Measurement of a Cosmographic Distance Ratio with Galaxy and CMB Lensing
Hironao Miyatake
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 USA and
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

Mathew S. Madhavacheril and Neelima Sehgal
Physics and Astronomy Department, Stony Brook University, Stony Brook, NY 11794, USA

Anže Slosar
Physics Department, Brookhaven National Laboratory, Brookhaven, NY 11973, USA

David N. Spergel
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 USA

Blake Sherwin
Berkeley Center for Cosmological Physics, LBL and Department of Physics,
University of California, Berkeley, CA, 94720 USA

Alexander van Engelen
Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON, Canada M5S 3H8(Dated: May 17, 2016)
We measure the gravitational lensing shear signal around dark matter halos hosting CMASS
galaxies using light sources at z ∼ 1 (background galaxies) and at the surface of last scattering atz ∼ 1100 (the cosmic microwave background). The galaxy shear measurement uses data from the
CFHTLenS survey, and the microwave background shear measurement uses data from the Plancksatellite. The ratio of shears from these cross-correlations provides a purely geometric distancemeasurement across the longest possible cosmological lever arm. This is because the matter distribution around the halos, including uncertainties in galaxy bias and systematic errors such as
miscentering, cancels in the ratio exactly. We measure this distance ratio in three different redshift
slices of the CMASS sample, and combine them to obtain a 15% measurement of the distance ratio,r = 0.344 ± 0.052 at an effective redshift of z = 0.54. This is consistent with the predicted ratio
from the Planck best-fit ΛCDM cosmology of r = 0.410.

INTRODUCTION

Cross-correlating optical weak lensing and cosmic microwave background (CMB) lensing is emerging as a
powerful tool for measuring cosmological parametersand quantifying systematic uncertainties. In particular,cross-correlations between optical and CMB lensing are
sensitive to structure growth, and thus dark energy properties and modifications to General Relativity on large
scales [1–4]. These cross-correlations can also isolate
systematic effects such as, for example, multiplicativeand photo-z biases in optical weak lensing measurements[5, 6]. Recently cross-correlations using CMB lensing
data from ACT, SPT, and Planck and optical lensing
data from the CFHTLenS and DES surveys have beenpresented with detections of modest significance [5, 7–13].
However, the precision of these measurements is expected
to increase rapidly with newer data from, e.g., ACTPol,
SPTpol, CMB-S4, HSC, DES, KiDS, and LSST.
In this work, we present the first measurement of aparticularly useful cross-correlation between optical and
CMB lensing: the cosmographic distance ratio. Thismeasurement is obtained by measuring the gravitational
lensing shear around a particular set of dark matter halos, first using background galaxies as the lensed source
plane and then using the CMB as the lensed source plane.
Taking the ratio of these shear measurements results ina purely geometric distance measurement that is insensitive to the details of the mass distribution around thelensing halos, their galaxy bias, or potential miscentering[14–16]. The ratio is given byr=dA (z c )dA (z L , z g )
γto
∼
γtc
dA (z g )dA (z L , z c )(1)

where γto and γtc are the optical and CMB tangential
shear, dA is the angular diameter distance, and z c , z g ,and z L are the redshifts to the CMB, the backgroundgalaxy source plane, and the lensing structure respectively [17, 18]. This ratio has been measured previously
when both source planes have been background galaxies
with z < 2 [19–22]. However, the advantage of using
the CMB as the second source plane is that it provides
the longest lever arm for distance ratios, which can result in an order of magnitude higher sensitivity to dark

2
TABLE I. Number of CMASS Galaxies Used
Redshift
Galaxy Density Optical CMB
Range(per arcmin2 ) Analysis Analysis
0.43 < z < 0.51
0.007
2,895 211,441
0.51 < z < 0.57
0.007
2,896 213,497
0.57 < z < 0.7
0.008
3,108 229,341
0.43 < z < 0.7
0.021
8,899 654,279

energy parameters [17, 18]. In this Letter, we present
the first measurement of such a ratio using data from
Planck, CFHTLenS, and the BOSS CMASS galaxy sample. The CFHTLenS measurement is made for 8,899
CMASS galaxies spanning an area of 105 square degrees,
and the Planck measurement is made for 654,279 CMASS
galaxies spanning an area of 8,502 square degrees.
DATA & METHOD
The Lenses: BOSS CMASS Galaxies

For the foreground lens sample, we use the CMASS
selection of galaxies from the DR11 release of the BOSS
spectroscopic survey. These mostly red galaxies constitute an approximately volume-limited selection of luminous galaxies from SDSS-III that span a redshift rangeof 0.4 < z < 0.7. They are very often (90%) at thecenter of their host halos [23] with masses of aroundM200 = 2 × 1013 M , measured both from optical [24]and CMB lensing [8]. As such, they are excellent tracers of massive halos that lens background sources. The
entire sample covers roughly 20% of the sky.
In both the optical and CMB analyses, each CMASS
lens galaxy is weighted as follows,
wl = (wnoz + wcp − 1)wsee wstar(2)

so as to account for redshift failures (wnoz ), fiber collisions (wcp ), effects of seeing (wsee ) and stars (wstar )[25]. To reduce systematics associated with the width
in redshift of the sample, we divide the sample into three
redshift slices (see Table I) and perform the analysis separately in each redshift slice, combining the results only
when calculating the final distance ratio at an effective
redshift (see Results Section). For completeness, we also
perform the analysis on the full sample in one wide redshift bin (see Figure 3), but do not obtain cosmologicalconstraints from this.
Source Plane 1: CFHTLenS Galaxies

We use the public CFHTLenS catalog [26, 27] for calculating the optical tangential shear. The total area of
the CFHTLenS survey is 154 deg2 in four distinct fields.

The overlapping area with the SDSS DR11 data is 105
deg2 which contains 8,899 CMASS galaxies.
The catalog has galaxy shapes, which were measured
by a Bayesian model-fitting method called lensfit[28],and photometric-redshifts (photo-z’s) which were estimated with the BPZ code [29, 30] by using point-spreadfunction (PSF) matched photometry [31]. The effective number density of CFHTLenS source galaxies is
14 arcmin−2 .
The tangential shear in the i-th radial bin is measured
by stacking galaxy shapes of lens-source pairs;
Pls
Ri wls eto,(3)hγt (Ri )i = P
Ri wlswhere et is the tangential component of galaxy shapes,
wls is a weight which is the product of the CMASS galaxy
weight wl given by Eq. (2) and the inverse-varianceweight for galaxy shapes ws provided by the CFHTLenS
catalog that is estimated from the intrinsic galaxy shapeand measurement error due to photon noise. Here thesource galaxies are selected so that the best-fit photo-z
is greater than the lens redshift.
The covariance matrix of the tangential shear is estimated by measuring the tangential shear around realistic mocks of the CMASS sample generated from N-bodysimulations [32, 33]. Using these CMASS mocks, we cannaturally include sample variance, which can be important given the small area of the CFHTLenS suvey. We
use 150 realizations of mocks to reduce the uncertainty of
the covariance. At the scales used for this distance ratio
analysis, the uncertainty due to lens shot noise and sample variance dominates the statistical uncertainty; it isabout 1.5 times larger than the statistical uncertainty due
to intrinsic galaxy shapes and becomes as large as a factor
of four in the largest radial bin. The noise due to sample
variance also induces correlations between neighboring> 10 h−1 Mpc
bins, which are typically ∼ 0.5 for the R ∼bins. Note that we could have canceled this sample variance exactly, by using exactly the same subset of galaxies
to measure lensing of the CMB. However, given the large
noise in the Planck convergence map, our overall statistical uncertainty would have increased.
If the PSF correction is imperfect, it can contaminate
the tangential shear. To estimate this effect, we calculate the tangential shear around random points. We
use 50 realizations of random points to reduce statistical uncertainties [34]. The random signal is non-zero for> 20h−1 Mpc. We then make a PSF correction by subR∼
tracting this random signal from the lensing signal. If
the correction works, the 45-degree-rotated shear should
be consistent with zero. Figure 1 shows the 45-degreerotated shear after the correction for each radial bin ineach redshift slice. The R ∼ 40 h−1 Mpc radial bins are
consistently smaller than zero for all the redshift slices,and thus we do not use them. The p-value based on the

30.000150.43 <z <0.51
0.51 <z <0.57
0.57 <z <0.7

0.000100.00000­

γ ×(R)®

0.00005

0.00005

0.000100.000155

10

15

20

25

30

R(h 1 Mpc)

35

40

45

FIG. 1. Null test of optical lensing signal. The R ∼40 h−1 Mpc bins are consistently smaller than zero for all
the redshift slices, and thus we do not use them. The
p-value based on the χ2 per degree of freedom of the 12< 30 h−1 Mpc bins over the redshift slices is 0.82, whichR∼
is within a 95%CL region. Thus we use these 12 data points
for the distance ratio analysis.
< 30 h−1 Mpc raχ2 per degree of freedom of the 12 R ∼dial bins over the redshift slices is 0.82, which is within a
95%CL region. Thus we use these 12 data points for the
distance ratio analysis shown in Figure 4.

Source Plane 2: Planck CMB Map

To extract a corresponding shear profile of CMASS
halos using the CMB as the background light source, we
prepare a HEALPIX map [35] of the CMASS galaxy overdensity (with nside = 1024) for each redshift slice andcross-correlate it with the Planck reconstructed lensing
convergence κ map [36]. Thus we obtain an estimate ofκδCl g in Fourier-space, which we then convert to a realspace shear estimate, hγtc (R)i, as discussed below.
To create the galaxy overdensity map of CMASS galaxies, for each HEALPIX pixel x, we assign a number given
by
P

wi
−1
i wi

P
δg (x) = 1 i∈x
N

(4)

P
where
i∈x wi sums over the weights of each CMASSP
galaxy i that falls in that pixel x, and where N1 i wi
sums over the weights of all CMASS galaxies in all unmasked pixels and then divides by the total number of unmasked pixels N . Here the weight wi = wl ws (z), where
wl is the BOSS systematic weight given in Eq. (2) andws (z) is an effective CFHTLens weight. We include the
CFHTLens weights here, which have been interpolated
as a function of lens redshift, because in the optical analysis they change the median redshift of the lens galaxies

within a redshift slice. When comparing with the CMB
signal, it is important that the median redshift of the lenssample is the same since galaxy properties could evolve
as a function of redshift. Although the effect of such an
evolution is mitigated by our use of thin redshift slices,
we still weight the lens galaxies in the CMB analysis consistently with the optical analysis.
The mask used in this analysis is a combination of amask derived from the completeness of the BOSS galaxies, where we exclude regions where the completeness isbelow 70%, and the convergence mask provided with the
Planck 2015 lensing data release. For the CMASS mask,
we have checked that decreasing the minimum completeness to 10% has a negligible impact on the results sincemost of the survey area is close to 100% complete. For
the Planck convergence mask, we note that it masks out
galaxy clusters identified through the thermal SunyaevZeldovich effect.
We obtain a Cl estimate of the cross-correlation bysumming over spherical harmonic transform coefficientsof the galaxy overdensity and CMB kappa maps, with theappropriate correction for fractional sky coverage (fsky =0.206 for 8,501 deg2 ),κδĈl g =lX1
δlm κlm .
κδ(2l + 1)fsky
m=−l

(5)

We then convert the cross-correlation estimate in Fourierspace to the real-space tangential shear of the CMB associated with CMASS galaxies, hγtc (R)i, via a Hankel
transform (e.g Eq.2 in [37]),hγtc (R)i =

1
2π

Z

κδ

`d`J2 (`R/χ)C` g .

(6)Note that this is exact only in the flat-sky limit, however we do not probe radial scales large enough that weshould be sensitive to the effects of a curved sky. Usingκδ
Simpson’s rule on the discrete set of Cl g ’s, this integral
is calculated at 5000 radial points and averaged in radial bins R corresponding to the optical analysis. Notethat the errors are uncorrelated between l bins to a very
good approximation in Fourier space, and are highly correlated between radial bins in real space. The latter isappropriately accounted for as described below.
To generate an expected theory curve we compute theκδshear transform in Eq. (6) using an input Cl g curve
generated with a linear matter power spectrum from CAMB
Sources [38–40] with a linear galaxy bias of 2. This isshown in Figure 2 both as the unbinned blue curve and as
the black crosses binned identically to the data. We alsoκδshow here the result of restricting the Cl g to the range
40 < L < 2000, which is the L range of the Planck κ-map
used in this analysis. (Modes with L < 40 can be affectedby the treatment of the mask, and Planck does not report4linear theory (2 <L <8000)
binned theory (2 <L <8000)
binned theory (40 <L <2000)
data (40 <L <2000)

CMB shear theory
Optical shear theory
CMB shear
Optical shear

0.00100.0008γt (R)

γtc (R)

®

®

10 3

0.0012­0.0006­0.00040.0002

0

5

10

15

20 25 30
R(h 1 Mpc)

35

40

FIG. 2. Theory expectation of CMB tangential shear using anκδinput Cl g curve from 2 < L < 8000 generated with a linear
matter power spectrum from CAMB Sources [38–40] with alinear galaxy bias of 2. We also show the effect of restrictingκδ
the Cl g to the range 40 < L < 2000, which is the L rangeof the Planck κ-map. We do not use radial bins that havea mismatch between these two (red shaded regions) as thatwould make the optical and CMB analyses inconsistent. The
green points show the shear from the data, and where thosepoints deviate from the theory at small scales is where there
is sensitivity to the one-halo term from the CMASS galaxy
halos themselves.

0.000010

We use 600 realizations of the CMASS mocks to makethe covariance matrix and repeat the procedure above,
cross-correlating a galaxy overdensity map generated
from each mock with the Planck data κ-map, and thentransforming that into a shear estimate. We note that
there is no correlated structure between the Planck data
map and the CMASS mocks, so that the resulting covariance matrix does not include sample variance from thiscorrelated structure. However, this effect is expected tobe negligible since the noise in the CMB κ-map is expected to dominate. We check this by calculating Fisherκδmatrix theory errors with and without this Cl g term(see, e.g., Eq. 15 in [11]), and find agreement to within
1% between the two.

20
1
R(h Mpc)

25

30

FIG. 3. CMB and optical shear around CMASS halos in theredshift range 0.43 < z < 0.7. The dashed blue curve shows atheory fit to the optical data, which includes both the 1-haloand 2-halo terms. This red curve is given by scaling up theblue curve to the CMB source redshift.
RESULTS

Shear profiles, γt (R), are related
R to the underlying projected mass density, Σ(R) = dχρ(R, χ), through therelationγt (R) =modes with L > 2048). Including 2000 < L < 8000
corresponds better to the resolution of the CFHTLenS
survey, and in Figure 2 we show a significant difference
at R ∼ 5 h−1 Mpc between L < 2000 and L < 8000.
Thus we do not include this bin in our distance ratio
analysis. For a similar reason, we exclude the radial bin
at R ∼ 40h−1 Mpc. The green points in Figure 2 show thereal-space shear from the data, and where those points
deviate from the theory curve at small scales indicateswhere the measurement is sensitive to the one-halo termfrom the CMASS galaxy halos themselves (which is notincluded in the theory curve).

15

∆Σ(R)
Σ̄(< R) − Σ(R)
=
Σcr
Σcr

(7)

where Σ̄(< R) is the average mass density within a circleof radius R, and Σcr is the critical surface mass density.
We note that ∆Σ(R) depends only on the total matter
distribution of the lens, and Σcr is a purely geometricquantity since it depends only on the distances to thelens and background sources. Since the criteria used to
select the lensing galaxies (the CMASS sample selection)is the same in the regions where the optical and CMB
analyses are performed, we assume that the underlying∆Σ(R) is identical in both cases. This allows us to write
the expected distance ratio asr({cp }) =γto
ΣCMB ({cp })= croptcγtΣcr ({cp })(8)where the dependence on the cosmological parameters,{cp }, enters through the distance-redshift relations. Herethe numerator is the critical surface density for CMB
lensing, which is calculated as
ΣCMB
=
cr

−1
−1
(z , z; {c })
ls wls Σcr
P l CMB pPls wls(9)where zCMB = 1100 is the redshift to the surface of lastscattering, and the sum is over CMASS lenses. The critical surface density Σ−1
cr is related to the comoving angular
diameter distances as,5Distance Ratio

1.5

Distance Ratio

0.43 <z <0.7
0.43 <z <0.51
0.51 <z <0.57
0.57 <z <0.7

1.0
0.5
0.0
5

10

15

20
1
R(h Mpc)

25

30

FIG. 4. Measured distance ratio for each radial bin and redshift slice of CMASS galaxies. Here the error bars are derived by Monte Carloing the covariance matrices for opticaland CMB measurements, taking the ratio for each realization,and showing the 68% CL region around the mean ratio. The
dashed line and error band show r = 0.344±0.052, the best-fit
value coadding all the radial bins and simultaneously fitting
to the three redshift slices.

Σ−1
cr =

4πG dA (zl , zs )dA (zl )(1 + zl )2.
c2
dA (zs )

P

−1
−1
stacked (zs |zl )Σcr (zl , zs ; {cp })ls wl PPls wl Pstacked (zs |zl ),(11)
where Pstacked (zs |zl ) is the photo-z PDF stacked oversource galaxies whose best-fit photo-z is greater than agiven lens redshift zl , using the weight provided by the
CFHTLenS catalog. ThusP
ws Ps (z|zl )Pstacked (z|zl ) = s P
.
(12)
s ws
Note that the dilution effect due to foreground galaxies
selected as source galaxies is effectively corrected for in
Eq. (11).
Cosmological Constraints: To constrain cosmologicalparameters, we minimize the following quantity,XXχ2 ({cp }) =di Cov−1(13)
ij dj ,
αPlanck ΛCDM
Planck wCDM
Distance ratio from z bin
Combined distance ratio

0.55

Redshift

0.60

0.65

0.70

FIG. 5. Comparison of the measured distance ratio with thatpredicted from different cosmological models. The thin crosspoints show the measured distance ratio fitted separately foreach redshift slice. The thick dot point shows the distanceratio fitted to all the redshift slices simultaneously assuming
linear dependence of the ratio on redshift (see text for details).
The black solid and dashed curves show the ratio for the bestfit ΛCDM and wCDM models respectively from the Planck
TT + lowP spectra [41]. The thin solid curves show deviations
from the best-fit Planck ΛCDM model as indicated.

and the covariance is given by(10)

Here dA (zs ), dA (zl ), and dA (zl , zs ) are the comoving angular diameter distances to the source, lens, and between
the source and lens respectively.
In the denominator is the equivalent expression for optical lensing,
Σopt
cr =

0.65
0.60
0.55 w = 2
k =0.50.50
0.45 w =00.40
k = 0.5
0.35
0.30
0.25
0.20
0.40 0.45 0.50

ij

where di = γ o (Ri ) − r({cp })γ c (Ri ) for the ith radial bin,

Covij = Cov(γ o (Ri ), γ o (Rj ))−2rCov(γ o (Ri ), γ c (Rj ))+r2 Cov(γ c (Ri ), γ c (Rj )).

(14)

We ignore the second term in Eq. 14 because the overlapping region for the two measurements is less than 2%of the region used in our CMB analysis. The index α inEq. 13 runs over the three redshift bins of the CMASS
sample shown in Table I. Correlations between z-bins due
to sample variance are not included because the contribution from clustering of CMASS galaxies was found to be
subdominant to the contributions from CMB lensing reconstruction noise, Poisson noise of CMASS counts, andshape noise of CFHTLens galaxies.
In Fig. 4, we show the measured distance ratio foreach radial bin and redshift slice of CMASS galaxies.
Fig. 5 shows the coadded distance ratio for each redshift slice. We also include the distance ratio simultaneously fitted to the three redshift slices. In doing this,
we assume the ratio linearly depends on redshift, i.e.,r(z) = r0 + r0 (z − zp ), where zp is the “pivot” redshiftdetermined so that the errors on r0 and r0 are uncorrelated. This yields r = 0.344 ± 0.052 at a pivot redshift of zp = 0.54, a 15% measurement of distance ratio. Measurements of r0 are very poor due to the limited
redshift span and were included in this solely to determine the pivot redshift. In Fig. 5, we also show the ratio
predicted for different cosmological models as a function

6

Planck wCDM
Planck wCDM
+ Distance Ratio

0.5

w0

1.0

fixed to the Planck best-fit ΛCDM cosmology. The obtained constraint is m = −0.14±0.13. Second, we parameterize the photo-z bias as a shift of photo-z PDF, i.e.,
P (z) → P (z + bz ). To avoid calculating the optical lensing signal with a new source galaxy selection every timebz is updated, we calculate the lensing signal without any
source galaxy selection, which means all the dilution correction is put into computing Σopt
cr (see Eq.11). With thefixed cosmology, we obtain bz = −0.08 ± 0.08. These results indicate (under the assumption of standard ΛCDM
cosmology) that there is no significant evidence of systematic uncertainties in our optical shear measurement.

1.5
2.0

DISCUSSION

0.2 0.3 0.4 0.5

2.0 1.5 1.0 0.5

m0

w0

FIG. 6. Constraints on Ωm0 and w0 . In the w0 versus Ωm0panel, the white and shaded regions show less than and morethan 68% CL respectively from the distance ratio alone. The
red curves show the 68% and 95% CL from the Planck TT +lowP spectra [41], and the blue curves show constraints fromthe combination of this distance ratio plus Planck TT + lowP.

of lens redshift using Eq. (8). The solid/dashed curves
show the ratio for the best-fit ΛCDM/wCDM models
from the Planck TT + lowP spectra [41]. The ratio
between ΛCDM and wCDM models changes within asmaller range compared to our statistical uncertainty,
which means it is difficult to place tight constraints inspite of the 15% accuracy of our measurement.
Nonetheless, we show cosmological constraints withthis distance ratio measurement combined with Planck
TT + lowP data. The constraints varying only Ωm0 and
w0 are shown in Fig. 6. The constraint from the distanceratio alone is fairly weak as expected from the discussion above; in the w0 versus Ωm0 panel, the white andshaded regions show less than and more than 68% CL
respectively from the distance ratio alone. The Planck
constraints alone from the TT + lowP spectra, which
prefer w0 < −1 and Ωm0 < 0.3, are shown as red contours. These contours are largely unchanged after adding
our distance ratio measurement, shown by the blue contours.
As possible systematic uncertainties of the optical
shear analysis, we explore the effect of possible multiplicative shear bias and photo-z bias on the optical measurement. Since these biases affect the overall amplitude of the lensing signal, they are totally degenerate.
Thus we investigate these biases separately. First, weparametrize multiplicative bias as hgobs i = (1+m)hgtrue i,and fit the distance ratio with cosmological parameters

In this work we have for the first time computed the
distance ratio using optical and CMB weak lensing, yielding a 15% measurement. We have used BOSS CMASS
galaxies for the lensing galaxies, and CFHTLenS galaxy
shapes and the Planck convergence map for optical and
CMB background sources, respectively. The distance ratio extracts a purely geometrical factor by canceling out
the matter distribution around halos, and thus we arefree from systematic uncertainties arising from modelinggalaxy bias and miscentering. Our distance ratio is consistent with the predicted ratio from the Planck best-fit
ΛCDM cosmology.
Separation of the lenses into thin redshift slices,which is enabled by the spectroscopic information in the
CMASS sample, (a) allows us to make independent measurements of the distance ratio at three different redshifts, providing consistency checks, (b) makes the measurement less sensitive to variations in the mass distribution as a function of redshift, and (c) naturally avoids
loss of signal-to-noise due to weighting of CMASS galaxies by CFHTLenS weights when applying these weights
in the CMB analysis, although the latter effect is almost
negligible.
In our CMB shear anlaysis, the dominant contribution
to the noise is from the noise in the Planck reconstructed
lens map. We note that our analysis includes CMB lensing angular scales in the range 400 < L < 2000, which region was excluded from the Planck lensing autospectrum
analysis [36]. The reason for this exclusion was due to a
failure of the curl null test around L ∼ 700. While there
may be a systematic affecting the autospectrum analysis,
in general, one would expect many systematics to not bepresent in a cross-correlation analysis. However, as thecause of the autospectrum systematic is unknown, we flag
this as a caveat to the above analysis.
In our optical shear analysis, sample variance and shotnoise of the CMASS subsample dominates the statistical uncertainty. This is because the CFHTLenS survey
consists of four small fields far apart from each other.
This fact demonstrates the importance of correct covari-7
ance estimation for a survey with patchy configuration offields.
Optical surveys such as HSC, DES, KiDS, LSST,WFIRST and Euclid are expected to provide orders ofmagnitude larger samples of background sources as wellas large foreground samples with accurate photometricredshifts from red sequence calibration. In addition,
datasets from surveys like DESI and PFS will provide
large foreground samples with spectroscopic redshifts.
Combining this with wide and deep high-resolution
maps of CMB lensing from AdvancedACT, SPT3G,
the Simons Observatory, and eventually CMB Stage-4,
the coming decade will allow for measurements of the
distance ratio to within 1% making it a competitiveand complementary probe of curvature and cosmic
acceleration.
We thank Shadab Alam and Rachel Mandelbaum fordiscussions. MM would like to thank Alexie Leauthaud
and Paul Stankus for very useful conversations. HM is
supported in part by Japan Society for the Promotionof Science (JSPS) Research Fellowships for Young Scientists and by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the
National Aeronautics and Space Administration. NS acknowledges support from NSF grant number 1513618.
This work is based on observations obtained with
MegaPrime/MegaCam, a joint project of CFHT and
CEA/IRFU, at the Canada-France-Hawaii Telescope(CFHT) which is operated by the National ResearchCouncil (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche
Scientifique (CNRS) of France, and the University of
Hawaii. This research used the facilities of the Canadian
Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. CFHTLenS data processing was
made possible thanks to significant computing support
from the NSERC Research Tools and Instruments grant
program.

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