# Combining cosmic shear data with correlated photo-$z$ uncertainties:
constraints from DESY1 and HSC-DR1

Carlos García-García  David Alonso  Pedro G. Ferreira  Boryana Hadzhiyska
Andrina Nicola  Carles Sánchez  Anže Slosar

###### Abstract

An accurate calibration of the source redshift distribution $p(z)$ is a key
aspect in the analysis of cosmic shear data. This, one way or another,
requires the use of spectroscopic or high-quality photometric samples.
However, the difficulty to obtain colour-complete spectroscopic samples
matching the depth of weak lensing catalogs means that the analyses of
different cosmic shear datasets often use the same samples for redshift
calibration. This introduces a source of statistical and systematic
uncertainty that is highly correlated across different weak lensing datasets,
and which must be accurately characterised and propagated in order to obtain
robust cosmological constraints from their combination. In this paper we
introduce a method to quantify and propagate the uncertainties on the source
redshift distribution in two different surveys sharing the same calibrating
sample. The method is based on an approximate analytical marginalisation of
the $p(z)$ statistical uncertainties and the correlated marginalisation of
residual systematics. We apply this method to the combined analysis of cosmic
shear data from the DESY1 data release and the HSC-DR1 data, using the COSMOS
30-band catalog as a common redshift calibration sample. We find that,
although there is significant correlation in the uncertainties on the redshift
distributions of both samples, this does not change the final constraints on
cosmological parameters significantly. The same is true also for the impact of
residual systematic uncertainties from the errors in the COSMOS 30-band
photometric redshifts. Additionally, we show that these effects will still be
negligible in Stage-IV datasets. Finally, the combination of DESY1 and HSC-DR1
allows us to constrain the “clumpiness” parameter to
$S_{8}=0.768^{+0.021}_{-0.017}$. This corresponds to a $\sim\sqrt{2}$
improvement in uncertainties with respect to either DES or HSC alone.

##  1 Introduction

The gravitational lensing of background galaxies is one of the most reliable
probes to study the late-time growth of structure, as well as the geometry of
our Universe. Imaging weak lensing surveys such as the Dark Energy Survey
(DES) [1, 2], the Hyper Suprime-Cam (HSC) [3, 4], and the Kilo-Degree Survey
(KiDS) [5], have produced increasingly tight direct constraints on the
amplitude of matter density inhomogeneities, and on the abundance of non-
relativistic matter at $z\lesssim 1$. Although qualitatively in line with the
standard cosmological model ($\Lambda$CDM) favoured by observations of the CMB
at early times, these measurements are quantitatively in tension with
constraints from Planck [6, 7] at the $\sim 3\sigma$ level [8, 9, 10]. If
real, this tension could be a smoking gun signature of new physics [11, 12,
13, 14, 15], although it might also be explained by a poor understanding of
baryonic effects [16, 17], systematic uncertainties in the characterisation of
the source redshift distribution [8], or unknown systematic effect in the CMB
measurements. These experiments will be superseded within the next decade by
so-called “Stage-IV” surveys, such as the ground-based Rubin Observatory
Legacy Survey of Space and Time (LSST) [18, 19], the Euclid satellite mission
[20], and the Nancy Grace Roman Space Telescope [21]. These surveys will
improve the width, depth, and angular resolution of current experiments, and
will be vital in ascertaining the source of this tension.

Carrying out combined analyses that add the statistical power of different
experiments [22, 8, 10, 23, 16, 24, 25] is key to improve their individual
cosmological constraints, and to test the robustness of these constraints to
instrumental or observational systematics. This is often challenging when
surveys targeting potentially different source populations share part of the
sky, due to the non-trivial survey geometry effects, and the need to
characterise the noise properties of the sources common to both datasets.
However, even disjoint observations may share correlated systematic and
statistical uncertainties due to the use of common external datasets, which
must be carefully modelled and propagated. This is the case of photometric
redshift uncertainties.

A precise calibration of the redshift distribution $p(z)$ of the source sample
is a key component of the cosmological analysis. To achieve it, weak lensing
surveys rely on the use of external spectroscopic or high-quality photometric
datasets. The redshift distribution can then be estimated via direct
calibration or through a clustering redshifts approach. Direct calibration
involves using directly the redshift distribution of the spectroscopic sample.
Since the spectroscopic sample is often not representative of the sample used
to measure the weak lensing signal, care must be taken to re-weight the former
to correct for its different color-magnitude distribution. A variety of
methods have been proposed to do so [26, 27]. In the clustering redshifts
method, the weak lensing sample is cross-correlated with the spectroscopic
sample in narrow bins of redshift. The amplitude of the resulting cross-
correlation is proportional to the redshift distribution of the photometric
sample. Care must be taken then to account for the evolution of the
photometric sample’s bias within the redshift range of interest, and to
quantify the potential contamination from magnification bias in the cross-
correlation [28, 29, 30, 31, 32]. Regardless of how the $p(z)$ is measured,
since large spectroscopic samples of the required depth are often difficult to
come by, it is likely that different weak lensing surveys will make use of the
same (or at least partially overlapping) spectroscopic samples. Thus, the
statistical uncertainties in the determination of redshift distributions, as
well as any systematic trends in the shared spectroscopic sample, will be
correlated between different cosmic shear analyses. As an example of this, the
analysis of the first-year data from the Dark Energy Survey (DESY1 hereafter),
and the first data release from HSC (HSC-DR1 hereafter), both relied heavily
on data from the COSMOS 30-band photometric catalog of [33] (COSMOS2015
hereafter) for $p(z)$ calibration. Likewise, it is reasonable to expect that
future surveys will aim to use the best spectroscopic dataset available at the
time of their analysis (e.g. from the Dark Energy Spectroscopic Instrument
[34]), and thus the correlation in the $p(z)$ uncertainties of different
datasets will need to be carefully quantified when comparing or combining
their results.

In this paper, we make use of the analytical marginalisation scheme proposed
in [35] to address this problem, developing a method to consistently propagate
$p(z)$ uncertainties for cosmic shear datasets that are otherwise independent.
We apply this method to public data from DESY1 and HSC DR1 in order to combine
the analysis of both datasets, quantifying the impact of correlated redshift
calibration uncertainties.

The paper is structured as follows. Section 2 presents the cosmological theory
background underlying the analysis of cosmic shear surveys, as well as the
method used here to calibrate redshift distributions and propagate their
uncertainties. Section 3 briefly describes the datasets used in our analysis.
The main results from the analysis, both in terms of the impact of correlated
$p(z)$ uncertainties, and the final combined cosmological constraints from
DESY1 and HSC DR1 are presented in Section 4, whereas in Section 5 we study
the expected effect in Stage-IV surveys. Finally, we conclude in Section 6.

##  2 Methods

###  2.1 Theory

The perturbation to observed ellipticities of a sample of galaxies with
redshift distribution $p_{\alpha}(z)$ due to weak gravitational lensing is
captured by the spin-2 “cosmic shear” field $\bm{\gamma}^{\alpha}_{G}$. The
$E$-mode component of this field is then related to the three-dimensional
matter overdensity via a line-of-sight projection with a radial kernel given
by [36]

$q^{\alpha}_{\bm{\gamma}_{G}}(\ell,\chi)=\frac{3}{2}H_{0}^{2}\Omega_{\rm
m}\frac{\chi}{a(\chi)}\int_{z(\chi)}^{\infty}\mathrm{d}z^{\prime}p_{\alpha}(z^{\prime})\frac{\chi(z^{\prime})-\chi}{\chi(z^{\prime})},$
(2.1)

where $H_{0}$ the expansion rate, and $\Omega_{\rm m}$ is the fractional
energy density of non-relativistic matter, both evaluated today. $z(\chi)$ is
the redshift corresponding to a source at comoving radial distance $\chi$.

The angular power spectrum between the cosmic shear signal from two different
samples, $\alpha$ and $\beta$, is then related to the matter power spectrum
$P(k,z)$ via

$C_{\ell}^{\alpha\beta}=G_{\ell}^{2}\int\frac{d\chi}{\chi^{2}}q_{\bm{\gamma}_{G}}^{\alpha}(\chi)q_{\bm{\gamma}_{G}}^{\beta}(\chi)\,P\left(k=\frac{\ell+1/2}{\chi},z(\chi)\right),$
(2.2)

where we have made use of the Limber approximation [37], which holds for broad
radial kernels (as is the case for cosmic shear). The scale-dependent
prefactor

$G_{\ell}\equiv\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}\frac{1}{(\ell+1/2)^{2}}$
(2.3)

accounts for the relation between the cosmic shear field and the angular
Hessian (as opposed to the three-dimensional Laplacian) of the Newtonian
gravitational potential [38].

The intrinsic correlation between true galaxy shapes, caused by local
interactions, is a potential contaminant to the cosmic shear signal from
gravitational lensing. Here we will model this “intrinsic alignment” (IA)
contribution, $\bm{\gamma}_{I}$, using the “linear-non-linear alignment model”
of [39, 40], where the perturbed galaxy ellipticity is proportional to the
local gravitational tidal forces. This amounts to including an additive
contribution to the radial kernel of the form:

$q_{\bm{\gamma}_{I}}^{\alpha}(\chi)=-A_{\rm
IA}(z)p_{\alpha}(z)\frac{dz}{d\chi}.$ (2.4)

We parametrise the IA amplitude as in [41, 42]

$A_{\rm IA}=A_{{\rm IA},0}\left(\frac{1+z}{1+z_{0}}\right)^{\eta_{\rm
IA}}\frac{0.0139\Omega_{\rm m}}{D(z)}.$ (2.5)

Here, $D(z)$ is the linear growth factor normalised to $D(0)=1$, $z_{0}$ is a
pivot redshift that we fix to $z_{0}=0.62$ as in [41, 42], and $A_{{\rm
IA},0}$ and $\eta_{\rm IA}$ are free parameters.

We use the Core Cosmology Library (CCL) [43] to compute all angular power
spectra, and CAMB [44] with HALOFIT [45] to compute the non-linear matter
power spectrum.

###  2.2 Measuring the $p(z)$

A calibrated model for the source redshift distribution $p(z)$ is a central
component of the theoretical prediction for the measured shear power spectra.
Here we use the direct calibration (DIR) method of [26]. Consider two galaxy
samples: a photometric sample $P$ containing the photometry ${\bf m}$
(magnitudes measured in a set of $N_{b}$ bands) of each source, and a
spectroscopic sample $S$, which contains accurate redshift measurements in
addition to ${\bf m}$ for each source. We will assume that $P$ is
representative of the sample of galaxies used in the cosmic shear analysis
(i.e. both follow the same distribution of ${\bf m}$). In practice, $S$ may be
initially an external catalog that does not contain the same photometry as the
photometric sample. In that case, photometry information is obtained by
spatially cross-matching $S$ with another sample (not necessarily the same as
$P$) containing the desired magnitude measurements. The DIR method then
proceeds as follows:

  1. 1.

For each object $i$ in $S$, we find its $N_{S}$ nearest neighbors in $S$, in
the $N_{b}$-dimensional space of ${\bf m}$ using a Euclidean metric to
determine distances. We then record the distance $d$ to the furthest neighbor.
Our fiducial results use $N_{S}=20$ nearest neighbors but we have made sure
they are not affected by this choice.

  2. 2.

For the same object in $S$, we find the number of objects $N_{P,i}$ in $P$
that lie within an $N_{b}$-sphere of the same radius $d$. If any weights were
applied to the cosmic shear catalog in the cosmological analysis (e.g. shape
measurement weights), $N_{P,i}$ should be the sum of weights.

  3. 3.

We then assign a weight $w_{i}=r\,N_{P,i}/N_{S}$ to each object in $S$, where
$r$ is the ratio between the total number of objects in $S$ and the total
(weighted) number of objects in $P$. $r$ is simply a normalisation constant
that does not affect the final inferred redshift distribution.

  4. 4.

Finally, the redshift distribution of $P$ is estimated as the redshift
distribution of $S$ where each source is weighted by $w_{i}$ defined above.

It is worth noting that the DIR method is only guaranteed to provide an
unbiased estimate of the redshift distribution under somewhat idealised
conditions. First, although $S$ need not follow exactly the same color
distribution as the target sample $P$, it must contain a sufficiently high
number of objects in all regions of magnitude space where $P$ has support.
Secondly, the redshift measurements in $S$ are assumed to be precise and
accurate. In particular, for fixed photometry, the redshift success rate
should not depend on the redhift itself. This is rarely true in practice,
particularly at high redshifts and for faint sources. For example, the
COSMOS2015 catalog we will use in this work, uses photometric redshift
measurements estimated from 30-band photometry. Although the resulting
redshifts are significantly more accurate than photo-$z$s estimated from
$O(4-5)$ bands, we can expect a number of catastrophic outliers, particularly
at high redshifts [8]. Finally, if $S$ is limited to a small sky patch (as is
also the case for COSMOS2015), the $p(z)$ estimated via DIR will be subject to
significant sample variance uncertainties that may be difficult to model in
practice. In summary, even after estimating the source redshift distributions
and their statistical uncertainties, we must allow for potential residual
systematic deviations in the $p(z)$.

###  2.3 Marginalisation over $p(z)$ statistical uncertainties

We marginalize analytically over the uncertainty on the $p(z)$ following [35].
The method is based on linearising the dependence of the model on the
deviations with respect to a fiducial $p(z)$, which then allows us to
marginalise over these deviations analytically. This marginalisation results
in an additive contribution to the covariance matrix of the data vector (i.e.
the set of angular power spectra), which effectively increases the uncertainty
of the modes that are most affected by $p(z)$ fluctuations.

The method proceeds as follows. Let ${\bf N}$ be a vector of parameters
consisting of the amplitudes of the redshift distributions for all the galaxy
samples analysed (e.g. the histogram heights for each $p(z)$ estimated via
DIR). Let $\bar{\bf N}$ be the best-fit measurements of ${\bf N}$, and ${\sf
P}$ the covariance matrix of these measurements. ${\sf P}$ may incorporate
both statistical uncertainties and any prior constraints (e.g. imposing
smoothness of the underlying redshift distribution). We will only consider
statistical uncertainties here. Finally, let ${\sf T}$ be the matrix
characterising the response of the data vector to variations in ${\bf N}$:

${\sf T}\equiv\frac{d{\bf C}}{d{\bf N}},$ (2.6)

where ${\bf C}$ is the vector containing all angular power spectra being
analysed. Then, analytically marginalising over ${\bf N}$, assuming a linear
dependence on the deviations with respect to $\bar{\bf N}$, is equivalent to
modifying the covariance matrix of ${\bf C}$ as:

${\sf Cov}_{\bf C}\rightarrow{\sf Cov}_{\bf C}+{\sf T}{\sf P}{\sf T}^{T}.$
(2.7)

As in [46], we use an analytical estimate of the covariance of ${\bf N}$. The
matrix ${\sf P}$ has two contributions: the covariance ${\sf Cov}^{\delta}$
due to fluctuations in the matter density traced by the galaxies in the
observed patch and a shot-noise contribution ${\sf Cov}^{S}$; i.e.

${\sf P}={\sf Cov}^{\delta}+{\sf Cov}^{S}.$ (2.8)

The shot noise part is simply given by

${\sf Cov}^{S}_{ij}=\delta_{ij}N_{i},$ (2.9)

where $N_{i}$ is the number of sources in the corresponding histogram bin. The
sample variance contribution is

${\sf
Cov}^{\delta}_{ij}=\frac{N_{i}N_{j}}{2\pi^{2}}\int_{0}^{\infty}dk_{\parallel}\cos(k_{\parallel}(\chi_{i}-\chi_{j}))\int_{0}^{\infty}dk_{\perp}k_{\perp}W_{i}(k_{\parallel},k_{\perp})W_{j}(k_{\parallel},k_{\perp})P_{gg}({\bf
k})\,.$ (2.10)

In this expression, $k_{\parallel}$ and $k_{\perp}$ are the components of the
Fourier-space wavevector ${\bf k}$ parallel and perpendicular to the line of
sight, and $\chi_{i}$ is the radial comoving distance to redshift $z_{i}$.
$W_{i}$ is the Fourier-space window function for the 3D volume covered by each
redshift interval over which ${\bf N}$ has been measured. Finally, $P_{gg}$ is
the galaxy power spectrum in redshift space, which we model as [47]

$P_{gg}({\bf
k},z)=\left(b_{g}(z)+f(z)\frac{k_{\parallel}^{2}}{k^{2}}\right)^{2}\,P_{mm}(k,z).$
(2.11)

Here, $f(z)$ is the linear growth rate and $b_{g}(z)$ is the linear galaxy
bias. For $b_{g}$, we use the fit of [48] for flux-limited samples, which
takes the form

$b_{g}(z)=(b_{1}(m_{\rm lim}-24)+b_{0})D(z)^{\alpha},$ (2.12)

with $b_{1}=-0.0624$, $b_{0}=0.8346$, and $\alpha=-1.30$. $m_{\rm lim}$ is the
limiting magnitude in the $i$ band. We use $m_{\rm lim}=24.5$ for HSC and
$m_{\rm lim}=22.5$ for DES. This fitting function was derived from the HSC
galaxy clustering data, but we have verified that using this fit or fitting
the bias directly to the galaxy clustering auto-correlation of the DES sample
leads to negligible differences in our final analysis.

We model the window function $W_{i}$ as that of a top-hat cylinder with a
comoving width $\Delta\chi$ given by the redshift interval used to define the
redshift distribution histogram, and a radius $R=\chi\theta_{\rm sky}$, where
$A_{\rm sky}=\pi\theta_{\rm sky}^{2}=2\,{\rm deg}^{2}$ is the area of the
COSMOS field used in our analysis. The resulting functional form for $W_{i}$
is

$W_{i}(k_{\parallel},k_{\perp})=j_{0}(k_{\parallel}\Delta\chi_{i}/2)\,\frac{2J_{1}(k_{\perp}\chi_{i}\theta_{\rm
sky})}{k_{\perp}\chi_{i}\theta_{\rm sky}},$ (2.13)

where $j_{0}$ is the zeroth-order spherical Bessel function and $J_{1}$ is the
first-order cylindrical Bessel function. Since $\theta_{{\rm
sky}}\propto\sqrt{A_{{\rm sky}}}$, a small change on $A_{\rm sky}$ barely
changes $\theta_{\rm sky}$ and, therefore, $W_{i}$.

Finally, in order to avoid underestimating the errors due to the analytical
treatment used here, we allow ourselves the possibility to enlarge the
diagonal of ${\sf P}$ by a factor $f_{P}$. The role of $f_{P}$ will be
discussed in Section 4.1.

###  2.4 Marginalisation over $p(z)$ systematic uncertainties

As we argued earlier, the fiducial redshift distribution $\bar{\bf N}$ may be
subject to systematic uncertainties due to imperfections in the calibration
sample. A particular concern in our case is the use of photometric redshifts
in the COSMOS2015 catalog. Refs. [8, 49] have argued that this could cause a
coherent shift in the means of all redshift distributions for a given survey.
If a model for these residual systematics can be obtained (e.g. through the
use of clustering redshifts, or from a higher-quality spectroscopic sample),
their impact can be taken into account by marginalising over the free
parameters of this model. A specific example of this will be described in
Section 4.2, where we will use the COSMOS-PAUS catalog of [50] to characterise
a coherent redshift shift in the COSMOS2015 catalog.

When combining two different surveys calibrated with the same sample, these
systematics will also be correlated. The effect, however, may not be the same
for both surveys, since their source samples will in general cover different
regions of color space. We can, however, explore the two limiting regimes of
perfect correlation and complete independence. In the first case, we use the
same nuisance parameters to characterise the systematic shift in the redshift
distributions of both samples, whereas in the latter we use an independent set
of parameters for each survey. If the final cosmological constraints are
sensitive to this choice, a more sophisticated model for the correlation
between the systematic effect in both surveys is necessary. In practice, we
will show that this is not the case.

###  2.5 Likelihood

To extract parameter constraints from measurements of the cosmic shear power
spectra in different surveys, we make use of a Gaussian likelihood. This
choice is justified by the central limit theorem on the scales used in this
analysis [51]. The power spectrum measurements and corresponding covariance
are described in Section 3.2.

Our model will depend on the 5 flat $\Lambda$CDM parameters
$\omega_{c}=\Omega_{c}h^{2}$, $\omega_{b}=\Omega_{b}h^{2}$, $h$, $\log(A_{s})$
and $n_{s}$. Here $\Omega_{c}$ and $\Omega_{b}$ are the fractional abundances
of cold dark matter and baryons respectively, $h\equiv H_{0}/(100\,{\rm
km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1})$, $A_{s}$ is the primordial amplitude of
scalar perturbations, and $n_{s}$ is the primordial scalar spectral index. The
optical depth to reionisation was set to $\tau=0.08$, and we only considered
massless neutrinos. Beside these cosmological parameters, we marginalised over
4 intrinsic alignment parameters, $\\{A_{{\rm IA},0}^{\rm DES},A_{{\rm
IA},0}^{\rm HSC},\eta_{\rm IA}^{\rm DES},\eta_{\rm IA}^{\rm HSC}\\}$,
characterising the effect of IAs in the DES and HSC samples according to Eq.
2.5. We also marginalise over a multiplicative bias parameter for each
redshift bin, using the shape measurement priors quoted in [52] and [3] for
DESY1 and HSC-DR1 respectively. When marginalising over residual systematic
uncertainties in the calibrated redshift distributions, we will make use of a
two-parameter linear model. The details, including the procedure used to
obtain a prior on the parameters of the model, is described in Section 4.2.
Finally, when comparing with the official DESY1 and HSC-DR1 constraints, we
will marginalise over shifts in the mean of the redshift distribution of each
redshift bin. The priors used on these nuisance parameters are listed in Table
1. For the cosmological parameter priors, we follow the HSC-DR1 analysis [3].
We found that this results in more conservative constraints (i.e. broader
posteriors) than using the DESY1 priors [42].

We use Cobaya [53, 54] to sample the parameter space with a Monte-Carlo
Markov-Chain (MCMC) method. In particular, we use the Metropolis-Hasting [55,
56] algorithm and check convergence requiring the Gelman-Rubin [57] parameter
to satisfy $R-1<0.01$ in the diagonalised parameter space.

Parameter | Prior
---|---
$\omega_{\rm c}$ | $U(0.03,0.7)$
$\omega_{\rm b}$ | $U(0.019,0.026)$
$h$ | $U(0.6,0.9)$
$\log(10^{10}A_{\rm s})$ | $U(1.5,6.0)$
$n_{\rm s}$ | $U(0.87,1.07)$
$\tau$ | 0.08
$\sum m_{\nu}$ | 0

| DES | HSC
---|---|---
$m_{i}$ | $\mathcal{N}(0.012,0.023)$ | $\mathcal{N}(0,01)$
$A_{{\rm IA},0}$ | $U(-5,5)$ | $U(-5,5)$
$\eta_{\rm IA}$ | $U(-5,5)$ | $U(-5,5)$
$\Delta z_{0}$ | $\mathcal{N}(0.,0.016)$ | $\mathcal{N}(0.,0.037)$
$\Delta z_{1}$ | $\mathcal{N}(0.,0.017)$ | $\mathcal{N}(0.,0.017)$
$\Delta z_{3}$ | $\mathcal{N}(0.,0.013)$ | $\mathcal{N}(0.,0.021)$
$\Delta z_{4}$ | $\mathcal{N}(0.,0.015)$ | $\mathcal{N}(0.,0.017)$

Table 1: Prior distributions for the cosmological (left) and nuisance (right)
parameters. $U(a,b)$ and $\mathcal{N}(\mu,\sigma)$ are a uniform distribution
with boundaries $(a,b)$, and a Gaussian distribution with mean $\mu$ and
variance $\sigma$, respectively. The $\Delta z_{i}$ have been calibrated for
our $p(z)$ sampling from a multivariate Gaussian with mean and covariance
${\bf N}$ and $P$ (see Eq. 2.8).

##  3 Data

###  3.1 DES and HSC

In this paper we combine the weak lensing datasets from the first year of
observations of the Dark Energy Survey (DES) and Hyper Suprime-Cam surveys. We
use the data produced in [58] with two modifications: we calibrate the DES
sample, instead of using the official $p(z)$, and we analytically marginalize
over the $p(z)$ uncertainty at the covariance level as described in Section
2.3. The spatial distribution of the two datasets are shown in Fig. 1. As can
be seen, the samples used for the cosmological analysis are completely
disjoint on the sky, and are therefore uncorrelated except for their use of a
common redshift calibration sample.

Figure 1: Sky footprint of the DESY1 (purple) and HSC-DR1 (yellow) samples. As
they occupy disjoint sky regions, all statistical correlations between them
come from the use of a common sample to obtain calibrated redshift
distributions.

The Hyper Suprime-Cam Subary Strategic Program (HSC SSP) is a wide-field
imaging survey that will eventually cover $1000\,\mathrm{deg^{2}}$ in five
filter bands ($grizy$). In this work we use the data from the first year of
observation (HSC DR1) of the Wide layer, which covers $\sim
108\,\mathrm{deg^{2}}$ and has a limiting magnitude of $m_{\rm lim}=26.4$ and
a median $i$-band seeing of $\sim 0.6\,\mathrm{arcsec}$. We choose cosmic
shear sample, with a magnitude limit of $m_{\rm lim}=24.5$, as defined in Ref.
[59]. We use the photo-$z$ estimates using the Ephor_AB method to split the
data into four $z$-bins with edges [0.3, 0.6, 0.9, 1.2, 1.5] (as done in the
HSC-DR1 analysis). We use the official galaxy distributions $p(z)$ used in
[3]. These were calibrated with COSMOS2015 and a DIR method with weights given
by self-organizing maps and can be found in Figure 2.

The Dark Energy Survey is a 5-year survey that has observed
$5000\,\mathrm{deg^{2}}$ in five different filter bands ($grizY$).
Observations were carried out from the Cerro Tololo Inter-American Observatory
(CTIO) with the 4m Blanco Telescope, using the 570-Mpix Dark Energy Camera
(DECam). The first data release used in this paper [60] covers
$1786\,\mathrm{deg^{2}}$ before masking. We use the Metacalibration weak
lensing sample, divided in the four different redshift bins defined in [61].
As in the case of HSC-DR1, DESY1 also used COSMOS15 in the calibration
process, for which they used a modified version of the BPZ method [62, 61]. In
order to use a consistent methodology to obtain calibrated measurements of the
redshift distribution across both HSC and DES, we made alternative
measurements of the DESY1 redshift distributions via DIR using the COSMOS2015
catalog, as described in Section 4.1.

###  3.2 Power spectrum measurements

Our analysis is based on the shear power spectrum measurements of the DESY1
and HSC-DR1 datasets, presented and described in detail in [58]. The power
spectra were calculated via the the pseudo-$C_{\ell}$ method using NaMaster
[63]. The DES power spectra were estimated from maps of the shear field
constructed using the HEALPix pixelisation scheme with resolution parameter
$N_{\rm side}=4096$ (corresponding to a pixel size of about 1 arcmin). Given
the smaller footprint size, the HSC power spectra were computed for each field
independently using the flat-sky approximation, and co-added afterwards. The
shape noise contribution to the auto-correlations was subtracted analytically
as described in [58].

The covariance matrix of these power spectra was estimated analytically, using
the methods described in [64, 58]. The disconnected (Gaussian) component was
calculated using the improved narrow-kernel approximation (NKA) [58], which is
able to account for the impact of survey geometry on both the signal and noise
components of the covariance. To the disconnected component we added the
connected non-Gaussian part and super-sample contributions, estimated
analytically using the halo model. Since they cover disjoint regions of the
sky, we assumed no covariance between the HSC and DES power spectra. As we
will see in Section 4, the use of the same calibration sample (COSMOS2015)
induces a covariance between both measurements after marginalising over the
$p(z)$ uncertainties.

When obtaining cosmological constraints, we use the angular power spectra up
to $\ell=2000$, and remove $\ell<30$ for DESY1, and $\ell<300$ for HSC-DR1.
These scales were shown to be free of systematics in [58] for DESY1 and in the
official analysis of HSC-DR1 [3]. The power spectra and covariance matrices
produced in [58] and used in this analysis are publicly
available111https://github.com/xC-ell/ShearCl.

##  4 Results

###  4.1 $p(z)$ calibration from COSMOS

Figure 2: Redshift distributions for the DES (blue) and HSC (orange) samples
calibrated with COSMOS and using DIR. In red, the official DES $p(z)$ and in
black, those of HSC. Note that the difference between the two $N(z)$ for HSC
is just the binning. The errors on the $p(z)$ have been estimated following
Section 2.3. We find good agreement with the official results, except for the
fourth bin where the shape changes significantly. As will be discussed in
Section 4.3, this leads to a small change in the inferred cosmological
constraints. The mean redshifts can be found in Table 2.

In order to ensure consistency in the analysis of the DES and HSC samples, we
use the DIR method, as described in Section 2.2, to estimate their redshift
distributions and their associated uncertainties. As a calibrating sample ($S$
in the notation of Section 2.2), we use the COSMOS2015 catalog [33]. This
sample was also used in the analysis of DESY1, in combination with clustering
redshifts and pdf stacking, and for HSC-DR1, using self-organising map to
obtain colour-space weights.

In the case of HSC-DR1, a cross-matched catalog over the COSMOS field is
provided with the public data release, including the lensing weights
associated with each galaxy, as well as the SOM weights that correct for the
different color distributions. We use this catalog directly when estimating
the HSC $p(z)$s.

In the case of DES, we first cross-match the COSMOS2015 catalog with the DES
data over the same patch to associate each cross-matched source with fluxes
measured in the DES bands. We used the four griz bands and also checked that
using the three riz bands yielded negligible changes in the redshift
distribution. We then apply the DIR procedure using the cross-matched catalog
as $S$, and the main sample used for the cosmic shear analysis as $P$. It is
worth noting that the colour distribution of the DESY1 sample in the COSMOS
field is noticeably different from that of the main sample, which results in a
non-negligible shift in the inferred redshift distributions if we were to use
it as $P$.

Bin | DIR | Official
---|---|---
0 | $0.369\pm 0.016$ | $0.389\pm 0.016$
1 | $0.503\pm 0.017$ | $0.507\pm 0.013$
2 | $0.774\pm 0.013$ | $0.753\pm 0.011$
3 | $1.004\pm 0.015$ | $0.950\pm 0.022$
Table 2: DES mean redshifts. We find that our calibration yields results
compatible with the official ones.

The resulting redshift distributions are shown in Fig. 2 in blue (DES) and
orange (HSC), together with their statistical uncertainties, estimated as
described in Section 2.3. The figure also shows the official redshift
distributions used in the DESY1 (red)222Taken from https://desdr-
server.ncsa.illinois.edu/despublic/y1a1_files/ and HSC-DR1 (black)333Taken
from http://gfarm.ipmu.jp/~surhud/#hamana analysis. The figure shows
appreciable differences between the DIR estimate of the $p(z)$ and the
official DESY1 estimate, which combined pdf stacking and clustering
redshifts.Looking at the mean redshifts in Table 2 we see deviations on the
higher $z$-bins of order $\sim 2\sigma$ with shifts $\Delta z\simeq 0.02$ and
$\simeq 0.05$, respectively. As we will see in Section 4.3, this results in a
non-negligible shift in the posterior constraints on cosmological parameters.
This effect was pointed out in [49], and will be discussed in further detail
later. On the other hand, the differences in the HSC $p(z)$ are just caused by
a different binning choice and has no impact on our constraints (see Section
4.3).

In order to compare our constraints with those found in the official DES and
HSC analyses, we need to ensure that the level of uncertainty in the redshift
distributions is commensurate in both analyses. As has become standard in the
analysis of cosmic shear data, the uncertainty on the $p(z)$ calibration was
quantified in the official analyses of both datasets by introducing a free
nuisance parameter for each redshift bin, characterising a shift $\Delta z$ in
the mean of each redshift distribution, so that $p(z)\rightarrow p(z+\Delta
z)$. This likely incorporates the main response of the weak lensing kernel to
variations in the $p(z)$ [61, 3]. To associate our estimate of the redshift
distribution uncertainties with an effective prior uncertainty on $\Delta z$,
we generate $10^{4}$ samples of the $p(z)$, sampling from a multivariate
Gaussian distribution with mean $\bar{{\bf N}}$ and covariance ${\sf P}$ (see
Section 2.3). For each of these, we compute the mean redshift $\langle
z\rangle$, and the prior on $\Delta z$ is then given by the standard deviation
of $\langle z\rangle$ over all samples. The mean and standard deviation of
$\langle z\rangle$ for each redshift bin in the DES sample are compared with
the priors used in the official DES analysis in Table 1. In order to match the
level of prior uncertainty used by DES, we adjusted our $p(z)$ uncertainties
by a factor $2$ (i.e. $f_{P}=4$ in Section 2.3).

Figure 3: Left. Redshift distribution correlation matrix for the third and
second redshift bins of DES and HSC, respectively. The off-diagonal terms show
that the calibration of the data sets of both surveys with the same sample
(COSMOS) introduce correlations between the different $p(z)$. Right.
Covariance matrix of all angular power spectra including the contribution from
the analytical marginalization of the $p(z)$. This introduces the visible
correlations between HSC and DES data.

The left panel of Fig. 3 shows the correlation matrix (see Eq. 2.8) for the
redshift distributions of the third DES bin and the second HSC bin, which
overlap significantly in redshift. There is a non-negligible correlation
between the uncertainties of both redshift distributions, due to the use of a
common calibrating sample. This then leads to a non-trivial correlation
between the DES and HSC data vectors (see Eq. 2.7) after marginalising over
the $p(z)$ uncertainties. This is shown in the right panel of the same figure.
The cross-terms between the DES and HSC blocks of the covariance are entirely
due to the uncertainties in the redshift distribution estimation.

###  4.2 Characterising residual $p(z)$ systematics

It has been argued that, in spite of their high accuracy, the photometric
redshifts of the COSMOS2015 catalog may induce a coherent shift in the
redshift distribution of samples calibrated from it, particularly at high
redshifts [50]. This could cause a bias in the inferred cosmological
parameters, and therefore this residual systematic must be modelled and
marginalised over.

Figure 4: Redshift measurement difference between PAU and COSMOS15 with the
best fit model (Eq. 4.1) and its 68% C.L. region. The points at $z\sim 0.5$
mainly drive the fit given the small error they have. These small errors are
caused by a better agreement between PAU and COSMOS2015 and a larger number of
common galaxies lying at those redshifts.

To characterise this effect in the redshift distribution, we used the COSMOS
catalog of [50], which combines the 30-band photometry of the original
COSMOS2015 catalog with 66 additional bands from the PAU survey [65]. We
retrieve the redshift obtained from the original COSMOS2015 photometry
($z_{30}$) and the redshift using the combined photometry ($z_{96}$) for all
cross-matched sources, and use them to measure the mean difference $\Delta
z=z_{96}-z_{30}$ in 6 bins of redshift between $z=0$ and $z=1.6$. We estimate
the uncertainty on the mean $\Delta z$ from the scatter of objects in each bin
(normalised by $\sqrt{N}$, where $N$ is the number of objects). We then use
these measurements to fit a model of the form

$\Delta z=A+Bz_{96},$ (4.1)

to the measured $\Delta z$. The covariance of the two free parameters $(A,B)$
is then estimated using bootstrap resampling. The result is $A=-0.0025\pm
0.0019$ and $B=0.0177\pm 0.0035$, with a correlation coefficient
$r_{AB}=-0.96$444Note that we could have fitted $\Delta z=A+B(z_{96}-0.5)$ to
reduce the correlation between the coefficients to $r_{AB}\sim-0.3$ and the
error of $A$. With this new fit $A=0.00629\pm 0.00047$, but the slope is left
unchanged with $B=0.0177\pm 0.0038$. For simplicity we used the
parametrization of Eq. 4.1. This result is illustrated in Fig. 4, which shows
the measured values of $\Delta z$ as a function of redshift, together with the
best-fit linear model and the bootstrap-estimated errors. We notice that the
fit is mainly driven by the points at $z\sim 0.5$ which have small error bars
due to a combination of better agreement between PAU and COSMOS2015 and a
larger number of common galaxies in those bins. Furthermore, as noted in [50],
the redshift of sources in COSMOS15 is biased low at high redshifts, and
slightly high at low redshifts.

To marginalise over this uncertainty, we proceed as in the standard cosmic
shear analyses, shifting the redshift distributions of all samples by $\Delta
z$, which now is a function of $z$. Note that the normalisation of the shifted
probability distribution changes by a constant factor of $1-B$.

We will explore the impact on our final results of various choices when
marginalising over $(A,B)$. First, since the DESY1 and HSC-DR1 samples are not
the same galaxy population, the systematic shift in their redshift
distributions is not necessarily 100% correlated. We will therefore produce
constraints using the same $(A,B)$ parameters to describe the shift in DES and
HSC (the limit of perfect correlation), and using two sets of parameters to
describe each shift independently (the limit of no correlation). Secondly, we
will study the effect on the final constraints of broadening the prior on
$(A,B)$ obtained above.

###  4.3 Validation of the individual surveys

Figure 5: HSC posterior distributions. The lower triangle shows the effect of
using the priors in Table 1. In purple, the result marginalizing at the
likelihood level over redshift shifts $\Delta z$. In red, when the uncertainty
on the $p(z)$ is analytically incorporated into the covariance. The agreement
between both is very good. The reason why the contours with the shifts are
slightly broader is probably due to a convergence issue, since they are not as
smooth as in the analytically marginalized case. The upper triangle focuses on
the origin of the $S_{8}$ shift. In order to explore the origin of the shift
we substitute, at a time, our $C_{\ell}$ or $p(z)$ by the official ones. The
results are shown in blue (official $C_{\ell}$ but our $p(z)$) and gray (our
$C_{\ell}$ but the official $p(z)$). Effectively, we find that changing the
$p(z)$ does not affect the constraints, whereas using the official $C_{\ell}$
largely removes the shift. The remaining difference probably comes from the
different modeling of the bandpower window functions applied to the theory
$C_{\ell}$ in our likelihood, the different choice of sampler (Nested Sampling
in their case and Metropolis-Hasting in ours), or the impact of the small PSF
leakage terms in the theory prediction that we ignored here.

Before trying to obtain combined constraints from DESY1 and HSC-DR1, we
compare the results we obtain on each of these datasets individually with the
official results published by both collaborations.

In the case of HSC, most of our analysis choices are relatively similar to
those made in the analysis of [3]. We make use of angular power spectra
calculated on the flat-sky, albeit binned into different bandpowers,
calculated from maps generated with different pixelisation, and using a
different (but mathematically equivalent) method to estimate the noise bias.
We use the re-weighted COSMOS catalog made available by HSC to calibrate the
redshift distributions of all four bins, although we sample them at a
different number of redshifts (shown in Fig. 2). We apply the same
cosmological parameter priors used in [3] with two exceptions: we allow for
the multiplicative bias of each redshift bin to vary (as opposed to a single
effective multiplicative parameter $m_{\rm eff}$ in [3]), and we do not
marginalise over parameters characterising the PSF leakage (although the
effect of these was found to be small in [3]). Fig. 5 shows the constraints on
$\Omega_{\rm m}$, $\sigma_{8}$, and $S_{8}$ in five cases:

  * •

The red contours show the constraints obtained from our data using the
analytical marginalisation over the redshift distribution uncertainties
described in the previous sections.

  * •

In purple, we show the constraints obtained from our data marginalising over a
shift in the mean of the redshift distribution of each redshift bin, using the
priors prescribed in [3].

  * •

The black contours show the official constraints of [3].

  * •

In gray, the constraints when we use our $C_{\ell}$ but the official $p(z)$

  * •

In blue, the constraints with the official $C_{\ell}$ but our $p(z)$

Our constraints are broadly compatible with the official ones, although we
observe a small upwards shift on $S_{8}$, which aligns them more with the
real-space results of HSC-DR1 [4, 66]. The size of this shift555We estimate
the shifts in the paper by $\Delta\sigma_{12}=|x_{1}-x_{2}|/\sigma_{x_{1}}$,
where $x_{1}$ is the mean value of a given parameter (e.g. $S_{8}$) for the
measurement 1 and $\sigma_{x_{1}}$ is its error estimated as the standard
deviation of the parameter $x_{1}$ (we assume the posterior is Gaussian). For
the case of estimating tensions where the measurements are uncorrelated, we
use
$\Delta\sigma_{12}=|x_{1}-x_{2}|/\sqrt{\sigma_{x_{1}}^{2}+\sigma_{x_{2}}^{2}}$.
is $\sim 1\sigma$ and comes from the differences in the $C_{\ell}$ binning
described above. In order to show this, we run two different test cases: one
with our $C_{\ell}$ and the official $p(z)$ and another with our $p(z)$ but
the official $C_{\ell}$666Obtained from gfarm.ipmu.jp/s̃urhud/#hikage. The
results are shown in the upper triangle of Figure 5. We see that using the
official $p(z)$ or ours does not change the posterior distributions, whereas
the shift largely disappears when using the original $C_{\ell}$. The remaining
difference probably comes from the different modeling choices at the level of
the bandpower window functions that are applied to the theory $C_{\ell}$ in
the likelihood, the use of different sampling methods (Nested Sampling [67,
68, 69] in their case and Metropolis-Hasting in ours), or the impact of the
PSF leakage contribution to the theoretical predictions that we have neglected
here. However, as shown in Figure 6, our $C_{\ell}$ are broadly in agreement
with the official ones, and the differences between them do not display any
specific trend. Furthermore, we see that the best fit $C_{\ell}$ do only
significantly differ at lower redshift where the error bars are larger.
Overall, we find that the bestfit $\chi^{2}=71.5$ ($p=0.11$) with our data
vector, in contrast to $\chi^{2}=48.8$ ($p=0.80$) with the official $C_{\ell}$
and our $p(z)$.

Figure 6: HSC data and bestfit angular power spectra. We see that our
$C_{\ell}$ (orange) are compatible with the official ones (blue). Furthermore,
the difference between bestfit power spectra obtained with the official data
(black) and our data (red) appears mainly at the lower redshift bins, where
the error bars are larger. In both cases we have a good fit to the data with
$p=0.80$ with the official $C_{\ell}$ and $p=0.11$ with our $C_{\ell}$.

More interestingly, we find that the constraints found marginalising over the
mean shifts $\Delta z$ are essentially equivalent to those found after fully
marginalising over the $p(z)$ uncertainties using the method of [35]. This
reinforces the conclusion that the most relevant $p(z)$ uncertainty mode for
cosmic shear is the redshift distribution mean, which affects the width and
amplitude of the lensing kernel [70, 71].

In the case of DES, our analysis choices differ more from those of [42].
First, we use power spectra instead of real-space correlation functions.
Secondly, we follow the cosmological parameter priors of the HSC analysis. The
most significant differences in this case are that we use flat priors on
$\log_{10}A_{s}$ instead of $A_{s}$, and that we sample over the physical
dark-matter and baryon densities ($\Omega_{c/b}h^{2}$) instead of the total
matter and baryon fractions ($\Omega_{m/b}$). Finally, and most importantly,
we use fiducial redshift distributions estimated from the COSMOS2015 catalog
using direct calibration, rather than the official redshift distributions made
available by [61]. The official redshift distributions were obtained by
sampling the individual photo-$z$ probability densities of individual
galaxies. Uncertainties on this fiducial distribution were then parametrised
in terms of a shift parameter $\Delta z$, with a prior estimated from a
combination of the reweighted COSMOS2015 catalog and clustering redshifts. As
discussed in detail in the re-analysis of the DESY1 data carried out by [8],
estimating the redshift distributions of the sample using direct calibration
on various spectroscopic samples can lead to significant downward shifts in
the posterior distribution of $S_{8}$. In [8], this was ascribed to potential
systematics in the photometric redshifts of COSMOS2015, which could
systematically underestimate the true shift of the fiducial DESY1
distributions. Irrespective of this discussion, we chose to re-estimate the
redshift distributions of the DESY1 sample from COSMOS2015 alone using DIR in
order to be able to estimate consistently the statistical uncertainties of the
distributions for both DES and HSC, including the correlations between both.

Figure 7: DESY1 posterior distributions for different analysis choices. The
lower corner shows the effect of the calibration of DES done in this work
(blue). For comparison, we have plotted the official results (in black) and
the contours obtained in using the data vector of this work but with the
official $p(z)$ and marginalizing over a redshift shift at the likelihood
level, as in Ref. [10] . Since the latter two agree, the shift must come from
the calibration of this work. Note that, however, this shift is consistent
with what was found using spectroscopic samples in [8]. The upper corner shows
constraints using the HSC priors (see Table 1). These are in orange and red.
The orange contours are the fiducial result of this work and are from the case
with the $p(z)$ uncertainty incorporated into the covariance matrix. In the
case of the red contours, these are the result of marginalizing at the
likelihood level over redshift shifts $\Delta z$. We see that these manage to
encapsulate most of the $p(z)$ uncertainty but, still, the constraints are
slightly tighter than those with the analytical marginalization.

In order to distinguish between differences due to the prior choice and those
due to the difference in the redshift distributions, we have first produced
cosmological parameter constraints using the same priors as the official DESY1
analysis. These are shown in the lower-left half of Fig. 7. The black contours
show the official constraints777Taken from https://desdr-
server.ncsa.illinois.edu/despublic/y1a1_files/. The gray contours show the
constraints found from the power spectrum measurements used here but employing
the official DESY1 redshift distributions and marginalising over the shift
parameters as in [10]. Finally, the blue contours show the constraints found
from our power spectrum measurements and the redshift distributions found via
DIR from COSMOS2015, marginalising over $\Delta z$, as in the two previous
cases. When using the same redshift distributions, we find that our
constraints are in good agreement with those found by [8]. In turn, using the
DIR $p(z)$s from COSMOS2015 leads to a non-negligible downward shift of $\sim
1.3\sigma$ in $S_{8}$. This is in agreement with the results of [8], and in
fact we find that the DIR redshift distributions used here are in qualitative
good agreement with those found by the authors of that paper using solely
spectroscopic data888Private communication with Shahab Joudaki.. It is worth
noting that we carried out extensive tests to verify that our results were not
sensitive to various choices made when estimating the DIR $p(z)$, including
the number of nearest neighbors used in the spectroscopic catalog, the choice
of photometric sample $P$ to derive weights (e.g. choosing the sample
overlapping the COSMOS2015 field as opposed to the cosmic shear sample),
including the Metacalibration responsivity as a weight when estimating the
redshift distribution, and varying the quality cuts made on COSMOS2015 to
define the $S$ sample.

This leads us to two conclusions: firstly, the constraints from DESY1 are
highly sensitive to the methodology used to estimate the redshift distribution
of the source sample. Secondly, the interpretation of the shift in $S_{8}$
found by [8] in terms of photometric redshift systematics in COSMOS2015 may be
partially inaccurate, since the DIR $p(z)$s estimated from COSMOS2015 are
qualitatively similar to those of [8], and lead to a shift in $S_{8}$ similar
to that found in the same paper. The origin of this disagreement then lies
either in the method used to calibrate the official DESY1 $p(z)$s, through
stacking and clustering redshifts, or in the DIR approach used here to obtain
calibrated $p(z)$s from the COSMOS2015 data. As pointed out by [49], the fact
that one has only four bands available might cause direct calibration methods
to yield shifted distributions. Further work examining and quantifying the
potential pitfalls of both methods will therefore be important for future
analyses.

The upper-right half of Fig. 7 then explores the impact of the different
choice of cosmological parameter priors used here. The red contours show the
constraints found using the priors of Table 1, while still marginalising over
$\Delta z$. Finally, the orange contours show the constraints found after
analytically marginalising over the $p(z)$ uncertainties. As before, we find
that the analytical marginalisation and the $\Delta z$ parametrisation lead to
equivalent constraints. More importantly, the impact of the choice of priors
is to marginally increase the left tail of the posterior $S_{8}$ distribution,
and to shift the marginalised posterior distribution of $\Omega_{m}$
downwards.

Figure 8: DESY1 posterior distributions considering only some of the redshift
bins in the analysis. This exercise shows that the shift in $S_{8}$ is not
coming from the calibration of the $p(z)$ of a particular tomographic bin, but
all prefer lower values. This is consistent with what was found using
spectroscopic samples in [8].

Before ending this section, we have carried out additional tests to try to
ascertain the origin of the downwards shift in in $S_{8}$ in terms of the
differences between the fiducial $p(z)$s and the DIR estimates used here, as
shown in Fig. 2. Fig. 8 shows, on the left-hand side, the constraints on
$S_{8}$ and $\Omega_{m}$ found after removing each one of the four redshift
bins (cyan, yellow, gray and pink contours in order of ascending redshift),
together with the constraints found here (orange), and the fiducial DESY1
constraints (black). We can see that the third redshift bin carries most of
the constraining power, since the uncertainty on $S_{8}$ increases
significantly when removing it. However, in all cases we find a posterior
$S_{8}$ that is consistently lower than the official constraints. The right
panel of the same figure shows the same constraints using only the first two
(purple) or the last two (brown) redshift bins. The shift in $S_{8}$ is mostly
caused by the two higher redshift bins, which also carry most of the
constraining power. This result is in qualitative agreement with Fig. 2, which
shows that the DIR $p(z)$s in the two high redshift bins are more heavily
weighted towards higher redshifts than the fiducial ones, and by table 2,
which shows that the mean redshift of both beans is consistently higher (by
$\Delta z\simeq 0.02$ and $\simeq 0.05$ respectively) than the official
$p(z)$s.

###  4.4 Combining DES and HSC

Figure 9: Posterior distributions for the DES (orange), HSC (purple) and their
combination (pink and gray). The DES and HSC results differ by $\sim
2.3\sigma$, with DES prefering lower values of $S_{8}$ and $\sigma_{8}$. The
gray contours show the combination of DES and HSC without taking into account
the correlations from the redshift distributions calibration, whereas the pink
contours do. Both constraints agree very well, showing that the correlation is
a small, likely negligible effect.

We obtain combined constraints from DESY1 and HSC-DR1 by combining both data
vectors together with their joint covariance matrix after analytically
marginalising over the $p(z)$ uncertainties (see right panel of Fig. 3). The
constraints resulting from this combination, together with the individual
constraints of both surveys is shown in Fig. 9, and the constraints on $S_{8}$
from different analysis choices are listed in Table 3.

| $N_{d}$ | $\chi^{2}$ | $p$ | $S_{8}$
---|---|---|---|---
DES ($\Delta z$) | 240 | 234.48 | 0.55 | $0.720^{+0.036}_{-0.022}$
DES (full $p(z)$) | 240 | 235.15 | 0.54 | $0.720^{+0.036}_{-0.024}$
HSC ($\Delta z$) | 60 | 71.47 | 0.11 | $0.812^{+0.028}_{-0.025}$
HSC (full $p(z)$) | 60 | 71.80 | 0.11 | $0.813^{+0.030}_{-0.024}$
DES+HSC | 300 | 311.70 | 0.28 | $0.768^{+0.021}_{-0.017}$
DES+HSC as independent | 300 | 311.93 | 0.28 | $0.767^{+0.021}_{-0.018}$
DES+HSC with systematic | 300 | 312.09 | 0.28 | $0.767^{+0.021}_{-0.018}$
DES+HSC with independent systematic | 300 | 311.92 | 0.28 | $0.767^{+0.021}_{-0.018}$
DES+HSC with systematic (prior x4) | 300 | 311.33 | 0.29 | $0.767^{+0.021}_{-0.017}$
Planck 2018 | – | – | – | $0.832\pm 0.013$
Table 3: Main results. All results are shown with 68% C.L. errors. The
effective number of degrees of freedom is estimated as $N_{\rm
dof,eff}=N_{d}-2$, where $N_{d}$ is the total number of data points.

As noted in the previous section, the DESY1 data favour a lower $S_{8}$ than
HSC-DR1. Taking the central values and standard deviations of both datasets,
and dividing by their errors added in quadrature, we find that the two
estimates are in tension at the $\sim 2.3\sigma$ level. This is traditionally
not considered strong statistical evidence of tension for two virtually
independent datasets, and therefore we will proceed to combine them. However,
this tension is sufficiently significant to warrant a careful comparison
between the results from both datasets in future data releases, in order to
determine whether the difference is due to a statistical fluctuation, or
evidence of potential systematics. We must note that, as shown in Table 3, the
joint best-fit model is able to fit the DESY1 and HSC-DR1 data simultaneously
with a reasonable $\chi^{2}$ probability (28%), which reinforces the
conclusion that both datasets are not significantly incompatible.

The combined constraint, shown as yellow contours in Fig. 9, lies in between
both posteriors, and yields

$S_{8}=0.768^{+0.021}_{-0.017}.$ (4.2)

The figure also shows the joint constraints found under the assumption that
both datasets are completely independent (i.e. nulling the elements of the
joint covariance matrix corresponding to power spectra from different
surveys). We find that the constraints are almost completely unchanged
($S_{8}=0.767^{+0.021}_{-0.018}$). Thus, although present, the correlation
between both datasets induced by their use of a common $p(z)$ calibration
sample is completely negligible in the final cosmological constraints. As will
be shown in Section 5, this will still be the case for Stage-IV surveys even
though the final parameter uncertainties will greatly shrink.

These results do not take into account the existence of a correlated
systematic bias due to errors on the COSMOS2015 photo-$z$. We account for
these, as described in Section 4.2, by marginalising over $A$ and $B$ as
defined in Eq. 4.1 as additional nuisance parameters. The result is shown in
Figure 10. The yellow contours show the constraints in the absence of this
additional systematic, while marginalising over $(A,B)$, assuming the same
systematic shift in DESY1 and HSC-DR1 yields the blue contours. The
constraints are almost equivalent, and lead only to a marginal ($<6\%$)
increase in the uncertainties on $S_{8}$. Since the DESY1 and HSC-DR1 samples
are not equivalent, the photo-$z$ systematic may not be the same in both, and
therefore we consider the case where $A$ and $B$ are different parameters for
each survey (i.e. we double the number of photo-$z$ nuisance parameters). The
results are shown as orange contours, and are indistinguishable from the
fully-correlated case. As in the case of the statistical uncertainties, we
therefore conclude that the correlation between both samples due to the common
use of COSMOS2015 is negligible.

Figure 10: Left. Posterior distributions for the combination of DES and HSC
with (blue) and without (yellow) marginalizing over the calibration systematic
bias. The effect is negligible with slightly broader contours in the cases
with more parameters. This true even in the case we multiply the covariance
matrix of the parameters systematic parameters by 4 (green) or when we use
each survey has its own systematic bias (orange). Right. Posterior
distributions for the calibration systematic bias parameters. The legend shows
the color for the parameters constraints obtained in the case DES and HSC
systematic parameters are different in the combined run (although it also
includes the results from the cases on the left). The result is completely
prior dominated in all cases, even for the case with the enlarged prior. The
posteriors for all the different cases (except for the enlarged prior case)
are the same and cannot be distinguished. This means that the calibration bias
effect is small and that the cosmology information is not powerful enough to
inform about possible redshift errors in the training sample.

In order to verify that this result is not a consequence of an under-estimate
of the prior uncertainties on $A$ and $B$, we repeat the analysis doubling the
width of the prior on these parameters (i.e. multiplying their covariance
matrix by $4$). As in the previous case, this has no appreciable impact on the
final constraints on $S_{8}$ (see green contours in Fig. 10). As shown in Fig.
4, the size of this systematic is comparable to the shift in the mean of the
redshift distributions allowed by the statistical uncertainties (see Table 2).
Thus, the impact of this systematic may to some extent be already absorbed by
the analytical marginalisation over the $p(z)$ statistical uncertainties.

Our final combined constraint on $S_{8}$ is therefore given in Eq. 4.2. This
corresponds to a $\sim\sqrt{2}$ improvement in the statistical uncertainties
compared to the errors of either survey. The constraints are in good agreement
with those found by the latest cosmic shear analysis from the KiDS
collaboration, and in tension with the value inferred by Planck at the
$2.7\sigma$ level. This result, together with the constraints obtained from
all other variations of our analysis, is also shown in Fig. 11.

Figure 11: Cosmological parameter constraints for the studied cases here and
Planck, for comparison. The error bars are the 68%C.L. 1D-posterior intervals.
We find that the lower values of $S_{8}$ respect to Planck, in this work, are
mainly due to lower values of $\Omega_{\rm m}$. This is partially a volume
effect, at least in the case of DES, since the sampling parameters in Table 1
allow much lower values of $\Omega_{\rm m}$ than the DES priors (see Figure
7).

##  5 Effect of correlated $p(z)$ uncertainties in future surveys

We have shown in the previous Section that the correlations induced by the
redshift distribution calibration using a common spectroscopic sample have a
negligible impact on the combined analysis of DESY1 and HSC-DR1. This result,
however, may no longer be valid with the advent of the Stage-IV surveys such
as LSST [18, 19] or Euclid [20] which are expected to achieve percent-level
accuracy on cosmological parameters and, thus, will become sensitive to
modeling effects that can now be neglected.

In this Section we explore the effect that the correlation between two future
non-overlapping surveys will have in the parameter constraints. In order to
model such a scenario, we produce a fiducial data vector given by the best-fit
prediction found in the combined DESY1+HSC-DR1 analysis, and artificially
reduce the errors for these two surveys to match the expected observed area of
$\sim 10,000\,\mathrm{deg^{2}}$. This area is a factor of $\sim$10 and
$\sim$100 larger than that covered by DESY1 and HSC-DR1 respectively. Taking
into account that ${\sf Cov}\propto f_{\rm sky}^{-1}$, we modify the
covariance matrices (before adding the analytically marginalized $p(z)$) so
that

${\sf Cov}^{\text{DES}}\rightarrow{\sf Cov}^{\text{DES}}/10\qquad{\sf
Cov}^{\text{HSC}}\rightarrow{\sf Cov}^{\text{HSC}}/100\,.$ (5.1)

To this modified covariance we then add the contribution from $p(z)$
uncertainties (Eq. 2.7), assuming the same prior uncertainty on the $p(z)$s of
both experiments. This is likely a conservative assumption, since by the time
these Stage-IV have finished taking their data, better calibrating
spectroscopic samples will likely be available. Nevertheless, this allows us
to quantify the potential impact of cross-survey correlations due to the
$p(z)$ calibration in a conservative scenario.

Figure 12: Posterior distributions for the combination of two non-overlapping
LSST/Euclid-like surveys taking into account (red) or not (black) the
correlations induced by the $p(z)$ calibration and including the systematic
effect (Eq. 4.1, green dashed line). We see that the effect of both the
$p(z)$-induced correlations and the systematic is negligible with only
slightly broader contours when the correlations are taken into account.

In order to test for the effect of the induced correlations we constrain the
parameter space in two different scenarios: one with the covariance as
explained above and another where we set the correlations between the two
surveys induced by the $p(z)$ calibration to zero, while leaving the $p(z)$
contribution to the individual covariances intact. As can be seen in Fig. 12,
both cases produce largely equivalent constraints We find that this is also
the case after accounting for the impact of correlated $p(z)$ systematics (Eq.
4.1), which barely change the constraints. Therefore, we conclude that even in
the case of Stage-IV surveys the effects of the correlated statistical
uncertainties induced by the $p(z)$ calibration, and the systematic bias in
the training sample will have a negligible effect. It is worth noting that
this result is true for the redshift bins explored in the DESY1 and HSC-DR1
analyses, and other galaxy samples may display different levels of correlation
between bins. The strong correlations present in the DESY1-HSC-DR1 case (see
Fig. 3) suggest this will likely not change the results found here.

##  6 Conclusions

In this paper we have addressed the problem of combining independent weak
lensing datasets in the presence of correlated systematic uncertainties from
the use of common external datasets for the purpose of redshift distribution
calibration. To do so, we have proposed a method that combines the analytical
marginalisation over statistical uncertainties in the reconstruction of the
$p(z)$ from the common sample, followed by a marginalisation, at the
likelihood level, over residual systematics in it. We have then applied this
methodology to the DESY1 and HSC-DR1 samples with redshift distributions
estimated from the COSMOS2015 catalog, obtaining cosmological constraints from
their joint analysis.

The methodology used to characterise the correlated statistical uncertainties
in the recovered redshift distribution, and to analytically marginalise over
them, is based on the work of [35], proposed for the analysis of projected
galaxy clustering, and which we have applied to cosmic shear data for the
first time. In applying this methodology to HSC and DES, we have first
recalculated the fiducial redshift distributions for the DESY1 sample using
direct reweighting (DIR) of the COSMOS2015 catalog. This was done in order to
obtain consistent $p(z)$ measurements and uncertainties for both datasets. The
DIR redshift distributions thus obtained have noticeable differences with
respect to the fiducial distributions used by DESY1, particularly in the two
highest redshift bins, which dominate the cosmic shear signal. As noted in
previous work [8], this leads to a sizeable shift in the posterior
distribution of the “clumping” parameter $S_{8}$. Although this was originally
ascribed to the use of COSMOS2015 to calibrate the allowable shift in the mean
of the fiducial redshift distributions used in the DESY1 analysis, our result
seems to hint towards a different motivation. Ultimately, the cause these
differences may lie in the methods used to calibrate the $p(z)$s (pdf
stacking, clustering redshifts, or the use of direct calibration with
information from only 4 bands [49]), rather than the dataset they were applied
to. The constraints from DESY1 found here thus differ from the official ones
by a $1.3\sigma$ downward shift in $S_{8}$.

The individual constraints from HSC-DR1 and DESY1 are found to be in mild
$\sim 2.3\sigma$ tension, although a joint best-fit model can be found that is
a good fit to both datasets. After combining both datasets, the uncertainties
on $S_{8}$ improve by a factor of $\sim\sqrt{2}$, and the posterior
constraints (Eq. 4.2) are in tension with the Planck value for this parameter
at the $2.7\sigma$ level. We have shown that the impact of the correlation
between both datasets induced by the use of a common redshift calibration
sample (COSMOS2015), is completely negligible in the current analysis, as is
the additional marginalisation over a systematic shift in the mean redshifts
of the COSMOS2015 sample calibrated from the COSMOS-PAU catalog, regardless of
whether the systematic is taken to be fully correlated or fully uncorrelated
in HSC-DR1 and DESY1. Additionally, we have shown that this will also be the
case for next-generation Stage-IV datasets. This will largely simplify their
analysis when different experiments make use of the same external catalogs for
the purposes of redshift calibration.

## Acknowledgements

We would like to thank Jo Dunkley, Shahab Joudaki, Emily Phillips Longley, and
David N. Spergel for useful comments and discussions.

CGG and PGF acknowledge support from European Research Council Grant No:
693024 and the Beecroft Trust. We made extensive use of computational
resources at the University of Oxford Department of Physics, funded by the
John Fell Oxford University Press Research Fund. DA is supported by the
Science and Technology Facilities Council through an Ernest Rutherford
Fellowship, grant reference ST/P004474. AN is supported through NSF grants
AST-1814971 and AST- 2108126. CS is supported by a Junior Leader fellowship
from the ”la Caixa” Foundation (ID 100010434), with code LCF/BQ/PI22/11910018,
and by grants AST-1615555 from the U.S. National Science Foundation, and DE-
SC0007901 from the U.S. Department of Energy (DOE). For the purpose of Open
Access, the author has applied a CC BY public copyright licence to any Author
Accepted Manuscript version arising from this submission.

Software: We made extensive use of the numpy [72, 73], scipy [74], astropy
[75, 76], healpy [77], matplotlib [78] and GetDist [79] python packages. The
codes used to produce these results can be found in
https://gitlab.com/carlosggarcia/DESxHSC.

This paper makes use of software developed for the Large Synoptic Survey
Telescope. We thank the LSST Project for making their code available as free
software at http://dm.lsst.org.

Data: This project used public archival data from the Dark Energy Survey
(DES). Funding for the DES Projects has been provided by the U.S. Department
of Energy, the U.S. National Science Foundation, the Ministry of Science and
Education of Spain, the Science and Technology Facilities Council of the
United Kingdom, the Higher Education Funding Council for England, the National
Center for Supercomputing Applications at the University of Illinois at
Urbana-Champaign, the Kavli Institute of Cosmological Physics at the
University of Chicago, the Center for Cosmology and Astro-Particle Physics at
the Ohio State University, the Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University, Financiadora de Estudos e Projetos,
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro,
Conselho Nacional de Desenvolvimento Científico e Tecnológico and the
Ministério da Ciência, Tecnologia e Inovação, the Deutsche
Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy
Survey.

The Collaborating Institutions are Argonne National Laboratory, the University
of California at Santa Cruz, the University of Cambridge, Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the
University of Chicago, University College London, the DES-Brazil Consortium,
the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH)
Zürich, Fermi National Accelerator Laboratory, the University of Illinois at
Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the
Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the associated Excellence
Cluster Universe, the University of Michigan, the National Optical Astronomy
Observatory, the University of Nottingham, The Ohio State University, the
OzDES Membership Consortium, the University of Pennsylvania, the University of
Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the
University of Sussex, and Texas A&M University.

Based in part on observations at Cerro Tololo Inter-American Observatory,
National Optical Astronomy Observatory, which is operated by the Association
of Universities for Research in Astronomy (AURA) under a cooperative agreement
with the National Science Foundation.

The Hyper Suprime-Cam (HSC) collaboration includes the astronomical
communities of Japan and Taiwan, and Princeton University. The HSC
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from Japanese Cabinet Office, the
Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and
Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK,
ASIAA, and Princeton University.

The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of
the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project
Office, the Max-Planck Society and its participating institutes, the Max
Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for
Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham
University, the University of Edinburgh, Queen’s University Belfast, the
Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory
Global Telescope Network Incorporated, the National Central University of
Taiwan, the Space Telescope Science Institute, the National Aeronautics and
Space Administration under Grant No. NNX08AR22G issued through the Planetary
Science Division of the NASA Science Mission Directorate, the National Science
Foundation under Grant No. AST-1238877, the University of Maryland, and Eotvos
Lorand University (ELTE) and the Los Alamos National Laboratory.

Based in part on data collected at the Subaru Telescope and retrieved from the
HSC data archive system, which is operated by Subaru Telescope and Astronomy
Data Center at National Astronomical Observatory of Japan.

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