# Cosmology with galaxy cluster weak lensing: statistical limits and
experimental design

Hao-Yi Wu Department of Physics, Boise State University, Boise, ID 83725, USA
Department of Physics, Department of Astronomy, and Center for Cosmology and
Astro-Particle Physics,
The Ohio State University, Columbus, OH 43210, USA  David H. Weinberg
Department of Physics, Department of Astronomy, and Center for Cosmology and
Astro-Particle Physics,
The Ohio State University, Columbus, OH 43210, USA  Institute for Advanced
Study, Princeton, NJ 08540, USA  Andrés N. Salcedo Department of Physics,
Department of Astronomy, and Center for Cosmology and Astro-Particle Physics,
The Ohio State University, Columbus, OH 43210, USA  Benjamin D. Wibking
Australian National University, Mount Stromlo Observatory, Cotter Road, Weston
Creek, ACT 2611 Australia

###### Abstract

We forecast constraints on the amplitude of matter clustering $\sigma_{8}(z)$
achievable with the combination of cluster weak lensing and number counts, in
current and next-generation weak lensing surveys. We advocate an approach,
analogous to galaxy–galaxy lensing, in which the observables in each redshift
bin are the mean number counts and the mean weak lensing profile of clusters
above a mass proxy threshold. The primary astrophysical nuisance parameter is
the logarithmic scatter $\sigma_{\rm lnM}$ between the mass proxy and true
mass near the threshold. For surveys similar to the Dark Energy Survey (DES),
the Roman Space Telescope High Latitude Survey (HLS), and the Rubin
Observatory Legacy Survey of Space and Time (LSST), we forecast aggregate
precision on $\sigma_{8}$ of 0.26%, 0.24%, and 0.10%, respectively, if the
mass–observable scatter is known externally to $\Delta\sigma_{\rm lnM}\leq
0.01$. These constraints would be degraded by about 20% for $\Delta\sigma_{\rm
lnM}=0.05$ in the case of DES or HLS and for $\Delta\sigma_{\rm lnM}=0.016$
for LSST. A one-month observing program with Roman Space Telescope targeting
$\sim 2500$ massive clusters could achieve a $\sim 0.5\%$ constraint on
$\sigma_{8}(z=0.7)$ on its own, or a $\sim 0.33\%$ constraint in combination
with the HLS. Realizing the constraining power of clusters requires accurate
knowledge of the mass–observable relation and stringent control of
systematics. We provide analytic approximations to our numerical results that
allow easy scaling to other survey assumptions or other methods of cluster
mass estimation.

Galaxy clusters (584), Cosmology (343), Weak gravitational lensing (1797),
Cosmological parameters (339), Sigma8 (1455)

##  1 Introduction

The masses of galaxy clusters have long been recognized as a powerful probe of
matter clustering in the low redshift universe (Evrard 1989; Peebles et al.
1989; see Allen et al. 2011 for a review). In an influential study combining
analytic approximations and numerical simulations, White et al. (1993) showed
that predicted cluster masses depend most strongly on the matter density
parameter $\Omega_{\rm M}$ and the power spectrum amplitude $\sigma_{8}$. The
parameter combination best constrained by low redshift cluster masses is
approximately $\sigma_{8}\Omega_{\rm M}^{0.5}$, and White et al. (1993) used
observationally estimated masses of clusters to conclude that either
$\Omega_{\rm M}$ was substantially lower than unity or the matter
$\sigma_{8}\approx 0.6$ was substantially lower than the corresponding value
for galaxies. Many subsequent studies have focused on the evolution of cluster
abundances, with redshift evolution providing a further sensitive probe of
cosmology (e.g., Holder et al., 2001; Haiman et al., 2001; Miller et al.,
2001; Kravtsov & Borgani, 2012). The mass function of dark matter halos in
comoving coordinates tracks the growth rate of matter fluctuations, and the
total number of clusters in a survey volume further depends on the cosmic
expansion history. Supernovae and baryon acoustic oscillations also afford
precise measurements of the cosmic expansion history, but galaxy clusters
remain one of the most powerful probes of matter clustering evolution, and
thus a valuable tool for distinguishing between dark energy and modified
gravity as explanations of cosmic acceleration (see Weinberg et al. 2013 for a
comprehensive review). In this paper, we focus on the structure growth
constraints that can be derived from cluster populations in large area weak
lensing surveys.

Many recent studies formulate the cluster cosmology problem as follows: (1)
measure the number counts of clusters as a function of an observable mass
proxy such as optical richness or X-ray luminosity; (2) use a dynamically
sensitive observable such as velocity dispersion, X-ray temperature, Sunyaev-
Zeldovich (SZ) decrement, or gravitational lensing to calibrate the
mass–observable relation; (3) combine these measurements to infer the cluster
mass function and use it to constrain $\sigma_{8}$, $\Omega_{\rm M}$, or (in
combination with external data sets) other cosmological parameters (e.g.,
Vikhlinin et al., 2009; Rozo et al., 2010; Mantz et al., 2010; Benson et al.,
2013; Planck Collaboration et al., 2014; Mantz et al., 2014; Planck
Collaboration et al., 2016; Bocquet et al., 2019; Abbott et al., 2020). The
second of these three steps is the most challenging, since one must reduce
systematic errors in the cluster mass scale to the $\sim 1\%$ level or below
to remain competitive with current and future constraints from other probes
such as cosmic shear or galaxy redshift-space distortions. Even the best X-ray
observables such as hydrostatic mass (e.g., Vikhlinin et al. 2009; Applegate
et al. 2016) or the “X-ray SZ parameter” $Y_{X}$ (Kravtsov et al., 2006), if
predicted directly from theory, have likely systematics at the few percent
level because of uncertain corrections for non-thermal pressure support and
departures from hydrostatic equilibrium.

We regard weak gravitational lensing as the most promising cluster mass probe
for precision cosmological investigations, in part because the relation
between the projected mass distribution and the weak lensing signal can be
predicted without theoretical uncertainty (provided one assumes that General
Relativity is correct), in part because weak lensing can probe the outskirts
of clusters where baryonic effects on the mass distribution are less
uncertain, and in part because the current and future generations of large
weak lensing surveys provide ideal data sets for cluster weak lensing
measurements. One can still view weak lensing as calibrating the
mass–observable relation — i.e., executing step (2) of the outline above — and
this is essentially the approach taken by the “Weighing the Giants” program
(von der Linden et al., 2014) and several other recent studies (e.g., Melchior
et al., 2017; Simet et al., 2017; McClintock et al., 2019). Alternatively, one
can view cluster weak lensing as analogous to galaxy–galaxy weak lensing
(e.g., McKay et al., 2001; Mandelbaum et al., 2006, 2013), where the basic
observables are the number counts of the cluster sample and the excess surface
density profile that governs the measured tangential shear. The constraining
power of optical cluster surveys is then determined by the accuracy of the
weak lensing measurements. Next generation optical imaging surveys will
provide unprecedented accuracy for cluster lensing studies, using either
clusters identified within the survey or clusters from extended X-ray or SZ
surveys.

In this work, instead of the more common one-point statistics perspective
(calibrating the mass–observable relation with weak lensing and inferring the
halo mass function), we adopt a two-point statistics perspective, with cluster
shear profiles as the main observable and the cluster number counts used to
break degeneracy. This perspective can be generalized to encompass other two
point statistics such as cluster auto-correlations, the cluster–galaxy cross-
correlation, and galaxy auto-correlation (e.g., Salcedo et al., 2020; Chiu et
al., 2020; To et al., 2020a, b).

We focus specifically on the ability of cluster weak lensing studies to
constrain $\sigma_{8}(z)$, the amplitude of matter fluctuations in redshift
bins, when the value of $\Omega_{\rm M}$ and the shape of the matter power
spectrum are known from external constraints. While the degeneracy between
$\sigma_{8}$ and $\Omega_{\rm M}$ is likely to remain important in
observational analyses, the ability to break this degeneracy depends strongly
on one’s assumptions about external data sets. It is thus simplest to
characterize the constraining power of cluster weak lensing samples in terms
of a single number at each redshift.

For testing some dark energy or modified gravity models, what matters is the
ability to pin down the shape of $\sigma_{8}(z)$, tracking changes in the rate
of structure growth at low redshift. In other cases, the most stringent test
comes from comparing the overall amplitude of low redshift matter clustering
to the cosmic microwave background predictions for $\Lambda$CDM (cosmological
constant and cold dark matter) or an alternative dark energy model. If we
assume $\Lambda$CDM and treat errors in different redshift bins as
independent, then we can combine the $\sigma_{8}(z)$ constraints from
different redshift bins by summing the inverse square to obtain the aggregate
uncertainty in $\sigma_{8}$ at $z=0$:

$\Delta\ln\sigma_{8}=\left(\sum_{i}\left[\Delta\ln\sigma_{8}(z_{i})\right]^{-2}\right)^{-1/2}\
.$ (1)

The goal for observational analyses is to have this aggregate precision
limited by the statistical uncertainties of the data set rather than by the
systematic uncertainties. For cluster weak lensing, the two main sources of
statistical uncertainty are the finite size of the cluster sample and the weak
lensing shape noise.

In the spirit of our galaxy–galaxy lensing analogy, we concentrate our
analyses on constraints from the full population of clusters above some
threshold in the observable mass proxy $M_{\rm obs}$. In this treatment the
“nuisance parameters” for inferring $\sigma_{8}(z)$ in a given redshift bin
are the true mass $M_{\rm true}$ that corresponds to $M_{\rm obs}$ and the
r.m.s. scatter $\sigma_{\rm lnM}$ between ${\ln M_{\rm obs}}$ and $\ln M_{\rm
true}$ at this threshold. We view this scatter as a quantity to be predicted
from simulations or constrained from external data, such as high-quality X-ray
observations of a subset of clusters in the weak lensing survey. The value of
$\sigma_{8}(z)$ and $M_{\rm true}$ can then be constrained using the mean weak
lensing profile and number counts of clusters above the threshold. We examine
what knowledge of scatter is needed to avoid degrading statistical constraints
on $\sigma_{8}(z)$. In Salcedo et al. (2020), we show that combining cluster
weak lensing with cluster–galaxy cross-correlations and galaxy auto-
correlations, measurable in the same data used for cluster identification and
weak lensing analysis, can constrain the scatter and break its degeneracy with
the mass threshold and $\sigma_{8}$.

In principle, an analysis that uses multiple $M_{\rm obs}$ thresholds or
divides clusters into $M_{\rm obs}$ bins could obtain additional cosmological
constraining power by probing the shape of the halo mass function in addition
to the mass and scatter at the $M_{\rm obs}$ threshold. However, in our tests
(see Appendix B), we find that the gains from a multi-bin analysis are limited
at best with the most general assumption for the mass–observable relation.
When considering multiple bins or multiple thresholds, one must introduce
additional mass and scatter nuisance parameters at each boundary. Much if not
all of the additional observational information goes to constraining these
nuisance parameters rather than cosmology, and the modeling is more
complicated because of the larger number of parameters and the need to
accurately estimate the covariance of measurement errors and systematic errors
across bins. One can try to reduce the number of parameters by assuming, e.g.,
that the mass–observable relation and the scatter follow a power-law
dependence on $M$ or $(1+z)$, as in Murata et al. (2019) or Eifler et al.
(2020a). However, this approach risks deriving spurious cosmological
constraints from the cluster counts that are actually an artifact of the
restrictive model. We therefore focus our attention on the single-threshold
analysis and advocate caution in using $M_{\rm obs}$ bins or multiple $M_{\rm
obs}$ thresholds at a given redshift.

Although we focus on the $\sigma_{8}(z)$ constraints from the combination of
cluster lensing and number counts, we also find it illuminating to compress
the cluster lensing signal into the constraints on the mean cluster mass. We
will show that such data compression leads to slight loss of information, but
it provides an easier way to assess the relative importance of the
uncertainties of number counts (related to cluster sample size), uncertainties
of mean mass (related to lensing measurements), and the prior on scatter (from
external data). In this work we focus on the statistical uncertainties, but
our formulae for parameter sensitivities are also applicable to systematic
uncertainties or to alternative methods for calibrating cluster masses (e.g.,
multi-wavelength, two-point statistics).

This paper is organized as follows. Section 2.1 discusses the signal, noise,
and parameter sensitivity of cluster number counts, while Section 2.2
discusses those for cluster lensing. Section 3 presents the general
constraining power from combining cluster number counts and lensing, while
Section 4 incorporates specific survey assumptions. In Section 5 we discuss
implications of our results, with particular attention to the control of
mass–observable scatter and systematic errors needed to realize the
statistical power of present and future surveys. We summarize our results in
Section 6.

We adopt a flat $\Lambda$CDM cosmology with the following cosmological
parameters: $w=-1$; $\Omega_{k}$ = 0; $\Omega_{\rm M}$ = 0.314;
$\Omega_{\Lambda}$ = 0.686; $h$ = 0.67; $\sigma_{8}$ = 0.83. These parameters
follow those of the simulations from the Abacus Cosmos suite
Abacus_Cosmos_720box_planck (Garrison et al., 2018), which are used for
calculating the small-scale lensing covariance matrices. Throughout this work,
cosmological distances are in comoving units, and halo masses are based on a
spherical density of 200 times the background density ($M_{200m}$). We use the
halo mass function and halo bias formulae provided by Tinker et al. (2008,
2010).

##  2 Galaxy cluster number counts and gravitational lensing

###  2.1 Cluster number counts

When constructing a cluster sample, we set a threshold on the observed
property $M_{\rm obs}$ (e.g., optical richness). At a given mass $M$, let
$M_{\rm obs}$ be the mass proxy re-scaled such that,

$\left\langle{{\ln M_{\rm obs}}|\rm lnM}\right\rangle=\rm lnM~{}.$ (2)

The number density above the observable threshold $M_{\rm obs}^{\rm th}$ is
given by

$\displaystyle\bar{n}$ $\displaystyle=\int_{M_{\rm obs}^{\rm
th}}^{\infty}dM_{\rm obs}\int_{0}^{\infty}dM\frac{dn}{dM}P(M_{\rm obs}|M)$ (3)
$\displaystyle=\int dM\frac{dn}{dM}\phi(M_{\rm obs}^{\rm th}|M)\ .$

The nuisance parameters are included in the selection function $\phi$. If the
mass–observable scatter is negligible, $\phi$ is described by a step function
$H(M>M_{\rm obs}^{\rm th})$. In our calculation, we assume that $\phi$ is
described by a complementary error function

$\phi(M_{\rm obs}^{\rm th}|M)=\frac{1}{2}{\rm erfc}\bigg{(}\frac{\ln M_{\rm
obs}^{\rm th}-\rm lnM}{\sqrt{2\uppi{\sigma_{\rm lnM}^{2}}}}\bigg{)}~{},$ (4)

which corresponds to assuming that ${\ln M_{\rm obs}}$ near the selection
threshold follows a normal distribution centered on $\rm lnM$ with standard
deviation $\sigma_{\rm lnM}$.111Here $\sigma_{\rm lnM}$ is a shorthand
$\sigma_{{\ln M_{\rm obs}}|M}$ and is our definition of scatter throughout
this paper. In other work sometimes the scatter is defined as the standard
deviation of $\rm lnM$ at a given $M_{\rm obs}$, i.e., $\sigma_{\rm lnM|M_{\rm
obs}}$, which differs from our definition by the slope of the observable–mass
relation. In practice, the exact function form of $\phi$ makes little
difference as long as it allows a gradual transition from 0 to 1 (e.g., a ramp
function), reflecting the scatter between ${\ln M_{\rm obs}}$ and $\rm lnM$.

In this paper, we choose $M_{\rm obs}^{\rm th}$ and $\sigma_{\rm lnM}$ as our
nuisance parameters. For a given scatter, we choose a $M_{\rm obs}^{\rm th}$
so that the cluster number counts are the same as in the case of zero scatter.
For example, for $\sigma_{\rm lnM}=0.2$, we use $M_{\rm obs}^{\rm
th}=1.03\times 10^{14}~{}h^{-1}\rm M_{\odot}$, which gives the same cluster
number counts as $\sigma_{\rm lnM}=0$ and $M\geq 10^{14}~{}h^{-1}\rm
M_{\odot}$. Hereafter we denote such selection as $M_{\rm eq}^{\rm
th}=10^{14}~{}h^{-1}\rm M_{\odot}$ (having equal counts as $M\geq
10^{14}~{}h^{-1}\rm M_{\odot}$ in the case of no scatter).

For a given survey volume $V$, the expected total number counts are given by
$\left\langle{N}\right\rangle=\bar{n}V$. For the convenience of readers, in
Appendix A we provide survey volumes and predicted cluster counts for various
assumptions about survey properties.

The noise of number counts is contributed by the Poisson noise ($\bar{n}V$)
and the halo sample variance (Hu & Kravtsov, 2003):

${\rm Var}[N]=\bar{n}V+V^{2}{\bar{n}}^{2}{\bar{b}}^{2}\sigma_{V}^{2}~{},$ (5)

where $\bar{b}$ is the mean halo bias associated with the cluster sample,

$\bar{b}=\frac{1}{\bar{n}}\int dM\frac{dn}{dM}b(M)\phi(M_{\rm obs}^{\rm
th}|M)~{},$ (6)

and $\sigma_{V}$ is the r.m.s. density fluctuation in the survey volume,

$\sigma_{V}^{2}=\int\frac{k^{2}dk}{2\uppi^{2}}W^{2}(k,R)P_{\rm lin}(k)~{}.$
(7)

Here $P_{\rm lin}(k)$ is the linear matter power spectrum, $W(k,R)$ is the
window function of the survey, and $R=(3V/4\uppi)^{1/3}$. We use a real-space
top-hat window function

$W(k,R)=\frac{3[\sin(kR)-kR\cos(kR)]}{(kR)^{3}}~{}.$ (8)

For a large survey volume, $\sigma_{V}^{2}\propto 1/V$.

The observed cluster number counts depend on $M_{\rm obs}^{\rm th}$ and
$\sigma_{\rm lnM}$, both of which are degenerate with cosmological parameters.
For example, a larger number count can be achieved by a higher $\sigma_{8}$, a
lower halo mass at the selection threshold, or a larger scatter. Therefore, to
constrain cosmological parameters, we need external observables to break the
degeneracy. In this work we use stacked weak gravitational lensing, which can
be interpreted as calibrating the mean mass of clusters above the $M_{\rm
obs}$ threshold. We assume that the scatter $\sigma_{\rm lnM}$ is known with
some uncertainty, and we investigate the sensitivity of cosmological
constraints to the uncertainty in $\sigma_{\rm lnM}$. Additional observables
that could be used to constrain $\sigma_{\rm lnM}$ include the cluster auto-
correlation or cross-correlation with galaxies (Baxter et al., 2016; Salcedo
et al., 2020), or comparisons of mass estimates from different observables
(Bleem et al., 2020).

###  2.2 Weak gravitational lensing of galaxy clusters

Figure 1:  Left: Stacked cluster lensing signal (points) and noise per cluster
(solid curves), for a selection threshold $10^{14}~{}h^{-1}\rm M_{\odot}$ at
$z=0.5$. The noise is contributed by density fluctuations (black) and shape
noise (blue and orange, which correspond to source densities of 10 and 50
arcmin2 at $z=1.25$). In a sample of $N_{\rm cl}$ clusters, each of the noise
curves would be lower by $N_{\rm cl}^{-1/2}$. Right: Derivatives of the mean
lensing signal with respect to $\ln\sigma_{8}$, $\ln M_{\rm obs}^{\rm th}$,
and $\sigma_{\rm lnM}$. The scale dependence of these derivatives is very
similar, indicating the strong degeneracy between these parameters in
determining the stacked cluster lensing signal.  Figure 2: Constraints on the
cluster mean mass using stacked lensing, with fixed $\sigma_{8}$ and scatter.
The left-hand panel corresponds to fixing the smallest scale and progressively
including larger scales, while the right-hand panel corresponds to fixing the
largest scale and progressively including smaller scales. The blue and orange
curves correspond to source densities of 10 and 50 arcmin-2, while the black
curve corresponds to an infinite source density (no shape noise). The curves
on the left-hand panel drop more rapidly than those on the right-hand panel,
indicating that the 1-halo regime ($<3\ h^{-1}\rm Mpc$) contains most of the
information. The right-hand panel shows that omitting scales below $r_{\rm
p}=0.5\ h^{-1}\rm Mpc$ would lead to negligible loss of information.

The weak lensing signal generated by stacking a sample of clusters can be
expressed in terms of the correlation function between the clusters and the
matter density field, $\xi_{\rm cm}$:

$\displaystyle\left\langle{\Delta\Sigma(r_{\rm p})}\right\rangle=$
$\displaystyle\bar{\rho}\bigg{[}\frac{4}{r_{\rm p}^{2}}\int_{0}^{r_{\rm
p}}{r_{\rm p}^{\prime}}{\rm d}{r_{\rm p}^{\prime}}\int_{0}^{\infty}{\rm
d}\chi\ \xi_{\rm cm}\left(\sqrt{{r_{\rm p}^{\prime}}^{2}+\chi^{2}}\right)$ (9)
$\displaystyle-2\int_{0}^{\infty}{\rm d}\chi\ \xi_{\rm cm}\left(\sqrt{r_{\rm
p}^{2}+\chi^{2}}\right)\bigg{]}\ ,$

where

$\xi_{\rm cm}(r)=\int dM\frac{dn}{dM}\ \xi_{\rm hm}(r|M)\ \phi(M_{\rm
obs}^{\rm th}|M)~{},$ (10)

and $\xi_{\rm hm}$ is the correlation function between dark matter and halos
of mass $M$. We approximate $\xi_{\rm hm}$ by the maximum of the 1-halo and
2-halo contribution (e.g., Hayashi & White, 2008):

$\xi_{\rm hm}={\rm max}(\xi_{\rm 1h},\xi_{\rm 2h})~{},$ (11)

where

$\displaystyle\xi_{\rm 1h}$ $\displaystyle=\frac{\rho_{\rm
NFW}(M,r)}{\bar{\rho}}-1$ (12) $\displaystyle\xi_{\rm 2h}$
$\displaystyle=b(M)\xi_{\rm lin}(r)~{},$

and $\rho_{\rm NFW}(r)$ is the halo density profile described by Navarro et
al. (1997). Here $b(M)$ is the halo bias, and $\xi_{\rm lin}$ is the matter
correlation function calculated from the linear matter power spectrum. We
adopt the concentration–mass relation from Correa et al. (2015), which is
cosmology independent.

We have provided a detailed treatment of cluster lensing covariance matrices
in Wu et al. (2019). To summarize, the covariance matrices for cluster
gravitational lensing are contributed by

  * •

Shape noise, which is inversely proportional to the number of source galaxies.

  * •

Large-scale structure noise contributed by the line-of-sight structure outside
galaxy clusters, which dominates the noise at large scales.

  * •

Intrinsic variation of halo density profiles, which contributes to the noise
at small scales.

For most Stage III optical imaging surveys, the last component is subdominant,
but for upcoming surveys like Roman Space Telescope’s HLS, this component will
be significant.

We analytically calculate the covariance matrices assuming Gaussian random
fields (Equation (22) in Wu et al. 2019), which account for the shape noise
and the large-scale structure noise. The small-scale contribution from halos
is calculated using the Abacus simulations (Garrison et al., 2018). To
validate the combination of analytical Gaussian calculations and the numerical
non-Gaussian calculations, we use ray-tracing simulations provided by
Takahashi et al. (2017).

The left-hand panel of Figure 1 shows the stacked cluster lensing signal vs.
noise for our baseline example: $z_{\rm lens}=0.5$, $M_{\rm eq}^{\rm
th}=10^{14}~{}h^{-1}\rm M_{\odot}$, and $\sigma_{\rm lnM}=0.2$. The points
show the signal, and the solid curves show the noise per cluster. For a sample
of $N_{\rm cl}$ clusters, the noise level scales approximately with
$1/\sqrt{N_{\rm cl}}$. We use 5 logarithmic radial bins per decade, and we
assume the source galaxies are located at $z_{\rm src}=1.25$. The black curve
shows the contribution of density fluctuations (the sum of large-scale
structure and small-scale halo density profiles). The blue and orange curves
show the shape noise for source densities ${\rm n_{src}}$ of 10 and 50
arcmin-2, which approximate the capabilities of Stage III and Stage IV
experiments. This color scheme will be used repeatedly in this paper.

If the shape noise is negligible (black curve), the noise from density
fluctuations is still higher than the signal of a single cluster at large
scales. For a Stage-IV level of shape noise (orange), we will have sufficient
signal-to-noise ratio for individual massive clusters at small scales, but for
the majority of the clusters and large scales we will need to stack the
signals of multiple clusters. In Section 4, we will implement the detailed
survey assumptions, which can be calculated by re-scaling the various
components in the covariance matrix. For example, the shape noise is
proportional to $\rm\left\langle{\Sigma_{crit}^{2}}\right\rangle/{\rm
n_{src}}$ (which can be calculated from the source redshift distribution), the
contribution of density fluctuation is approximately independent of the source
redshift, and the entire covariance matrix is inversely proportional to the
survey area.

The right-hand panel of Figure 1 shows the derivatives of
$\ln\left\langle{\Delta\Sigma}\right\rangle$ with respect to the model
parameters: $\ln\sigma_{8}$, $\ln M_{\rm obs}^{\rm th}$, and $\sigma_{\rm
lnM}$. The derivative with respect to $\ln M_{\rm obs}^{\rm th}$ is the
easiest to understand: increasing $\ln M_{\rm obs}^{\rm th}$ increases the
mass scale at the threshold, and thus $\Delta\Sigma$ increases at all scales.
The 1-halo regime reflects the change of the density profile with mass and
concentration; since $\Delta\Sigma$ is affected by the total mass density
inside the aperture, its sensitivity peaks approximately at the outer edge of
the 1-halo regime. The sensitivity in the 2-halo regime is associated with the
mass-dependence of halo bias.

The derivative with respect to $\ln\sigma_{8}$ is very similar to that with
respect to $\ln M_{\rm obs}^{\rm th}$: increasing $\ln\sigma_{8}$ also
increases the mass of all halos. We choose a concentration–mass relation that
is independent of cosmology; if the concentration–mass relation depends on
cosmology, the small-scale profile will have larger differences. On the other
hand, increasing the scatter decreases the mass of halos near the selection
threshold and thus leads to a lower $\Delta\Sigma$. The similarity between the
derivatives indicates that these parameters are degenerate with each other,
and thus lensing alone cannot constrain $\sigma_{8}$. We will show that with
an external prior on scatter, combining lensing and number density can break
the degeneracy between $\sigma_{8}$ and mass scales.

##  3 Constraining power per comoving volume

###  3.1 Constraining cluster mean mass using gravitational lensing

In cluster lensing analyses, a common strategy is to use gravitational lensing
to constrain the mean mass of clusters, and then combine the mean mass and and
number density to constrain cosmological and nuisance parameters (e.g.,
Costanzi et al., 2019; Abbott et al., 2020). As shown in Figure 1, the
derivatives of $\Delta\Sigma$ with respect to threshold, scatter, and
$\sigma_{8}$ have similar shapes; therefore, we cannot constrain these
parameters simultaneously.

We first examine a case in which we fix $\sigma_{8}$ and scatter and constrain
the mean mass. The constraints on the mean mass can be estimated by

$\big{(}\Delta\left\langle{\rm
lnM}\right\rangle\big{)}^{-2}=\frac{\partial\left\langle{\Delta\Sigma}\right\rangle^{T}}{\partial\left\langle{\rm
lnM}\right\rangle}{\mathcal{C}}^{-1}\frac{\partial\left\langle{\Delta\Sigma}\right\rangle}{\partial\left\langle{\rm
lnM}\right\rangle}~{},$ (13)

where ${\mathcal{C}}$ is the lensing covariance matrix, and

$\frac{\partial\left\langle{\Delta\Sigma}\right\rangle}{\partial\left\langle{\rm
lnM}\right\rangle}=\frac{\partial\left\langle{\Delta\Sigma}\right\rangle}{\partial\ln
M_{\rm obs}^{\rm th}}\frac{\partial\ln M_{\rm obs}^{\rm
th}}{\partial\left\langle{\rm lnM}\right\rangle}~{}.$ (14)

Figure 2 shows $\Delta\left\langle{\rm lnM}\right\rangle$ using small vs.
large scales. We again use our baseline example, $z_{\rm lens}=0.5$, $M_{\rm
eq}^{\rm th}=10^{14}~{}h^{-1}\rm M_{\odot}$, $z_{\rm src}=1.25$, survey volume
1 $(h^{-1}\rm Gpc)^{3}$. The blue and orange curves correspond to source
densities of 10 and 50 arcmin-2, and the black curves correspond to an
infinite source density (no shape noise). A source density of 50 arcmin-2 is
very close to the case without shape noise. In the left-hand panel, we fix the
smallest radius to 0.1 $h^{-1}\rm Mpc$ and progressively increase the largest
scale used, denoted by ${\rm r_{p}^{max}}$. When ${\rm r_{p}^{max}}$
increases, the constraints improve up to $\sim 1\ h^{-1}\rm Mpc$, above which
including larger scale information does not add further information. This
indicates that most of the information comes from the 1-halo regime.

In the right-hand panel, we fix the largest radius to 100 $h^{-1}\rm Mpc$ and
progressively include small-scale information down to ${\rm r_{p}^{min}}$.
When ${\rm r_{p}^{min}}$ decreases, the constraints improve quickly, again
indicating small scales contain more information. Below 0.5 $h^{-1}\rm Mpc$,
the information saturates, which indicates that discarding scales smaller than
$\sim$ 0.5 $h^{-1}\rm Mpc$ would lead to little loss of information. This is
encouraging because using scales below 0.5 $h^{-1}\rm Mpc$ is more challenging
due to systematic uncertainties such as baryonic effects, cluster mis-
centering, and weak lensing boost factors. By construction, the right end of
the left-hand panel matches the left end of the right-hand panel, both
corresponding to including information from all scales. Using scales $r_{\rm
p}\gtrsim 3\ h^{-1}\rm Mpc$ yields an error roughly twice that of scales
$r_{\rm p}\lesssim 3\ h^{-1}\rm Mpc$; therefore, although the 2-halo regime is
less constraining, it is competitive enough to allow a useful check from
scales where the underlying physics and observational systematics are quite
different from those in the 1-halo regime.

In practice, fitting the mean lensing profile to a mean mass can lead to small
bias, which can be corrected for using simulations (e.g., McClintock et al.,
2019). In addition, one sometimes fits the mean mass and concentration
simultaneously (e.g., Applegate et al., 2014). We find that we only need a
modest prior on concentration to avoid degradation in the mass constraint, and
therefore we adopt a fixed concentration–mass relation in this work.

###  3.2 Combining cluster number density and lensing to constrain
$\sigma_{8}$

We investigate the constraining power on $\sigma_{8}$ by combining number
counts and gravitational lensing observed at different scales. We start by
assuming a survey volume of 1 $(h^{-1}\rm Gpc)^{3}$, a galaxy cluster sample
selected with $M_{\rm eq}^{\rm th}=10^{14}~{}h^{-1}\rm M_{\odot}$ in a narrow
lens redshift bin centered on $z_{\rm lens}$, and a narrow source redshift bin
at $z_{\rm src}=2.5z_{\rm lens}$.

We calculate the Fisher matrix for the data vector ${\bf
D}=(N,\left\langle{\Delta\Sigma}\right\rangle)$, and model parameters $p$ =
($\ln\sigma_{8}$, $\ln M_{\rm obs}^{\rm th}$, $\sigma_{\rm lnM}$)

$F_{ij}=\frac{\partial{\bf D}}{\partial
p_{i}}{\mathcal{C}}^{-1}\frac{\partial{\bf D}}{\partial p_{j}}\ .$ (15)

The marginalized constraint on parameter $p_{i}$ is given by

$\sigma(p_{i})=\sqrt{(F^{-1})_{ii}}\ .$ (16)

We assume no covariance between number density and lensing. We have verified
this assumption using the Abacus simulations.

Figure 3:  Error bars of $\ln N$ (left-hand), $\left\langle{\rm
lnM}\right\rangle$ (middle), and $\ln\sigma_{8}$ (right-hand) from a 1
$(h^{-1}\rm Gpc)^{3}$ volume at various redshifts, for clusters selected with
$M_{\rm eq}^{\rm th}=10^{14}~{}h^{-1}\rm M_{\odot}$. In the left-hand panel,
we show a dashed curve associated with a fixed area (to be compared with
Figure 6). In the middle panel, we use the lensing signal between 0.1 and 100
$h^{-1}\rm Mpc$ and assume three levels of shape noise. In the right-hand
panel, we combine the information from both number counts and lensing,
additionally assuming three levels of prior on the scatter. A source density
of 50 arcmin-2 and a prior of 0.01 scatter is very close to the statistical
limit of no shape noise and perfectly known scatter. The constraining power at
a fixed volume is slightly better at low redshift because of the higher
cluster comoving density.

Figure 3 demonstrates the constraining power of a 1 $(h^{-1}\rm Gpc)^{3}$
volume survey at various redshifts. This is not a direct forecast for a sky
survey, in which different redshift ranges have different volumes. The left-
hand panel corresponds to the uncertainties in cluster number counts, which
are contributed by Poisson noise and the sample variance (Section 2.1) and are
determined by the mass and the volume. The solid curve corresponds to a fixed
volume, 1 $(h^{-1}\rm Gpc)^{3}$, which is also used in the other two panels.
To compare with the calculations with fixed area (Figure 6 below), we add a
dashed curve corresponding to 5000 $\deg^{2}$, $\Delta z=0.1$, at all
redshifts.

The middle panel shows the constraints on the mean mass using the lensing
signal from 0.1 to 100 $h^{-1}\rm Mpc$, assuming three levels of shape noise
(indicated by different colors). The right-hand panel shows the constraints on
$\sigma_{8}$ combining number density and lensing, for three levels of shape
noise (different colors) and three levels of prior on scatter (different line
styles). For a fixed volume, the constraining power on $\sigma_{8}$ is
slightly better at low redshifts due to the higher cluster density and larger
source density for lensing. Nevertheless, the trend is very weak, indicating
that the constraining power per volume is approximately constant with
redshift. In all cases the constraint on $\sigma_{8}$ for $\Delta\sigma_{\rm
lnM}=0.01$ is very close to that of a perfectly known scatter (not shown).

Figure 4: Dependence of $\sigma_{8}$ constraints on the prior on scatter
$\Delta\sigma_{\rm lnM}$, for $z=0.5$, $M_{\rm eq}^{\rm
th}=10^{14}~{}h^{-1}\rm M_{\odot}$, and $V=1~{}(h^{-1}\rm Gpc)^{3}$, for three
levels of shape noise. The solid curves correspond to the number count
uncertainties associated with a 1 $(h^{-1}\rm Gpc)^{3}$ volume, while the
dashed curves correspond to negligible number count uncertainty. When
$\Delta\sigma_{\rm lnM}\approx$ 0.05, improving the prior on scatter plays a
more important role than either increasing the cluster sample or increasing
the source density.

Figure 4 shows the constraints on $\ln\sigma_{8}$ as a function of the prior
on scatter ($\Delta\sigma_{\rm lnM}$), for 3 levels of lensing uncertainties
represented by different colors, for our baseline example $z=0.5$, $M_{\rm
eq}^{\rm th}=10^{14}~{}h^{-1}\rm M_{\odot}$, $V=1\ (h^{-1}\rm Gpc)^{3}$. The
solid curves assume the number count uncertainties associated with the 1
$(h^{-1}\rm Gpc)^{3}$ volume, while the dashed curves assume negligible number
count uncertainties ($\Delta N=0$). The latter would be relevant if we have a
very large volume but only a small subset of clusters have high-quality
lensing measurements. When the scatter is well-constrained (left-hand side of
the solid curves), increasing the source number density from 10 to 50 arcmin-2
produces a modest (13%) improvement in the $\sigma_{8}$ constraint, while
eliminating shape noise entirely produces little further gain. If $\Delta N=0$
(dashed curves) the effect of shape noise is somewhat larger. For
$\Delta\sigma_{\rm lnM}\lesssim 0.01$, number count and lensing uncertainties
dominate the $\sigma_{8}$ uncertainty. For $\Delta\sigma_{\rm lnM}\gtrsim
0.02$, the uncertainty in scatter significantly inflates the $\sigma_{8}$
error, and reducing weak lensing or number count uncertainties has little
impact unless $\Delta\sigma_{\rm lnM}$ can also be reduced.

##  4 Dependence of parameter constraints on survey conditions

Figure 5:  Surface number density of source galaxies above a given lens
redshift, for various survey assumptions. The solid curves correspond to the
total galaxy number densities above the lens redshift, while the dashed curves
are the number densities weighted by the lensing efficiency.  Figure 6:
Constraining power for three assumptions about survey properties (see text),
per redshift bin of $\Delta z=0.1$. Left: Fractional uncertainties of cluster
number counts, which are inversely proportional to the square root of the
survey area. Middle: Fractional uncertainties of cluster mean mass constrained
by weak lensing, which are determined by both survey area and source redshift
distribution. Right: Fractional constraints on $\sigma_{8}(z)$, combining
number density and lensing, with priors on $\sigma_{\rm lnM}$ of 0.01 (solid)
or 0.05 (dashed).

Figure 7: Relative importance of the uncertainties related to cluster number
counts, lensing lensing, and scatter (external prior). We show their impacts
on $\Delta\ln\sigma_{8}$ as a function of survey area, for $z=0.5$ (upper) and
$z=0.7$ (lower), and selection thresholds $10^{14}$ (left) and $2\times
10^{14}~{}h^{-1}\rm M_{\odot}$ (right). Here we focus on one source of
uncertainty at a time and assume the other two are negligible. The
intersection between lines indicates when the impact of the uncertainties
becomes comparable. For example, in the upper left panel for Stage IV deep
(green), number count and weak lensing errors contribute almost equally to the
$\sigma_{8}$ uncertainty, and for a $1000$ deg2 survey they have the same
impact as $\Delta\sigma_{\rm lnM}=0.05$. In all panels, the blue, green, and
orange squares show our assumed survey areas for Stage III wide, Stage IV
deep, and Stage IV wide, respectively, and the solid lines show forecasts that
assume the same source densities but different survey areas.  Figure 8:
Aggregate constraints on $\sigma_{8}$ combining $0.2\leq z\leq 1.1$, assuming
the information in each redshift bin of $\Delta z=0.1$ is independent. We
assume the source density for various survey assumptions while varying the
survey area ($x$-axis); squares mark our assumed survey areas. Different line
styles correspond to different levels of prior on scatter. A tight prior is
required so that the constraining power improves with increasing survey area.

In this section, we calculate the constraining power of various surveys on
$\sigma_{8}(z)$. We adopt the following illustrative survey parameters:

  * •

Stage III wide: 5000 $\deg^{2}$, ${\rm n_{src}}$=10 arcmin-2

  * •

Stage IV wide: 18000 $\deg^{2}$, ${\rm n_{src}}$=27 arcmin-2

  * •

Stage IV deep: 2000 $\deg^{2}$, ${\rm n_{src}}$=51 arcmin-2

These assumptions are motivated by the survey designs of DES, LSST, and Roman
Space Telescope’s HLS, respectively. For Stage III wide and Stage IV wide, we
assume that the source distribution is given by

$p_{\rm src}(z)\propto z^{2}\exp(-(z/z_{0})^{\alpha})$ (17)

with $z_{0}=0.11$ and $\alpha=0.68$ for Stage III wide, and $z_{0}=0.5$ and
$\alpha=1.4$ for Stage IV wide. For Stage IV deep, we use Figure 2 in Eifler
et al. (2020b). We normalize the source distribution such that

${\rm n_{src}}=\int_{0}^{\infty}p_{\rm src}(z)dz~{}.$ (18)

Figure 5 shows the cumulative surface number density of source galaxies above
a given lens redshift. The solid curves correspond to directly integrating
$p_{\rm src}$ above $z_{\rm lens}$, while the dashed curves correspond to
weighting $p_{\rm src}$ by the lensing efficiency,

$q(\chi)=\int_{\chi}^{\infty}d\chi^{\prime}p_{\rm
src}(\chi^{\prime})\frac{\chi^{\prime}-\chi}{\chi^{\prime}}\ ,$ (19)

where $\chi$ denotes the comoving distance (see, e.g., Kilbinger, 2015, for a
review). The lensing efficiency characterizes the lensing strength at
different lens redshifts for a given source distribution. Here we normalize
$p_{\rm src}$ to ${\rm n_{src}}$ instead of 1. For example, for Stage IV deep
(green), the source density above $z_{\rm lens}=0.5$ is $\approx$ 45 arcmin-2
(green solid), but their lensing signal would be equivalent to $\approx$ 20
arcmin-2 for sources at a single $z_{\rm src}\gg z_{\rm lens}$ (green dashed).

We assume that all these surveys have the same $M_{\rm obs}^{\rm th}$ and that
the uncertainties in lensing measurements are determined only by the source
redshift distribution. Therefore, these assumptions are somewhat optimistic
for wide surveys like DES and LSST but potentially achievable with deep
surveys like HLS because of the high source galaxy density and better
deblending of source galaxies. Figure 6 shows the fractional uncertainties on
the number counts, the mean mass, and $\sigma_{8}$, per redshift bin of
$\Delta z=0.1$. The points indicate the middle of the bin. We assume an
independent set of model parameters in each redshift bin. This assumption is
observationally realistic; because of the evolution of cluster properties, the
same $M_{\rm obs}$ threshold may not correspond to the same true mass
threshold at different redshifts, and the corresponding mass–observable
scatter may not be the same either. For a specified cosmological model one can
combine $\sigma_{8}(z)$ estimates to obtain a tighter constraint on
$\sigma_{8}(z=0)$, while the errors in $\sigma_{8}(z)$ indicate the power of
the survey to map out the low-redshift growth history.

The left-hand panel shows the error bars on number counts. Since we assume the
same cluster selection for these surveys, the uncertainty of number counts is
inversely proportional to the square root of the survey area. The three survey
assumptions have an area ratio of 1: 2.5: 9, and thus the fractional count
error scales as 3: 1.9: 1.

The middle panel shows the uncertainty of mean mass, which is determined by
the lensing covariance matrix. The elements of the covariance matrix are
inversely proportional to the number of clusters multiplied by the sum of
density noise and shape noise. When we compare Stage IV wide vs. deep, the
larger number of clusters of the wide survey outperform the higher source
density of the deep survey. The right-hand panel shows the resulting
constraints on $\ln\sigma_{8}$. We assume two levels of prior on the scatter.
With a tight prior on scatter (solid curves), the constraints closely follow
the mass uncertainties, while a weak prior significantly degrades the
constraints.

Figure 7 isolates the roles of number count uncertainties (dashed), lensing
uncertainties (solid), and uncertainties on scatter (dotted horizontal) for
determining the constraints on $\sigma_{8}$. The uncertainties in number
counts and lensing both decrease with area, while the uncertainty on scatter
is assumed to come from external experiments and is thus independent of area.
We show two redshifts, $z_{\rm lens}=0.5$ (upper) and 0.7 (lower), and two
selection thresholds, $M_{\rm eq}^{\rm th}=10^{14}~{}h^{-1}\rm M_{\odot}$
(left) and $2\times 10^{14}~{}h^{-1}\rm M_{\odot}$ (right). We consider one
source of uncertainty at a time and assume the other two are negligible. The
intersections between lines indicate comparable contributions from these
sources of uncertainties. For example, in the upper left panel
($10^{14}~{}h^{-1}\rm M_{\odot}$), for Stage IV wide, the contributions from
number count uncertainties (dashed) and lensing uncertainties (green) are
comparable for any survey area. For a 1000 deg2 survey area,
$\Delta\sigma_{\rm lnM}$ would exceed the other contributions for
$\Delta\sigma_{\rm lnM}>0.05$ and be smaller for $\Delta\sigma_{\rm
lnM}<0.05$. For a 6000 deg2 survey area, $\Delta\sigma_{\rm lnM}$ must be
smaller than 0.02 to avoid degrading the $\sigma_{8}(z)$ constraints.

For a higher mass threshold, the number of clusters at a fixed survey area is
smaller. The number count and weak lensing uncertainties both increase, but
the increase for number counts is smaller because the sample variance
contribution is almost unchanged (see Equation (5)). For lensing
uncertainties, at $z=0.5$ the greater depth of Stage IV wide produces
significant (64%) gains over Stage III wide, but the change to Stage IV deep
produces little further gain (7%). At $z=0.7$, the benefit of the greater
depth is somewhat larger (30%).

Figure 8 shows the aggregate constraints on $\sigma_{8}$, assuming the
constraints on $\sigma_{8}(z)$ for $0.2\leq z\leq 1.1$ at each redshift bin of
$\Delta z=0.1$ are independent of each other. For comparing the amplitude of
low redshift clustering to extrapolation from the cosmic microwave background
assuming $\Lambda$CDM, this aggregate $\sigma_{8}$ precision is the relevant
measure of survey power. It also indicates the level of weak lensing
systematics control required; roughly speaking, the impact of a 0.5% change in
$\sigma_{8}$ is comparable to that of a 0.5% multiplicative shear bias. We use
the source redshift distributions of the three survey assumptions and vary the
survey area (the $x$-axis). We show four assumptions for the prior on scatter,
again allowing independent values of $\sigma_{\rm lnM}$ in each redshift bin.
When the external prior on scatter at each redshift is well constrained (solid
curves), the constraining power on $\sigma_{8}$ keeps improving with
increasing area. When the scatter is not well constrained, the precision on
$\sigma_{8}$ is limited by the prior on scatter; for example, if
$\Delta\sigma_{\rm lnM}=0.1$, increasing the survey area beyond 2000 deg2
yields limited improvement.

##  5 Discussion

###  5.1 Statistical uncertainties

Our results highlight a number of key points. First, realizing the statistical
precision achievable with current and future cluster weak lensing surveys
requires good knowledge of the scatter $\sigma_{\rm lnM}$ between mass proxies
and true mass. For a DES-like (Stage III wide, $5000\deg^{2}$) or HLS-like
(Stage IV deep, $2000\deg^{2}$) survey, uncertainty at the level of
$\Delta\sigma_{\rm lnM}=0.05$ would degrade the aggregate precision on
$\sigma_{8}$ by $\approx$ 20% relative to a much tighter (0.01) prior (Figure
8). For an LSST-like (Stage IV wide, $18,000\deg^{2}$) survey, a 20%
degradation of $\sigma_{8}$ precision occurs for $\Delta\sigma_{\rm lnM}=$
0.016. The weak lensing calibration from these surveys constrains the mean
cluster mass at the 1-2% level (DES-like, HLS-like) or the 0.5-1% level (LSST-
like) in each of the $\Delta z=0.1$ bins that contributes substantially to the
cosmological constraints (Figure 6). An alternative mass calibration
technique, e.g., based on cluster X-ray properties, would need to be this
accurate or better to achieve the cosmological constraints forecasted here.

One approach to constraining $\sigma_{\rm lnM}$ is to use simulations, which
may give a reliable estimate of mass–observable scatter even if they cannot
predict the mean relation at the required accuracy. For example, hydrodynamic
simulations have shown that gas-based mass proxies have a scatter of
$10-15\%$, depending on the radius and dynamic state (e.g., Nagai, 2006;
Kravtsov et al., 2006; Stanek et al., 2010; Wu et al., 2015; Yu et al., 2015;
Hahn et al., 2017; Le Brun et al., 2017), with gas mass, X-ray temperature,
and the combination of the two achieving the smallest scatter. Optical mass
tracers tend to have a larger scatter, approximately $30-50\%$ for richness
and $20-30\%$ for stellar mass (e.g., Pillepich et al., 2018; Bradshaw et al.,
2020; Anbajagane et al., 2020), depending on the selection of cluster members.

Another approach is to compare one observable (e.g., optical galaxy richness)
to another (e.g., X-ray temperature) that is expected to have substantially
smaller scatter (e.g., Rozo & Rykoff, 2014). The scatter of the first
observable can be estimated by subtracting in quadrature the estimated scatter
of the more precise observable, which can in turn be inferred from
simulations. Applying this approach to X-ray observations of $\sim 200$
optical clusters from DES, Farahi et al. (2019) infer $\sigma_{\rm
lnM|\lambda}=0.30\pm 0.04_{(\rm stat)}\pm 0.09_{(\rm sys)}$ for clusters
selected by richness $\lambda$ from the redMaPPer algorithm. An uncertainty
$\Delta\sigma_{\rm lnM}=0.1$ degrades the achievable DES $\sigma_{8}$
precision by a factor of 1.4 relative to $\Delta\sigma_{\rm lnM}=0.05$, and
therefore improved determination of $\sigma_{\rm lnM}$ is needed to realize
the potential of DES measurements. Even with $\Delta\sigma_{\rm lnM}=0.1$, our
forecast of the achievable precision on $\sigma_{8}$ from the final DES
cluster weak lensing data set is $\approx 0.36\%$, a large improvement over
the current $\sim 3\%$ precision from DES Year 1 galaxy clustering and weak
lensing probes (Abbott et al., 2018). If the effective mass threshold for
clusters in the cosmological analysis is $2\times 10^{14}~{}h^{-1}\rm
M_{\odot}$ instead of $1\times 10^{14}~{}h^{-1}\rm M_{\odot}$ then the
forecast $\sigma_{8}$ precision is 0.58% for $\Delta\sigma_{\rm lnM}=0.1$ and
0.43% for $\Delta\sigma_{\rm lnM}=0.05$. Looking to the future, data from the
eRosita survey and continuing targeted studies with Chandra and XMM will be
valuable for sharpening knowledge of $\sigma_{\rm lnM}$, and SZ data from the
Simons Observatory and eventually CMB-S4 will be valuable for higher mass
clusters.

In Salcedo et al. (2020) we propose an alternative approach to constraining
$\sigma_{\rm lnM}$ based on a combination of the projected cluster–galaxy
cross-correlation function $w_{\rm cg}(r)$ and the projected galaxy auto-
correlation function $w_{\rm gg}(r)$. For a DES-like survey with a cluster
mass threshold of $1\times 10^{14}~{}h^{-1}\rm M_{\odot}$, we forecast an
error of $\Delta\sigma_{\rm lnM}\approx 0.1$ in a redshift bin $z=0.35-0.55$.
In combination with weak lensing $\Delta\Sigma(r)$ measurement, we forecast a
$\sigma_{8}$ precision of 0.95% from this redshift bin and 0.81% when adding a
$z=0.15-0.35$ redshift bin. These can be compared to this paper’s forecast of
0.26% for the case of perfectly known $\sigma_{\rm lnM}$ and all redshift
bins. Compared to this paper’s number-count plus $\Delta\Sigma(r)$ approach,
the $w_{\rm cg}+w_{\rm gg}+\Delta\Sigma$ analysis proposed in Salcedo et al.
(2020) is much less sensitive to errors in the cluster space density, and
perhaps to the projection systematics discussed below, so it provides a useful
complement even if $\sigma_{\rm lnM}$ can be adequately constrained by other
means.

A second point highlighted by our results is that for source densities higher
than the Stage III value ${\rm n_{src}}\sim 10\ {\rm arcmin}^{-2}$ the gains
in $\sigma_{8}(z)$ precision begin to grow slower than $\sqrt{{\rm n_{src}}}$.
This is true even if one considers weak lensing errors alone, and at Stage IV
depth the number count uncertainties become comparable to weak lensing
uncertainties (Figure 8). For these reasons, the larger area of a Stage IV
wide survey wins over the greater depth of a Stage IV deep survey, if both
surveys map clusters down to the same mass threshold and are statistics
limited.

This brings us to a third, critical point: the achievable precision for
statistics-limited cluster weak lensing surveys is high. Our forecast of
aggregate $\sigma_{8}$ precision with $\Delta\sigma_{\rm lnM}=0.01$ is 0.10%
for Stage IV wide, 0.24% for Stage IV deep, and 0.26% for Stage III wide.
Oguri & Takada (2011) and Weinberg et al. (2013) conclude that cosmological
constraints from cluster weak lensing are competitive with those from cosmic
shear using the same weak lensing data set. We do not re-examine this point
here, but we note that Spergel et al. (2015, §2.2.3.2, forecast by C. Hirata)
project 0.12% precision for $\sigma_{8}$ from cosmic shear with the WFIRST
(now Roman) HLS, treating $\sigma_{8}$ as the single free cosmological
parameter as we do here. The 0.24% precision that we forecast for cluster weak
lensing is weaker than cosmic shear; however, because these two probes have
different sensitivity to other cosmological parameters and to modified gravity
scenarios, the combination should be significantly more powerful than either
alone.

###  5.2 Systematic uncertainties

Realizing this high precision requires excellent control of observational
systematics in addition to tight constraints on $\sigma_{\rm lnM}$. For
example, multiplicative shear biases from shear calibration uncertainty or
photo-$z$ uncertainties must be controlled at about the same fractional level
as the aggregate $\sigma_{8}$ uncertainty to avoid dominating the cosmological
parameter error. These effects can be estimated by direct calibration methods.
A complementary approach is to describe them by nuisance parameters and
constrain them using multiple observables. This strategy may gain
significantly by combining cluster weak lensing with galaxy–galaxy lensing and
cosmic shear, especially when using multiple redshift bins and measurements
that traverse the linear and non-linear clustering regimes. We will explore
this topic in future work.

There are several systematic effects that are specific to cluster weak
lensing. One of these is cluster mis-centering. The central galaxy is often
taken as the center ($r_{\rm p}=0$) for computing tangential shear profiles.
If some central galaxies are misidentified or do not reside at the center of
the cluster’s dark matter halo, then shear profiles will be shallower than
model predictions that ignore mis-centering. Zhang et al. (2019) quantified
the distribution of mis-centering of DES clusters by comparing peaks of X-ray
emission with centers determined by the redMaPPer cluster finder. This
distribution is in turn incorporated into model predictions when fitting
cosmological models to the DES cluster data (McClintock et al., 2019; Abbott
et al., 2020). Our use of a wide $r_{\rm p}$ range reduces the impact of mis-
centering, which only affects the signal on small scales, and it allows
consistency checks between the large scale and small scale regimes.
Nonetheless, controlling mis-centering systematics at the level of our
forecast statistical precision will require further multi-wavelength cluster
studies and simulation studies to accurately quantify mis-centering effects.
Hydrodynamic, radiative cooling, and feedback effects alter the distribution
of baryonic matter in clusters relative to dark matter (Henson et al., 2017).
Uncertainties in these effects introduce a theoretical systematic in
predicting weak lensing profiles, with impact that qualitatively resembles
that of mis-centering. Continued investigation using hydrodynamic cosmological
simulations with varied feedback prescriptions, and development of
parameterized models that can be tested against SZ and X-ray data (e.g., Mead
et al. 2020) will be essential to mitigating this systematic in future cluster
cosmology analyses.

Another cluster-specific observational systematic is the dilution of the weak
lensing signal by apparent sources that are in fact associated with the
cluster itself. Put differently, the true redshift distribution of galaxies in
a given photo-$z$ bin will be distorted by the presence of a cluster at the
center of the field. The correction for this dilution is often called the
“boost factor.” For the DES Year 1 analysis, the uncertainty of the boost
factor is approximately 2% (Varga et al., 2019). Stage IV experiments should
have better photometric redshifts and thus more accurate boost factors
(Hemmati et al., 2019). Boost factor systematics can be checked by
investigating radial trends of the weak lensing relative to model expectations
and by constructing source galaxy samples with colors that firmly exclude the
cluster redshift.

Incompleteness and contamination of cluster catalogs add non-Gaussian tails to
the distribution of $P(\ln M_{\rm obs}|\ln M)$. Accurate modeling of these
tails using simulations and multi-wavelength selection comparisons is
necessary for making high accuracy predictions of number counts and weak
lensing signals (Aguena & Lima, 2018).

The single most challenging systematic for cluster weak lensing cosmology is
anisotropic selection bias, which arises because triaxial halos oriented along
the line of sight or halos projected against correlated large scale structure
can be incorrectly boosted above the $M_{\rm obs}$ threshold. These clusters
also have stacked weak lensing signals that are higher than expected for
random halos of the same mass, an effect that extends to large scales in the
2-halo regime (Dietrich et al. 2014; Osato et al. 2018; Sunayama et al. 2020;
Wu et al., in prep.). Simulations and observations imply that the effect on
weak lensing signals can be as large as 20-30% on some scales (see the above
references and Herbonnet et al. 2019; Abbott et al. 2020). Correcting this
bias requires simulations that accurately model the entire cluster selection
process, with realistic galaxy populations embedded in realistic large scale
structure. Creating and analyzing these simulations is a challenging task, and
uncertainties in the correction for anisotropic selection bias could well be
the long-term limiting factor in cluster cosmology.

The impact of this systematic can be reduced by using a cluster selection
observable that is less affected by projection, such as X-ray luminosity or
temperature. For this reason, weak lensing of eRosita selected clusters may
yield the most powerful Stage IV cosmology results even if the effective mass
threshold is higher than that achievable with optical cluster selection. SZ
selection may also be a valuable tool, though it can also be affected by
anisotropy to some degree. In optical surveys, mass proxies weighted towards
central galaxies may be less sensitive to projection than the richness
estimators in current use, an approach worth further investigation with
simulated catalogs.

Systematics associated with cluster selection are somewhat mitigated by our
use of a single $M_{\rm obs}$ threshold instead of multiple bins, since
clusters with $M_{\rm obs}$ far above threshold will remain in the sample
regardless of fluctuations from noise, projection, or anisotropy. However, a
large fraction of clusters are close to the threshold, so these effects still
matter. In Appendix B we examine the impact of splitting a mass threshold
sample into two bins, with the addition of nuisance parameters describing
$M_{\rm obs}$ and $\sigma_{\rm lnM}$ at the bin boundary. We find that there
is no gain in cosmological parameter precision. Examination of multiple
thresholds or multiple bins may be valuable for identifying and diagnosing
systematics — at a minimum, cosmological parameters derived from samples with
different $M_{\rm obs}$ thresholds should be mutually consistent. We are not
convinced, however, that there is significant cosmological information in the
use of multiple $M_{\rm obs}$ bins, unless one adopts an unjustifiably
restrictive form for the mass–observable relation such as a power law with
constant scatter.

###  5.3 Ground vs. Space

Space-based observations from Roman or Euclid offer some significant
advantages relative to ground-based imaging such as LSST. The sharp and stable
PSF available from a space platform allows tighter control of shear
calibration uncertainties and better deblending of sources, an effect that may
be particularly important in cluster environments. The combination of space-
based IR and ground-based optical photometry allows better photo-$z$
determinations than either alone. The baseline design of the Roman HLS focuses
on weak lensing systematics control as the primary requirement, choosing
depth, multi-band coverage, and multiple dithers and roll angles over the
greater statistical power that would come from a wider and shallower survey.
The most effective cosmological studies may well come from analyses that use
overlap of the HLS with LSST and Euclid to calibrate systematic effects, and
then leverage the statistics of the larger area surveys. Eifler et al. (2020b)
discuss survey strategies that would enable Roman to cover the entire LSST
footprint with either a 5-month shallow W-band survey that would support
deblending and source modeling or a 1.5-year deep H-band survey that would
enable independent weak lensing shape measurements.

In this spirit, we note that medium-scale Guest Observer programs with Roman
could complement the HLS and substantially amplify its power for cluster weak
lensing cosmology. The key factors that motivate such a program are (1) that a
single Roman field of view is large enough to encompass several Mpc around a
moderate redshift cluster, and (2) that the typical angular separation of
clusters is much larger than this field of view. In the concept described by
Akeson et al. (2019), the HLS observations consist of 8 or 9 exposures of 140
seconds in each band (Y, J, H, F184). The characteristic exposure depth used
for forecasts (including ours) is taken to be $5\times 140=700$ seconds, as
individual points on the sky fall into chip gaps on some exposures. An
individual cluster can therefore be observed to the HLS depth in the H-band in
approximately 1000 seconds, depending on overheads, dither patterns, and other
details. These observations would cover a roughly 0.25 deg radius around each
cluster, corresponding to 3.1, 5.8, and 7.7 comoving $h^{-1}\rm Mpc$ at
$z=0.25$, 0.5, and 0.7, respectively. Because the cosmological analysis uses
the stacked weak lensing signal, homogeneous coverage around each individual
cluster is not necessary.

An observing program totaling 2.6 Msec (1 month) could obtain weak lensing
observations of 2600 clusters. These could, for example, be the eRosita
selected clusters above an $M_{\rm obs}$ threshold corresponding to $2\times
10^{14}~{}h^{-1}\rm M_{\odot}$ in the redshift range $z=0.65-0.75$, over
$14,000\deg^{2}$ (see Figure 11 below). If we add these clusters to our
$2000\deg^{2}$ Stage IV deep forecast, then the constraint on
$\sigma_{8}(z=0.7)$ improves from 0.60% to 0.33%, assuming a mass threshold of
$1\times 10^{14}~{}h^{-1}\rm M_{\odot}$ in the HLS region and that
$\Delta\sigma_{\rm lnM}=0.01$ for both samples. This would be a substantial
gain for a program requiring only about 1/15 of the HLS observing time. The
impact on the aggregate $\sigma_{8}$ precision is small (0.24% vs. 0.22%)
because the HLS observes $M_{\rm obs}\geq 1\times 10^{14}~{}h^{-1}\rm
M_{\odot}$ clusters at all redshifts (23000 total). However, a virtue of
cluster weak lensing relative to cosmic shear is the ability to target a
specific redshift range (or halo mass range) that could be highlighted for
investigation by future data. If tensions with cosmological models appear to
peak at low redshift, then the most effective use of a targeted 1-month survey
might be to observe all 2600 $M_{\rm obs}\geq 4\times 10^{14}~{}h^{-1}\rm
M_{\odot}$ eRosita clusters with $z=0.25-0.45$ over $20,000\deg^{2}$. Such a
survey would sharpen the precision on $\sigma_{8}(z=0.25-0.45)$ from HLS
clusters from 0.63% to 0.35%. A longer duration cluster survey could achieve
correspondingly larger gains.

Figure 9 shows the improved $\sigma_{8}$ constraints vs. the number of extra
clusters observed at $z=0.7$. We show three different $M_{\rm obs}$ thresholds
corresponding to $M_{\rm eq}^{\rm th}=10^{14}$, $2\times 10^{14}$, and
$4\times 10^{14}~{}h^{-1}\rm M_{\odot}$. The $4\times 10^{14}$ selection has
approximately 1500 clusters in the entire sky in $0.65<z<0.75$. If we target
clusters individually, a higher mass threshold would lead to larger
improvement in $\sigma_{8}$ constraint. The vertical lines indicate 2600 and
5000 clusters. We also add a set of dashed lines that show the constraints
achievable with targeted observations alone, before consideration of the HLS.

While even Stage III surveys have the imaging depth needed to identify optical
clusters with $M_{\rm obs}\approx 10^{14}~{}h^{-1}\rm M_{\odot}$ (see Weinberg
et al. 2013, Figure 27), future studies of systematic effects in cluster
selection may show that the threshold for cluster populations that can be
effectively used for precision cosmology is higher. In this case the expected
$\sigma_{8}(z)$ precision from the Roman HLS is lower (e.g., Figure 7), but
the relative gains from a targeted survey are larger.

Figure 9: Improvement of $\sigma_{8}(z=0.7)$ constraints by adding an extended
targeted cluster survey using Roman on top of HLS, assuming $\Delta\sigma_{\rm
lnM}$ = 0.01 for both HLS and targeted observations. Targeting 2600 clusters
above $10^{14}~{}h^{-1}\rm M_{\odot}$ (approximately 2.6 Msec) can improve the
$\sigma_{8}$ constraints by approximately a factor of 2 at this redshift.
Dashed lines show the constraints from the target observations alone. Vertical
lines mark 2600 and 5000 clusters.

##  6 Summary

Current and upcoming optical imaging surveys will provide unprecedented
precision for weak gravitational lensing, making the combination of cluster
weak lensing and cluster abundances one of the most powerful probes of the
growth of structure. In this work, we explore systematically how the
cosmological constraining power of a cluster weak lensing survey depends on
the cluster sample size, the lensing source density, and the prior on the
scatter between cluster observables and mass. In contrast to most treatments
of cluster cosmology, we advocate an approach, analogous to that of
galaxy–galaxy lensing, in which the observables are the mean cluster space
density $\bar{n}$ and the mean weak lensing profile $\Delta\Sigma(r_{\rm p})$
of clusters above a threshold in an observable mass proxy $M_{\rm obs}$, which
could be galaxy richness, X-ray temperature, SZ decrement, or some other
property. In this formulation, the primary astrophysical nuisance parameter is
the scatter $\sigma_{\rm lnM}$ between ${\ln M_{\rm obs}}$ and the true halo
mass $\rm lnM$ near the threshold. This scatter may vary with redshift, but
focusing on a single thresholded sample in each redshift bin avoids the need
to adopt a parameterized form of the mean $M_{\rm obs}-M$ relation or the
dependence of scatter on $M_{\rm obs}$. In Appendix B we argue that any
statistical gain from considering multiple mass bins or thresholds is minimal
at best.

To characterize the constraining power of cluster weak lensing surveys, we
focus on the precision in measuring the matter fluctuation amplitude
$\sigma_{8}(z)$ with other cosmological parameters held fixed. In our approach
one goes directly from the observed $\bar{n}$ and $\Delta\Sigma(r_{\rm p})$
and a prior on $\sigma_{\rm lnM}$ to cosmological parameters such as
$\sigma_{8}$. The mass corresponding to the threshold $M_{\rm obs}$ and the
mean mass of clusters above the threshold are useful by-products of the
analysis. To build intuition we first examine expected constraints from a
fixed comoving survey volume of $(1h^{-1}\rm Gpc)^{3}$ at various redshifts,
for different assumptions about weak lensing shape noise and the prior on
$\sigma_{\rm lnM}$ (Figures 3, 4). We then forecast statistical constraints
(Figures 6-8) for three different survey scenarios that correspond
approximately to DES (Stage III wide), the Roman HLS (Stage IV deep), and LSST
(Stage IV wide). Appendix C provides analytic results that approximately
reproduce our numerical results and allow rapid back of the envelope estimates
for alternative survey assumptions or even different methods of estimating
cluster mass.

If the scatter is known to $\Delta\sigma_{\rm lnM}=0.01$, then a survey of
$M\geq 10^{14}~{}h^{-1}\rm M_{\odot}$ clusters yields an aggregate precision
on $\sigma_{8}$ of 0.26%, 0.24%, and 0.10% for DES-like, HLS-like, and LSST-
like survey parameters, respectively. For a single $\Delta z=0.1$ redshift bin
centered at $z=0.7$, the corresponding errors on $\sigma_{8}(z=0.7)$ are
0.91%, 0.83%, and 0.60%. The errors on the mean cluster mass in this redshift
bin are 1.9%, 1.3%, and 0.57%. If the cluster mass threshold is $2\times
10^{14}~{}h^{-1}\rm M_{\odot}$ instead of $1\times 10^{14}~{}h^{-1}\rm
M_{\odot}$, then the constraints on $\sigma_{8}(z)$ degrade by roughly a
factor of two.

For a statistics-limited survey with the same cluster mass threshold, the
greater area of an LSST-like survey (or a Euclid-like survey) wins easily over
the greater depth of an HLS-like survey. (See Eifler et al. 2020b for a
similar comparison of cosmic shear and a detailed discussion of Roman+LSST
synergies and of possible alternative strategies for the HLS.) However,
realizing this advantage requires accurate external knowledge of $\sigma_{\rm
lnM}$. For a DES-like or HLS-like survey, a prior with $\Delta\sigma_{\rm
lnM}=0.05$ is sufficient for limited (20%) degradation of the aggregate
$\sigma_{8}$ constraint, but for an LSST-like survey the corresponding
requirement is $\Delta\sigma_{\rm lnM}=0.016$. We emphasize that our forecasts
allow a different, independent value of $\sigma_{\rm lnM}$ in each redshift
bin and do not assume any specific form of redshift evolution.

Cluster weak lensing surveys can yield stringent tests of cosmological model
predictions of matter clustering, and thus of dark energy and modified gravity
theories. Our results highlight the importance of constraining $\sigma_{\rm
lnM}$, which will require careful combination of multi-wavelength data sets
and numerical simulations. For a DES-like survey, Salcedo et al. (2020)
forecast that the combination of cluster–galaxy cross-correlation and galaxy
auto-correlation can achieve $\Delta\sigma_{\rm lnM}\approx 0.1$, which is
already sufficient for an 0.5% constraint on $\sigma_{8}$ (see Figure 8).

Realizing the potential of cluster cosmology will require strict control of a
variety of observational and theoretical systematics (Section 5.2), of which
anisotropic cluster selection is perhaps the most challenging. Baryonic
physics impacts on cluster mass profiles are an important theoretical
uncertainty, but limiting the weak lensing analysis to $r_{\rm p}>0.5\
h^{-1}\rm Mpc$ or even $1.0\ h^{-1}\rm Mpc$ produces little degradation of
statistical precision (Figure 2), which suggests that the effects of baryons
and cluster mis-centering can be adequately mitigated by including
$\Delta\Sigma(r_{\rm p})$ measurements out to large $r_{\rm p}$. Combinations
of space-based and ground-based data will be valuable for controlling other
observational systematics while leveraging the large area of ground-based
surveys. We also note (Section 5.3) that relatively short cluster observing
programs with Roman Space Telescope can achieve tight constraints on
$\sigma_{8}$ in a targeted redshift range, which could be valuable for
validating other results or for developing deeper understanding of departures
from $\Lambda$CDM predictions. For example, a roughly 1-month observing
program could sharpen the HLS constraint on $\sigma_{8}(z=0.7)$ by about a
factor of two, or make a 0.5% measurement of $\sigma_{8}(z=0.7)$ on its own.

If statistical limits can be achieved, then the expected constraints on cosmic
structure growth from cluster weak lensing are competitive with and
complementary to those from cosmic shear with the same weak lensing data
(Oguri & Takada, 2011; Weinberg et al., 2013). The value of cluster weak
lensing as an independent probe becomes much higher still in models with
modified gravity or decaying dark matter, which predict distinctive signatures
in matter clustering as a function of environment, scale, and redshift. If we
live in such a universe, then cluster weak lensing will be a critical tool for
characterising its rich phenomenology and revealing its underlying physics.

We thank Lehman Garrison for providing the Abacus Cosmos simulation suite, and
Ying Zu and Chun-Hao To for helpful discussions and comments. This work was
supported in part by NSF Grants AST-1516997 and AST-2009735 and NASA Grant
15-WFIRST15-0008. DHW acknowledges additional support at the IAS from the W.
M. Keck Foundation. The computations in this paper were run on the CCAPP condo
of the Pitzer Cluster at the Ohio Supercomputer Center (1987).

##  Appendix A Survey volume and number of galaxy clusters

Figure 10: Comoving volume per redshift bin of $\Delta z=0.1$, for various
survey areas. Figure 11: Halo number counts as a function of redshift, for
various mass thresholds. Left: Number counts per 1 $(h^{-1}\rm Gpc)^{3}$
Right: Number counts in redshift bins of $\Delta z=0.1$ in the entire sky.

Here we show a few basic numbers for survey volume and cluster counts that are
useful for approximate calculations. Figure 10 shows the comoving volume as a
function of redshift, in redshift bins $\Delta z=0.1$, for three survey areas:
2000, 5000, and 18000 deg2. Figure 11 shows the halo number counts (zero
scatter) above a given mass threshold: the left-hand panel shows the number
counts per comoving volume of 1 $(h^{-1}\rm Gpc)^{3}$, and the right-hand
panel shows the number counts per $\Delta z=0.1$ in the entire sky.

##  Appendix B Thresholded vs. binned samples

Figure 12: Constraints from thresholded vs. binned samples. The black
horizontal line corresponds to the baseline assumption $z=0.5$, $M_{\rm
eq}^{\rm th}=10^{14}~{}h^{-1}\rm M_{\odot}$, and $\Delta\sigma_{\rm lnM}=0.1$,
with a 1 $(h^{-1}\rm Gpc)^{3}$ survey volume. We split the sample into two
according to the $M_{\rm split}$ (indicated by the $x$-axis). The blue curve
corresponds to the sample below $M_{\rm split}$, and the orange curve
corresponds to the sample above $M_{\rm split}$. The green curve represents
the result from combining the two bins. Splitting the sample into two bins
requires extra nuisance parameters and does not present an advantage over
using the entire sample.

In this work we combine all clusters above a threshold of the observable
$M_{\rm obs}^{\rm th}$. However, many publications have put clusters into
multiple bins of observables (e.g., Abbott et al., 2020). In this section, we
demonstrate that under the most general assumption for the mass–observable
relation (that is, the mass scaling and scatter at the threshold or the bin
edges), introducing more bins would lead to more nuisance parameters and
present no advantage over a thresholded sample.

We compare the following two cases:

  1. 1.

Case 1: one observable threshold, which corresponds to the (unknown) mass
scale $M_{1}$. The nuisance parameters are $M_{1}$ and $\sigma_{\rm lnM,1}$,
with $\sigma_{8}$ the desired cosmological parameter.

  2. 2.

Case 2: two bins. The first sample corresponds to mass between $M_{1}$ and
$M_{2}$, and the second sample corresponds to clusters above mass $M_{2}$. The
nuisance parameters are ($M_{1}$, $M_{2}$, $\sigma_{\rm lnM,1}$, $\sigma_{\rm
lnM,2}$).

For the first case, we only need to constrain the nuisance parameters near the
selection threshold, while in the second case we need to constrain the
nuisance parameters at the two selection boundaries. For the latter, we assume
that statistical errors in $\bar{n}$ and $\Delta\Sigma(r_{\rm p})$ for the two
bins are uncorrelated with each other. Any correlation between the two samples
will further weaken the constraints.

Figure 12 demonstrates the constraints on $\sigma_{8}$ using one thresholded
sample vs. two binned samples. The black horizontal line corresponds to the
constraint from the thresholded sample (Case 1). The blue and orange curves
correspond to the $\sigma_{8}$ constraints from the first and the second bins
in Case 2, respectively, and the $x$-axis $M_{\rm split}$ corresponds to where
we split the sample into two bins (the changing value of $M_{2}$). The first
bin (blue curve) has a binning equivalent to $(10^{14},\ M_{\rm split})$, and
the second bin (orange curves) has a binning equivalent to $(M_{\rm split},\
\infty)$.

For the largest $M_{\rm split}$, the first bin is equivalent to the
thresholded case, and the second bin has little information; for the smallest
$M_{\rm split}$, the second bin is equivalent to the thresholded case, and the
first bin provides little information. For the thresholded case, we set the
prior on scatter $\Delta\sigma_{\rm lnM}=0.1$; for the 2-bin case, we set the
prior $\Delta\sigma_{\rm lnM,1}=\Delta\sigma_{\rm
lnM,2}=\sqrt{2}\times\Delta\sigma_{\rm lnM}$. When we combine the two bins,
the result is very close to the thresholded case.

When $M_{\rm split}$ is large, $\sigma_{\rm lnM,1}$ in Case 2 plays the same
role as $\sigma_{\rm lnM}$ in Case 1, and because we assume $\Delta\sigma_{\rm
lnM,1}=\sqrt{2}\Delta\sigma_{\rm lnM}$, Case 2 shows a slightly weaker
constraint than a single threshold. When $M_{\rm split}$ is small, both
$\sigma_{\rm lnM,1}$ and $\sigma_{\rm lnM,2}$ in Case 2 describe the scatter
near $10^{14}~{}h^{-1}\rm M_{\odot}$, and the combination of the two is
equivalent to the prior on $\Delta\sigma_{\rm lnM}$.

Our results show that, if we use the most general assumption for the
mass–observable relation, splitting the sample into bins would introduce extra
nuisance parameters and does not improve the constraining power. If one adopts
a specific form for the mass–observable relation, such as a power-law mean
relation with constant logarithmic scatter, then a binned analysis may appear
to yield a stronger cosmological constraint. However, this constraint should
be suspect unless the assumed form can be robustly justified. A useful test
for spurious information is to see whether number counts yield a cosmological
constraint without calibration of the mean mass–observable relation (e.g.,
from $\Delta\Sigma$). Unless the form of $P(M_{\rm obs}|M)$ is known almost
perfectly, number counts alone have no useful cosmological information because
for any reasonable cosmology one could reproduce the observed counts by
abundance matching.

##  Appendix C Analytical estimate of $\sigma_{8}$ constraints

Figure 13:  Coefficients for analytically calculating the $\sigma_{8}$
constraints (Equation (C3)), which can be expressed as the linear combination
of the square of the uncertainties of number density, mean mass, and scatter,
with the weights given by these coefficients. The weight on mass uncertainty
($C_{M}$) is particularly large, indicating the crucial role of mass
uncertainties in determining the constraining power of clusters.  Figure 14:
Example of trade-off between the error bars on cluster counts, mean mass, and
scatter, based on Equation (C3), for $z=0.5$, $10^{14}~{}h^{-1}\rm M_{\odot}$
threshold, and 1 $(h^{-1}\rm Gpc)^{3}$. Each panel corresponds a fixed error
bar in mean mass and a lower limit of $\Delta\ln\sigma_{8}$, and different
contours correspond to the $\Delta\ln\sigma_{8}$ associated with given error
bars on number counts ($x$-axis) and scatter ($y$-axis). For example, if
$\Delta\sigma_{\rm lnM}=0.03$ (middle panel), the best possible constraint is
$\Delta\ln\sigma_{8}=0.012$, but errors of $\sigma_{N}=0.1$ or
$\sigma_{\sigma}=0.1$ degrade the constraint to $\Delta\ln\sigma_{8}=0.022$.
Figure 15:  Fractional difference between $\sigma_{8}$ constraints calculated
using mean mass and using the full lensing profile. Both cases include the
number density information. The mean mass calculation results in slightly
weaker constraints than the full lensing calculation, indicating the
information lost due to compressing lensing information to mean mass. The
difference is larger when the prior on scatter is weaker, but in general the
mean-mass calculation is a good approximation and can be easily calculated
using the analytical formula (C3).

In the main text, we present the constraining power of lensing on the mean
mass of the sample. In this appendix, we will show that the process of
compressing the lensing information from all scales into the constraints on
mean mass results in little loss of information for most cases, and one can
usually fit the number density and mean mass simultaneously to obtain
$\sigma_{8}$ constraints. The advantage of the latter approach is that we can
estimate the constraints on $\sigma_{8}$ analytically using the error bars on
number density, mean mass, and scatter. Let us denote

$\displaystyle N_{1}=\frac{d\ln\langle n\rangle}{d\ln\sigma_{8}}~{},\quad
N_{2}=\frac{d\ln\langle n\rangle}{d\ln M_{\rm obs}^{\rm th}}~{},\quad
N_{3}=\frac{d\ln\langle n\rangle}{d\sigma_{\rm lnM}}~{},$ (C1) $\displaystyle
M_{1}=\frac{d\left\langle{\rm lnM}\right\rangle}{d\ln\sigma_{8}}~{},\quad
M_{2}=\frac{d\left\langle{\rm lnM}\right\rangle}{d\ln M_{\rm obs}^{\rm
th}}~{},\quad M_{3}=\frac{d\left\langle{\rm lnM}\right\rangle}{d\sigma_{\rm
lnM}}~{},$

and

$\sigma_{N}=\Delta\ln\langle
n\rangle~{},\quad\sigma_{M}=\Delta\left\langle{\rm
lnM}\right\rangle~{},\quad\sigma_{\sigma}=\Delta\sigma_{\rm lnM}\ .$ (C2)

We can calculate the 3$\times$3 Fisher matrix and its inverse analytically.
The constraint on $\ln\sigma_{8}$ is given by

$\displaystyle(\Delta\ln\sigma_{8})^{2}$
$\displaystyle=\frac{N_{2}^{2}\sigma_{M}^{2}+M_{2}^{2}\sigma_{N}^{2}+(N_{2}M_{3}-M_{2}N_{3})^{2}\sigma_{\sigma}^{2}}{(N_{1}M_{2}-M_{1}N_{2})^{2}}$
(C3)
$\displaystyle=C_{N}\sigma_{N}^{2}+C_{M}\sigma_{M}^{2}+C_{\sigma}\sigma_{\sigma}^{2}\
.$

That is, $\Delta\ln\sigma_{8}$ is the weighted sum of the square of the
uncertainties of number counts, mean mass, and scatter. Figure 13 shows the
coefficients $C_{N}$, $C_{M}$, and $C_{\sigma}$, as a function of redshift
($x$-axis), for various mass thresholds (different lines). The coefficients
$C_{M}$ are always much larger than $C_{N}$ and $C_{\sigma}$, indicating that
the uncertainties of mass typically dominate the error budget. Both $C_{N}$
and $C_{M}$ decrease with mass, indicating that high-mass clusters are more
sensitive to the reduction of uncertainties. Nevertheless, since low-mass
clusters have much smaller uncertainties, they have the highest constraining
power.

Figure 14 shows an example of estimating the constraining power for a survey
using this formula. We use $z=0.5$, $M_{\rm eq}^{\rm th}=10^{14}~{}h^{-1}\rm
M_{\odot}$, and 1 $(h^{-1}\rm Gpc)^{3}$. For each panel, we choose a value of
$\sigma_{M}$, which leads to a lower limit of $\Delta\ln\sigma_{8}$
corresponding to negligible $\sigma_{N}$ and $\sigma_{\sigma}$. In each panel,
the contours show $\Delta\ln\sigma_{8}$ for given $\sigma_{N}$ ($x$-axis) and
$\sigma_{\sigma}$ ($y$-axis) values. Comparing the contours with the same
value among these panels shows us the trade-off between these uncertainties.

How well does Equation (C3) work? Figure 15 compares the analytical results
(equivalent to compressing all lensing signals into mean mass constraints) and
the full lensing calculation. The former always gives a slightly larger
$\ln\sigma_{8}$ error bar, indicating some loss of information due to this
compression. We show the fractional difference between these two approaches,
for various levels for shape noise and prior on scatter. When the scatter is
tightly constrained (solid), the difference is less than 10%. When the prior
on scatter is weak, the analytical formula loses more constraining power in
this data compression. The derivative of lensing with respect to scatter
depends on radius (Figure 1), and when the prior is not well constrained this
scale dependence can provide extra information on scatter compared with using
the mean mass constraints alone. Figure 15 might not look impressive, but it
shows that the analytical formula we provide (Equation (C3)) can be used for
reasonably accurate back of the envelope calculations for estimating the
constraining power on $\sigma_{8}$ of a given survey condition, without
computing the full Fisher matrix. It also allows comparison of our results to
those from other methods of mass calibration, e.g., based on X-ray or SZ
observables. Equation (C3) can also be used to make an approximate estimate
when a systematic error on the mean mass scale, number counts, or
mass–observable scatter would influence results at a statistically important
level.

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