Measuring redshift-space distortions with future SKA surveys

0.34 MB·English

by  Alvise Raccanelli

Measuring redshift-space distortions with future
SKA surveys
Alvise Raccanelli1,2,3, Philip Bull4, Stefano Camera5,6, David Bacon7, Chris Blake8,
Olivier Doré2,3, Pedro Ferreira9, Roy Maartens10,7, Mario Santos10,11,6, Matteo
Viel12,13, Gong-bo Zhao14,7
1DepartmentofPhysics&Astronomy,JohnsHopkinsUniversity,3400N.CharlesSt.,
5
Baltimore,MD21218,USA
1
0 2JetPropulsionLaboratory,CaliforniaInstituteofTechnology,PasadenaCA91109,USA
2 3CaliforniaInstituteofTechnology,PasadenaCA91125,USA

n 4InstituteofTheoreticalAstrophysics,UniversityofOslo,P.O.Box1029Blindern,N-0315Oslo,
a
Norway
J
5JodrellBankCentreforAstrophysics,TheUniversityofManchester,ManchesterM139PL,UK
5
1 6CENTRA,InstitutoSuperiorTécnico,UniversidadedeLisboa,Lisboa1049-001,Portugal
7InstituteofCosmology&Gravitation,UniversityofPortsmouth,PortsmouthPO13FX,UK

]
O 8CentreforAstrophysics&Supercomputing,SwinburneUniversityofTechnology,POBox218,
Hawthorn,VIC3122,Australia
C
9Astrophysics,UniversityofOxford,DWB,KebleRoad,OxfordOX13RH,UK
.
h
10PhysicsDepartment,UniversityoftheWesternCape,CapeTown7535,SouthAfrica
p
- 11SKASA,4rdFloor,ThePark,ParkRoad,Pinelands,7405,SouthAfrica
o
12INAF-OsservatorioAstronomicodiTrieste,ViaTiepolo11,34143,Trieste,Italy
r
t 13INFNsez. Trieste,ViaValerio2,34127Trieste,Italy
s
a 14NationalAstronomyObservatories,ChineseAcademyofScience,Beijing,100012,P.R.China
[

1 E-mail: [email protected]
v
1 Thepeculiarmotionofgalaxiescanbeaparticularlysensitiveprobeofgravitationalcollapse. As
2 such, itcanbeusedtomeasurethedynamicsofdarkmatteranddarkenergyaswellthenature
8
3 ofthegravitationallawsatplayoncosmologicalscales. Peculiarmotionsmanifestthemselvesas
0 anoverall anisotropyinthemeasured clusteringsignalas afunctionofthe angletothe line-of-
.
1 sight,knownasredshift-spacedistortion(RSD).Limitingfactorsinthismeasurementincludeour
0
5 abilityto modelnon-lineargalaxy motionsonsmall scalesandthe complexitiesofgalaxy bias.
1 Theanisotropyinthemeasuredclusteringpatterninredshift-spaceisalsodrivenbytheunknown
:
v distancefactorsattheredshiftinquestion,theAlcock-Paczynskidistortion.Thisweakensgrowth
i
X rate measurements, but permits an extra geometric probe of the Hubble expansion rate. In this
r chapter we will briefly describe the scientific background to the RSD technique, and forecast
a
the potential of the SKA phase 1 and the SKA2 to measure the growth rate using both galaxy
cataloguesandintensitymapping,assessingtheircompetitivenesswithcurrentandfutureoptical
galaxysurveys.
AdvancingAstrophysicswiththeSquareKilometreArray
June8-13,2014
GiardiniNaxos,Italy
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
RSDwiththeSKA
1. Introduction
Oneofthebiggestchallengesofmoderncosmologyistounderstandtheacceleratedexpansion
rateoftheUniverse. Anumberofproposalshavebeenputforward,themostnotableofwhichare
the presence of dark energy or, alternatively, a modification to general relativity on cosmological
scales. Radio surveys have been used to test these different hypotheses in the past, mainly using
the integrated Sachs-Wolfe effect and the galaxy angular power spectrum (see e.g. Nolta et al.
2004; Raccanelli et al. 2008; Xia et al. 2010). With the new generation of radio arrays such as
LOFAR(Rottgeringetal.2011)andASKAP(Johnstonetal.2008),andinpreparationtotheSKA,
there is a growing interest in understanding how future radio surveys can constrain cosmological
parameters,andmaybediscriminatebetweenthetwoscenariosdescribedabove. Someexamplesof
theseinvestigationscanbefoundinRaccanellietal.(2012);Cameraetal.(2012);Raccanellietal.
(2014). HerewefocusonhowtheproposedSKAsurveyswillbeabletoprovidemeasurementsof
Redshift-SpaceDistortions(RSD),allowinginthiswaymeasurementsofcosmologicalparameters,
andinparticularthegrowthofstructures.
In this chapter, we will briefly review the physics of Redshift-Space Distortions and how we
modeltheireffectonthepowerspectrum. Weinvestigatepotentialissuesandsystematicsandthen
wepresentforecastsforthemeasurementswecanperformusingthedifferentproposedSKAsur-
veys,focusinginparticularonmodelsdescribingthegrowthofstructures. Intwocomplementary
chapters (Bull et al. 2015; Camera et al. 2015) are presented forecasts for the BAO and large-
scale measurements, investigating how those can constrain dark energy models and primordial
non-Gaussianity.
2. Thephysicsofthegrowthrate
The presence of a dark energy component in the energy-density of the Universe (or the fact
thatourtheoryofgravityneedstobemodifiedonlargescales),modifiesthegravitationalgrowthof
large-scalestructures. Thelarge-scalestructureweseetracedbythedistributionofgalaxiesarises
throughgravitationalinstability,whichamplifiesprimordialfluctuationsthatoriginatedinthevery
earlyUniverse. Therateatwhichstructuregrowsfromsmallperturbationsoffersakeydiscriminant
between cosmological models, as different models predict measurable differences in the growth
rate of large-scale structure with cosmic time (e.g. Jain & Zhang 2008; Song & Percival 2008;
Song&Koyama2008). Forinstance,darkenergymodelsinwhichgeneralrelativityisunmodified
predict different large-scale structure formation compared to Modified Gravity models with the
same background expansion (e.g. Dvali et al. 2000; Carroll et al. 2004; Brans 2000; Yamamoto
etal.2008,2010).
Thegrowthrate, f(a),asafunctionofscalefactoraisdefinedas:
dlnδ (a)
M
f(a) ≡ , (2.1)
dlna
whereδ (a)istheamplitudeofthegrowingmodeofmatterdensityperturbations. Intheconfor-
M
malNewtoniangaugetheevolutionequationsforthevelocitypotentialθ andδ are:
M
δ˙ = 3(Φ˙ +H Ψ)−(cid:2)k2+3(H 2−H˙)(cid:3)θ , (2.2)
M M
θ˙ = −H θ +Ψ. (2.3)
M M
2
RSDwiththeSKA
Weusedotstodenotederivativeswithrespecttoconformaltime,andourconventionsforthemetric
potentialsaredisplayedintheperturbedlineelement: ds2=a2(η)(cid:2)−(1+2Ψ)dη2+(1−2Φ)dxidx(cid:3)
i
whichsatisfy(inthequasi-staticregime)thefieldequations: 2∇2Φ=κa2µ(a,k)ρ¯ ∆ andΦ/Ψ=
M M
γ(a,k). Theparametersµ andγ encapsulatealltheinformationaboutdeviationfromGRformetric
theoriesofgravity(Bakeretal.2011).
Again,inthequasistaticregime,theevolutionequationforδ simplifiesto:
M
3
δ¨ +H δ˙ − H 2Ω ξδ =0, (2.4)
M M M M
2
where we have defined ξ ≡ µ/γ (which is equal to 1 in GR). Using x=lna as the independent
variablewehave:
(cid:32) (cid:33)
H (cid:48) 3
(cid:48)(cid:48) (cid:48)
δ + 1+ δ − Ω ξδ =0. (2.5)
M H M 2 M M
Primesdenotederivativeswithrespecttox. Wecanconvertthisintoanevolutionequationfor f :
f(cid:48)+q(x)f + f2= 3Ω ξ, (2.6)
2 M
where q(x)= 1[1−3ω(x)(1−Ω (x))]. The effect of the expansion rate (via q(x)) and modified
2 M
gravity(viaξ)areexplicitinthetimeevolutionof f.
3. RedshiftSpaceDistortions
MeasurementsofRSDplayedanimportantroleindevelopingthecurrentcosmologicalmodel,
anditwillbeafundamentalpartofseveralfuturecosmologicalexperiments,becauseobservations
of RSD in galaxy surveys are a powerful way to study the pattern and the evolution of the Large
Scale Structure of the Universe (Kaiser 1987; Hamilton 1997), as they provide constraints on the
amplitude of peculiar velocities induced by structure growth, thereby allowing tests of the theory
ofgravitygoverningthegrowthofthoseperturbations. RSDhavebeenmeasuredusingtechniques
based on both correlation functions and power-spectra (e.g. Peacock et al. 2001; Percival et al.
2004; Tegmark et al. 2006; Guzzo et al. 2008; Samushia et al. 2012; Samushia et al. 2013; Reid
et al. 2010; Reid et al. 2012; Sánchez et al. 2012; Blake et al. 2010, 2011, 2012); the most recent
analysescomefromBOSSDR11andGAMA(Samushiaetal.2014;Blakeetal.2013).
3.1 Formalism
RSD arise because we infer galaxy distances from their redshifts using the Hubble law: the
radialcomponentofthepeculiarvelocityofindividualgalaxieswillcontributetoeachredshiftand
will be misinterpreted as being cosmological in origin, thus altering our estimate of the distances
to them. The correction due to peculiar velocities can be used to set constraints on cosmological
modelsandparameters,asitdependsonthecoherentlargescaleinfallofmattertowardoverdense
regions. Therelationbetweentheredshift-spacepositionsandreal-spacepositionris:
s(r)=r+v (r)rˆ, (3.1)
r
3
RSDwiththeSKA
wherev isthevelocityintheradialdirection.
r
TheRedshift-SpaceDistortions(RSD)correctionscomefromthefactthatthereal-spaceposi-
tionofasourceintheradialdirectioninmodifiedbypeculiarvelocitiesduetolocaloverdensities;
thiseffectcanbemodeledas(Kaiser1987;Hamilton1997):
δs(k)=(cid:0)1+βµ2(cid:1)δr(k), (3.2)
where,inthelinearregime,β isthequantitythatsolvesthelinearizedcontinuityequation:
βδ+∇¯ ·v¯=0. (3.3)
Here β = f/b, where b is the bias relating the visible to the underlying matter distribution (see
Section3.3formoredetailsonit).
For this reason, measuring f from RSD allows us to set constraints on cosmological models
andparameters.
3.2 Thepowerspectrum
The matter power spectrum depends on a variety of cosmological parameters, and for this
reasonitsmeasurementhasbeenused(togetherwithitsFouriertransform,thecorrelationfunction)
toconstraine.g. darkenergyparameters(Samushiaetal.2012),modelsofgravity(Raccanellietal.
2013),neutrinomass(dePutteretal.2012;Zhaoetal.2012),darkmattermodels(Cyr-Racineetal.
2014; Dvorkin et al. 2014), the growth of structures (Samushia et al. 2013; Reid et al. 2012), and
non-Gaussianity(Rossetal.2013).
Wedefinethepowerspectrumas:
Ps(k,µ,z)=(cid:2)b(z)+ f(z)µ2(cid:3)2Pr(k,z)+P (z), (3.4)
g m shot
wherethesuperscriptsrandsindicaterealandredshift-space,respectively,andthesubscripts and
m
stands for matter and galaxies; µ is the angle with the line of sight. The shot noise contribution
g
istakentobe:
1
P (z)= . (3.5)
shot
n¯ (z)
g
The standard analysis of RSD makes use of the so-called Kaiser formalism (Equation 3.4),
that relies on several simplifying assumptions, including considering only the linear regime and
thedistantobserverapproximation;inSection3.4webrieflymentionsomepossibleextensionsof
thismodel. InthisChapterwewillmakeuseoftheKaiserformula,butforadetaileddataanalysis
somefurtherinvestigationswillbeneeded.
3.3 Bias
While the distribution of galaxies is the observed quantity, the cosmological model directly
predicts the statistical distribution of (dark) matter. The simplest assumption is that the galaxy
distribution is a biased version of the underlying matter field, the so-called linear bias model, at
positionx:
δ (x)=bδ (x), (3.6)
galaxies matter
4
RSDwiththeSKA
with b a constant bias factor independent of a given smoothing scale R over which the density
fields are calculated. This model is motivated by the fact that rare peaks in the density field (e.g.
clustersofgalaxies)havetobemorestronglyclustered(i.e. biased)thanmatteritself (e.g.Kaiser
1984). Thisisasimplifyingassumptionandtherelationshipbetweengalaxiesandmatterismore
complex: in fact, clustering properties of galaxies do depend on galaxies’ intrinsic features. For
example the relation could be scale-dependent, nonlinear, stochastic, non-local, a function of the
particularsampleofgalaxieschosen,afunctionofcosmictimeordependentonmanyotherphys-
ical quantities (such as the gas temperature, environment, merging history, etc.). Thus, the above
equationcanbegeneralizedtoamorecomplexform:
δ (x)= f(δ (x)+ε), (3.7)
galaxies matter
withε embeddingallthedependenciesonphysicalquantitiesotherthandarkmatterdensity.
Currently, analytical efforts to model the bias are first attempting to model the halo-matter
bias by relying on: the peak background split formalism in a coarse grained perturbation theory
framework (e.g. Schmidt et al. 2013, and references therein); the excursion set approach also for
non gaussian initial conditions as in Musso et al. (2012); perturbation theories (Bernardeau et al.
2001). Particular emphasis is also put on unveiling the scale dependence of the bias, out to the
largestscales,thatcanbeapowerfulprobefortestinginitialconditionsand/orthenatureofgravity
and it has been recently shown that also in the standard cosmological model (ΛCDM) the halo
bias is scale dependent due to general relativitistic effects and not only to non-gaussianities (e.g.
Baldaufetal.2011).
A comprehensive analysis of halo bias is presented in Smith et al. (2006) by comparing the
resultsofN-bodysimulationswithsemi-analyticalprescriptionsbasedonperturbationtheoryand
thehalomodel,alsorelyingonthecross-spectrumbetweenmatterandhaloes. Intheworkabove,
itisshownthatthenon-linearitiesofthebiasaredeterminednotonlybythenon-linearevolutionof
thepowerspectrumbutalsobythefactthathaloesofdifferentmassesarebiasedinadifferentway.
InarecentstudybasedonN-bodysimulationscomplementedbyagalaxyformationmodel,Crocce
etal.(2013)foundanearlyscaleindependentbiasatthelevelof∼2−5%atscaleslargerthan20
Mpc/hforamockLuminousRedGalaxiessample.
Overall, simulations show that on scales larger than ≈30 Mpc/h the simplest linear parame-
terization works reasonably well, so for the purposes of this Chapter, we will assume the bias to
belinearandconstantatthescalesofinterestsforSKAforecastsasinEquation3.6;however,this
assumptionshouldbecarefullytestedinthefuture.
3.4 BeyondtheKaisermodel
Equation 3.4 is valid only on linear scales, assumes the plane-parallel approximation and is
derivedusingNewtonianphysics;thisapproximationisvalidwhenconsideringpairseparationsin
alimitedrangeofscales,largeenoughtoavoidnon-lineareffect(i.e. (cid:38)30Mpc)andarerelatively
small (up to ∼200 Mpc). If one wants to extend analyses of RSD to smaller and larger scales,
therearemodificationstothestandardformalismtotakeintoaccount.
5
RSDwiththeSKA
3.4.1 Non-linearities
Within dark matter haloes, peculiar velocities of galaxies are highly non-linear, and these
velocitiescaninduceRSDthatarelargerthanthereal-spacedistancebetweengalaxieswithinthe
halo. For this reason, on small scales we observe the so-called Fingers of God (FOG) effect –
strong elongation of structures along the line of sight (Jackson 1972). This results in a damping
of the power spectrum on small scales compared to the predictions of the linear model, and is
usually modeled by multiplying the linear power-spectrum by a function F(σ ,k,µ), where σ is
v v
theaveragevelocitydispersionofgalaxieswithintherelevanthaloes.
Modeling of non-linearities has been investigated numerous times (e.g. Scoccimarro 2004;
Taruya et al. 2009, 2010; Reid et al. 2010; Anselmi et al. 2010; Anselmi & Pietroni 2012; Kwan
etal.2011;Neyrincketal.2009,2011;Jennings2012;Carron&Szapudi2013).
InthisChapterwelookonlyatlinearscales;extensionstothequasi-andnon-linearregimes
willhelpgivingmoreconstrainingpower,buttheyrequireinvestigationsthatarebeyondthescope
ofthispaper.
3.4.2 Largescaleeffects
Whenconsideringwidesurveysandgalaxypairswithlargeseparation,amorepreciseanalysis
involving wide-angle and GR corrections should be used (see e.g. Szalay et al. 1997; Matsubara
1999;Szapudi2004;Papai&Szapudi2008;Raccanellietal.2010;Samushiaetal.2012;Monta-
nari&Durrer2012;Bertaccaetal.2012;Raccanellietal.2013;Raccanellietal.2013a). Moreover,
onverylargescales,themodelingforthepowerspectrumneedstotakeintoaccountGeneralRel-
ativity(GR)effectsthatwillbeimportantonscalescomparabletothehorizon(seee.g.Yoo2010;
Bonvin&Durrer2011;Challinor&Lewis2011;Yooetal.2012;Jeongetal.2011;Bertaccaetal.
2012;Raccanellietal.2013b;Dioetal.2014).
However,includingwide-angleandGRcorrectionsinthepowerspectrumisbeyondthescope
of this Chapter. A more detailed analysis of large scale effects for the SKA is carried out in the
SKAChapter“CosmologyontheLargestScales”(Cameraetal.2015).
3.5 Alcock-PaczynskiEffect
Positions of galaxies are given in terms of angular positions and redshifts; angular diame-
ter distances and Hubble expansion rates as functions of redshift are required in order to convert
angular and redshift separations into physical distances. Those functions depend on the adopted
cosmological model. If the real cosmology is significantly different from the fiducial one, this
difference will introduce additional anisotropies in the correlation function through the Alcock-
Paczynski effect. This can significantly bias the measurements of growth (see e.g. Ballinger et al.
1996;Simpson&Peacock2010;Samushiaetal.2012;Montanari&Durrer2012).
InthepresenceofAlcock-Paczynskieffect,theredshift-spacepower-spectrumis:
(cid:34) (cid:115) (cid:35)
(b+µ(cid:48)2f)2 k(cid:48) (cid:18) 1 (cid:19)
Ps(k(cid:48),µ(cid:48),α ,α ,p)= Pr 1+µ(cid:48)2 −1 , (3.8)
⊥ || α2α α F2
⊥ || ⊥
where p are standard cosmological parameters determining the shape of the real-space power-
spectrum, k(cid:48) and µ(cid:48) are the observed wavevector and angle, related to the real quantities by k(cid:48) =
||
6
RSDwiththeSKA
k(cid:48)
α k , k(cid:48) =α k , µ(cid:48) = || , where F =α /α , with α and α being the ratios of angular
|| || ⊥ ⊥ ⊥ (cid:113)k(cid:48)+k(cid:48) || ⊥ || ⊥
|| ⊥
andradialdistancesbetweenfiducialandrealcosmologies,α = Hfid ,α = Dreal.
|| Hreal ⊥ Dfid
IgnoringtheAPeffectisequivalenttoassumingthatα factorsareequaltounityinEq.(3.8).
4. SKASurveys
TheSquareKilometreArray(SKA)projectisaninternationalefforttobuildtheworld’slargest
radio telescope, several times more sensitive than any existing radio telescope and capable of ad-
dressing fundamental questions about the Universe (Carilli & Rawlings 2004). The SKA will be
developed in two stages. The first stage currently encompasses two mid-frequency facilities (∼ 1
GHz)operatingwithinSouthAfrica(SKA1-Mid)andAustralia(SKA1-Sur). Alowfrequencyar-
ray(SKA1-Low∼100MHz)willalsobesetinAustralia. WerefertoDewdneyetal.(2009)fora
descriptionofthesetups. InthesecondstageoftheSKA,theplanistoextendthearraybyabouta
factorof10,bothincollectingareaandprimarybeam(fieldofview),thussignificantlyincreasing
the survey power of the facility. In the following sections, we consider two types of surveys that
canbeusedtoprobetheredshiftspacedistortions.
4.1 HIsurveys
The most straightforward way to go after the RSD signal is through a line galaxy survey. In
theradio,thesolutionistousetheHI21cmlinewhich,bymeasuringitscharacteristicshape,will
allowdeterminationofveryaccurateredshifts(δz<1.0×10−4). Theadvantageofsuchthreshold
surveysisthatwecanbeconfidenttobefreeofanyforegroundcontamination. Thedisadvantage
isthatitrequireshighsensitivitiestodetectHIgalaxiesatnon-localredshifts(thehighestredshift
HIgalaxiesdetecteduptodatewasatz∼0.14withArecibo(Freudlingetal.2011)).
Cosmological applications will require detecting enough galaxies to beat shot noise and over
alargeenoughareatoreducecosmicvariance. WiththesensitivitiesforSKA1andtaking10,000
hours of observation time, the optimal survey area will be around 5,000 deg2. This will allow the
detectionofabout107galaxiesusingband2fromSKA1-MidorSKA1-Sur,whileSKA1-Midband
1shoulddetectlessgalaxies(∼104)sinceitwillbeconstrainedtohigherredshifts(0.4(cid:46)z(cid:46)3).
SKA2, on the other hand, should be capable of detecting about 109 galaxies over a 30,000 deg2
area, up to z∼2.0, making it the largest galaxy redshift survey ever. The noise calculations and
parameters for this HI galaxy survey can be found in the HI simulations chapter (Santos et al.
2015a). InFigure1weplottheredshiftdistributionsandbiasforthedifferentSKAconfigurations
describedabove.
4.2 Late-timeHIintensitymapping
A relatively new alternative to large galaxy redshift surveys is 21 cm intensity mapping.
Galaxy surveys need high signal-to-noise detections of many millions of individual sources, re-
quiring high flux sensitivity and long, dedicated surveys to reach z∼1. Intensity mapping (IM)
attempts to circumvent these requirements by performing fast, low angular resolution surveys of
the redshifted 21 cm emission line from neutral hydrogen (HI) integrated over many unresolved
galaxies. For a more extensive discussion on HI intensity mapping, particularly in the context of
7
RSDwiththeSKA
107 SKA1 MID B1 16 SKA1 MID B1
SKA1 MID B2 SKA1 MID B2
SKA1 SUR B1 14 SKA1 SUR B1
106 SKA1 SUR B2 SKA1 SUR B2
SKA 2 12 SKA 2
105
10
N(z)104 b(z) 8
103
6
102
4
101 2
1 0
0 0.5 1.0 1.5 2.0 0 1 2 3
z z
Figure1: Redshiftdistributions(leftpanel)andbias(rightpanel)fortheSKA1andSKA2surveysusedin
thiswork.
the SKA, we refer to Santos et al. (2015b). If we assume that, after reionisation, all the neutral
hydrogeniscontainedwithingalaxies,ashostgalaxiesarebiasedtracersofthecosmologicallarge
scale structure, so too is the integrated HI emission. Much of the cosmological information of
interest (e.g. RSDs and BAOs) is found at large scales, so the lack of resolution is tolerable, and
asthesignalisfromanemissionline,redshiftinformationisautomaticallyprovidedaswell. This
allowslargesurveystobeperformedextremelyrapidly,efficientlyrecoveringthe3Dredshift-space
matter power spectrum on large scales. An intensity mapping survey on SKA1-MID or SUR will
be able to measure BAOs and RSDs over 25,000 deg2 on the sky from 0(cid:46)z(cid:46)2.5, for example
(Santosetal.2015b).
One way of thinking about an IM survey, then, is as a galaxy survey with the small angular
scales averaged out. Information in the radial direction is mostly preserved, as modern radio re-
ceivershavesufficientlynarrowfrequencychannelbandwidthsthathighredshiftresolutioncanbe
obtained. The model for the RSD signal in intensity maps is therefore quite similar to that for a
galaxysurvey,exceptthattheobservableisthepowerspectrumofHIbrightnesstemperaturefluc-
tuations,(cid:104)δT∗δT (cid:105)∝T2P(k),whereT isthemeanHIbrightnesstemperature. Notethattheshot
b b b b
noise contribution has to be replaced by a more complicated direction-dependent noise term (see
e.g.Bulletal.2014). InthischapterwewillfocusontheRSDconstraintsthatcanbeachievedby
measuringtheanisotropicpowerspectrumwithIMsurveysonSKA1-MIDandSUR.Ourforecasts
arefor10,000hourautocorrelationsurveysover25,000deg2 onbands1and2ofbotharrays.
5. Forecast
In this Section we forecast the cosmological measurements that will be performed using the
SKAusingtheconfigurationpresentedinSection4;wepresentforecastsonparametersdescribing
modelsforthegrowthofstructuresmentionedinSection5.2.
5.1 FisherAnalysis
Inordertopredicttheprecisioninthemeasurementsofcosmologicalparameters,weperform
aFishermatrixanalysis(Fisher1935;Tegmarketal.1997);wewritethecurvatureorFishermatrix
8
RSDwiththeSKA
forthepowerspectruminthefollowingway:
(cid:90) zmax (cid:90) kmax (cid:90) +1 (cid:20) n¯g(z)P(k,µ,z) (cid:21)2 Vs(z)k2 ∂P(k,µ,z)∂P(k,µ,z)
F = dz dk dµ B ,
αβ zmin kmin −1 1+n¯g(z)P(k,µ,z) 8π2[P(k,µ,z)]2 ∂ϑα ∂ϑβ nl
(5.1)
where ϑ is the α(β)-th cosmological parameter,V is the volume of the survey and n¯ is the
α(β) s g
meancomovingnumberdensityofgalaxies. Thelasttermaccountsforthenon-linearitiesinduced
bytheBAOpeak(Seo&Eisenstein2007):
(cid:16) (cid:17)
−k2Σ2−k2µ2 Σ2−Σ2
B =e ⊥ || ⊥ , (5.2)
nl
andΣ =Σ D,Σ =Σ (1+f)D,whereΣ isaconstantphenomenologicallydescribingthenonlin-
⊥ 0 || 0 0
eardiffusionoftheBAOpeakduetononlinearevolution. FromN-bodysimulationsitsnumerical
valueis12.4h−1Mpcandseemstodependlinearlyonσ ,butonlyweaklyonkandcosmological
8
parameters. Theintegralinkisperformedineachredshiftbinusing(Smithetal.2003):
2π
k = ; (5.3)
min 1/3
V
bin
k =k (1+z)2/(2+ns). (5.4)
max NL,0
IntherestofthisSectionwepresentthemodelsweinvestigate: wefocusonwaystoexplain
thecosmicacceleration,eitherviadynamicaldarkenergyormodificationsofthemodelforgravity.
5.2 Growthofstructures
WestudyhowtheSKAcouldconstrainparametersdescribingthegrowthofstructures. There
areseveral modelsfor itbased ondifferentexplanations forthe acceleratedexpansionof theUni-
verse;theycanbedividedintotwomaincategories: darkenergyandmodifiedgrowth. Measuring
RSD allows us to test different cosmological models and provides a good discriminant between
modifiedgravityanddarkenergymodels(seee.g.Linder2005,2007;Guzzoetal.2008).
5.2.1 DarkEnergyModels
In the standard ΛCDM model, the accelerated expansion of the universe is caused by a dark
energy component that behaves like a cosmological constant, but alternative models have been
proposedandarestillallowedbydata.
Dynamical models can be distinguished from the cosmological constant by considering the
evolutionoftheequationofstateofdarkenergy,w= p/ρ,where pandρ arethepressuredensity
and energy density of the fluid, respectively. In the cosmological constant model, w=−1, while
fordynamicalmodelsw=w(a). ItistheusefultoconsideraTaylorexpansionoftheequationof
state(Linder2003):
w(a)=w +w (1−a); (5.5)
0 a
intheΛCDMmodelwehavew =−1andw =0. Ifadeviationfromthesevalueswillbedetected
0 a
(inparticularifw (cid:54)=0),thenthiswouldsuggestthatthecorrectmodelisonewherethedarkenergy
a
componentoftheuniverseisevolvingwithtime.
9
RSDwiththeSKA
In Figure 2 we plot constraints on the parameters {w ,w } (including Planck+BOSS priors),
0 a
for the different SKA1 and SKA2 surveys, comparing results with predictions for the Euclid ex-
periment.
0.4
SKA1-MID B1 (IM)
SKA1-SUR B1 (IM)
SKA2 (gal.)
0.2 Euclid (gal.)
a 0.0
w
0.2
0.4
1.1 1.0 0.9
w
0
Figure2: PredictedconstraintsfromSKAondynamicaldarkenergyparameters. Weshowpredictedcon-
straintsfromSKAIMandSKA2,comparedwithpredictionsforEuclid.
SKA1 HI surveys will not be able to provide competitive constraints on these parameters, so
wedon’tshowthem,butresultsfromtheIMsurveyswillbecompetitive,andtheSKA2galaxysur-
veyshouldbeabletoallowimprovementsonmeasurementsofdynamicaldarkenergyparameters
overthepredictedEuclidgalaxysurvey.
5.2.2 ModifiedGrowthModels
Measuringthemattervelocityfieldatthelocationsofthegalaxiesgivesanunbiasedmeasure-
mentof fσ ,providedthatthedistributionofgalaxiesrandomlysamplesmattervelocities,where
8m
f is given by Equation 2.1 and σ quantifies the amplitude of fluctuations in the matter density
8m
field. Thegrowthfactorissometimesparameterizedas(Linder2005):
D(a)=aexp(cid:20)(cid:90) a(cid:2)Ωγ(a(cid:48))−1(cid:3)da(cid:48)(cid:21), (5.6)
m a(cid:48)
0
whichleadstothefollowingexpressionfor f:
f =[Ω (a)]γ , (5.7)
m
where:
Ω a−3
m
Ω (a)= , (5.8)
m (cid:104) (cid:105)
∑iΩi exp 3(cid:82)a1[wi(a(cid:48))+1]daa(cid:48)(cid:48)
10