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