Cosmology on the Largest Scales with the SKA 5 1 StefanoCamera,ab AlviseRaccanelli,cde Philip Bull,f DanieleBertacca,g Xuelei 0 ∗ 2 Chen,h PedroG.Ferreira,i Martin Kunz,jk RoyMaartens,gl Yi Mao,m Mário G. n Santos,gnb PaulR. Shapiro,o MatteoVielpq andYidongXuh a aJodrellBankCentreforAstrophysics,UniversityofManchester,ManchesterM139PL,UK; J bCENTRA,IST,UniversidadedeLisboa,1049-001Lisboa,Portugal;cDepartmentofPhysics& 9 1 Astronomy,JohnsHopkinsUniversity,Baltimore,MD21218,USA;dJetPropulsionLaboratory, CaliforniaInstituteofTechnology,Pasadena,CA91109,USA;eCaliforniaInstituteof ] O Technology,Pasadena,CA91125,USA; fInstituteofTheoreticalAstrophysics,Universityof C Oslo,0315Oslo,Norway;gDepartmentofPhysics,UniversityoftheWesternCape,CapeTown . 7535,SouthAfrica;hNationalAstronomicalObservatories,ChineseAcademyofSciences, h Beijing100012,China;iAstrophysics,PhysicsDepartment,UniversityofOxford,OxfordOX1 p - 3RH,UK; jDépartementdePhysiqueThéoriqueandCenterforAstroparticlePhysics, o UniversitédeGenève,CH1211Genève4,Switzerland;kAfricanInstituteforMathematical r t Sciences,CapeTown7945,SouthAfrica;lInstituteofCosmology&Gravitation,Universityof s a Portsmouth,PortsmouthP013FX,UK;mInstitutd’AstrophysiquedeParis,InstitutLagrangede [ Paris,CNRS,UPMCUniv.Paris06,UMR7095,F-75014Paris,France;nSKASA,4thFloor, 2 ThePark,ParkRoad,Pinelands7405,SouthAfrica;oDepartmentofAstronomyandTexas v CosmologyCenter,UniversityofTexasatAustin,Austin,TX78712,USA;pINAF,Astronomical 1 5 ObservatoryofTrieste,viaTiepolo11,34131Trieste,Italy;qINFN,SezionediTrieste,34100 8 Trieste,Italy 3 E-mail: [email protected] 0 . 1 Thestudyofthe Universeonultra-largescales isoneofthe majorscience casesfortheSquare 0 KilometreArray(SKA).TheSKAwillbeabletoprobeavastvolumeofthecosmos,thusrepre- 5 1 sentingauniqueinstrument,amongstnext-generationcosmologicalexperiments,forscrutinising : v the Universe’s propertieson the largestcosmic scales. Probing cosmic structureson extremely i X largescaleswillhavemanyadvantages. Forinstance,thegrowthofperturbationsiswellunder- r stoodforthosemodes,sinceitfallsfullywithinthelinearrégime.Also,suchscalesareunaffected a bythepoorlyunderstoodfeedbackofbaryonicphysics.Onultra-largecosmicscales,twokeyef- fectsbecomesignificant:primordialnon-Gaussianityandrelativisticcorrectionstocosmological observables.Moreover,iflate-timeaccelerationisdrivennotbydarkenergybutbymodifications togeneralrelativity,thensuchmodificationsshouldbecomeapparentnearandabovethehorizon scale. Asaresult,theSKAisforecasttodelivertransformationalconstraintsonnon-Gaussianity andtoprobegravityonsuper-horizonscalesforthefirsttime. AdvancingAstrophysicswiththeSquareKilometreArray June8-13,2014 GiardiniNaxos,Italy Speaker. ∗ (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ CosmologyontheLargestScales StefanoCamera 1. Introduction Anunchartedareaofphysicalandobservationalcosmologyisthephysicsofultra-largescales. By this we mean length scales which are near or beyond the cosmic horizon. Most large-scale surveys have been limited to sampling length scales of order hundreds of megaparsecs, whilst deeper surveys(i.e. higherredshift surveys)arenotyetgoodenoughtoaccurately measuremodes which are of order the horizon. Yet, the cosmological information hidden in the extremely large scales is of great importance for our comprehension of the Universe. For a start, the growth of cosmological perturbations iswellwithinthelinearregimeonthoseultra-large scales. Thisallows forbetterchecksofourtheoreticalmodel,sincewecantheresafelydisregardthenon-lineargrowth ofstructures—whichalwaysrequiressomedegreeofadhocmodelling. Furthermore,onverylarge scalesthemostrelevantphysicsissimplygivenbythegravitationalinteraction,thusfreeingusfrom thenecessity oftakingastrophysical processes intoaccount. Theoretical knowledge of the Universe on the largest cosmic scales can be very precise, but ultra-largescaleshavebeenmostdifficulttoaccess. Thisismainlyduetotwoobservationalissues. Firstly, if we want to probe very large scales—i.e. very large wavelengths or very small Fourier momenta—a wide field of view is not sufficient. For a wide-field, shallow survey the transverse components, k ,ofthephysical wavevector kmaybearbitrarily small,butthesmallest measured ⊥ wavenumber, k k = k2 +k2,islimitedbythefactthatthesurvey probes onlyathin shell— ≡| | q ⊥ k thus providing us with a large parallel component, k . In other words, we need to observe a very k large ‘cube’ of Universe. This is a challenging task for conventional galaxy redshift surveys, for it isextremely hard toachieve therequired sensitivity athigh redshift overalarge areaof thesky. Closely related tothisisthesecond observational problem, namelycosmicvariance. Eventhough the full SKAwillbe ableto detect HIgalaxies asfaraway asz 2over roughly three quarters of ≃ the celestial dome—thus actually observing a huge cosmic cube—the relatively small number of sources withultra-large separation canheavily limittheconstraining potential ofthesurvey. The SKA is in a unique position to change this state of affairs and push forward the study of ultra-large scale modes. To tackle successfully the two observational issues described above, the SKAwill employ two newly developed techniques, one being HI intensity mapping (IM) and the other the so-called multi-tracer approach. Since we can exploit IM both during and after the epoch of reionisation (EoR), the SKA will be able to probe exceptionally large redshifts. When mappinglarge-scalestructureaftertheEoR,itispossibletoefficientlycollectinformationonlarge wavelengths without having to worry about resolving small-scale features or individual galaxies. WhenmappingtheEoR,onecaneffectively samplealargenumberofhorizon-size volumesatthe time. In addition, the multi-tracer technique will allow us to beat down cosmic variance. This ap- proach isbased on galaxies being biased but not stochastic tracers of the underlying density field. Thus,byobservingdifferentgalaxypopulations—namely, differenttracers—wecanconstructsev- eralrealisations oftheclustered halodistribution. Althoughmeasurements ofglobalcosmological propertieslikethetotalmatterdensityW ortheHubbleconstantH willstillbeaffectedbycosmic m 0 variance, wewillbeabletoconstrain haloproperties toamuchhigherdegreeofaccuracy. InthisChapter,weoutlinethesciencetobedonebyusingtheSKAtomapthematterdistribu- tion onthelargest cosmicscales. Wefirstmakethecaseforwhyultra-large scales areinteresting, 2 CosmologyontheLargestScales StefanoCamera focussing on relativistic effects, modified gravity (MG) and scale-dependent biasing arising from primordial non-Gaussianity (PNG). Then, we address how different methods will constrain these various physical phenomena, showing thatindeed theSKAwillbetransformational forconstrain- ingPNG,anduniqueforprobinggravityonsuper-horizon scales. Lastly,wediscusstheissuesthat need to be faced when trying to probe the largest scales. We refer the reader to Chapters Santos etal.(2014), Abdallaetal.(2014)andJarvisetal.(2014) fordetailsonthephasesoftheSKA,its observation modesandthedifferentmethodswearediscussing here. 2. The Importance ofProbing Ultra-Large CosmicScales The study of the biggest volumes ever of cosmic large-scale structure will allow a major ad- vance intackling twoofthemostfundamental questions incontemporary cosmology: Whatisthe physical mechanism thatgenerated theinflationary expansion intheearlyUniverse? Doesgeneral relativity (GR)holdonthelargestscales? Most models of inflation do predict some degree of non-Gaussianity in the distribution of primordial density fluctuations, but the simplest models predict negligible PNG. A detection of PNG will be vital for ruling out classes of models and advancing our understanding of inflation. Currently, themoststringentconstraintsonPNGcomefromthePlancksatellite,butfuturecosmic microwavebackground(CMB)experimentsareunlikelytoimprovetheseresultssignificantly. The newfrontier ofconstraining PNGislarge-volume surveysofthematterdistribution. Similarly, tests of GR on cosmological scales are based on observations of the large-scale structure. Current constraints are weak, but with its huge surveyed volumes and multiple probes, theSKAwilltake theleadinthenextgeneration oftests. Additionally, wecantighten thecurrent constraints on dark energy (DE) and MG models by including much larger scales, thus increas- ing the statistical power of the observations and improving the sensitivity to any scale-dependent deviation fromGR. InthisSectionwereviewsuchultra-large scalephenomena, highlighting theirimportance for ourunderstanding oftheUniverse. 2.1 GeneralRelativistic Effects Probing ultra-large scales involves a theoretical challenge that has been recognised only re- cently. GR effects arise from observing on the past light-cone, which distorts the number counts and brightness temperature fluctuations on very large scales. So far, most analyses have been performed using a Newtonian approximation, with the flat-sky redshift-space distortions (RSDs) grafted on as a special relativistic effect. This is the ‘standard’ relativistic correction to the New- tonian approximation, whichissignificant onsub-horizon scales. Sometimes,theeffectoflensing convergence isalsoincluded. Thisisadequate forpast andpresent surveys, whichanalyse galaxy clusteringonscaleswellbelowthehorizon. SometimesthisNewtonian-likeapproachalsoincludes the contribution of weak lensing magnification to the matter over-density. Thelensing effects can be significant on small scales. But future wide and deep surveys will need to employ a more pre- cise modelling, accounting not only for RSDs and lensing, but for all geometric and relativistic corrections. Thefull relativistic analysis includes terms that are suppressed on sub-horizon scales such as velocity (or Doppler) terms, Sachs-Wolfe (SW)and integrated SW(ISW)type terms, and 3 CosmologyontheLargestScales StefanoCamera time-delay contributions (e.g. Yoo 2010; Jeong et al. 2012; Bonvin & Durrer 2011; Challinor & Lewis2011;Brunietal.2012). Onecanworkwiththeredshift-space correlationfunctionsandincludeallGRandwide-angle contributions(Bertaccaetal.2012;Raccanellietal.2014a,2013;Bonvinetal.2014). Byanalysing this, we can potentially extract more information about the structure of galaxy clustering. For instance, theoddLegendre multipoles ofthecorrelation function vanish intheNewtonian flat-sky approximation, whereastheyareingeneralnon-zeroforwide-angleseparations andinthefullGR analysis (Raccanelli et al. 2014a; Bonvin et al. 2014). Alternatively, the use of the angular power spectrum also avoids the flat-sky assumption, which is important for full-sky surveys with many wide-angle correlations (Bonvin&Durrer2011;Challinor &Lewis2011;DiDioetal.2013). To give a flavour of how GR corrections alter the Newtonian prediction, we show in Fig. 1 the ratio between a galaxy clustering angular power spectrum in which some GR effect has been switched on and the simple Newtonian result. We factorise the various corrections as follows: RSDs (top, left panel); velocity (top, right panel), where we include terms proportional to the velocity along the line of sight; weak lensing convergence (bottom, left panel); and potentials (bottom,rightpanel),whichaccountfortheeffectofgravitationalpotentialsatthesource,SW,ISW and time-delay terms (see e.g. Challinor & Lewis 2011). The two sets of curves refer to different surveys, all of them assumed to observe a certain population of galaxies with constant bias 1.5, distributed according toaGaussianwindowcentredatredshiftz andwithwidths . Specifically, m w we compare a shallow survey with z = 0.5 (red curves) to a deep survey with z = 2.0 (blue m m curves). Moreover, weconsider both thecases ofanarrow (solid lines)andabroad (dashed lines) window,viz.s =0.01and0.1fortheshallow surveyands =0.05and0.5forthedeepsurvey. w w From Fig. 1, it is apparent that, although RSDsand lensing are the dominant, the other terms can alsobecomeimportant, particularly forasdeepasurveyastheSKA. The GR corrections to the power spectrum should be observable for certain SKA surveys, using the multi-tracer technique—the task of forecasting this capacity is under way. Evenifthese corrections are not directly detectable, it is essential to include them in theoretical analysis of the powerspectrum, fortworelatedreasons: 1. to avoid biasing parameter estimations through a theoretical systematic of incorrect mod- elling; 2. tocorrectly extractmaximalinformation fromthelargestscales. 2.2 PrimordialNon-Gaussianity The largest cosmic scales are a crucial source of information about the physical processes at play during the early stages of the Universe’s evolution. The standard model of inflation and its generalisations predict seed primordial density fluctuations with somelevel of non-Gaussianity in the probability distribution (e.g. Bartolo et al. 2004). We can parameterise the non-Gaussianity in Bardeen’s gauge invariant potential F as the sum of a linear Gaussian term f and a quadratic correction (Verdeetal.2000;Komatsu&Spergel2001), i.e. F =f + f f 2 f 2 . (2.1) NL −h i (cid:0) (cid:1) 4 CosmologyontheLargestScales StefanoCamera RSD Velocity 2.5 1.05 1.04 2.0 wt 1.03 e N (cid:1)(cid:2) 1.02 1.5 / (cid:2) 1.01 (cid:1) 1.0 1.00 0.99 2 5 10 20 50 2 5 10 20 50 Lensing Potentials 2.5 1.01 1.00 2.0 wt 0.99 e N (cid:2)(cid:1) 0.98 1.5 / (cid:1) 0.97 (cid:2) 1.0 0.96 0.95 2 5 10 20 50 2 5 10 20 50 (cid:1) (cid:1) Figure1:RatiosofangularpowerspectraincludingsomekindofGRcorrectionstothestandardNewtonian approximation, CNewt. Red (blue) curves refer to a shallow (deep) survey, while solid (dashed) lines to ℓ narrow(broad)sourceredshiftdistributions.(Seetextfordetails.) If the distribution of primordial density perturbations isnot Gaussian, itcannot be fully described by apower spectrum; werather need higher-order moments such asthe bispectrum. In particular, differentmodelsofinflationgiverisetodifferentbispectrumshapes,thusmakingthestudyofPNG valuable forobtaining adeeperknowledgeofthephysicsofinflation. Thestandardsingle-fieldinflationaryscenariogeneratesnegligiblysmalldeviationsfromGaus- sianity. Thesedeviations aresaidtobeofthelocalshape, andtherelated bispectrum ofBardeen’s potential is maximised for squeezed configurations, where one of the three wavenumbers has a much smaller magnitude than the other two. In this case, the PNG parameter, f , is expected NL to be of the same order as the slow-roll parameters, namely very close to zero (Falk et al. 1993). However, this does not mean that f 0. Due to the inherent non-linearity of GR, it is not pos- NL ≈ sible for Gaussianity intheprimordial curvature perturbation tobereflected exactly inthe density perturbation. Anon-linearGRcorrectiontotheinitialconditionsleadstoaneffective fGR= 5/3 NL − inlarge-scale structure (Verde&Matarrese2009). Local-shape PNG can also be generated when an additional light scalar field other than the inflatoncontributes totheobserved curvature perturbations (Bartoloetal.2004). Thishappens for instance in curvaton models (Sasaki et al. 2006; Assadullahi et al. 2007) or in multi-field mod- els (Bartolo et al. 2002; Bernardeau & Uzan 2002; Huang 2009). Other than local-type PNG, there areinflationary models inwhich thekinetic term ofthe inflaton Lagrangian isnon-standard, 5 CosmologyontheLargestScales StefanoCamera containing higher-order derivatives of the field itself. Onesignificant example ofthis is the ‘DBI’ model(Alishahihaetal.2004;Arkani-Hamedetal.2004;Seery&Lidsey2005),wheretheprimor- dial bispectrum is maximised for configurations where the three wavevectors have approximately the same amplitude—the so-called equilateral-type PNG (Creminelli et al. 2007). Otherwise, for deviations from Gaussianity evaluated in the regular Bunch-Davies vacuum state, the primordial potential bispectrum is of local or equilateral shape, depending on whether or not higher-order derivativesplayasignificantrôleintheevolutionoftheinflatonfield. IftheBunch-Daviesvacuum hypothesisisdropped,theresultingbispectrumismaximalforsquashedconfigurations(Chenetal. 2007; Holman & Tolley 2008). Lastly, another shape of the bispectrum can be constructed that is nearlyorthogonal toboththelocalandequilateral forms(Senatoreetal.2010). The best probe of PNG up to now has relied on measuring the CMB temperature anisotropy bispectrum(seeAdeetal.2014,forthelatestresults). However,ithasbeendemonstratedthatPNG also induces an additional, peculiar scale and redshift dependence in a biased tracer of the under- lying matter distribution (Dalal et al. 2008; Matarrese & Verde 2008; Schmidt & Kamionkowski 2010; Desjacques etal.2011). Themodification D b (z,k)totheGaussian large-scale biasb ofa X X biasedtracerX inducedbylocal-type PNGis W H2d D b (z,k)=b (z)+3[b (z) 1] m 0 c f , (2.2) X X X − k2T(k)D (z) NL + where b (z) is assumed scale-independent, d is the critical value of the matter over-density at X c collapse, the transfer function T(k) 1 on large scales, and D (z) is the linear growth factor of + → density perturbations normalised tounitytoday.1 It is clear from this equation that the effect of PNG on the power spectrum grows on large scales, as k 2. These are the same scales on which GR corrections are becoming significant. − Therefore, one needs to incorporate the GR corrections in theoretical analysis in order to make accurate (and non-biased) predictions and estimates of PNG (Camera et al. 2014b). Similarly, an incorrect treatment of PNG on scales where its effects are not negligible may threaten future cosmological experimentaccuracy (e.g.Cameraetal.2014a). 2.3 ModifiedGravity Einstein’stheoryofGRhasbeentestedtoexquisite precisionon‘small’scales,namelyinthe laboratory,theSolarSystemandwiththehelpofpulsars. Onthecontrary,testsonscalesapproach- ing the cosmological horizon are still rather weak. However, there is great interest in testing GR on very large scales in the context of DE, as the cosmological constant suffers from a plethora of theoretical problems. Since the accelerated expansion of the Universe isa late-time phenomenon, it is natural tolook for hints concerning the physical nature ofthe underlying mechanism on very largescales. Thelandscape of MGtheories is rather heterogeneous, and there is little reason to prefer any onemodelovertheothers. Attemptstoparameterise thislandscape inasuitably generic wayhave beenonlypartiallysuccessful(Hu&Sawicki2007;Linder&Cahn2007;Bakeretal.2013;Battye 1Notethatweadoptthelarge-scalestructure(LSS)convention,whichimplies fLSS 1.3fCMB(seee.g.Dalaletal. NL ≈ NL 2008;Afshordi&Tolley2008,formoredetails). 6 CosmologyontheLargestScales StefanoCamera & Pearson 2012) although methods based on effective field theory approaches provide a possi- ble avenue for the large class of scalar field models with second-order equations of motion, like Horndeski-type theories (Piazza & Vernizzi 2013; Bellini & Sawicki 2014). Phenomenological parameterisations can in principle describe the full range of deviations from GR relevant for cos- mology and thus fully exploit the information contained in the data by effectively modelling the metricperturbations (seeKunz2012,andreferencestherein). However,phenomenological models are mostly useful tocapture the evolution of linear perturbations, which effectively limits them to largescales. Many MG theories modify the expansion history of the Universe—often by construction, as theyareintendedasalternatives toDE.However,ithasbeenshownthatmanyDEmodelsarealso able to produce arbitrary w(z), and MG models can be tuned to mimic the cosmological constant value of w(z)= 1. Therefore, the expansion history is not a smoking gun for MG.On the other − hand, MG theories in general also alter the growth history of the Universe, i.e. the evolution of the metricperturbations and thusthe growthofthematter density contrast too. ManyMGmodels modifytheeffectivestrengthofgravityasafunctionofscaleand/orredshift,breaktheequivalence principle and so forth, thus yielding anumber of possible observational signatures. On large (lin- ear) scales, the linear growth rate, f(z), is the most sensitive probe of the growth history, and can be well-measured using RSDsfrom galaxy redshift surveys and IM (see Halletal. 2013, for IM). Another important quantity in this context is the scalar anisotropic stress (or gravitational slip), whichisgenerallynon-zero inMGtheories, especially onverylargescales(Saltas&Kunz2011). Theanisotropicstressisanattractivetestfordeviationsfromthestandardmodelbecauseitismea- surable without any assumptions on the initial power spectrum or the bias by combining peculiar velocities (e.g. RSDsfrom IM)withweaklensing measurements (Motta etal. 2013). Itspresence appearstobealsolinkeddirectlytoamodificationofthepropagationofgravitationalwaves,which canbeusedasawaytodefinewhatMGmeans(Saltasetal.2014;Camera&Nishizawa2013). Goingtoverylargescalesisimportantformanyreasons. Generallyspeaking,thesmallscales whereperturbationsarehighlynon-lineartendtobelessusefulforcosmologicaltestsofGR,asMG theories generically needtobescreened onsmallscalestoavoidviolatingthestrongSolarSystem constraints (although the mildlynon-linear region isinteresting). Inaddition, accurate predictions onnon-linearscalesrequirededicatedN-bodysimulationsforeachtheory,andasmentionedabove, itisunclear howtousethe unifying phenomenological framework onnon-linear scales. Baryonic effects on small scales are also badly understood and add a systematic error which effectively renders thesescales unusable forprecision cosmology forthetimebeing. Then,astheeffects that wearelooking foraresmall,weneedasmanymodesaswecanget,whichalsopushes ustolarge volumes. Lastbutnotleast, thescale dependence ofdeviations from GRisacrucial observable to distinguishbetweendifferentmodels,andfirstweneedawiderangeofscalestobeabletoobserve ascaledependence, andsecondlythehorizon scaleisanaturalplacewheretolookforthiseffect. Eventually, another way to probe MG at early times provides a link to PNG. According to Bartolo et al. (2014), modifications of GR during inflation can create a non-zero quasi-local bis- pectrum with a non-negligible amplitude that can be probed with the methods described in this Chapter. Furthermore,theGReffectsthatbecomerelevantonverylargescalesareactuallybenefi- cialfortestinggravityonthosescales,astheycontainadditionalinformationabouttheevolutionof perturbations. Howtoextract thisadditional information bestiscurrently anactiveresearch topic, 7 CosmologyontheLargestScales StefanoCamera buttheSKAisuniquely suitedtotakeadvantage ofthisnewsourceofconstraints. 3. Accessing Ultra-Large CosmicScales In this Section, we review the two envisaged techniques that the SKA will exploit to tackle the difficultproblem ofaccessing thelargest cosmic scales, viz. IM(both fromand aftertheEoR) and galaxy multi-tracing. In Sec. 3.4, we also discuss the most important systematics that we shall have to deal with for a fruitful exploitation of such methods. As a figure of merit, we adopt the achievable accuracy s (f ) on a forecast measurement of the PNG parameter. However, the NL techniquespresentedherewillingeneralenableustoinvestigateallthepeculiarphenomenawhich occur onextremely largecosmicscales. Forexample, dedicated analyses aimingatstudying other ultra-large scale effects such as GR corrections are being performed at the time of the editing of thisChapter. 3.1 IntensityMappingAftertheEpochofReionisation IMis analternative approach for probing thedensity field ofthelarge-scale structure (Battye etal.2004;Wyithe&Loeb2008;Changetal.2008;Petersonetal.2009). Itinvolvesmappingout the combined emission of the 21 cm, orHI, line from unresolved galaxies. In doing so, the large- scalestructureisdetectedinthreedimensions. Ifoneforegoes identifying individual galaxies, one cangreatlyspeeduptheobservation anddetectionofthelarge-scale structure. IMexperimentsare sensitive to structures at a redshift range that is observationally difficult to span for ground-based optical surveys (Seo et al. 2010). SKA IM surveys will cover a large fraction of the sky (around 25,000 sq. deg.) over an extremely broad redshift range, making it possible to access the largest cosmic scales inthelate Universe. Here, weoutline thebasics oflate-time IM,but refertoSantos etal.(2014)foramoredetailed description. Inordertomodelthe21cmpowerspectrum,itiscrucialtoquantifythebiasfunctionbetween thematterandHIpowerspectra,b (k,z),whichallowsustowrite HI PHI(k,z)=b2 (k,z)Pd (k,z), (3.1) HI d where P (k,z) isthe underlying matter power spectrum. Severaltechniques can beadopted inor- der to extract mock HI power spectra from realistic hydrodynamic or N-body simulations of the cosmic large-scale structure. In one of the simplest approaches (Bagla et al. 2010; Guha Sarkar etal.2012;Bharadwajetal.2001),theHIisassignedtoparticlesthatbelongtodarkmatterhaloes as identified in the simulated cosmological volume and the HI content of each halo of mass M, M (M),needstobecharacterised. Other, morerecentapproaches areinsteadbased onaparticle- HI by-particle method where the HI is assigned to gas particles according to more refined physical prescriptions which take into account self-shielding effects and the conversion to molecular hy- drogen (e.g. Davé et al. 2013; Popping et al. 2009; Rahmati et al. 2013; Duffy et al. 2012; Mao et al. 2012). A comprehensive comparison of several different methods in terms of 21 cm power spectrumisredshiftspace,performedinthepost-reionisationeraatz 2 4,hasbeenrecentlypre- ≃ − sentedbyVillaescusa-Navarro etal.(2014),whousedhighresolutioncosmologicalhydrodynamic simulations. They show that the b (k,z) approaches aconstant value to a good approximation at HI 8 CosmologyontheLargestScales StefanoCamera scales k.1hMpc 1,thatgalactic feedback intheform ofwindsisnotaffecting thesignal onthe − largest scales and that the signal is dominated atleast atz<6by HIresiding inhaloes. Although a full characterisation of the bias function in the (k,z)-plane is difficult to make, it is reassuring that onextremely large scales theoverall HIbiasisflat,thus allowing ustoperform cosmological studies of MG and PNG. It is also worth mentioning that the other key quantity is the total HI fraction, W (see Padmanabhan etal. 2014), which is in fact measured by observations of quasar HI absorptionlinesoftheLyman-a forestanddampedLyman-a systemsatz>2(Tescarietal.2009; Zafaretal.2013). Bymeansofthisdatawecanthenconstrain theHIbias. 3.1.1 PrimordialNon-GaussianityProbedbyScale-DependentBias Cameraetal.(2013)exploredthismethodforconstrainingthePNGofprimordialfluctuations. IM experiments seem ideally suited for this goal (see Joudaki et al. 2011; Hazra & Sarkar 2012; D’Aloisio et al. 2013; Lidz et al. 2013; Mao et al. 2013, for proposals for doing so in the EoR). As argued before, the tightest constraints on f will be obtained for large and deep surveys. NL Therefore, the volume of the survey determines the ability to probe below f of O(1). For such NL a method to be successful, we need a deep survey with a large bandwidth accessing frequencies of400MHzandbelow. Assumingthatline-of-sight scattering andself-absorption phenomena can be neglected, the HIline radiation can be related to the differential number counts of halo objects (e.g.Challinor &Lewis2011), from whichwecanestimate b . Crucially, givenourfundamental HI ignorance about the redshift evolution of the bias, we need to span a wide range of redshifts to capture b (k,z)=1. Thisisbecause, asitisclearfromEq.(2.2),noPNGeffect istobedetected HI 6 ifthetracerwearelookingatisunbiasedwithrespecttotheunderlyingmatterdensitydistribution. Fig. 2 shows s (floc) contours in the plane of the surveyed area and total observation time NL obtained with an HI IM experiment with bandwidth between 250 and 1000 MHz, corresponding to 0.5.z.4.5, subdivided into 75 frequency bins of width D n =10 MHz(Camera et al. 2013). Abscissas roughly cover from a 15 15deg2 survey to half sky. The three top panels stand for × the dish survey case, where the y-axis actually shows total observation time, t , multiplied by TOT the number of dishes, N . Three maximum angular modes are presented, ℓ =25, 60, and 300, d max corresponding to dish diameters of 5, 15 and 80 m at z 3. For higher angular resolution, inter- ≈ ferometers may be a better option. In the bottom panels of Fig. 2, we show s (floc) for such a NL possibility using 1, 10 and 100 pointings. Choosing D 80 m as the diameter for the array, the a ≃ resolution is set at ℓ 300. Here, the main design parameter is the field of view, FoV, which max ≃ sets ℓ =2p /√FoV and is fixed by the effective size of each element, d l /√FoV. For a min eff ∼ ‘dense array’, this is related to the number of elements, N D2/d2 . Given that the maximum e ∼ a eff angular scaleissetbytheFoV,byadding morepointings, wesimplydiminish thesignal variance byN ,though thenoiseincreases too,becausetheobservation timet t /N . p obs obs p → SKA1woulduse15mdishes, viz.ℓ 60. Specifically, 254dishesofSKA1-MID,10,000 max ≃ hours and a survey area around 30,000 sq. deg. will correspond to s (floc) 2, as can be seen NL ∼ in Fig. 2. On the one hand, SKA1-MID will only go down to 350 MHz, whereas the minimum frequency assumed in Fig. 2 is 250 MHz. On the other hand, Camera et al. (2013) adopted a more conservative value for the system temperature, i.e. 30 K, whilst SKA1-MID will do better, with a system as cold as 20 K. Otherwise, SKA1-SUR will have a band from 900 to 350 MHz (0.58 <z<3.06) and another from 1400 MHz to 700 MHz (0<z<1.1). Temperature will be 9 CosmologyontheLargestScales StefanoCamera Figure2: Forecast1s marginalerrorcontourson floc asafunctionofsurveyedskyandtotalobservation NL time, for a dish survey with N dishes (upper panels) and an interferometer making N pointings (lower d p panels)(fromCameraetal.2013). slightlyhigher,at50K,fortheformerand30Kforthelatter. Theeffectivenumberofdisheswould beredshiftdependent,mostlikely64 n forthefirstbandwith(z,n )=(0.6,30),(1.0,20),(2.0,9) × and (3.0,5), whereas for the second band itwould be 94 n with (z,n )=(0.0,23), (0.5,10) and × (1.0,6). For interferometric surveys, we would need to wait for SKA phase 2, with the proposed ‘aperture array’ system working below 1GHz. It should be possible to achieve FoV 1000deg2, ∼ thus reaching thes (floc).1limit. SeeSantosetal.(2014) foradetailed description ofSKAIM NL specifics. 3.1.2 PrimordialNon-GaussianityProbedbytheBispectrum Compared toobservable galaxies, HIisaweaklier biased tracer ofthe underlying matter dis- tribution,exceptforthelargestscaleswherethescale-dependent correctionduetoPNGdominates. Therefore, on moderately large scales, we may model the HI bispectrum as the tree-level matter bispectrum modified by linear and non-linear bias factors, b and b . Here, we focus on the HI,1 HI,2 reduced HI bispectrum in redshift space, QHI, which is much less sensitive to other cosmological s parameters (Sefusatti&Komatsu2007). Wehave aB(b ) 1 b QHI(k ,k ,k )= 0 Qtree(k ,k ,k )+ HI,2 , (3.2) s 1 2 3 aP(b ) 2(cid:20)bHI,1 1 2 3 (bHI,1)2(cid:21) 0 (cid:2) (cid:3) where Qtree is the reduced matter bispectrum predicted by the second-order perturbation theory, aP(b )=1+2b /3+b 2/5 and aB(b )=1+2b /3+b 2/9, with b f(z)/b the linear Kaiser 0 0 ≡ HI,1 factor. TherelativeimportanceofPNGintheHIbispectrum increasestowardshigherredshifts. This is very promising for measuring the primordial component from the HI bispectrum with SKA1- MID, which covers the whole redshift range from z 0 to z 3 (350 1420 MHz). In addition, ∼ ∼ − 10