SKA Weak Lensing I: Cosmological Forecasts and the Power of Radio-Optical Cross-Correlations

0.9 MB·

by  Ian Harrison

Mon.Not.R.Astron.Soc.000,1–14(2016) Printed26October2016 (MNLATEXstylefilev2.2)
SKA Weak Lensing I: Cosmological Forecasts and the
Power of Radio-Optical Cross-Correlations
Ian Harrison(cid:63), Stefano Camera, Joe Zuntz & Michael L. Brown
Jodrell Bank Centre for Astrophysics, School of Physics & Astronomy, The University of Manchester, Manchester M13 9PL, UK
26October2016
6
1 ABSTRACT
0
Weconstructforecastsforcosmologicalparameterconstraintsfromweakgravitational
2
lensingsurveysinvolvingtheSquareKilometreArray(SKA).Consideringmattercon-

t tent,darkenergyandmodifiedgravityparameters,weshowthatthefirstphaseofthe
c
SKA (SKA1) can be competitive with other Stage III experiments such as the Dark
O
Energy Survey (DES) and that the full SKA (SKA2) can potentially form tighter

5 constraints than Stage IV optical weak lensing experiments, such as those that will
2 beconductedwithLSST,WFIRST-AFTAorEuclid-likefacilities.Usingweaklensing
alone, going from SKA1 to SKA2 represents improvements by factors of ∼10 in mat-

] ter, ∼ 10 in dark energy and ∼ 5 in modified gravity parameters. We also show, for
O
thefirsttime,thepowerfulresultthatcomparablytightconstraints(within∼5%)for
C both Stage III and Stage IV experiments, can be gained from cross-correlating shear
. maps between the optical and radio wavebands, a process which can also eliminate a
h
number of potential sources of systematic errors which can otherwise limit the utility
p
- of weak lensing cosmology.
o
r Key words: gravitational lensing: weak, dark energy, dark matter, large-scale struc-
st ture of Universe, radio continuum, galaxies
a
[

2
v 1 INTRODUCTION subsequentparameterconfidenceregions.Thesesystematics
7
include(butarenotlimitedto)telescopesystematics,galaxy
4 Mapping the cosmic shear signal with weak gravitational
intrinsic alignments (see e.g. Joachimi et al. 2015), image
9 lensing has long been regarded as an excellent probe of
analysis algorithm errors and uncertainties associated with
3 cosmology (see e.g. Kilbinger 2015, for a recent review).
modelling the non-linearity of matter clustering on small
0 In particular, future weak lensing measurements are one of
. physical scales.
1 themostpromisingobservablesforconstrainingthehistory
Inthispaperwewillconsiderinparticularthepromise
0 of the growth of cosmic structure (and the physics which
of future weak lensing experiments involving the Square
6 causedit)throughdirectsensitivitytothetotalmassalong
Kilometre Array (SKA)1 radio interferometer telescope,
1 a line of sight (e.g. Weinberg et al. 2013).
both alone and in cross-correlation with representative op-
: Fromearlydetections(Baconetal.2000;Wittmanetal.
v ticalweaklensingsurveys.TheSKAhasuniquevaluebyit-
i 2000;VanWaerbekeetal.2000;Kaiseretal.2000),progress self,theexactextentofwhichwilldependontheproperties
X has been made to the point whereby current experiments
ofthefaintradiosourcepopulationwhichwillbeprobedby
r (Heymans et al. 2013; Jee et al. 2016; The Dark Energy
a surveys with SKA pathfinders and precursors. In an ideal
Survey Collaboration et al. 2015) are able to provide mat-
scenario, the properties of this population will contain a
ter contents and dark energy constraints comparable with
long-tailed source redshift distributions, expected for the
the best available from other probes such as the Cosmic
star-forming galaxy (SFG) population that will dominate
Microwave Background (CMB, Planck Collaboration et al.
theSKAsurveys,andadduniqueadditionalinformationon
2015) and galaxy clustering (Parkinson et al. 2012; de la
thelensingshearsignalfromradiopolarisationandresolved
Torre et al. 2013; Anderson et al. 2014). As the depth and
spectrallineobservations(seeBrownetal.2015,forasum-
sky area of these and future experiments increases, uncer-
mary).Evenwithouttheadditionofmoreinformation,extra
taintiesontheseconstraintswillbegintobecomedominated
advantagescanalsobegainedbycross-correlatingtheshear
by the numerous systematic effects which come into play
maps produced from SKA data with shear maps generated
whenturningtherawastronomicaldataintoshearmapsand
(cid:63) E-mail:[email protected] 1 http://www.skatelescope.org
(cid:13)c 2016RAS
2 Harrison et al.
by other experiments in different wavebands, as recently lensingexperiment,andchooseafiducialexperimentalcon-
demonstratedbyDemetroullas&Brown(2016).Inthispro- figurationfortheSKAweaklensingsurveys.Inacompanion
cedure, any spurious shear generated by systematics which paper(Bonaldietal.2016,hereafterPaperII)weconstruct
areuncorrelatedbetweenthewavebandsshouldbeinstantly a sophisticated simulation pipeline to produce mock weak
eliminated(e.g.Pateletal.2010).Inparticular,contamina- lensing catalogues for future SKA surveys which we also
tionfromanincorrectlydeconvolvedspatiallyvaryingPoint process through a tomographic weak lensing power spec-
Spread Function (PSF) and errors from algorithms used to trum analysis. We then use this pipeline to explore the op-
measure the shapes of individual galaxies to infer the shear timal instrumental configuration for performing SKA weak
should be uncorrelated between the different experiments. lensing surveys in the presence of real-world effects such as
When measuring an observed shear map γ made in wave- signal-to-noise dependent shape measurement errors, real-
(cid:101)
bandX,theobservedsignalreceivescontributionsfromthe istic distributions in galaxy sizes, fluxes and redshifts and
truegravitationalshearingγ (whichisachromaticandiden- ionospheric distortions.
tical in both wavebands), the intrinsic shape of the galaxy Theoutlineofthispaperisasfollows.Wefirstprovide
γint and spurious shear from incorrectly deconvolved PSF a brief review of radio weak lensing in Section 2. In Sec-
or shape measurement error γsys. The cross-correlation of tion3wethendescribetheexperimentalsurveysconsidered
shear maps in different wavebands then has terms: fortheforecastsanddescribeourmethodologyforconstruc-
tion of cross-experiment shear power spectra. In Section 4
(cid:104)γ γ (cid:105)=(cid:104)γγ(cid:105)+(cid:104)γintγ(cid:105)+(cid:104)γintγ(cid:105)
(cid:101)X(cid:101)Y X Y we describe the methods used in producing our forecasts.
(1)
+(cid:104)γintγint(cid:105)+(cid:104)γsysγsys(cid:105). Then,inSection5weshowresultsforcosmologicalparame-
X Y X Y
terconstraintsusingSKA,StageIIIoptical(DES),StageIV
The first term is the cosmological signal that we are inter-
optical (Euclid-like) and cross-correlations, demonstrating
estedin,thefollowingthreetermsarecontaminating‘intrin-
the power of using optical and radio experiments together.
sicalignment’terms(seeJoachimietal.2015;Kiesslingetal.
Finally in Section 6 we discuss these results and conclude.
2015;Kirketal.2015,forarecentreview)andthefinalterm
isasystematicsterm(wehaveignoredtermscorrelatingsys-
tematicswithsignalsonthesky).Anycontributionstothese
systematics terms which are uncorrelated between different 2 WEAK LENSING COSMOLOGY
experimentsandwavebandswillbesuppressedbythecross-
We refer the reader to Bartelmann & Schneider (2001) for
correlation, greatly increasing the robustness of cosmologi-
acomprehensiveoverviewofweaklensingcosmology,which
calconstraints.Ifpolarisedandneutralhydrogen(HI)21cm
we will briefly introduce here. Weak lensing analyses typ-
lineemissionfractionsfromhighredshiftsourcesprovetobe
ically involve the measurement of the individual shapes of
high enough, radio weak lensing experiments can also pro-
largenumbersofgalaxiesonthesky.Foralargenumberden-
vide useful information on intrinsic alignment systematics
sityofsourcesinasinglepatchofsky,theestimatedchange
throughpolarisation(Brown&Battye2011)androtational
in shape due to the cosmic shear along the line of sight to
velocityinformation(Blain2002;Morales2006),thoughwe
thatpatch(γˆ)canbeestimatedbytakingasimpleaverage
donotconsidersuchapproachesintheseforecasts.Instead,
overtheobservedellipticityofthegalaxies((cid:15)obs),assuming
we consider what can be achieved with ‘vanilla’ SKA weak
that the intrinsic shapes before shearing are uncorrelated:
lensingsurveysinwhichcosmologicalinformationcomefrom
forming shear power spectra from measured galaxy elliptic- N
ities, just as in typical optical experiments. Adopting the γˆ= 1 (cid:88)(cid:15)obs. (2)
N i
surveycategorisationschemeoftheDarkEnergyTaskForce i=1
(DETF,Albrechtetal.2006),wewillshowthatsurveyscon- The two-point statistics of this observed shear field, such
ductedwiththefirstphaseoftheSKA(SKA1)willbecom- as the power spectrum C(cid:101)(cid:96), can then be related to the un-
petitivewith‘StageIII’opticalweaklensingsurveyssuchas derlyingmatterpowerspectrumP ,whichcanbepredicted
δ
DES2, KiDS3 and HSC4, and that full SKA (SKA2) weak theoretically for different cosmological models. For sources
lensing surveys can provide ‘Stage IV’ constraints similar confined to a thin shell in redshift, the C(cid:101)(cid:96) are sensitive to
to those achievable with the weak lensing components of theintegratedmatterpowerspectrumouttothisredshift.In
the Euclid5, WFIRST-AFTA6 and LSST7 surveys. We will practice, sources are distributed across a range of redshifts
also show that constraints obtained from cross-power spec- dn /dz (which is in turn affected by imprecise knowledge
gal
tra measured between shear maps made in different wave- oftheredshiftsofindividualsources)andextrainformation
bandswillprovidemeasurementswhicharestilljustastight is gained about the growth of structures along the line of
aseachexperimentbyitself,butshouldbefreeofanywave- sight by constructing the auto- and cross-power spectra of
length dependent systematics. shearmapsmadeusingsourcesdividedintodifferenttomo-
Here we make forecasts using simple prescriptions for graphic redshift bins.
the noise spectra and covariance matrices within a weak The full relation for the power spectrum between two
different tomographic bins i,j is given by (Bartelmann &
Schneider 2001):
2 http://www.darkenergysurvey.org
3 http://kids.strw.leidenuniv.nl Cij = 9H04Ω2m (cid:90) χhdχgi(χ)gj(χ)P (cid:18) (cid:96) ,χ(cid:19) . (3)
4 http://subarutelescope.org/Projects/HSC (cid:96) 4c4 a2(χ) δ f (χ)
0 K
5 http://euclid-ec.org
6 http://wfirst.gsfc.nasa.gov Here, H0 is the Hubble constant, Ωm is the (total) matter
7 http://www.lsst.org density, c is the speed of light, a(χ) is the scale factor of
(cid:13)c 2016RAS,MNRAS000,1–14
SKA Weak Lensing I: Forecasts 3
the Universe at co-moving distance χ, f (χ) is the angu- ThismodelrepresentsthefirstorderterminaTaylorexpan-
K
lar diameter distance (given simply by f (χ) = χ in a flat sion of a generally evolving equation of state. We consider
K
Universe), P (k,χ) is the matter power spectrum and the these parameters in ϑ =ϑ +{w ,w }.
δ w ΛCDM 0 a
functions gi(χ) are the lensing kernels for the redshift bins
in question. The lensing kernels are given by:
2.1.3 Modified Gravity
gi(χ)=(cid:90) χhdχ(cid:48)ni(χ(cid:48))fKf(χ((cid:48)χ−(cid:48))χ). (4) Wealsoconsidermodificationstogravityasparametrisedin
χ K Dossett et al. (2011, 2015). In General Relativity, from the
Thenumberdensitydistributionsn (χ)givethenormalised perturbed Friedmann-Lemaitre-Robertson-Walker (FLRW)
i
number of galaxies with radial co-ordinate χ in this tomo- metric in the conformal Newtonian gauge:
graphic bin. For single experiment weak lensing cosmology, ds2 =a2(η)(cid:2)−(1+2Ψ)dη2+(1−2Φ)dxadx (cid:3), (6)
the i,j label different tomographic redshift bins and the a
uncertainty on the power spectrum depends on ngal, the we define the Newtonian gravitational potential Ψ felt by
number density of detected galaxies on the sky and σg, matterandthelensingpotentialΦwhichisalsofeltbyrela-
the variance of the distribution of galaxy ellipticities (or tivisticparticles.Wenowdefinemodifiedgravityparameters
‘shape noise’). We will generalise these measurement and Q ,whichmodifiesthepotentialΦintherelativisticPoisson
0
noise terms to include cross-experiment power spectra in equation:
Section 2.4.
k2Φ=−4πGa2ρ∆Q (7)
0
andthegravitationalslipRwhich,inthecaseofanisotropic
2.1 Cosmological Parameters stress, gives the ratio between the two potentials:
In this paper we will consider the ability of weak lensing Ψ
R= . (8)
experimentstomeasureabasesix-parameterΛCDMmodel Φ
and two well-motivated extensions: dynamical dark energy As R is degenerate with Q it is convenient to define the
0
and a phenomenological modification to Einstein’s gravity. derived parameter Σ = Q (1+R)/2 and our constraints
0 0
We note that these choices are merely common parametri- are given in terms of this. Weak lensing probes the sum of
sations of these extensions and are not specifically tailored potentials Φ+Ψ and is hence extremely effective at con-
to the strengths of SKA weak lensing. Different parametri- straining Σ but much less sensitive to Q . Combination
0 0
sations (for example, non-parametric dark energy equation with probes for which the opposite is true (i.e. which are
ofstatereconstructionwhichequallyweightsinformationat sensitivetotheNewtonianpotential),suchasredshiftspace
all redshifts) may more optimally use the information from distortions, is then capable of breaking the degeneracy in-
these experiments for model selection, but are not consid- herent in each probe individually (see e.g. Simpson et al.
ered here. 2013;Leonardetal.2015).Weconsidertheseparametersin
ϑ =ϑ +{Σ ,Q }.
mg ΛCDM 0 0
2.1.1 Base ΛCDM
2.2 Weak Lensing Systematics
For our base cosmology we consider six parameters: to-
tal matter content Ω , baryonic matter content Ω , am- Whilst the statistical error on a weak lensing measurement
m b
plitude of matter fluctuations σ , Hubble expansion pa- of a cosmological parameter can be beaten down through
8
rameter h0, scalar fluctuation spectral index ns and reion- increasing the number density of galaxies ngal with mea-
isation optical depth τ. Unless otherwise stated, all con- sured shapes on the sky (or by selecting a population with
straints presented are marginalised over the first five of a smaller intrinsic shape dispersion σg), forthcoming Stage
these parameters (with τ kept fixed) with central values of IIIandStageIVexperimentswillbegintoentertheregime
ϑ = {Ω ,Ω ,σ ,h ,n } = {0.3,0.04,0.8,0.72,0.96}. wherethecontributionfromsystematicerrorsonshearmea-
ΛCDM m b 8 0 s
Weak lensing is highly effective at probing the overall am- surement will become comparable to, and larger than, the
plitude of the matter power spectrum, which depends on a statistical noise. Here we provide a brief overview of many
degenerate combination of the total matter Ω and clus- (although not all) of these systematics, whereas a more de-
m
tering strength σ ; we will therefore present constraints in tailed analysis of their effects and ways to overcome them
8
these two parameters only. willbeprovidedinacompanionpaper(Cameraetal.2016,
hereafter Paper III).
• PSF uncertainties. The light from all sources used in
2.1.2 Dark Energy weak lensing is convolved with the telescope point spread
function. This convolution will induce changes in the size
AsoneextensiontoΛCDM,wewillconsidermeasuringthe
and ellipticity of the apparent galaxy shape in the image
parametersinasimplemodelofevolvingdarkenergywhere
data, and must be accounted for when estimating the true
the equation of state w evolves as a linear function of the
observedellipticity.Typically,amodeliscreatedforthePSF
scale factor a (known as the Chevallier-Polarski-Linder pa-
which is then deconvolved during shear measurement. For
rameterisation, see Chevallier & Polarski 2001 and Linder
ground-based optical experiments, the primary systematic
2003):
isresidual,un-modelledPSFshapedistortionsduetoinsta-
w(a)=w +w (1−a). (5) bilities in the atmosphere above the telescope (i.e. seeing).
0 a
(cid:13)c 2016RAS,MNRAS000,1–14
4 Harrison et al.
For space-based telescopes the atmosphere is not a consid- spectroscopiccalibrationandnoisydata.Foradiscussionof
eration, but other effects from detectors and telescope op- theseissuesseeBonnettetal.(2016)andreferencestherein.
ticscanstillcreateananisotropicandtime-varyingPSF.In
addition, the deterministic nature of the changes in inter-
ferometer dirty beam shape with observing frequency may 2.3 Radio Weak Lensing
potentially avoid issues with shear bias from colour gradi-
Performing weak lensing experiments in the radio band of-
ents in source galaxies (see e.g. Voigt et al. 2012, for a full
fers a number of potential advantages compared to using
description of the problem). However, care will need to be
optical telescopes alone. In addition to opening the door
taken to ensure the primary beam of each antenna is well-
to powerful cross-correlation techniques (which we consider
characterisedenoughtoavoidthereturnofshearbiasesorig-
in more detail in the following subsection), the radio band
inating from the beam.
has the potential to bring unique added value to this area
• Shear measurement uncertainties (see Mandelbaum
of cosmology by way of new approaches to measuring the
etal.2014,andreferencesthereinforanoverview).Usingthe
weak lensing signal using polarisation and rotational veloc-
observedgalaxyellipticityasashearestimatorasinEq.(2)
ity observations. Here we summarise the key benefits that
depends on having a reliable, unbiased estimator of the el-
radioweaklensingexperimentscanofferandhighlightsome
lipticity. Whilst in the noise-free case, (cid:15) can be defined as a
ofthechallengeswhichneedtobemet.Wereferthereader
simplefunctionofthequadrupolemomentsoftheimage,sig-
to Brown et al. (2015) for more information.
nificantcomplicationsarisewhenevernoiseispresentasthe
un-weightedquadrupoleswilldiverge.Ingeneral,maximum • Weak lensing surveys conducted with radio telescopes
likelihoodestimatorsforellipticitywillbecomeincreasingly are, in principle, much less susceptible to instrumental sys-
biased at lower signal-to-noise ratios (as ellipticity is a ra- tematic effects associated with residual PSF anisotropies.
tioofquadrupolemoments),andsomustbecalibrated(e.g. The image-plane PSF (or ‘dirty beam’) is set by the base-
Refregieretal.2012).Shearestimatorswhichmeasure(cid:15)us- line distribution and time and frequency sampling of the
ingparametrisedmodelswithellipticalisophotesalsosuffer telescope, all of which are deterministic and known to the
from ‘model bias’ caused by under-fitting of real galaxy in- observer and may be controlled. An anisotropic PSF can
tensity profiles (Voigt & Bridle 2010). Accounting for these mimicthesought-aftercosmicshearsignalandareoneofthe
biases correctly, through either explicit calibration or ap- most worrisome systematic effects in optical lensing analy-
plication of correct Bayesian priors, is a major step in the ses. Whilst the turbulent ionosphere can cause similar ef-
analysispipelineformostsurveysandrequiressophisticated, fectsintheradio,theseeffectsscalestronglywithfrequency,
large scale simulations which correctly reflect the observa- meaning at the high frequency considered here (1.355GHz,
tions. seePaperIIforafulldiscussion)thisislessofaconcernfor
• IntrinsicAlignment(IA)contamination.Akeyassump- radio weak lensing.
tion in Eq. (2) is that intrinsic galaxy shapes are uncor- • However, whilst the dirty beam is precisely known and
related and so any coherent shape must be due to cosmic highlydeterministic,theincompletesamplingoftheFourier
shear. However, in reality there are two other astrophysical plane by the finite number of interferometer baselines leads
effects which contaminate the shear signal. Galaxies which to significant sidelobes which may extend across the entire
are nearby on the sky form within the same large scale visible sky. Deconvolving this PSF then becomes a com-
structureenvironmentasoneanother,creatingspurious‘II’ plicated non-local problem as flux from widely-separated
(Intrinsic-Intrinsic)correlations.Inaddition,galaxieswhich sources is mixed together and traditional methods (such as
are local in redshift to an overdensity will develop intrinsic the CLEAN algorithm, Ho¨gbom 1974) have been shown to
shapes in anti-correlation with the shearing of background beinadequateforpreservingmorphologytothedegreenec-
galaxiesbythatsameoverdensity–the‘GI’(Gravitational- essary for weak lensing.
Intrinsic)alignment.Typically,thesealignmentscanbemit- • The SFGs which are expected to dominate the deep,
igatedthroughmodellingtheireffectonthepowerspectrum, wide-field surveys to be undertaken with the SKA are also
or discounting galaxies which are expected to be most af- expected to be widely distributed in redshift space (see
fected (such as close pairs on the sky or redder galaxies). Wilman et al. 2008, and Paper II). In particular, a high-
An overiew of IA effects can be found in Joachimi et al. redshifttailofsignificantnumbersofsuchgalaxies,extend-
(2015), Kiessling et al. (2015) and Kirk et al. (2015). ing beyond z ∼ 1 would provide an additional high-z bin
• Non-linearevolutionandbaryonicfeedbackeffects.Cos- to what is already accessible with optical surveys. See the
mology with cosmic shear relies on the comparison be- end of Section 4.2 for a demonstration of the increase in
tween an observed shear power spectrum and a theoreti- cosmologicalconstrainingpowerfromtheinclusion ofthese
cally predicted one. However, outside of the regime of lin- high-redshift sources. The details of the flux and size dis-
ear evolution of large scale structures (i.e. on smaller scales tributions of this population are still somewhat uncertain
k(cid:38)0.2hMpc−1),avarietyofphysicaleffectswillaffectthe (seePaperIIforafulldiscussion)andwillbenefitfromthe
shape of this power spectrum in uncertain ways which are efforts of SKA precursor and pathfinder surveys.
possibly degenerate with changes in cosmological parame- • The orientation of the integrated polarised emission
ters (e.g. Huterer & Takada 2005). fromSFGsisnotalteredbygravitationallensing.Ifthepo-
• Redshift uncertainty estimation. Placing sources into larisationorientationisalsorelatedtotheintrinsicstructure
tomographicbinsusuallyrequiresanestimateofthesource’s ofthehostgalaxythenthisprovidesapowerfulmethodfor
redshift from a small number of broad photometric bands. calibratingandcontrollingintrinsicgalaxyalignmentswhich
Significant biases may arise due to insufficient freedom in are the most worrying astrophysical systematic effect for
Spectral Energy Distribution (SED) templates, incorrect precisionweaklensingstudies(Brown&Battye2011;Whit-
(cid:13)c 2016RAS,MNRAS000,1–14
SKA Weak Lensing I: Forecasts 5
takeretal.2015).Again,thepolarisationfractionandangle The noise is a function of the number density of galaxies in
of scatter between position and polarisation angle is cur- each experiment individually nXi,nYj, the number of ob-
gal gal
rently subject to much uncertainty and have currently only jectswhicharecommontobothexperimentsnXiYj andthe
been tested on small low-redshift samples (Stil et al. 2009). gal
covariance of galaxy shapes between the two experiments
This result may not preserve in the high-redshift SFGs we
and redshift bins cov((cid:15) ,(cid:15) ). Note that this final term
are interested in here, but will become better informed by Xi Yj
cov((cid:15) ,(cid:15) )isingeneralafunctionofbothwavebandX,Y
other surveys leading up to the SKA. Xi Yj
and redshift bin i,j, describing how galaxy shapes are cor-
• Much like the polarisation technique, observations of
relatedbetweenthetwowavebandsandhowthiscorrelation
the rotation axis of disk galaxies also provides informa-
evolves with redshift. We can then write the expression for
tion on the original (un-lensed) galaxy shape (Blain 2002;
the noise on an observed shear power spectrum:
Morales2006;Huffetal.2013).Suchrotationaxismeasure-
ments may be available for significant numbers of galaxies NXiYj = 1 (cid:104)(cid:88) (cid:15) (cid:88) (cid:15) (cid:105)
with future SKA surveys through resolved 21 cm HI line (cid:96) nXinYj α β
gal gal α∈Xi β∈Yj
observations.
• HIlineobservationsalsoprovideanopportunitytoob- nXiYj
= gal cov((cid:15) ,(cid:15) ). (11)
tainspectroscopicredshiftsforsourcesusedinweaklensing nXinYj Xi Yj
gal gal
surveys(e.g.Yahyaetal.2015),greatlyimprovingthetomo-
graphicreconstructionforthesourcesforwhichspectraare For correlations between redshift bins in the same experi-
available. For SKA1 this will be a relatively small fraction ment this reduces to the familiar shape noise term (e.g. Hu
ofsources(∼10%)atlowredshifts(whicharelessusefulfor & Jain 2004):
gravitational lensing) but this will improve significantly for
σ2
SKA2. Nij =δij gi . (12)
(cid:96) ni
• BecauseGalacticradioemissionatrelevantfrequencies gal
issmooth,itis‘resolvedout’byradiointerferometers.This
If we make the simplifying assumption that for cross-
meansthatradiosurveyshaveaccesstomoreoftheskythan
experiment correlations, where redshift bins overlap, both
experiments in other wavebands, which cannot see through
experiments probe the same populations of galaxies which
the Galaxy because of dust obscuration effects.
havethesameshapeandshapevarianceinbothwavebands
A detection of a weak lensing signal in radio data was and across all redshift bins, the noise term becomes:
firstmadebyChangetal.(2004)inashallow,wide-areasur-
nXiYj
vey.MorerecentlyDemetroullas&Brown(2016)havemade NXiYj = gal σ2. (13)
a measurement in cross-correlation with optical data, and (cid:96) nXinYj g
gal gal
the SuperCLASS8 survey is currently gathering data with
Here, for the two sets of tomographic redshift bins for
the express purpose of pushing forward radio weak lensing
each experiment we consider the fraction of sources which
techniques.
may be expected to appear in both the radio and optical
shape catalogues. In reality, this overlap will be between a
2.4 Shear Cross-Correlations deep optical sample and a deep radio sample of SFGs on a
widearea.Datasetswiththiscombinationofareacoverage
Whilstradioweaklensingsurveyshaveworthinthemselves,
and depth do not as yet exist, but useful information can
asdiscussedabove,combiningshearmapsmadeatdifferent
begainedfromsomeshallowerornarrowerarchivalsurveys.
observational wavelengths has further potential to remove
Here we consider the large but shallow SDSS-DR10 optical
systematics which can otherwise overwhelm the cosmologi-
catalogue (Ahn et al. 2014) and the FIRST radio catalogue
calsignal.Hereweconstructaformalismforforecastingthe
(Beckeretal.1995;Changetal.2004);anddeepbutnarrow
precisionwithwhichcross-correlationpowerspectracanbe
observations of the COSMOS field using the Hubble Space
measured from shear maps obtained from two different ex-
Telescope (McCracken et al. 2010) in the optical and VLA
periments X,Y, which may be in different wavebands. We
intheradio(Schinnereretal.2010).TheSDSS-FIRSTover-
maystillsplitsourcesineachexperimentintodifferentred-
lapregioncontainsasignificantpart(∼10,000 deg2)ofthe
shift bins i,j, giving the cross power spectra:
northern sky, but the radio catalogue is shallow (a 10σ de-
CXiYj = 9H04Ω2m (cid:90) χhdχgXi(χ)gYj(χ)P (cid:18) (cid:96) ,χ(cid:19) . tection limit of 1.5mJy). The COSMOS overlap survey is
(cid:96) 4c4 a2(χ) δ f (χ) deep (a 10σ detection limit of 0.28mJy) but covers only
0 K
(9) 1 deg2. These data sets appear to indicate that matching
Herethebinscanbedefineddifferentlyforeachexperiment, fractions are low (< 10%) and do not evolve significantly
takingadvantageofe.g.highermedianredshiftdistributions with redshift. In addition, the optical and radio weak lens-
orbettermeasuredphotometricredshiftsinoneortheother ingsamplesconstructedbyPateletal.(2010)inan8.5(cid:48)×8.5(cid:48)
of the two experiments. fieldintheHDF-Nregioncontaina4.2%matchingfraction
When observed, each power spectrum also includes a across all redshifts.
noise power spectrum from the galaxy sample: To investigate how much a non-vanishing radio-optical
C(cid:101)XiYj =CXiYj +NXiYj. (10) matching fraction could degrade the radio-optical cross-
(cid:96) (cid:96) (cid:96) correlationconstrainingpowerforcosmology,weproceedas
follows.Weintroduceaparameterf ∈[0,1]quantifying
O−R
8 http://www.e-merlin.ac.uk/legacy/projects/superclass. thenumberofsourcesthatappearsinboththeradioandthe
html optical/near-infrared catalogues for a given combination of
(cid:13)c 2016RAS,MNRAS000,1–14
6 Harrison et al.
consisting of both:
 
C(cid:101)(cid:96)XX
d(cid:101)=C(cid:101)(cid:96)XY, (14)
C(cid:101)(cid:96)YY
we can also write the covariance matrix between two bins
indifferentexperiments(nowsuppressingthei,j forclarity
and with ν =δ(cid:96)(cid:96)(cid:48)/(2(cid:96)+1)fsky):
Γ(cid:101)(cid:96)(cid:96)(cid:48) = (15)
 
2(C(cid:101)(cid:96)XX)2 2C(cid:101)(cid:96)XXC(cid:101)(cid:96)XY 2(C(cid:101)(cid:96)XY)2
ν2C(cid:101)(cid:96)XXC(cid:101)(cid:96)XY (C(cid:101)(cid:96)XY)2+C(cid:101)(cid:96)XXC(cid:101)(cid:96)YY 2C(cid:101)(cid:96)XYC(cid:101)(cid:96)YY,
2(C(cid:101)(cid:96)XY)2 2C(cid:101)(cid:96)XYC(cid:101)(cid:96)YY 2(C(cid:101)(cid:96)YY)2
making the simplifying assumption that different (cid:96) modes
are uncorrelated and hence the covariance matrix is diago-
nal in (cid:96)−(cid:96)(cid:48). However, here we are interested in forecasting
constraints which can be gained which are free of system-
Figure 1. Ratio with respect to the case with no radio-optical atics caused by e.g. incorrect PSF deconvolution within an
matching fractions (fO−R =0) for dark energy FoMs as a func- experiment and so consider only cross-experiment spectra
tionoffO−R forthecross-correlationbetweenStageIII(dashed (as such systematics will be uncorrelated between the two
line) and Stage IV (solid line) experiments. The shaded regions experiments), giving data vector:
showstherangeofvaluesforfO−R forthedatasetsdiscussedin
(cid:16) (cid:17)
thetext. d(cid:101)= C(cid:101)(cid:96)XY , (16)
and covariance matrix:
(cid:16) (cid:17)
tomographicbins.Inotherwords,wekeepnXiYj fixedtothe Γ(cid:101)(cid:96)(cid:96)(cid:48) =ν (C(cid:101)(cid:96)XY)2+C(cid:101)(cid:96)XXC(cid:101)(cid:96)YY . (17)
gal
amount of sources present in the overlap between to given
Forecasts presented here for cross-correlation experiments
radio-opticalbinpairsX −Y .Wethenmultiplythisquan-
i j willbeofthiscross-onlyformandwithnoisetermsgivenby
tity by f and perform a Fisher matrix analysis letting
O−R Eq. (13).
f vary continuously between 0 and 1 (but identically
O−R
acrossallredshiftbins).Fig.1illustratesthedegradationof
the Dark Energy Task Force Figure of Merit (DETF FoM
– the inverse area of a Fisher ellipse in the w -w plane,
0 a
see Albrecht et al. 2006 and Eq. (22)) – as the fraction of
3 EXPERIMENTS CONSIDERED
matching radio-optical sources, f , increases (note that
O−R
for simplicity we assume cov((cid:15)Xi,(cid:15)Yj) = σ(cid:15)2). We show the A number of surveys across multiple wavebands are both
ratio between the DETF FoM for a non-vanishing radio- currentlytakingplaceandplannedforthenearfuturewhich
optical matching fraction fO−R and the same quantity for have weak lensing cosmology as a prominent science driver.
fO−R =0. It is easy to see that even if 100% of the sources We adopt the language of the Dark Energy Task Force
appeared in both catalogues, the degradation of the dark (DETF, Albrecht et al. 2006) in loosely grouping these ex-
energy FoM would be <5% for Stage III cosmic shear sur- perimentsinto‘StageIII’and‘StageIV’experiments,where
veys, and even lower for Stage IV experiments. If we then StageIIIreferstoexperimentswhichwereinthenearfuture
consider the available data as described previously in this whentheDETFdocumentwaspreparedcomparedtoStage
section,therangeofvaluesoffO−R forwhichareindicated IV experiments which follow these in time. The distinction
bytheshadedarea,wemayseetheminimalimpactofreal- can also be cast in terms of the expected level of constrain-
istic noise terms on the cross-correlation power spectra. ing power, with Stage III Weak Lensing alone experiments
In order to account for this in the following forecasts giving O(50%) constraints on the Dark Energy equation of
weconsidertheregimewhereoverlapfractionsarehighand statewandStageIVO(10%).Wepointoutthatwepresent
photometric redshifts are provided for the 85% and 50% of hereconstraints fromweak lensing analysesonly;inreality,
sources which do not have spectroscopic HI 21cm line red- significant improvements on constraints will be gained by
shifts in the case of SKA1 and SKA2 respectively (as de- both the SKA and optical surveys’ measurements of galaxy
scribed in Table 1). However, as mentioned in Section 3.4, clustering and other probes (such as supernovae and Inten-
itmaybepossibleforradiosurveysalonetoprovidesignifi- sity Mapping), as well as combination with external data
cantlymoreredshiftsthanthosefromonlyhigh-significance sets.
HI detections. Foreachstageweconsiderarepresentativeexperiment
In the regime where systematics are controlled, the from both the optical and the radio. We now give short
maximumamountofinformationisavailablebyusingboth background descriptions of the source populations assumed
crossandauto-experimentpowerspectra.Foradatavector and the particulars of each experiment considered.
(cid:13)c 2016RAS,MNRAS000,1–14
SKA Weak Lensing I: Forecasts 7
Experiment Asky[deg2] ngal[arcmin−2] zm α β γ fspec-z zspec-max σphoto-z zphoto-max σno-z
√
SKA1 5,000 2.7 1.1 2 2 1.25 0.15 0.6 0.05 2.0 0.3
√
DES 5,000 12 0.6 2 2 1.5 0.0 2.0 0.05 2.0 0.3
√
SKA2 30,000 10 1.3 2 2 1.25 0.5 2.0 0.03 2.0 0.3
√
Euclid-like 15,000 30 0.9 2 2 1.5 0.0 0.0 0.03 4.0 0.3
Table 1.Parametersusedinthecreationofsimulateddatasetsfortherepresentativeexperimentsconsideredinthispaper.
3.1 Source Populations
For the number density of sources in each tomographic bin
in each experiment we use a redshift number density distri-
bution of the form:
dn
gal =zβexp(−(z/z )γ), (18)
dz 0
where z = z /α (α is a scale parameter) and z is the
0 m m
medianredshiftofsources.FortheSKAexperimentsweuse
thesourcecountsintheSKADSS3-SEXsimulationofradio
sourcepopulations(Wilmanetal.2008);wehaveappliedre-
scalings of these populations in both size distributions and
number counts in order to match recent data (see Paper II
for a full description). Values of the parameters in Eq. (18)
are given in Table 1, including the best-fit parameters to
the SKADS S3-SEX distribution. The top panel of Fig. 2
shows these distributions for the experiments considered,
including the high-redshift tail present in the radio source
populations. For each experiment we then subdivide these
populationsintotentomographicredshiftbins,givingequal
numbersofgalaxiesineachbin.Wealsoaddredshifterrors,
spreadingtheedgesofeachredshiftbinandcausingthemto
overlap.Weassumeafractionofsourceswithspectroscopic
redshifts(i.e.withnoredshifterror)f uptoaredshift
spec-z
ofz .FortheremainingsourcesweassignaGaussian-
spec-max
distributed (with the prior z > 0) redshift error of width
(1+z)σ uptoaredshiftofz ,beyondwhichwe
photo-z photo-max
assumeno‘good’photometricredshiftestimateandassigna
fargreatererror(1+z)σ .Valuesfortheseparametersfor
no-z
eachrepresentativeexperimentareshowninTable1andthe
resultingbinneddistributionsforSKA2andtheEuclid-like
experiment (see Section 3.3 below) are shown in the lower
panelofFig.2.Wetakeanintrinsicgalaxyshapedispersion
ofσ =0.3forallredshiftbinsandexperiments,consistent
gi
with that found for the radio and optical lensing samples
used in previous radio weak lensing (Patel et al. 2010).
3.2 Stage III Experiments
3.2.1 SKA Phase 1 (SKA1) Figure 2. Source (top) and “observed” (bottom, split into
tentomographicbinsforeachexperiment)redshiftdistributions
The Square Kilometre Array (SKA) will be built in two dn /dz fortheEuclid-likeandSKA2experimentsdescribedin
gal
phases:thefirst(SKA1)willconsistofalowfrequencyaper- Section 3.3. The curves in both panels are normalised such that
turearrayinWesternAustralia(SKA1-LOW)andadishar- thetotalareaunderthecurvesisequaltothetotalngal foreach
ray to be built in South Africa (SKA1-MID) with expected experiment.
commencement of science observations in 2020. Of these,
it is SKA1-MID which will provide the necessary sensitiv-
ity and resolution to conduct weak lensing surveys. Here half maximum (FWHM). This experimental configuration
we have assumed source number densities expected to em- is expected to give a close-to-optimal combination of high
anate from a 5,000deg2 survey conducted at the centre of galaxy number density and quiescent ionosphere, as well as
observing Band 2 (1.355 GHz) and with baselines weighted maximise commensality with other SKA science goals (see
to give an image-plane PSF of size 0.5 arcsec full width at PaperIIandHarrison&Brown2015forfurtherdiscussion).
(cid:13)c 2016RAS,MNRAS000,1–14
8 Harrison et al.
Wethencalculatetheexpectedsensitivityoftheinstrument formanceoftheweaklensingcomponentoftheEuclid satel-
when used in this configuration using the curves from the lite(Laureijsetal.2011;Amendolaetal.2013)plannedfor
SKA1 Imaging Science Performance Memo (Braun 2014), launchin2020.WerefertothisrepresentativeStageIVop-
which assumes a two year survey, and including all sources tical weak lensing-only experiment as “Euclid-like ”.
which are resolved and detected at a signal-to-noise greater
than10.Wenotethatestimatesforthenumberdensitiesand
distributionofsizesforSFGsatmicro-Janskyfluxesarecur- 3.4 Cross-Correlations
rently somewhat uncertain. To arrive at our estimates, we
For cross-correlation experiments, we take combinations of
followtheproceduredescribedinPaperII.Inbrief,weonce
StageIIIexperiments(DESandSKA1)andStageIVexper-
againmakeuseoftheSKADSS3-SEXsimulation(Wilman
iments(Euclid-likeandSKA2).ForDES×SKA1weassume
et al. 2008) but we have re-calibrated the absolute num-
the5,000deg2skycoverageisthesameforbothsurveysand
bersandsizesofSFGsfoundinthatsimulationsothatthey
construct theoretical power spectra C with lensing kernels
matchthelatestobservationaldatafromdeepradiosurveys. (cid:96)
givenbygDESi andgSKA1i,withtentomographicbinsfrom
ForbothSKAexperimentswealsoincludefractionsofspec-
each experiment defined to have equal numbers of sources
troscopicredshifts,obtainedbydetectionofHIlineemission
in each bin (i.e. bin i for DES does not correspond to, but
from the source galaxies.
may overlap with, bin i for SKA1). For the noise power
spectra NXiYj we assume a limiting case in which there is
(cid:96)
3.2.2 Dark Energy Survey (DES) negligible overlap between the source populations probed
by the different experiments (as found in Demetroullas &
For our Stage III optical weak lensing survey we follow the
Brown2016)andforobjectswhichdoexistinbothsurveys,
performance specifications of the weak lensing component
shapesareuncorrelated,assuggestedbythefindingsofPatel
oftheDarkEnergySurvey(DES).DESisanopticalsurvey
etal.(2010),meaningthepopulationsinthetwentydifferent
with a primary focus on weak lensing cosmology, covering
5,000deg2oftheSouthernhemisphereskyusingthe4-metre binsaretreatedaswhollyindependent.Asdemonstratedin
Fig. 1, the relaxation of this assumption should not signif-
BlancotelescopeattheCerroTololoInter-AmericanObser-
icantly affect the achievable constraints. In the case where
vatory in Chile. It has already produced cosmological pa-
the samples are completely separate, redshift information
rametermeasurementsfromweaklensingwithScienceVer-
willbenecessaryfortheSKAsources,butcouldbeobtain-
ificationdata(TheDarkEnergySurveyCollaborationetal.
able from sub-threshold techniques which make use of the
2015)andrepresentsa‘StageIII’weaklensingsurveyalong
HI 21 cm line below the detection limit traditionally used
withcontemporariessuchastheKilo-DegreeSurvey(KiDS,
for spectroscopic redshifts (techniques we are exploring in
Kuijken et al. 2015) and Hyper Suprime Cam (HSC) weak
ongoing work), something which should be very capable in
lensing projects. Here we use the expected performance of
providingimperfectdn /dz(inthemannerofphotometric
thefullfiveyearsurveydata,withobservationsing,r,i,z,Y gal
redshifts) for tomographically binned sources.
bandsandalimitingmagnitudeof24.Theachievableweak
ForEuclid-like×SKA2weconsideronlythe15,000deg2
lensing source number densities and redshift distributions
survey region available to both experiments. Again, ten
considered here are drawn from (The Dark Energy Survey
equally populated tomographic redshift bins are chosen for
Collaboration 2005, 2016).
eachexperimentandobservedcross-spectraareformed.We
emphasisethatwearenotmerelyconsideringthelowestn
gal
3.3 Stage IV Experiments of the two experiments for the cross-correlations, but using
thefulldn /dz distributionsintwentybins,tenfromeach
gal
3.3.1 Full SKA (SKA2)
experiment, making use of all the galaxies present.
As described in Dewdney (2013), the full SKA (SKA2) will
be a significant expansion of SKA1, with the current plan
for SKA-MID increasing the number of dishes from 194 to
4 FORECASTING METHODS
∼2000(withtheinitial194integratedintothelargerarray)
andspreadinglongbaselinesoverSouthernAfrica,undergo- For forecasting constraints on cosmological parameters
ing construction between 2023 and 2030. As the sensitivity which will be possible with the SKA and cross-correlations
scaleswithapproximatelythetotalcollectingarea,forSKA2 we use two approaches: Markov Chain Monte Carlo
we assume a ten times increase in sensitivity of the instru- (MCMC) mapping of the likelihood distribution and the
mentandmakeourforecastsfora3πsteradiansurvey,again Fisher Matrix approximation. For a given likelihood func-
at the centre of observing Band 2 (1.355 GHz) and with a tion and covariance matrix, MCMC methods are accurate
0.5 arcsec PSF. andcapableoftracingcomplicatedposteriorprobabilitydis-
tributionsurfacesinmultipledimensions,butarecomputa-
tionally expensive. Here, we run MCMC chains for all of
3.3.2 Euclid-like
our experiments and use them as a calibration for Fisher
For a Stage IV optical weak lensing experiment we con- matrices, allowing the latter to be robustly used for future
sider as a reference a space-based survey capable of ob- similar work. The calculation of realistic covariance matri-
taining a galaxy number density of n = 30arcmin2 over cesbeyondtheapproximationinEq.(17)typicallyrequires
gal
15,000 deg2ofthesky,withmoreaccuratephotometricred- large-scale simulations of data of the type expected to be
shifts than the DES survey, but still no spectroscopic red- generated in an experiment; in Paper II we construct such
shiftmeasurements.Weexpectthistobesimilartotheper- simulations for a fiducial cosmology.
(cid:13)c 2016RAS,MNRAS000,1–14
SKA Weak Lensing I: Forecasts 9
4.1 Forecasts with CosmoSIS
Experiment σw0 MC,Fisher σwa MC,Fisher
ForourMCMCparameterconstraintforecastswemakeuse
SKA2-simple 0.0161,0.0168 0.0651,0.0660
of the CosmoSIS modular cosmological parameter estima-
Euclid-like-simple 0.0226,0.0236 0.104,0.108
tioncode(Zuntzetal.2015).Foragivensetofcosmological
parameters ϑ we calculate a non-linear matter power spec-
Table2.Onedimensionalparameterconstraintsfromcovariance
trum using CAMB (Lewis et al. 2000) (with modifications matricescalculatedusingfullMCMCchainsandtheFisherma-
from ISiTGR for the modified gravity models from Dossett trix formalism for the simplified weak lensing-only experiments
et al. 2011, 2015) and halofit (Smith et al. 2003; Takahashi described in Section 4.2, showing good agreement, as shown in
et al. 2012). This is then converted to a shear power spec- Fig. 3. The constraints for SKA2 correspond to a DETF figure-
trumusingEq.(3)andtheassumedn (z)fortherelevant of-meritof∼2500.
Xi
experiment and redshift bin.
TheseshearpowerspectraarecomparedinaGaussian
likelihood to an ‘observed’ data vector d(cid:101)(cid:96) and covariance ogfravpihewic,awnedcmaunltciopnosleidsepraecaechassaingplaerammoedteerC(cid:101)o(cid:96)XfYtheinthtoemoroy-.
matrix, calculated using the same method at our fiducial
Then,torecasttheFishermatrixinthespaceofthemodel
cosmological parameters:
parameters, ϑ, it is sufficient to multiply the inverse of the
−2lnL= covariancematrixbytheJacobianofthechangeofvariables,
(cid:96),(cid:96)(cid:96)(cid:48)(cid:88)m=a(cid:96)xmin(cid:16)C(cid:96)XY(ϑ)−d(cid:101)(cid:96)(cid:17)(cid:104)ΓX(cid:96)(cid:96)(cid:48)Y(cid:105)−1(cid:16)C(cid:96)X(cid:48)Y(ϑ)−d(cid:101)((cid:96)1(cid:48)(cid:17)9), viz. Fαβ =(cid:96),(cid:96)(cid:96)(cid:48)(cid:88)m=a(cid:96)xmin ∂∂CϑX(cid:96)αY (cid:104)ΓX(cid:96)(cid:96)(cid:48)Y(cid:105)−1 ∂∂CϑX(cid:96)(cid:48)βY , (21)
summing over all multipoles as ΓX(cid:96)(cid:96)(cid:48)Y is assumed to be di- where again we sum over all the multipoles because ΓXY is
agonal in (cid:96) and (cid:96)(cid:48). We then use the MultiNest (Feroz et al. here assumed to be diagonal in (cid:96) and (cid:96)(cid:48). (cid:96)(cid:96)(cid:48)
2013) code to sample over this parameter space and form
Fisher matrices can be quickly computed, requiring
the posterior confidence regions shown in our results plots.
computationofobservationalshearspectraonlyatthesetof
For all of our MCMC forecasts we include information up
points in parameter space necessary for approximating the
to a multipole of (cid:96) = 3000, capturing mildly non-linear
max derivative, rather than at enough points to create a good,
scales, dependent on the redshift being probed.
smoothapproximationtothetrueposterior.Thisallowsex-
plorationoftheimpactofdifferentsystematicsandanalysis
4.2 Comparison with Fisher Matrices choices on forecast parameter constraints, which we intend
toexploreinafollowingpaper.Here,wevalidatetheuseof
Whilst fully sampling the posterior distribution with
the Fisher approximation for such an exploration by com-
MarkovChainmethodsprovidesarobustandaccuratepre-
paring for a simple case the predictions from our MCMC
diction for parameter constraints, it is typically computa-
chains and Fisher matrices. We use simplified versions of
tionallyexpensiveandtimeconsuming.TheFishermatrixis
the SKA2 and Euclid-like experiments (intended to max-
an alternative approach for parameter estimation which as-
imise the Gaussianity of the contours and be quicker to
sumesthepresenceofalikelihoodfunctionL(ϑ)thatquan-
compute),inwhichweconsiderbothascoveringthefullsky
tifies the agreement between a certain set of experimental (A =41,253 deg2), only use information up to (cid:96)=1000
sky
data and the set of parameters of the model, ϑ = {ϑ }. It
α and cut off both redshift distributions at z = 4. For these
also assumes that the behaviour of the likelihood near its
simplified experiments we calculate the parameter covari-
maximum characterises the whole likelihood function suf-
ance matrix in the two parameters {w ,w } using both the
0 a
ficiently well to be used to estimate errors on the model
MCMCprocedureandviatheFishermatrixapproximation.
parameters(Jeffreys1961;Vogeley&Szalay1996;Tegmark
Figure 3 shows confidence region ellipses corresponding to
et al. 1997).
both these methods and Table 2 the associated one dimen-
Under the hypothesis of a Gaussian likelihood, the
sional parameter constraints, showing O(5%) agreement.
Fisher matrix is defined as the inverse of the parameter co-
As a demonstration of the usefulness of this approach,
variancematrix.Thence,itispossibletoinferthestatistical
we show the benefit of the high-redshift tail in the source
accuracywithwhichthedataencodedinthelikelihoodcan
distribution for SKA by calculating constraints in {w ,w }
0 a
measurethemodelparameters.Ifthedataistakentobethe
both including and excluding all sources above z = 2. For
expected measurements performed by future experiments,
SKA1,excludingthesesourcesleadstoa{3.63,4.04}factor
the Fisher matrix method can be used, as we do here, to
increase in the width of the uncertainties, whilst for SKA2
determineitsprospectsfordetectionandthecorresponding
the factors are {1.32,1.51}.
level of accuracy. The 1σ marginal error on parameter ϑ
α
reads
(cid:113)
σ(ϑα)= (cid:0)F−1(cid:1)αα, (20) 5 RESULTS
where F−1 is the inverse of the Fisher matrix, and no sum- InFigs.4to6weshowthetwodimensionalparametercon-
mation over equal indices is applied here. straintsfromourMCMCforecastsonmatter{σ ,Ω },dark
8 m
Ourexperimentaldatawillcomefromthemeasurement energy {w ,w } and modified gravity {Σ ,Q } parameter
0 a 0 0
of the (cross-)correlation angular power spectrum CXY be- pairs, each marginalised over the full base ΛCDM param-
(cid:96)
tweentheobservablesXandY.Fromanobservationalpoint eter set {Ω ,Ω ,σ ,h ,n }, with the light (dark) regions
m b 8 0 s
(cid:13)c 2016RAS,MNRAS000,1–14
10 Harrison et al.
CMB by the Planck satellite (Planck Collaboration et al.
2015)inFig.7.Forthis,were-weightourMCMCchainsus-
ing the plikHM-TTTEEE-lowTEB-BAO Planck likelihood
chain9, re-centred around our fiducial cosmology. We also
show the combined, marginalised parameter constraints for
both auto and cross-correlation experiments in Table 3.
Whilst these result in little difference in the matter param-
eters, the different degeneracy direction of the Planck con-
straintson(w ,w )allowsforasignificantlysmallerareain
0 a
thecontours,improvingtheDETFFoMbyafactor∼5for
each experiment and allowing O(10%) constraints on both
parameters.
6 CONCLUSIONS
In this paper we have presented forecasts for cosmological
parameter constraints from weak lensing experiments in-
volving the Square Kilometre Array (SKA), both in isola-
Figure3.Fisher(unfilledcontours)andMCMC(filledcontours) tion and in cross-correlation with comparable optical weak
predictionsforthesimplifiedweaklensing-onlyexperimentscon-
lensing surveys. We have shown that the first phase of the
sideredinSection4.2,showingagreementinbothsizeanddegen-
SKA (SKA1) can provide O(5%) constraints on matter pa-
eracydirection.Onedimensionaluncertaintiesforbothcasesare
rameters Ω and σ , O(50%) constraints on dark energy
showninTable2. m 8
equation of state parameters w and w , and O(10%) con-
0 a
straints on modified gravity parameters Σ and Q , com-
0 0
representing95%(68%)confidenceregionsfortheparameter petitivewiththeDarkEnergySurvey(DES).ThefullSKA
values, and Table 3 showing one dimensional 1σ confidence (SKA2)cansignificantlyimproveonalloftheseconstraints
regions for each parameter individually. Table 3 also shows andbecompetitivewiththesurveysplannedwithStageIV
the DETF Figure of Merit (FoM) for each experiment, cal- opticalweaklensingexperiments.Furthermore,wehaveex-
culated as the inverse area of a elliptical confidence region plored what may be achieved with weak lensing constraints
defined from the calculated parameter covariance matrix of from the cross-correlation power spectra between radio and
the simulated experiments: optical experiments. Such cross-correlation experiments are
(cid:16) (cid:112) (cid:17)−1 importantastheywillbefreeofwavelength-dependentsys-
FoM= σw0σwa 1−ρ2 (22) tematicswhichcanotherwisecauselargebiaseswhichdom-
inate statistical errors and can lead to erroneous cosmolog-
where ρ is the correlation coefficient and σ and σ are
w0 wa ical model selection. For both the Stage III (SKA1, DES)
the one dimensional parameter standard deviations.
and Stage IV (SKA2, Euclid-like) experiments, such sys-
The left column of Figs. 4 to 6 shows these for the
tematics are potentially larger than the statistical errors
three Stage III experiments: DES, SKA1 and their cross-
available from the number density of galaxies probed. We
correlation. SKA1 performs only slightly worse than DES,
haveshownthatparameterconstraintsmadeusingonlythe
tobeexpectedduetothesignificantlylowergalaxynumber
cross-waveband power spectra can be as powerful as tradi-
density, some of which deficit is made up for by the higher-
tional approaches considering each experiment separately,
medianredshiftdistribution,whichmaybeexpectedtopro-
but with the advantage of being more robust to systemat-
vide a stronger lensing signal. The DES×SKA1 contours,
ics. Such cross-correlation experiments represent significant
which make use of all of the galaxies in both experiments,
promise in allowing weak lensing to maximise its potential
outperform each experiment individually in the {σ ,Ω }
8 m in extracting cosmological information. At both Stage III
case.
and Stage IV, constraints on (w ,w ) are significantly im-
The right column of Figs. 4 to 6 shows the constraints 0 a
proved withthe addition of Cosmic Microwave Background
for Stage IV experiments. Here, SKA2, for which Galactic
priors from the Planck satellite, down to O(10%) in both
foregroundsarenotaconsiderationandhencehasaccessto
parameters for SKA2 + Planck.
a full 30,000 deg2, outperforms the Euclid-like experiment
The realisation of this promise in practice will rely on
in the {σ ,Ω } contours. The cross-correlation contours,
8 m a number of developments:
which only include galaxies in the 15,000 deg2 available to
bothexperimentsareslightlylargerthantheindividualex- • The accuracy and reliability of shape measurements of
periments, but may be expected to be significantly more galaxies from SKA data (which will arrive in the poorly-
robustduetotheremovalofwavelength-dependentsystem- sampledFourierplaneasvisibilities)willneedtomatchthat
atics. available from image-plane optical experiments (see Patel
et al. 2015, for further discussion).
5.1 Application of Planck Priors
Wealsoshowconstraintsobtainedbycombiningtheresults 9 ObtainedfromthePlanckLegacyArchivehttp://www.cosmos.
from our experiments with results from observations of the esa.int/web/planck/pla
(cid:13)c 2016RAS,MNRAS000,1–14