_geoeq_ano}\][2]

where g_αβ is a metric tensor, and the comma represents a partial derivative.

In an expanding Universe with inhomogeneous density fluctuations, the line element is given by

  

\begin{aligned} ds^2 &=-\left(1+\frac{2\Phi}{c^2}\right)c^2dt^2+a^2 \left(1-\frac{2\Psi}{c^2}\right)\nonumber\\ &\quad\times\left[d\chi^2+f_K^2(\chi)\omega_{ab}dx^adx^b\right], \label{eq:lenseq_metric_newtonian}\end{aligned}[3]

where a is the scale factor, Φ is the gravitational potential, Ψ is the curvature perturbation,

  

\[f_K(\chi) = \begin{cases} {\displaystyle \frac{1}{\sqrt{K}}}\sin\left(\sqrt{K}\chi\right), & (K>0)\\ \chi, & (K=0)\\ {\displaystyle \frac{1}{\sqrt{-K}}}\sinh\left(\sqrt{-K}\chi\right), & (K<0) \end{cases} \label{eq:geodesic_def_fk}\][4]

where K is the spatial curvature of the Universe, and

  

\[\omega_{ab}dx^adx^b = d\theta^2+\sin^2\theta d\phi^2. \label{eq:lenseq_def_metric_omega}\][5]

By plugging the metric tensor defined by Eq. [3] in Eq. [2] and computing the angular part (μ ＝ a) of the equation, we obtain

  

\[\frac{d}{d\chi}\left[f_K^2(\chi)\frac{dx^a}{d\chi}\right] +\frac{1}{c^2}\omega^{ab}\left(\Phi_{,b}+\Psi_{,b}\right)=0. \label{eq:lenseq_lenseq_diff_pre}\][6]

Since we have Φ ＝ Ψ from the Einstein equations without anisotropic stress, we always set Φ ＝ Ψ and rewrite Eq. [6] as

  

\[\frac{d}{d\chi}\left[f_K^2(\chi)\frac{dx^a}{d\chi}\right] +\frac{2}{c^2}\omega^{ab}\Phi_{,b}=0. \label{eq:lenseq_lenseq_diff}\][7]

Equation [6] can also be obtained by Fermat’s principle by considering the variation of the conformal time dη ＝ dt/a integrated along the line-of-sight and using the Euler-Lagrange equation.

The lens equation for a source at χ ＝ χ_s is obtained by integrating Eq. [7] twice

  

\[x^a(\chi_{\mathrm{s}})-x^a(0)= -\frac{2}{c^2}\int_0^{\chi_{\mathrm{s}}}d\chi \frac{f_K(\chi_{\mathrm{s}}-\chi)}{f_K(\chi)f_K(\chi_{\mathrm{s}})} \omega^{ab}\Phi_{,b}. \label{eq:lenseq_lenseq1}\][8]

The right-hand side of Eq. [8] represents the deflection angle. In cosmological gravitational lensing, the deflection angle is always very small compared with the radian, so a locally flat sky coordinate is often adopted for analysis. In addition, in weak gravitational lensing analysis, the Born approximation, in which the integration in Eq. [8] is evaluated along a straight line, is commonly adopted. With these approximations, Eq. [8] is recast into

  

\[\boldsymbol{\beta} =\boldsymbol{\theta}-\nabla_{\boldsymbol{\theta}}\psi, \label{eq:lenseq_lens_equation_4}\][9]

where β (＝ x^a(χ_s)) is the source position on the sky, θ (＝ x^a(0)) is the image position on the sky, ∇_θ is the gradient vector with respect to the image position θ, and Ψ is the lens potential defined by

  

\[\psi(\boldsymbol{\theta})=\frac{2}{c^2}\int_0^{\chi_{\mathrm{s}}}d\chi \frac{f_K(\chi_{\mathrm{s}}-\chi)}{f_K(\chi)f_K(\chi_{\mathrm{s}})} \Phi(\chi, \boldsymbol{\theta}). \label{eq:lenseq_lens_potential_1}\][10]

Equation [10] indicates that the lens potential is given by the gravitational potential integrated along the line-of-sight with a weight.

2.2. Convergence and shear.

Equation [9] suggests that the shape of the image δθ is related to that of the source δθ as

  

\[\delta\boldsymbol{\beta}=A(\boldsymbol{\theta})\delta\boldsymbol{\theta}, \label{eq:lensprop_image_deform}\][11]

where the Jacobi matrix A(θ) is expressed by the second derivatives of the lens potential in the image plane θ ＝ (θ₁, θ₂) and is given by

  

\[A(\boldsymbol{\theta}) = \begin{pmatrix} 1-\psi_{,\theta_1\theta_1} & -\psi_{,\theta_1\theta_2} \\ -\psi_{,\theta_1\theta_2} & 1-\psi_{,\theta_2\theta_2} \\ \end{pmatrix}, \label{eq:lensprop_jacobi_matrix2}\][12]

which is rewritten as follows:

  

\[A(\boldsymbol{\theta})= \begin{pmatrix} 1-\kappa-\gamma_1 & -\gamma_2 \\ -\gamma_2 & 1-\kappa+\gamma_1 \\ \end{pmatrix}, \label{eq:lensprop_jacobi_matrix3}\][13]

where \(\kappa=(\psi_{,\theta_1\theta_1}+\psi_{,\theta_2\theta_2})/2\) is convergence and \(\gamma_1=(\psi_{,\theta_1\theta_1}-\psi_{,\theta_2\theta_2})/2\) and \(\gamma_2=\psi_{,\theta_1\theta_2}\) are shear. From Eq. [10], it is shown that convergence can be connected with the matter density fluctuation \(\delta_{\mathrm{m}}=(\rho_{\mathrm{m}}-\bar{\rho}_{\mathrm{m}})/\bar{\rho}_{\mathrm{m}}\) as

  

\begin{aligned} \kappa(\boldsymbol{\theta}) &=\frac{4\pi G }{c^2}\int_0^{\chi_{\mathrm{s}}}d\chi \frac{f_K(\chi_{\mathrm{s}}-\chi)f_K(\chi)}{f_K(\chi_{\mathrm{s}})}\nonumber\\ &\quad\times\bar{\rho}_{\mathrm{m}}a^2\delta_{\mathrm{m}}(\chi, \boldsymbol{\theta}), \label{eq:lensprop_kappa_born_calc2}\end{aligned}[14]

using the Poisson’s equation for the gravitational potential. This indicates that convergence is essentially the matter density field projected along the line-of-sight with weight.

2.3. Mass map reconstruction.

In weak gravitational lensing, the average shear in some regions of the sky is estimated by averaging the shapes of the galaxies in the region, assuming that the intrinsic orientations of galaxies are random. Once the complex shear field γ(θ) ＝ γ₁(θ) ＋ iγ₂(θ) is constructed from the average shapes of galaxies, we can convert γ(θ) to κ(θ) as

  

\[\kappa(\boldsymbol{\theta})=\frac{1}{\pi}\int d\boldsymbol{\theta}' \gamma(\boldsymbol{\theta}') D^*(\boldsymbol{\theta}-\boldsymbol{\theta}'), \label{eq:wl_kaiser_squires}\][15]

  

\[D(\boldsymbol{\theta})= \frac{\theta_2^2-\theta_1^2-2i\theta_1\theta_2}{\left|\boldsymbol{\theta}\right|^4},\][16]

because convergence and shear are described by the second derivatives of the same lens potential.

However, in practice, the reconstructed convergence map is heavily affected by shot noise arising from the fact that we use a discrete galaxy sample to estimate the shear field. To suppress the shot noise, we first smooth the shear field using the Gaussian kernel:

  

\[W_{\mathrm{s}}(\boldsymbol{\theta})=\frac{1}{\pi\sigma_{\mathrm{s}}^2}\exp\left(-\frac{\left|\boldsymbol{\theta}\right|^2}{\sigma_{\mathrm{s}}^2}\right), \label{eq:wl_kernel_ws}\][17]

to obtain the smoothed shear field as

  

\[\gamma_{\mathrm{s}}(\boldsymbol{\theta})=\int d\boldsymbol{\theta}' \gamma(\boldsymbol{\theta}') W_{\mathrm{s}}(\boldsymbol{\theta}-\boldsymbol{\theta}'), \label{eq:wl_gamma_s}\][18]

and the smoothed shear field is used to reconstruct the convergence field. The resulting convergence field is a smoothed convergence field with the same kernel function, Eq. [17]. This method is called the Kaiser-Squires method.⁵⁾

It is known that Eqs. [15] and [18] are also described using the tangential shear⁶⁾

  

\[\gamma_+(\boldsymbol{\theta}';\,\boldsymbol{\theta}) = -\mathrm{Re}\left[\gamma(\boldsymbol{\theta}')\, e^{-2i\varphi_{\boldsymbol{\theta}'-\boldsymbol{\theta}}}\right],\][19]

where φ_θ′－θ is the polar angle of θ′－θ. The smoothed convergence field κ_s(θ) is given by

  

\begin{aligned} \kappa_{\mathrm{s}}(\boldsymbol{\theta})&=\int d\boldsymbol{\theta}' \kappa(\boldsymbol{\theta}')W_{\mathrm{s}}(\boldsymbol{\theta}-\boldsymbol{\theta}')\nonumber\\ &=\int d\boldsymbol{\theta}' \gamma_+(\boldsymbol{\theta}';\,\boldsymbol{\theta}) Q_{\mathrm{s}}(|\boldsymbol{\theta}-\boldsymbol{\theta}'|), \label{eq:def_kappa_s}\end{aligned}[20]

where

  

\[Q_{\mathrm{s}}(\theta)=\frac{1}{\pi\theta^2}\left[1-\left(1+\frac{\theta^2}{\sigma_{\mathrm{s}}^2}\right)\exp\left(-\frac{\theta^2}{\sigma_{\mathrm{s}}^2}\right)\right]. \label{eq:qs_gauss}\][21]

2.4. Aperture mass map.

We can generalize Eq. [20] to consider the aperture mass map,^6),7) which is the convergence field convolved with a kernel U(θ). Specifically, the aperture mass map M_ap(θ) is defined as follows:

  

\[M_{\mathrm{ap}}(\boldsymbol{\theta})=\int d\boldsymbol{\theta}' \kappa(\boldsymbol{\theta}')U(|\boldsymbol{\theta}-\boldsymbol{\theta}'|). \label{eq:map_def}\][22]

Assuming that the filter U(θ) is a compensated filter

  

\[\int_0^\infty d\theta\,\theta\,U(\theta)=0, \label{eq:u_comp}\][23]

Eq. [22] can also be described using the tangential shear as follows:

  

\[M_{\mathrm{ap}}(\boldsymbol{\theta})=\int d\boldsymbol{\theta}' \gamma_+(\boldsymbol{\theta}';\,\boldsymbol{\theta}) Q(|\boldsymbol{\theta}-\boldsymbol{\theta}'|), \label{eq:map_map_q}\][24]

where Q(θ) is related with U(θ) as

  

\[Q(\theta)=\frac{2}{\theta^2}\int_0^\theta d\theta'\theta'U(\theta')-U(\theta). \label{eq:from_u_to_q}\][25]

Inversely, the filter U(θ) can be obtained from Q(θ) as

  

\[U(\theta)=\int_\theta^\infty d\theta'\frac{2}{\theta'}Q(\theta')-Q(\theta).\][26]

As discussed later, by carefully selecting the kernel functions U(θ) and Q(θ) we can mitigate various systematic effects.

The locally flat sky was assumed for these calculations. The flay-sky approximation is valid when the area of the mass map is not large or less than hundreds of square degrees. The extension to the curved sky is possible by using e.g., spin-weighted spherical harmonics.⁸⁾

3. Searching for clusters of galaxies from mass map peaks

3.1. Construction of mass map.

Clusters of galaxies are quite efficient in producing weak lensing signals (see e.g. Ref. 9 for a review), and they appear as high signal-to-noise ratio peaks in weak lensing aperture mass maps, depending on the filter.^10)-14) In practice, we reconstruct the aperture mass map from a discrete-source galaxy sample. The estimator of Eq. [24] is given by

  

\[\hat{M}_{\mathrm{ap}}(\boldsymbol{\theta})=\frac{1}{\bar{n}}\sum_j \gamma_+(\boldsymbol{\theta}_j;\boldsymbol{\theta}) Q(|\boldsymbol{\theta}-\boldsymbol{\theta}_j|), \label{eq:map_est1}\][27]

where j runs over the galaxies in the source galaxy sample and \(\bar{n}\) is the average surface number density of the source galaxies.

Because the precision of shape measurements of individual galaxies varies depending on the size and magnitude of the galaxies, a weight is often introduced to optimize the weak lensing signal. In the presence of weight w_j in the j-th galaxy, Eq. [27] is modified as

  

\[\hat{M}_{\mathrm{ap}}(\boldsymbol{\theta})=\frac{1}{\bar{w}\bar{n}}\sum_j w_j\gamma_+(\boldsymbol{\theta}_j;\boldsymbol{\theta}) Q(|\boldsymbol{\theta}-\boldsymbol{\theta}_j|), \label{eq:map_est2}\][28]

where \(\bar{w}\) denotes the average weight value.

3.2. Signal-to-noise ratio.

The main source of noise in aperture mass maps is the intrinsic shapes of galaxies. By averaging the intrinsic shapes of N galaxies, we can reduce the noise of the estimated average shear by \(1/\sqrt{N}\) , suggesting that the number density of galaxies used for weak lensing determines the quality of the resulting mass map. Due to the consequence of the central limit theorem, intrinsic shape noise translates into random Gaussian noise of the weak lensing mass map, whose statistical property is well studied,¹⁵⁾ to a good approximation.¹⁶⁾ The significance of the aperture mass map peak at θ_peak is usually quantified by the signal-to-noise ratio

  

\[\nu=\frac{M_\mathrm{ap}(\boldsymbol{\theta}_{\mathrm{peak}})}{\sigma}, \label{eq:def_nu}\][29]

where σ is the 1σ noise of the aperture mass map. Suppose \(\sigma_e=\sqrt{\sigma_{e_1}^2+\sigma_{e_2}^2}\) is the intrinsic shape noise of the source galaxies, the variance of the tangential shear of each galaxy is dominated by the intrinsic shape noise and is given by

  

\[\langle \left\{\gamma_+(\boldsymbol{\theta}_j;\boldsymbol{\theta})\right\}^2\rangle = \frac{\sigma^2_e}{2}.\][30]

Thus, from e.g., Eq [28], the 1σ noise of the aperture mass map σ is computed as

  

\[\sigma^2=\frac{1}{2\bar{w}^2\bar{n}^2}\sum_j w_j^2\sigma_e^2 Q^2(|\boldsymbol{\theta}-\boldsymbol{\theta}_j|). \label{eq:map_sigma}\][31]

We note that the normalization of the aperture mass map \(\hat{M}_{\mathrm{ap}}\) or the kernel function Q(θ) is not important as long as we use the signal-to-noise ratio ν defined by Eq. [29] for selecting cluster candidates, as such normalization cancels out when calculating ν.

The construction of the mass map and the calculation of the signal-to-noise ratio assume a uniform distribution of source galaxies on the sky. However, in practice, the number density of resolved galaxies on the sky is quite inhomogeneous due to e.g., the variation of observing conditions such as seeing sizes. In addition, some parts of the sky are masked due to e.g., the presence of bright foreground stars. As evident from Eqs. [27] and [28], the aperture mass map value at any point on the sky is given by the shapes of the surrounding source galaxies and hence is significantly affected by the boundary effect.

One way to account for such inhomogeneity and masking effects in measurements is to estimate the noise σ as well as the average number density \(\bar{n}\) and weight \(\bar{w}\) locally as a function of the sky position θ. For example, the local value of \(\bar{w}\bar{n}\) at θ can be estimated as

  

\[\bar{w}\bar{n}\propto\sum_j w_jQ(|\boldsymbol{\theta}-\boldsymbol{\theta}_j|). \label{eq:map_wn}\][32]

Furthermore, a simple and powerful way to derive the noise σ locally at each position on the sky is to randomize the orientations of individual source galaxies and construct the aperture mass map, and repeat this many times to estimate σ from the variance of randomized mass map values.^17),18) There are advantages and disadvantages to such local estimations of the average density and noise,^19),20) which depend also on the choice of the kernel functions discussed below. It is worth emphasizing that such inhomogeneity must be properly considered in cosmological inference, which is one of the main challenges in using mass map peaks or shear-selected clusters to constrain cosmological parameters.

3.3. Choice of the kernel functions.

The selection of the kernel functions U(θ) and Q(θ) is important for efficient searching of clusters of galaxies from the peaks in the aperture mass maps. The signal-to-noise ratio was optimized by selecting a matched filter for which Q(θ) follows the expected tangential shear profile γ＋ of the lensing cluster. Since it is known that the density profile of clusters of galaxies is well approximated by the Navarro-Frenk-White (NFW) density profile,²¹⁾ from theoretical predictions assuming ΛCDM as well as from weak lensing observations,^22),23) the functional form of Q(θ) that resembles the tangential shear profile predicted by the NFW profile has been considered.^7),24)-27)

In addition, it is important to reduce contamination from large-scale structures (LSS). Matter fluctuations in front of and behind halos of interest can sometimes enhance the signal-to-noise significantly, and such LSS is thought to be another important source of noise when we construct a sample of massive clusters of galaxies from mass map peaks. In particular, aperture mass maps constructed with the simple Gaussian filter given by Eq. [21] are known to be significantly affected by LSS noise and thus are not ideal. To reduce LSS noise, one has to choose kernel functions that suppress density fluctuations on scales much larger than typical sizes of clusters of galaxies.^27),28)

Weak lensing signals near the center of clusters are subject to various systematic uncertainties, including cluster member dilution effect,²⁹⁾ intrinsic alignments of cluster member galaxies,^30),31) the non-linearity of shear,³²⁾ and the obscuration by cluster member galaxies.³³⁾ In this regard, choosing Q(θ) that is suppressed at small θ corresponding to the central region of clusters is advantageous because signals are less affected by these systematics effects.¹⁸⁾

Finally, it is also useful to consider kernel functions that are confined within a small finite radius because they are less affected by the boundary effect.⁶⁾

3.4. Examples of aperture mass maps.

Here, we present examples of aperture mass maps constructed using different kernel functions. In addition to the Gaussian smoothing kernel given by Eq. [21], we consider the power law with the outer exponential-cutoff³⁴⁾

  

\[Q(\theta) ={(\theta/\theta_{\mathrm{in}})^n \over {\theta_{\mathrm{in}}^2(1+a \theta /\theta_{\mathrm{in}})^{(2+n)}}} \exp\left(-{{\theta^2}\over {2 \theta_{\mathrm{out}}^2}}\right), \label{eq:def_q_pex}\][33]

with (n, a) ＝ (0, 0.25) and (1, 0.7) being named as PEX0 and PEX1, respectively. PEX0 mimics the matched filter for the NFW density profile proposed,²⁷⁾ whereas PEX1 approximates the kernel function that reduces the contribution of lensing signals from the inner part of clusters.²⁶⁾ For PEX0 and PEX1, we adopt an inner scale radius of θ_in ＝ 1.5′ and an outer scale radius of θ_out ＝ 7′.

In addition, we consider truncated isothermal kernel functions,¹⁸⁾ which adopts the following functional form⁶⁾

  

\[U(\theta)= \left\{ \begin{array}{ll} 1 & (\theta \leq \nu_1\theta_R),\\ U_1(\theta) & (\nu_1\theta_R \leq \theta \leq \nu_2\theta_R),\\ U_2(\theta) & (\nu_2\theta_R \leq \theta \leq \theta_R),\\ 0 & (\theta_R \leq \theta), \end{array} \right. \label{eq:def_u_ti}\][34]

  

\[U_1(\theta)= \frac{1}{1-c} \left( \frac{\nu_1\theta_R}{\sqrt{(\theta-\nu_1\theta_R)^2+(\nu_1\theta_R)^2}}-c\right),\][35]

  

\[U_2(\theta)= \frac{b}{\theta_R^3}(\theta_R-\theta)^2(\theta-\alpha\theta_R),\][36]

with c, b, α being determined from the requirement that U(θ) and its first derivative are continuous at θ ＝ ν₂θ_R as well as the condition of the compensation filter given by Eq. [23]. The parameters ν₁, ν₂, and θ_R are carefully chosen to maximize the signal-to-noise ratio. Specifically, for the TI05 setup ν₁ ＝ 0.027, ν₂ ＝ 0.36, and θ_R ＝ 18.5′ are adopted, and for the TI20 setup, ν₁ ＝ 0.121, ν₂ ＝ 0.36, and θ_R ＝ 16.6′ are adopted. We use Eq. [25] to convert U(θ) defined in Eq. [34] to Q(θ).

In Fig. 1, we compare the kernel functions U(θ) and Q(θ). As shown in Eq. [23], the kernel function U(θ) is a compensated filter, which is important to mitigate the contamination due to LSS, except for the case of the Gaussian filter. The peak of U(θ) is located at a few arcminutes, which corresponds to the typical angular size of massive clusters of galaxies on the sky. The corresponding kernel functions Q(θ) indicate that the tangential shear at θ ～ 3′－10′ typically contributes to the signal.

Fig. 1

The kernel functions U(θ) and Q(θ). See the text for the definitions of individual kernel functions.

In Fig. 2, we show aperture mass maps obtained using three different filters: Gaussian, PEX0, and TI20. The mass map for the Gaussian filter retained large- and small-scale structures. In contrast, in mass maps with PEX0 and TI20, the large-scale structure is removed by the compensated filter. For the case of TI20, the small-scale structure is removed because of the large inner boundary of Q(θ). The TI20 setup is used to minimize various systematic effects that are important near the centers of massive clusters.¹⁸⁾

Fig. 2

Examples of aperture mass maps from the Hyper Suprime-Cam Subaru Strategic Program data.¹⁸⁾ The aperture mass maps constructed with the Gaussian filter (Eq. [21]), PEX0 (Eq. [33]), and TI20 (Eq. [34]) are compared. The color scale indicates the signal-to-noise ratio defined by Eq. [29].

The selection efficiency and completeness of clusters from aperture mass maps can be improved by combining the results of several different mass maps constructed with different setups. For instance, we can combine mass maps with different source redshifts^18),20),26) or different kernel sizes.³⁵⁾ By fully utilizing source redshift information, it is in principle possible to reconstruct three-dimensional mass maps to select clusters from their peaks.^36)-38)

4. Cluster samples

The first cluster discoveries from weak lensing mass map peaks were reported in the early 2000s.^17),39) Since then, weak lensing shear-selected cluster samples have been constructed in various surveys.^40)-51) Table 1 summarizes the history of the construction of cluster samples from weak lensing mass map peaks, focusing on clusters with high signal-to-noise ratios ν ≥ 5. It is found that the cluster samples have been quite small until very recently, which is explained by the fact that the construction of a large sample of shear-selected cluster samples require deep and wide imaging survey observations. On one hand, the number density of shear-selected clusters on the sky is quite sensitive to the source number density used for weak lensing analysis (see e.g., Appendix 1 of Miyazaki et al.⁵¹⁾), while on the other hand clusters are rare objects.

Table 1

Compilation of weak lensing shear-selected cluster samples

Author	Data	Area	Number of clusters (ν ≥ 5)
Miyazaki et al. (2002)17)	Subaru/Suprime-Cam52)	2.1 deg2	～ 5
Wittman et al. (2006)41)	DLS53)	8.6 deg2	～ 2
Gavazzi & Soucail (2007)42)	CFHTLS Deep54)	4 deg2	～ 2
Miyazaki et al. (2007)43)	Subaru/Suprime-Cam52)	16.7 deg2	12
Shan et al. (2012)46)	CS8255)	64 deg2	～ 17
Liu et al. (2015)49)	CFHTLenS56)	130 deg2	～ 10
Miyazaki et al. (2018)51)	HSC-SSP57)	160 deg2	47
Hamana et al. (2020)20)	HSC-SSP57)	120 deg2	124
Oguri et al. (2021)18)	HSC-SSP57)	510 deg2	325

A breakthrough was achieved by the Hyper Suprime-Cam⁵⁸⁾ mounted on the Subaru 8.2-meter telescope.⁵⁹⁾ The Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP)¹⁸⁾ is a deep multi-band imaging survey covering more than 1000 deg² at the completion of the survey. The proposed method successfully finds \(\mathcal{O}(100)\) shear-selected clusters from the first-year HSC-SSP data covering ～ 160 deg².^20),51) From the HSC-SSP Year 3 data covering ～ 510 deg², more than 300 shear-selected clusters are found,¹⁸⁾ which represent the largest sample of shear-selected clusters constructed to date.

The history of the shear-selected cluster samples is also summarized in Fig. 3. We note that all these samples were constructed from deep optical imaging surveys with ground-based telescopes. Given the typical number density of source galaxies of ～ 20－30 arcmin^－2 for deep imaging with ground-based telescopes, the typical number density of shear-selected clusters on the sky from such imaging surveys is ～ 0.3－1 deg^－2. Thus, a larger sample of shear-selected clusters can be constructed by increasing the area for deep imaging.

Fig. 3

The history of shear-selected cluster samples. See Table 1 for more details on individual samples.

5. Applications to cluster astrophysics and cosmology

5.1. Baryonic properties of shear-selected clusters.

An interesting application of the shear-selected cluster samples is to study the baryonic properties of clusters of galaxies. Traditionally, clusters of galaxies are selected based on their baryonic properties, including cluster member galaxies and the intracluster medium (ICM). The concern is that such selection based on baryonic features may miss some clusters with unique baryonic properties. For instance, it is known that X-ray luminosities are significantly affected by cluster mergers,⁶⁰⁾ suggesting that X-ray selected clusters are biased in terms of their dynamical states.⁶¹⁾ Therefore, if there is a large population of underluminous X-ray clusters, a large fraction of those clusters are missed by traditional cluster searchers based on X-ray observations. This has a considerable impact on using cluster abundances to measure cosmological parameters and is thus important. Because shear-selected clusters do not rely on such baryonic features in constructing a cluster sample and the selection function is relatively straightforward to understand,^13),62),63) they allow us to test whether such populations of X-ray underluminous clusters exist or not.⁶⁴⁾

From a systematic X-ray study of shear-selected clusters from the Subaru Suprime-Cam weak lensing survey, it has been claimed that shear-selected clusters appear to be underluminous in X-rays by a factor of 2 or so compared with the expected X-ray luminosities from the known scaling relation between cluster masses and X-ray luminosities.⁶⁵⁾ A caveat is that the so-called Eddington bias is important when interpreting the X-ray luminosity of shear-selected clusters.^66),67) Due to the shape and LSS noise, some low-mass clusters are up-scattered and are detected as weak lensing mass map peaks. By correcting for the Eddington bias, it is argued that the observed X-ray luminosities of Subaru Suprime-Cam shear-selected clusters are consistent with the expectation from the scaling relation.⁶⁶⁾

A more complete analysis of the X-ray properties of the shear-selected clusters was conducted by comparing the shear-selected clusters from the HSC-SSP with the X-ray data from the eROSITA Final Equatorial-Depth Survey.⁶⁸⁾ By comparing weak lensing masses and X-ray luminosities of all 25 shear-selected clusters in an overlapping sky region covering ～ 90 deg², it is found that the normalization of the scaling relation is consistent between shear-selected and X-ray-selected clusters, as long as the selection bias is properly corrected for. This result indicates that there is no significant population of X-ray underluminous clusters, which is good news for using X-ray-selected cluster samples as an accurate cosmological probe.

In addition to the X-ray properties, it is of great interest to carefully examine the optical properties and the Sunyaev-Zel’dovich effect properties for a large sample of shear-selected clusters.

5.2. Cosmology with shear-selected clusters.

The abundance of clusters of galaxies is known as a sensitive probe of cosmology,³⁾ yet challenges lie in how to measure cluster masses and how to model the selection function. Cosmology with shear-selected clusters can potentially overcome these challenges because the cluster selection is purely based on gravitational effects, including those of dark matter. Bearing this advantage in mind, theoretical studies have been conducted to check their constraining power for cosmology^69)-79) and to develop analytic models to predict the abundance of shear-selected clusters for each cosmological model.^80)-87) It is found that the weak lensing signals of high ν peaks are indeed dominated by single massive halos corresponding to clusters of galaxies. However, the contribution of the LSS projected along the line-of-sight is also found to have a non-negligible impact on the abundance of weak lensing mass map peaks at high ν, which needs to be taken into account for cosmological analyses. An alternative approach is to predict the abundance of mass map peaks directly from ray-tracing simulations, for which the association of mass map peaks with clusters of galaxies is not necessarily needed.

Because of the small number of shear-selected clusters, as summarized in Table 1, cosmological constraints from the abundance of shear-selected clusters have not been very competitive until recently.^49),88)-90) With the significant advance achieved by HSC-SSP, it is now possible to tighten the constraints on cosmological parameters based on the abundance of shear-selected clusters.⁹¹⁾

However, to obtain accurate cosmological constraints, one must take proper account of various systematic effects. One such effect arises from the construction of weak lensing mass maps. The real galaxy catalogs used for weak lensing are quite inhomogeneous due to variations in observing conditions, and many regions, such as those around bright stars, are masked. The boundary and mask of the survey region, as well as the pixelation and flat sky projection, affect weak lensing signals in a complicated manner,^92)-96) and these effects need to be quantified for accurate cosmological analyses. Another important effect derived from source galaxy samples is intrinsic alignments, cluster member dilution effects, and source clustering effects.^18)-20),30) It is not clear how to model these effects in theoretical predictions of peak abundance; hence, they are potentially an important source of systematic effects in cosmological analyses. Furthermore, weak lensing signals near the centers of clusters of galaxies are subject to various systematic effects, such as the non-linearity of shear⁵¹⁾ and the modification of central density profiles of clusters due to baryonic effects.^97)-101)

Recently Chen et al.¹⁰²⁾ propose a new approach for deriving robust cosmological constraints from shear-selected cluster samples. To mitigate the various systematic effects summarized above, this approach adopts a conservative cut-off of the source galaxies to select galaxies at z ＞ 0.7, aiming at selecting galaxies located behind most of the shear-selected clusters, and chooses a kernel function that minimizes the contributions from the centers of the clusters of galaxies (specifically TI20 filter mentioned above). In addition, to take full account of the complicated uncertainties associated with the construction of weak lensing mass maps, such as the variation of observing conditions and masking effects and the resulting inhomogeneity of noise properties of mass maps, as discussed in Sec. 3.2, this approach conducts semi-analytical injection simulations for which NFW halos are injected into real weak lensing mass maps. Taking advantage of this new approach, Chiu et al.¹⁰³⁾ derive cosmological constraints from HSC-SSP Year 3 shear-selected cluster catalogs to find \(\hat{S}_8\equiv \sigma_8(\Omega_{\mathrm{m}}/0.3)^{0.25}=0.835^{+0.041}_{-0.044}\) . This study demonstrated that accurate cosmological constraints can indeed be obtained from shear-selected cluster catalogs.

5.3. Comparison with peak statistics.

A more popular approach to derive cosmological constraints from weak lensing mass maps is the so-called peak statistics for which low signal-to-noise ratio peaks are mainly used to derive cosmological constraints.^{30),104)-107)} In the peak statistics, mass map peaks are regarded as observables that can be predicted from cosmological models e.g., using ray-tracing simulations; therefore, no association of mass map peaks with astronomical objects, such as clusters of galaxies or galaxies are not assumed. We note that cosmology with shear-selected clusters has advantages over peak statistics. The fact that weak lensing signals are dominated by single massive clusters indicates that we have better control of systematics arising from e.g., the member dilution effect, intrinsic alignment, and baryonic effect by carefully choosing the setup to construct aperture mass maps. Theoretical modeling of the abundance of shear-selected clusters is more straightforward. Given the increasing importance of controlling various systematic effects in cosmological analyses, it is of great interest to carefully compare results from low ν and high ν peaks in weak lensing mass maps.

6. Conclusion

Peaks in weak lensing aperture mass maps provide a new route for identifying clusters of galaxies. This selection is based on the gravitational lensing effects of the total mass distribution, including dark matter, and is highly complementary to traditional approaches to selecting clusters of galaxies from member galaxies or ICM properties. Such shear-selected cluster samples have several important applications, including studying the baryonic properties of clusters and their cosmology.

The application of shear-selected clusters is hindered by the small number of such clusters. This is because of the high observation requirements. To construct a large sample of shear-selected clusters, it is essential to conduct deep and wide imaging surveys because the number density of shear-selected clusters is quite sensitive to the source number density of galaxies used for weak lensing analyses and because clusters are rare objects on the sky.

Recently, significant advances have been achieved by the HSC-SSP, in which more than 300 shear-selected clusters with a signal-to-noise ratio ν ≥ 5 have been discovered. It was demonstrated that accurate cosmological constraints can be derived from such cluster samples by carefully selecting kernel functions and background-source galaxy samples to mitigate various systematic effects.

In the next decade, we can expect rapid progress in increasing the number of shear-selected clusters, thanks to massive imaging surveys conducted by Euclid,¹⁰⁸⁾ the Rubin Observatory Legacy Survey of Space and Time,¹⁰⁹⁾ and the Nancy Grace Roman Space Telescope.¹¹⁰⁾

Acknowledgments

We thank the anonymous referees for their careful reading of the manuscript and constructive comments. This work was supported by JSPS KAKENHI Grant Nos. JP20H05856, JP22K21349, JP19KK0076, JP23K22531, JP24K00684.

Notes

Edited by Masanori IYE, M.J.A.

Correspondence should be addressed to: M. Oguri, Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan (e-mail: [e-mail]).

References

• 1)  Abdalla, E., Abellán, G.F., Aboubrahim, A., Agnello, A., Akarsu, Ö., Akrami, Y. et al. (2022) Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies. JHEAp 34, 49-211.
• 2)  Bartelmann, M. and Schneider, P. (2001) Weak gravitational lensing. PhR 340, 291-472.
• 3)  Allen, S.W., Evrard, A.E. and Mantz, A.B. (2011) Cosmological parameters from observations of galaxy clusters. ARA&A 49, 409-470.
• 4)  Voit, G.M. (2005) Tracing cosmic evolution with clusters of galaxies. RvMP 77, 207-258.
• 5)  Kaiser, N. and Squires, G. (1993) Mapping the dark matter with weak gravitational lensing. ApJ 404, 441.
• 6)  Schneider, P. (1996) Detection of (dark) matter concentrations via weak gravitational lensing. MNRAS 283, 837-853.
• 7)  Schneider, P., van Waerbeke, L., Jain, B. and Kruse, G. (1998) A new measure for cosmic shear. MNRAS 296, 873-892.
• 8)  Castro, P.G., Heavens, A.F. and Kitching, T.D. (2005) Weak lensing analysis in three dimensions. PhRvD 72, 023516.
• 9)  Umetsu, K. (2020) Cluster-galaxy weak lensing. A&ARv 28, 7.
• 10)  Reblinsky, K. and Bartelmann, M. (1999) Projection effects in mass-selected galaxy-cluster samples. A&A 345, 1-16.
• 11)  Kruse, G. and Schneider, P. (2000) The non-Gaussian tail of cosmic-shear statistics. MNRAS 318, 321-328.
• 12)  Jain, B. and Van Waerbeke, L. (2000) Statistics of dark matter halos from gravitational lensing. ApJL 530, L1-L4.
• 13)  Hamana, T., Takada, M. and Yoshida, N. (2004) Searching for massive clusters in weak lensing surveys. MNRAS 350, 893-913.
• 14)  Tang, J.Y. and Fan, Z.H. (2005) Effects of the complex mass distribution of dark matter halos on weak-lensing cluster surveys. ApJ 635, 60-72.
• 15)  Bardeen, J.M., Bond, J.R., Kaiser, N. and Szalay, A.S. (1986) The statistics of peaks of Gaussian random fields. ApJ 304, 15.
• 16)  van Waerbeke, L. (2000) Noise properties of gravitational lens mass reconstruction. MNRAS 313, 524-532.
• 17)  Miyazaki, S., Hamana, T., Shimasaku, K., Furusawa, H., Doi, M., Hamabe, M. et al. (2002) Searching for dark matter halos in the Suprime-Cam 2 square degree field. ApJL 580, L97-L100.
• 18)  Oguri, M., Miyazaki, S., Li, X., Luo, W., Mitsuishi, I., Miyatake, H. et al. (2021) Hundreds of weak lensing shear-selected clusters from the Hyper Suprime-Cam Subaru Strategic Program S19A data. PASJ 73, 817-829.
• 19)  Schmidt, F. and Rozo, E. (2011) Weak-lensing peak finding: Estimators, filters, and biases. ApJ 735, 119.
• 20)  Hamana, T., Shirasaki, M. and Lin, Y.-T. (2020) Weak-lensing clusters from HSC survey first-year data: Mitigating the dilution effect of foreground and cluster-member galaxies. PASJ 72, 78.
• 21)  Navarro, J.F., Frenk, C.S. and White, S.D.M. (1997) A universal density profile from hierarchical clustering. ApJ 490, 493-508.
• 22)  Umetsu, K., Broadhurst, T., Zitrin, A., Medezinski, E., Coe, D. and Postman, M. (2011) A precise cluster mass profile averaged from the highest-quality lensing data. ApJ 738, 41.
• 23)  Okabe, N., Smith, G.P., Umetsu, K., Takada, M. and Futamase, T. (2013) LoCuSS: The mass density profile of massive galaxy clusters at z = 0.2. ApJL 769, L35.
• 24)  Padmanabhan, N., Seljak, U. and Pen, U.L. (2003) Mining weak lensing surveys. NewA 8, 581-603.
• 25)  Schirmer, M., Erben, T., Schneider, P., Wolf, C. and Meisenheimer, K. (2004) GaBoDS: The Garching-Bonn Deep Survey. II. Confirmation of EIS cluster candidates by weak gravitational lensing. A&A 420, 75-78.
• 26)  Hennawi, J.F. and Spergel, D.N. (2005) Shear-selected cluster cosmology: Tomography and optimal filtering. ApJ 624, 59-79.
• 27)  Maturi, M., Meneghetti, M., Bartelmann, M., Dolag, K. and Moscardini, L. (2005) An optimal filter for the detection of galaxy clusters through weak lensing. A&A 442, 851-860.
• 28)  Marian, L., Smith, R.E., Hilbert, S. and Schneider, P. (2012) Optimized detection of shear peaks in weak lensing maps. MNRAS 423, 1711-1725.
• 29)  Medezinski, E., Oguri, M., Nishizawa, A.J., Speagle, J.S., Miyatake, H., Umetsu, K. et al. (2018) Source selection for cluster weak lensing measurements in the Hyper Suprime-Cam survey. PASJ 70, 30.
• 30)  Kacprzak, T., Kirk, D., Friedrich, O., Amara, A., Refregier, A., Marian, L. et al. (2016) Cosmology constraints from shear peak statistics in Dark Energy survey science verification data. MNRAS 463, 3653-3673.
• 31)  Zhang, T., Liu, X., Wei, C., Li, G., Luo, Y., Kang, X. et al. (2022) Effects of galaxy intrinsic alignment on weak lensing peak statistics. ApJ 940, 96
• 32)  Schneider, P. and Seitz, C. (1995) Steps towards nonlinear cluster inversion through gravitational distortions. I. Basic considerations and circular clusters. A&A 294, 411-431.
• 33)  Simet, M. and Mandelbaum, R. (2015) Background sky obscuration by cluster galaxies as a source of systematic error for weak lensing. MNRAS 449, 1259-1269.
• 34)  Hamana, T., Oguri, M., Shirasaki, M. and Sato, M. (2012) Scatter and bias in weak lensing selected clusters. MNRAS 425, 2287-2298.
• 35)  Leroy, G., Pires, S., Pratt, G.W. and Giocoli, C. (2023) Fast multi-scale galaxy cluster detection with weak lensing: Towards a mass-selected sample. A&A 678, A125.
• 36)  Leonard, A., Lanusse, F. and Starck, J.-L. (2015) Weak lensing reconstructions in 2D and 3D: Implications for cluster studies. MNRAS 449, 1146-1157.
• 37)  Li, X., Yoshida, N., Oguri, M., Ikeda, S. and Luo, W. (2021) Three-dimensional reconstruction of weak-lensing mass maps with a Sparsity Prior. I. Cluster Detection. ApJ 916, 67.
• 38)  Yang, S., Li, X. and Yoshida, N. (2023) Three-dimensional reconstruction of weak-lensing mass maps with a Sparsity Prior. II. Weighing Triaxial Cluster Halos. arXiv:2312.00309.
• 39)  Wittman, D., Tyson, J.A., Margoniner, V.E., Cohen, J.G. and Dell'Antonio, I.P. (2001) Discovery of a galaxy cluster via weak lensing. ApJL 557, L89-L92.
• 40)  Hetterscheidt, M., Erben, T., Schneider, P., Maoli, R., van Waerbeke, L. and Mellier, Y. (2005) Searching for galaxy clusters using the aperture mass statistics in 50 VLT fields. A&A 442, 43-61.
• 41)  Wittman, D., Dell'Antonio, I.P., Hughes, J.P., Margoniner, V.E., Tyson, J.A., Cohen, J.G. et al. (2006) First results on shear-selected clusters from the deep lens survey: Optical imaging, spectroscopy, and X-ray follow-up. ApJ 643, 128-143.
• 42)  Gavazzi, R. and Soucail, G. (2007) Weak lensing survey of galaxy clusters in the CFHTLS Deep. A&A 462, 459-471.
• 43)  Miyazaki, S., Hamana, T., Ellis, R.S., Kashikawa, N., Massey, R.J., Taylor, J. et al. (2007) A Subaru weak-lensing survey. I. Cluster Candidates and Spectroscopic Verification. ApJ 669, 714-728.
• 44)  Schirmer, M., Erben, T., Hetterscheidt, M. and Schneider, P. (2007) GaBoDS: The Garching-Bonn Deep Survey. IX. A sample of 158 shear-selected mass concentration candidates. A&A 462, 875-887.
• 45)  Hamana, T., Miyazaki, S., Kashikawa, N., Ellis, R.S., Massey, R.J., Refregier, A. et al. (2009) Subaru weak-lensing survey II: Multi-object spectroscopy and cluster masses. PASJ 61, 833.
• 46)  Shan, H., Kneib, J.-P., Tao, C., Fan, Z., Jauzac, M., Limousin, M. et al. (2012) Weak lensing measurement of galaxy clusters in the CFHTLS-wide survey. ApJ 748, 56.
• 47)  Kurtz, M.J., Geller, M.J., Utsumi, Y., Miyazaki, S., Dell'Antonio, I.P. and Fabricant, D.G. (2012) Testing weak-lensing maps with redshift surveys: A Subaru field. ApJ 750, 168.
• 48)  Shan, H.Y., Kneib, J.-P., Comparat, J., Jullo, E., Charbonnier, A., Erben, T. et al. (2014) Weak lensing mass map and peak statistics in Canada-France-Hawaii Telescope Stripe 82 survey. MNRAS 442, 2534-2542.
• 49)  Liu, J., Petri, A., Haiman, Z., Hui, L., Kratochvil, J.M. and May, M. (2015) Cosmology constraints from the weak lensing peak counts and the power spectrum in CFHTLenS data. PhRvD 91, 063507.
• 50)  Miyazaki, S., Oguri, M., Hamana, T., Tanaka, M., Miller, L., Utsumi, Y. et al. (2015) Properties of weak lensing clusters detected on Hyper Suprime-Cam's 2.3 deg² field. ApJ 807, 22.
• 51)  Miyazaki, S., Oguri, M., Hamana, T., Shirasaki, M., Koike, M., Komiyama, Y. et al. (2018) A large sample of shear-selected clusters from the Hyper Suprime-Cam Subaru Strategic Program S16A Wide field mass maps. PASJ 70, S27.
• 52)  Miyazaki, S., Komiyama, Y., Sekiguchi, M., Okamura, S., Doi, M., Furusawa, H. et al. (2002) Subaru prime focus camera ― Suprime-Cam. PASJ 54, 833-853.
• 53)  Wittman, D.M., Tyson, J.A., Dell'Antonio, I.P., Becker, A., Margoniner, V., Cohen, J.G. et al. (2002) Deep lens survey. SPIE 4836, 73-82.
• 54)  Ilbert, O., Arnouts, S., McCracken, H.J., Bolzonella, M., Bertin, E., Le Fèvre, O. et al. (2006) Accurate photometric redshifts for the CFHT legacy survey calibrated using the VIMOS VLT deep survey. A&A 457, 841-856.
• 55)  Moraes, B., Kneib, J.-P., Leauthaud, A., Makler, M., Van Waerbeke, L., Bundy, K. et al. (2014) The CFHT/MegaCam Stripe-82 Survey. RMxAC 44, 202-203.
• 56)  Heymans, C., Van Waerbeke, L., Miller, L., Erben, T., Hildebrandt, H., Hoekstra, H. et al. (2012) CFHTLenS: The Canada-France-Hawaii Telescope Lensing Survey. MNRAS 427, 146-166.
• 57)  Aihara, H., Arimoto, N., Armstrong, R., Arnouts, S., Bahcall, N.A., Bickerton, S. et al. (2018) The Hyper Suprime-Cam SSP Survey: Overview and survey design. PASJ 70, S4.
• 58)  Miyazaki, S., Komiyama, Y., Kawanomoto, S., Doi, Y., Furusawa, H., Hamana, T. et al. (2018) Hyper Suprime-Cam: System design and verification of image quality. PASJ 70, S1.
• 59)  Iye, M. (2021) Subaru Telescope ―History, active/adaptive optics, instruments, and scientific achievements―. Proc. Jpn. Acad. Ser. B 97, 337-370.
• 60)  Randall, S.W., Sarazin, C.L. and Ricker, P.M. (2002) The effect of merger boosts on the luminosity, temperature, and inferred mass functions of clusters of galaxies. ApJ 577, 579-594.
• 61)  Eckert, D., Molendi, S. and Paltani, S. (2011) The cool-core bias in X-ray galaxy cluster samples. I. Method and application to HIFLUGCS. A&A 526, A79.
• 62)  White, M., van Waerbeke, L. and Mackey, J. (2002) Completeness in weak-lensing searches for clusters. ApJ 575, 640-649.
• 63)  Pace, F., Maturi, M., Meneghetti, M., Bartelmann, M., Moscardini, L. and Dolag, K. (2007) Testing the reliability of weak lensing cluster detections. A&A 471, 731-742.
• 64)  Starikova, S., Jones, C., Forman, W.R., Vikhlinin, A., Kurtz, M.J., Geller, M.J. et al. (2014) Comparison of galaxy clusters selected by weak-lensing, optical spectroscopy, and X-rays in the deep lens survey F2 field. ApJ 786, 125.
• 65)  Giles, P.A., Maughan, B.J., Hamana, T., Miyazaki, S., Birkinshaw, M., Ellis, R.S. et al. (2015) The X-ray properties of weak-lensing-selected galaxy clusters. MNRAS 447, 3044-3059.
• 66)  Chen, K.-F., Oguri, M., Lin, Y.-T. and Miyazaki, S. (2020) Mass bias of weak-lensing shear-selected galaxy cluster samples. ApJ 891, 139.
• 67)  Hamana, T. (2023) An empirical method for mitigating an excess up-scattering mass bias on the weak lensing mass estimates for shear-selected cluster samples. PASJ 75, 14-29.
• 68)  Ramos-Ceja, M.E., Oguri, M., Miyazaki, S., Ghirardini, V., Chiu, I., Okabe, N. et al. (2022) The eROSITA Final Equatorial-Depth Survey (eFEDS). A complete census of X-ray properties of Subaru Hyper Suprime-Cam weak lensing shear-selected clusters in the eFEDS footprint. A&A 661, A14.
• 69)  Wang, S., Haiman, Z. and May, M. (2009) Constraining cosmology with high-convergence regions in weak lensing surveys. ApJ 691, 547-559.
• 70)  Marian, L., Smith, R.E. and Bernstein, G.M. (2009) The cosmology dependence of weak lensing cluster counts. ApJL 698, L33-L36.
• 71)  Kratochvil, J.M., Haiman, Z. and May, M. (2010) Probing cosmology with weak lensing peak counts. PhRvD 81, 043519.
• 72)  Dietrich, J.P. and Hartlap, J. (2010) Cosmology with the shear-peak statistics. MNRAS 402, 1049-1058.
• 73)  Bergé, J., Amara, A. and Réfrégier, A. (2010) Optimal capture of non-Gaussianity in weak-lensing surveys: Power spectrum, bispectrum, and halo counts. ApJ 712, 992-1002.
• 74)  Yang, X., Kratochvil, J.M., Wang, S., Lim, E.A., Haiman, Z. and May, M. (2011) Cosmological information in weak lensing peaks. PhRvD 84, 043529.
• 75)  Pires, S., Leonard, A. and Starck, J.-L. (2012) Cosmological constraints from the capture of non-Gaussianity in weak lensing data. MNRAS 423, 983-992.
• 76)  Marian, L., Smith, R.E., Hilbert, S. and Schneider, P. (2013) The cosmological information of shear peaks: Beyond the abundance. MNRAS 432, 1338-1350.
• 77)  Yuan, S., Pan, C., Liu, X., Wang, Q. and Fan, Z. (2019) Cosmological studies from tomographic weak lensing peak abundances and impacts of photo-Z errors. ApJ 884, 164.
• 78)  Davies, C.T., Cautun, M., Giblin, B., Li, B., Harnois-Déraps, J. and Cai, Y.-C. (2022) Cosmological forecasts with the clustering of weak lensing peaks. MNRAS 513, 4729-4746.
• 79)  Li, Z., Liu, X. and Fan, Z. (2023) Weak-lensing peak statistics - steepness versus height. MNRAS 520, 6382-6400.
• 80)  Maturi, M., Angrick, C., Pace, F. and Bartelmann, M. (2010) An analytic approach to number counts of weak-lensing peak detections. A&A 519, A23.
• 81)  Fan, Z., Shan, H. and Liu, J. (2010) Noisy weak-lensing convergence peak statistics near clusters of galaxies and beyond. ApJ 719, 1408-1420.
• 82)  Lin, C.-A. and Kilbinger, M. (2015) A new model to predict weak-lensing peak counts. I. Comparison with N-body simulations. A&A 576, A24.
• 83)  Lin, C.-A. and Kilbinger, M. (2015) A new model to predict weak-lensing peak counts. II. Parameter constraint strategies. A&A 583, A70.
• 84)  Lin, C.-A., Kilbinger, M. and Pires, S. (2016) A new model to predict weak-lensing peak counts. III. Filtering technique comparisons. A&A 593, A88.
• 85)  Zorrilla Matilla, J.M., Haiman, Z., Hsu, D., Gupta, A. and Petri, A. (2016) Do dark matter halos explain lensing peaks? PhRvD 94, 083506.
• 86)  Yuan, S., Liu, X., Pan, C., Wang, Q. and Fan, Z. (2018) Projection effects of large-scale structures on weak-lensing peak abundances. ApJ 857, 112.
• 87)  Wei, C., Li, G., Kang, X., Liu, X., Fan, Z., Yuan, S. et al. (2018) The correspondence between convergence peaks from weak lensing and massive dark matter haloes. MNRAS 478, 2987-2998.
• 88)  Liu, X., Pan, C., Li, R., Shan, H., Wang, Q., Fu, L. et al. (2015) Cosmological constraints from weak lensing peak statistics with Canada-France-Hawaii Telescope Stripe 82 Survey. MNRAS 450, 2888-2902.
• 89)  Hamana, T., Sakurai, J., Koike, M. and Miller, L. (2015) Cosmological constraints from Subaru weak lensing cluster counts. PASJ 67, 34.
• 90)  Shan, H., Liu, X., Hildebrandt, H., Pan, C., Martinet, N., Fan, Z. et al. (2018) KiDS-450: Cosmological constraints from weak lensing peak statistics - I. Inference from analytical prediction of high signal-to-noise ratio convergence peaks. MNRAS 474, 1116-1134.
• 91)  Liu, X., Yuan, S., Pan, C., Zhang, T., Wang, Q. and Fan, Z. (2023) Cosmological studies from HSC-SSP tomographic weak-lensing peak abundances. MNRAS 519, 594-612.
• 92)  Utsumi, Y., Miyazaki, S., Geller, M.J., Dell'Antonio, I.P., Oguri, M., Kurtz, M.J. et al. (2014) Reducing systematic error in weak lensing cluster surveys. ApJ 786, 93.
• 93)  Liu, X., Wang, Q., Pan, C. and Fan, Z. (2014) Mask effects on cosmological studies with weak-lensing peak statistics. ApJ 784, 31.
• 94)  Lin, C.-A. and Kilbinger, M. (2018) Quantifying systematics from the shear inversion on weak-lensing peak counts. A&A 614, A36.
• 95)  Pires, S., Vandenbussche, V., Kansal, V., Bender, R., Blot, L., Bonino, D. et al. (2020) Euclid: Reconstruction of weak-lensing mass maps for non-Gaussianity studies. A&A 638, A141.
• 96)  Ayçoberry, E., Ajani, V., Guinot, A., Kilbinger, M., Pettorino, V., Farrens, S. et al. (2023) UNIONS: The impact of systematic errors on weak-lensing peak counts. A&A 671, A17.
• 97)  Yang, X., Kratochvil, J.M., Huffenberger, K., Haiman, Z. and May, M. (2013) Baryon impact on weak lensing peaks and power spectrum: Low-bias statistics and self-calibration in future surveys. PhRvD 87, 023511.
• 98)  Osato, K., Shirasaki, M. and Yoshida, N. (2015) Impact of baryonic processes on weak-lensing cosmology: Power spectrum, nonlocal statistics, and parameter bias. ApJ 806, 186.
• 99)  Fong, M., Choi, M., Catlett, V., Lee, B., Peel, A., Bowyer, R. et al. (2019) The impact of baryonic physics and massive neutrinos on weak lensing peak statistics. MNRAS 488, 3340-3357.
• 100)  Coulton, W.R., Liu, J., McCarthy, I.G. and Osato, K. (2020) Weak lensing minima and peaks: Cosmological constraints and the impact of baryons. MNRAS 495, 2531-2542.
• 101)  Osato, K., Liu, J. and Haiman, Z. (2021) κTNG: effect of baryonic processes on weak lensing with IllustrisTNG simulations. MNRAS 502, 5593-5602.
• 102)  Chen, K.-F., Chiu, I.-N., Oguri, M., Lin, Y.-T., Miyatake, H., Miyazaki, S. et al. (2025) Weak-lensing shear-selected galaxy clusters from the Hyper Suprime-Cam Subaru Strategic Program: I. Cluster catalog, selection function and mass-observable relation. OJAp 8, 2.
• 103)  Chiu, I.-N., Chen, K.-F., Oguri, M., Rau, M.M., Hamana, T., Lin, Y.-T. et al. (2024) Weak-lensing shear-selected galaxy clusters from the Hyper Suprime-Cam Subaru Strategic Program: II. Cosmological Constraints from the Cluster Abundance. OJAp 7, 90.
• 104)  Harnois-Déraps, J., Martinet, N., Castro, T., Dolag, K., Giblin, B., Heymans, C. et al. (2021) Cosmic shear cosmology beyond two-point statistics: A combined peak count and correlation function analysis of DES-Y1. MNRAS 506, 1623-1650.
• 105)  Zürcher, D., Fluri, J., Sgier, R., Kacprzak, T., Gatti, M., Doux, C. et al. (2022) Dark energy survey year 3 results: Cosmology with peaks using an emulator approach. MNRAS 511, 2075-2104.
• 106)  Marques, G.A., Liu, J., Shirasaki, M., Thiele, L., Grandón, D., Huffenberger, K.M. et al. (2024) Cosmology from weak lensing peaks and minima with Subaru Hyper Suprime-Cam Survey first-year data. MNRAS 528, 4513-4527.
• 107)  Harnois-Deraps, J., Heydenreich, S., Giblin, B., Martinet, N., Troester, T., Asgari, M. et al. (2024) KiDS-1000 and DES-Y1 combined: Cosmology from peak count statistics. arXiv:2405.10312.
• 108)  Euclid Collaboration, Mellier, Y., Abdurro'uf, Acevedo Barroso, J.A., Achúcarro, A., Adamek, J. et al. (2024) Euclid. I. Overview of the Euclid mission. arXiv:2405.13491.
• 109)  Ivezić, Ž., Kahn, S.M., Tyson, J.A., Abel, B., Acosta, E., Allsman, R. et al. (2019) LSST: From science drivers to reference design and anticipated data products. ApJ 873, 111.
• 110)  Eifler, T., Miyatake, H., Krause, E., Heinrich, C., Miranda, V., Hirata, C. et al. (2021) Cosmology with the Roman Space Telescope - multiprobe strategies. MNRAS 507, 1746-1761.

Profile

Masamune Oguri was born in Ishikawa Prefecture in 1978. He graduated from the University of Tokyo in 2000 and received his Ph.D. from the University of Tokyo in 2004. After working as a post-doctoral fellow at Princeton University, Stanford University, and the National Astronomical Observatory of Japan, he moved to Kavli Institute for the Physics and Mathematics of the Universe at the University of Tokyo as a Project Assistant Professor in 2011. He then moved to the Department of Physics at the University of Tokyo as an Assistant Professor in 2013. Since 2022, he has been a Professor at the Center for Frontier Science, Chiba University. He has worked on a range of topics in theoretical and observational cosmology. He received the President's Prize of the University of Tokyo (2005), the Inoue Research Award for Young Scientists (2007), the Young Scientist Award from the Physical Society of Japan (2009), the Hayashi Chushiro Prize from the Astronomical Society of Japan (2019), and the Japan Academy Prize (2024).

Satoshi Miyazaki was born in Nagano Prefecture in 1965 and graduated from Chigusa High School, Nagoya City, Aichi Prefecture, where he was highly influenced by a physics teacher. He finished his graduate studies at the University of Tokyo as a Doctor of Science in 1993 in the field of astrophysics. He spent two years at the Institute for Astronomy, University of Hawai'i as a post-doctoral fellow supported by the JSPS. In 1996, he joined the National Astronomical Observatory of Japan. Since then, he has been working on developing optical image sensors and imagers for the Subaru Telescope. Using the cameras built by his team, he has been observing wide areas of the sky as part of observational cosmology. He received the Commendation Science and Technology Award (2015) from the Minister of Education, Culture, Sports, Science and Technology, the Hayashi Chushiro Prize in 2015, the Nishina Memorial Award in 2021, and the Japan Academy Prize in 2024.