Foreground Subtraction in Intensity Mapping with the SKA

0.5 MB·English

by  Laura Wolz

Foreground Subtraction in Intensity Mapping with
the SKA
Laura Wolz 1,2, Filipe B. Abdalla1,3, David Alonso2, Chris Blake4, Philip Bull5,
∗
Tzu-Ching Chang6, Pedro G. Ferreira2, Cheng-Yu Kuo6, Marios G. Santos7,8 and
5 Richard Shaw9
1
0 1DepartmentofPhysicsandAstronomy,UniversityCollegeLondon,LondonWC1E6BT,UK
2 2Astrophysics,UniversityofOxford,DWB,KebleRoad,OxfordOX13RH,UK
n 3DepartmentofPhysicsandElectronics,RhodesUniversity,POBox94,Grahamstown,6140
a SouthAfrica
J
4CentreforAstrophysics&Supercomputing,SwinburneUniversityofTechnology,P.O.Box218,
5
Hawthorn,VIC3122,Australia
1
5InstituteofTheoreticalAstrophysics,UniversityofOslo,P.O.Box1029Blindern,N-0315Oslo,

]
Norway
O
6AcademiaSinicaInstituteofAstronomyandAstrophysics,P.O.Box23-141,Taipei,10617
C
Taiwan
.
h 7DepartmentofPhysics,UniversityofWesternCape,CapeTown7535,SouthAfrica
p
8SKASA,4rdFloor,ThePark,ParkRoad,Pinelands,7405,SouthAfrica
-
o 9CanadianInstituteforTheoreticalAstrophysics,60St. GeorgeStreet,Toronto,ONM5S3H8,
r
t Canada
s
a
[
E-mail: [email protected]

1
v 21cmintensitymappingexperimentsaimtoobservethediffuseneutralhydrogen(HI)distribution
3
onlargescaleswhichtracestheCosmicstructure. TheSquareKilometreArray(SKA)willhave
2
8 thecapacitytomeasurethe21cmsignaloveralargefractionofthesky. However,theredshifted
3
21cmsignalintherespectivefrequenciesisfaintcomparedtotheGalacticforegroundsproduced
0
. by synchrotron and free-free electron emission. In this article, we review selected foreground
1
0 subtractionmethodssuggestedtoeffectivelyseparatethe21cmsignalfromtheforegroundswith
5
intensity mapping simulations or data. We simulate an intensity mapping experiment feasible
1
: withSKAphase1includingextragalacticandGalacticforegrounds. Wegiveanexampleofthe
v
i residualsoftheforegroundsubtractionwithaindependentcomponentanalysisandshowthatthe
X
angularpowerspectrumisrecoveredwithinthestatisticalerrorsonmostscales. Additionally,the
r
a
scaleoftheBaryonAcousticOscillationsisshowntobeunaffectedbyforegroundsubtraction.
AdvancingAstrophysicswiththeSquareKilometreArray
June8-13,2014
GiardiniNaxos,Italy
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ForegroundSubtractioninIntensityMapping LauraWolz
1. Introduction
Intensitymappingofneutralhydrogen(HI)isaverypromising,efficienttooltomeasurethelarge-
scalestructureoftheUniverse(Battyeetal.2004;Vujanovicetal.2009;Lidzetal.2011;Peterson
& Suarez 2012). The HI gas with spectral emission of λ =21cm caused by the spin-flip of the
valence electron is assumed to linearly trace the Dark Matter distribution. In intensity mapping,
theentireHIfluxismeasuredinlargeresolutionelementswhichstillallowobservationsofstruc-
ture on the Baryon Acoustic Oscillation scale (BAO) (Wyithe et al. 2008; Chang et al. 2008; Bull
etal.2014a). Theseexperimentsareoptimallyconductedbytelescopeswithalargefield-of-view
(FoV) through, for instance, multi-pixel feeds. The Green Bank telescope (GBT) has pioneered
the detection of the large scale structure with the 21cm signal (Chang et al. 2010; Masui et al.
2013;Switzeretal.2013). NearfutureexperimentsincludesingledishtelescopesuchasBINGO
(Battye et al. 2012) and interferometric designs such as CHIME (Bandura et al. 2014) or Tianlai
(Chen2012). TheSKAwillbeabletocarryoutvariousintensitymappingexperimentsatdifferent
redshifts either in auto-correlation mode, i.e. single dish observations, or interferometric mode.
ForamoredetaileddiscussionofintensitymappingwiththeSKA,werefertothechaptersSantos
etal.(2014)andBulletal.(2014b).
The drawback of intensity mapping observations are the foregrounds mainly due to the high
radio emission of our own Galaxy caused by synchrotron and free-free electron radiation. In the
consideredfrequencyrange,theGalacticforegroundssignificantlydominatetheacquiredmapsby
several orders of magnitude. The only full-sky observation of the Galactic radio emission is at
frequencyν =408MHz(Haslametal.1982). Thismapcombinedwithsmallersurveysisthebasis
of most studies to understand and fully model the Galactic radio emission, i.e. de Oliveira-Costa
et al. (2008); Jelic et al. (2008); Alonso et al. (2014). The varying spectral index with latitude
and the lack of small-scale observations impede the comprehension of the Galactic foregrounds.
The missing observational constraints on the Galactic foregrounds make the subtraction a crucial,
highly sensitive step in the intensity mapping analysis pipeline. Foreground residuals can cause
serious systematic effects which bias the cosmological analysis and therefore require careful con-
sideration (Wolz et al. 2014). Most statistical methods employed for this task are based on the
spectral smoothness of the Galactic foregrounds whereas the cosmological signal is expected to
have low correlations between frequency bins. Extragalactic foregrounds caused by bright point
sourceemissionareexpectedtoshowdifferentspatialstructureduetogravitationalclusteringand
be far less dominant than Galactic sources. Extragalactic contaminations are yet to be fully in-
corporated in realistic simulations, however, not considered to cause any significant systematic
effects.
In this article, we review three different foreground subtraction approaches: the Karhunen-
Loèvetransform(KL,seeShawetal.2014a,b),thesingularvaluedecomposition(SVD,seeMasui
et al. 2013) and the independent component analysis (FASTICA, see Wolz et al. 2014). The SVD
and FASTICA are blind search methods which decompose the data in principal or independent
components, respectively, whereas the KL transform separates foregrounds and 21cm signal by
modelingtheirstatisticalproperties. FortheKLtransform,wereconsidertheforegroundresiduals
ofthepowerspectrumofanearlierstudy. Wegiveanexampleofforegroundremovalbyapplying
the FASTICA toanSKAphase1simulationforauto-correlationmodewherewechosethesettings
2
ForegroundSubtractioninIntensityMapping LauraWolz
for SKA1-MID band 1. We emphasize that an SKA1-SUR band 2 experiment would have very
similarprospectsintermsofforegroundremovalandresultsaretransferabletosuchanexperiment.
ForSKA2theforegroundsubtractionforexperimentsinthelowredshiftregimez 1willbeless
≈
challenging due to higher signal-to-noise ratio. Intensity mapping of structure in higher redshifts
z 3,(whichisofparticularinterest),isexpectedtobesimilarlychallengingduetotheincreased
≈
sensitivity of the SKA2 experiment. We present the expected level of foreground residuals and
BAO scale measurements in the angular power spectrum for FASTICA. In this work, we focus on
methods which have been recently applied to intensity mapping simulations or data. In Chapman
etal.(2014),foregroundsubtractionmethodsarepresentedinthecontextofSKAexperimentson
theepochofreionizationwhichpotentiallycanbetransferredtotheintensitymappingframework.
The article is structured as follows. In Sec. 2, we describe the 21cm signal, Galactic fore-
groundsandtelescopenoisepropertiesincludingthesimulationdetails. Wealsoreviewtheeffect
of instrumental errors on the foreground subtraction. In Sec. 3, the removal techniques are estab-
lished and in the following Sec. 4, the performance of the FASTICA in foreground subtraction is
evaluatedbytheconsideringthesystematicerrors. TheconclusionsarepresentedinSec.5.
2. DescriptionoftheSignalComponents
Inthissection,wedescribethenatureofthecosmologicalsignalweexpecttoobserveandhowwe
simulate it. We briefly outline the expected instrumental noise levels and review possible instru-
mentalerrors.
2.1 21cmSignal
Theintensityinafrequencybinδν comingfromthe21cmemissionofanobjectatredshiftzwith
neutralhydrogenmassM ,subtendingasolidangleδΩisgivenbyAbdalla&Rawlings(2005)
HI
3h A 1 M
p 12 HI
I(ν,nˆ)= ν , (2.1)
16πm ((1+z)r(z))2δνδΩ 21
H
whereA istheEinsteincoefficientcorrespondingtotheemissionfromthe21cmhyperfinetran-
12
sition, h is Planck’s constant and m is the hydrogen atom mass. Here, r(z) is the comoving
p H
(cid:112) (cid:112)
curvature distance r(z) = csinn(H Ω χ(z)/c)/(H Ω ) and χ(z) is the radial comoving
0 k 0 k
| | | |
distanceχ(z)=(cid:82)zdz/H(z).
0 (cid:48) (cid:48)
ThisintensityI(ν,nˆ)canbewrittenintermsofablack-bodytemperatureintheRayleigh-Jeans
approximation T =Ic2/(2k ν2), where k is Boltzmann’s constant. Using this we can estimate
B B
themeanbrightnesstemperaturecomingfromredshiftzanditsfluctuationsintermsoftheneutral
hydrogendensity:
Ω h(1+z)2x (z)
b HI
T (z,nˆ)=(0.19055K) (1+δ ). (2.2)
21 (cid:112) HI
Ω (1+z)3+Ω
M Λ
Herex (z)istheneutralhydrogenfraction(i.e. fractionofthetotalbaryondensityinHI)andδ
HI HI
istheHIoverdensityfieldinredshiftspace(smoothedoverthevolumedefinedbyδν andδΩ).
Wesimulatedthecosmologicalsignalbygeneratingathree-dimensionalrealizationoftheneu-
tral hydrogen density in the lightcone using a lognormal field as described in section 3 of Alonso
3
ForegroundSubtractioninIntensityMapping LauraWolz
etal.(2014). ThiswasdoneusinggenericΛCDMcosmologicalparameters(Ω ,Ω ,Ω ,h,w ,w ,σ ,n )=
M b k 0 a 8 s
(0.3,0.049,0,0.67, 1,0,0.8,0.96), andwefurtherassumedaredshiftdependencefortheneutral
−
hydrogen fraction of x =0.008(1+z) and a clustering bias of 1 (δ =δ). Redshift-space dis-
HI HI
tortions were included using the radial velocity field inferred from the Gaussian overdensity field
used for the lognormal realization. The simulation box used is large enough to encompass a full-
sky volume to redshift 2.5 (with the observer placed at the centre), and has a spatial resolution of
about2.65Mpc/h.
Weinterpolatedthedensityfieldintosphericaltemperaturemapsatdifferentfrequenciesusing
Eq. 2.2. These maps were generated using the HEALPix package (Gorski et al. 2005) with a
resolutionparametern_side=512(δθ 0.11o),atfrequencyintervalsofδν=0.7MHzbetween
∼
405MHz(z 2.5)and945MHz(z 0.5),makingupatotalof770frequencybands.
∼ ∼
2.2 GalacticForegrounds
For the simulation used in this analysis we generated foreground realizations for Galactic syn-
chrotron emission and free-free emission (both Galactic and extragalactic). We used the method
described in section 4 of Alonso et al. (2014) and did not include any leakage of the polarized
synchrotronradiation.
Thefree-freeforegroundsweregeneratedasaGaussianrandomrealizationofthecorrespond-
ing power spectra modelled in Santos et al. (2005). This models these foregrounds as isotropic
processes, which is obviously not a good approximation for the Galactic case. However, due to
theirexceptionallysmoothfrequencydependenceandsubdominantamplitude,wedonotbelievea
moresophisticatedmodellingisrequiredatthisstage.
Galactic synchrotron is by far the largest foreground for intensity mapping, and a more com-
plex method was used to simulate it. The method is largely based on that of Shaw et al. (2014b),
andisalsosimilartothoseusedinotherstudies(Jelic´ etal.2010;Shawetal.2014a). Themethod
startsbyextrapolatingthe408MHzHaslammap(Haslametal.1982)tootherfrequenciesusinga
givenmodelforthedirection-dependentsynchrotronspectralindex(forwhichweusedthePlanck
Sky Model, see Delabrouille et al. 2013). Smaller-scale structure and frequency-decorrelation is
thenaddedontopofthisthroughaGaussianrealizationofthepowerspectrainSantosetal.(2005).
Furtherdetailscanbefoundinsection4.2ofAlonsoetal.(2014).
2.3 InstrumentalNoise
ThenoiseRMS(fluxsensitivity)forasingle-pointingobservationwithasingle-dishradiotelescope
isgivenby
2k T
B sys
σ = , (2.3)
S (cid:112)
A δνt
e p
where A is the effective collecting area of the dish, t is the duration of the observation, and we
e p
have assumed that the noise is Gaussian and uncorrelated. The total system temperature is T =
sys
T +T , where T few 10K depends on the noise characteristics of the receiver system,
inst sky inst
∼ ×
andT 60K(ν/300MHz) 2.55 accountsforthetemperatureoftheskyduetobackgroundradio
sky −
≈ (cid:112)
emission. ConvertingEq. (2.3)intoabrightnesstemperaturesensitivitygivesσ =T / δνt in
T sys p
theRayleigh-Jeanslimit.
4
ForegroundSubtractioninIntensityMapping LauraWolz
In our simulations, we assume that the various SKA configurations will be able to perform
10,000hourauto-correlationsurveysoveranareaof30,000sq. deg1. ForaninstrumentwithN
dish
independentdishes,thetotalsurveyareaΩ canbecoveredwithanobservationtimeperpointing
tot
oft =t (N Ω /Ω ), where Ω λ2/A is the beam solid angle of a single dish. Assuming
p tot dish B tot B e
≈
nooverlapbetweensurveypointings,thebrightnesstemperaturesensitivityperpointingbecomes
(cid:114)
T Ω
sys tot
σ = . (2.4)
T
√δνttot NdishΩB
(cid:112)
ThenoiseRMSperpixelinoursimulationsisthengivenbyσ =σ Ω δν/Ω δν .
px T B pix pix
We simulate an intensity mapping experiment which can be performed by an SKA1-MID
surveywithband1. WeassumeT =28KforN =190withadishdiameterofD =15m.
inst dish dish
2.4 InstrumentalEffects
Intheory,thespectrallysmoothforegroundscanbeverywellextractedbypresentedmethods,how-
ever,instrumentaleffectssignificantlycomplicatethesubtraction. Mostoftheforegroundsubtrac-
tionmethodsarenotcapableofdealingwithvaryingbeamswhichisrequiredforauto-correlation
observations. Theissuecanbemitigatedbydeconvolvingthedatatothelowestresolution,ashas
been done for our simulation and for example in Wolz et al. (2014). Scale-dependent foreground
removalmethodsarerequiredforfutureexperimentsinordertonotlosespatialinformation.
Themostprominentinstrumentalsystematicsarepolarizationleakageandfrequency-dependent
beamdistortionswhichcausespatialandspectralmodestomix. InAlonsoetal.(2014),afirststep
istakentoincludepolarizationleakageintheintensitymappingsimulationsuchthatfuturestudies
cantestthemodemixingeffects. Additionally,calibrationerrors,telescopepointingerrorsandRFI
adulterate signal processing. Existing methods need to be advanced to include these instrumental
effectsintheforegroundremovalstep.
3. ReviewofForegroundRemovalMethods
Inthefollowingsection,wereviewselectedforegroundremovalmethods,theSVD,KLtransform
andFASTICA. Eachofthesemethodshavebeendesignedandappliedtodifferenttypesofintensity
mapping data types and are part of power spectrum estimator pipelines. The KL transform is part
of a detailed interferometer simulation for CHIME (Shaw et al. 2014a,b). The SVD is a more
empirical approach utilized to subtract the foregrounds of GBT observations (Masui et al. 2013;
Switzeretal.2013). The FASTICA hasbeentestedonSKA-likesimulationsinWolzetal.(2014).
Eachofthesemethodsworkinadifferentmathematicalandexperimentalframeworkandwerefer
thereadertotherespectivearticlesformoredetailsonthebackgrounddescription.
3.1 Karhunen-LoèveTransform
The Karhunen-Loève transform has a long history in Cosmology (Bond 1995; Vogeley & Szalay
1996),butwasfirstsuggestedasamethodfor21cmforegroundremovalinShawetal.(2014a). It
1Wefocusonauto-correlationsurveysherebecauseoftheirbettersensitivitytorelevantcosmologicalscalesfor
theSKAPhase1configurations;seeSantosetal.(2014)foradiscussionoftherelativemeritsofauto-correlationand
interferometricsurveys.
5
ForegroundSubtractioninIntensityMapping LauraWolz
hasalsobeenusedfortheanalogousproblemofE/BmodeseparationforpolarisationoftheCMB
(Lewis et al. 2002; Bunn et al. 2003). In this section, we give an overview of the technique, and
discussitspracticalimplementationinShawetal.(2014a,b).
Ultimatelyforegroundcleaningissimplyamatteroffindingasubsetofourdatawithinwhich
there is significantly more 21cm signal than there are astrophysical foregrounds. However, in the
presence of mode mixing there is no immediately apparent representation which separates these
signals. The Karhunen-Loève transform (often called the Signal-to-Noise eigendecomposition),
givesanautomatedwayofderivingthisbasisfromthetwo-pointstatisticsofeachcomponent.
Thismethodrequiresmodelsforthetwo-pointstatisticsofboththesignalandtheforegrounds
on the sky. We denote the matrix representation of the signal power spectrum as C , whereas
21
the foreground model, which includes both the synchrotron emission from our galaxy, and the
contributionfromabackgroundofextragalacticpointsources,iswrittenasC . Appropriatemodels
f
aredescribedinShawetal.(2014b)
TheKarhunen-Loèvetransformseekstofindalineartransformationofthedatad =Pdsuch
(cid:48)
that the covariance matrices of the 21cm signal S = BC B† and foregrounds F = BC B† are
21 f
jointlydiagonalised. Thatis
S S =PSP†=ΛΛΛ, (3.1)
(cid:48)
→
and
F F =PFP†=I, (3.2)
(cid:48)
→
where ΛΛΛ is a diagonal matrix, and I is the identity. In this diagonal basis we can simply compare
the amount of power expected in each mode by the ratio of the diagonal elements (this is given
by the corresponding entries of ΛΛΛ), and identify the regions of the space with low foreground
contamination(largeentriesinΛΛΛ).
This transformation can be found by solving the generalised eigenvalue problem Sx=λFx.
Thisgivesasetofeigenvectorsx,andcorrespondingeigenvaluesλ. Writingtheeigenvectorsina
matrixP, row-wise, givesthetransformationmatrixtodiagonalisethecovariances. Theeigenval-
uesλ correspondingtoeacheigenvectorgivethediagonalmatrixΛΛΛ. Toisolatethe21cmsignal,we
can simply select modes with eigenvalue (signal-to-foreground power) greater than some thresh-
old. InFig.1weshowtheeffectofforegroundcleaningonthepowerspectrumerrorsofsimulated
cylindertelescopesimilartoBanduraetal.(2014).
TheKarhunen-Loèvetransformisageneralandveryeffectiveschemeforthecomponentsep-
aration. Unfortunately,thefullcovariancematricesaredense,andverylarge,O(N N ) 108–
pix freq
∼
109 onaside,makingthediagonalisation,whichisanO(N3)operation,impossibleinthegeneral
case. However, this technique can still be applied to the restricted domain of transit telescopes,
wherethesymmetriesofthesystemallowtheproblemtobebrokenupintomanysmallerproblems
whicharesignificantlymoretractable. Thisdecompositionisknownasthem-modeformalism,and
was first demonstrated in Shaw et al. (2014a). Overall the complexity is reduced to O(N2 N3 )
pix freq
savingafactorof 106 incomputation.
∼
3.2 SingularValueDecomposition
In this section, we present the foreground cleaning formalism in Switzer et al. (2013) for the red-
shiftedHIsurveywiththeGreenBankTelescope(GBT).Itisbasedontheideathatthe21cmsig-
6
ForegroundSubtractioninIntensityMapping LauraWolz
NoFG NoPol FullFG 1.00
0.25
0.70
0.20 0.50
0.30
1pc− 0.15
M 0.20
h
/
kk 0.15
0.10
0.10
0.05
0.07
0.05
0.00
0.00 0.04 0.08 0.00 0.04 0.08 0.00 0.04 0.08
k /hMpc−1
⊥
Figure1: Forecasterrorsonthepowerspectrumasafractionofitsfiducialvaluefora40m 40mcylinder
×
telescope similar to Bandura et al. (2014), observing between 400–500MHz. The three panels show the
predictederrorswithoutforegrounds(left),withunpolarisedforegrounds(centre),andwithfullypolarised
foregrounds (right). We can see that we lose sensitivity to the large scale line of sight mode. The dashed
line indicates the predicted bound of the ‘foreground wedge’, showing that with perfect knowledge of our
instrument,foregroundscanbesuccessfullycleanedwellintothisregion.
nalisalineemissionandhasstructureinthefrequency/redshiftdomain,whiletheforegroundsare
smoothonthosescales. Onecanthusseparatethetwocomponentsinamodelindependentwayby
making useof principalcomponent analysis. In this case, we arrangethe three-dimensionalmaps
intoafrequency-frequencycovariancematrixandusesingularvaluedecompositiontoidentifythe
dominant frequency modes. We label them as “foregrounds“ and project out their contribution to
obtainforegroundcleanedmaps.
Specifically, we rearrange the three-dimensional map into an N N matrix M, where N
ν θ θ
×
includesalltwo-dimensionalspatialpixels. Forthepurposeofthiscomparison,weignorethermal
noise in the map. The empirical ν ν covariance of the map is C=MMT/N , which contains
θ
−
both foregrounds and 21cm signal. Making use of the component separation idea, the matrix can
befactoredasC=UΛΛΛUT, whereΛΛΛisdiagonalandcontainstheeigenvaluesindescendingorder.
We tag the first few modes as ’foregrounds’, and from each line of sight, we can then subtract a
subset of the modes U that describe the largest components of the frequency variance through the
operation (1 USUT)M, where S is a selection matrix with 1 along the diagonal for modes to be
−
removedand0elsewhere.
In practice, the separation of foreground and signal modes is not perfect due to the imperfect
characterization of instrumental response. The choice of the selection matrix S is a compromise
between maximal foreground removal and minimal signal loss. To estimate the latter, we inject
simulated 21cm signal to the data stream to determine the transfer function T which describes
7
ForegroundSubtractioninIntensityMapping LauraWolz
loss of 21cm signal due to foreground removal. As a rule of thumb, T =P /P [(1
sig.out sig.in
∼ −
N /N )(1 N /N )]2,whereN isthenumberofmodesremoved,N isthenumberoffrequency
m ν m res m ν
−
channels and N is the number of angular resolution elements. A limited number of resolution
res
elements can greatly reduce the efficacy of the foreground cleaning at the expense of signal. The
details of foreground transfer function calculation and the signal compensation can be found in
Switzeretal.(2013).
Ourapproachtoforegroundremovalislimitedbytheamountofinformationinthemaps. The
fundamentallimitationhereisnotsimplyfromthenumberofdegreesoffreedomalongthelineof
sight,butinsteadislimitedbythesmallerofindependentangularorfrequencyresolutionelements
in the map (Nityananda 2009). To see why this is the case, notice that in the absence of noise,
our cleaning algorithm is equivalent to taking the SVD of the map directly: M=UΣΣΣVT and thus
C∝MMT =UΣΣΣ2UT,withthesamesetoffrequencymodesUappearinginbothdecompositions.
The rank of C coincides with the rank of M and is limited by the number of either angular or
frequencydegreesoffreedom.
3.3 IndependentComponentAnalysis
In the following subsection, the basic principles of the fast independent component analysis FAS-
TICA method (Hyvärinen 1999) are outlined. FASTICA has been successfully used for foreground
removalintheastrophysicalframework,seeforexampleMainoetal.(2002);Bottinoetal.(2010);
Chapmanetal.(2012).
FASTICAisamethoddesignedtodecomposemixedsignalsintotheirindependentconstituents.
StatisticalindependenceismeasuredbytheuseoftheCentralLimittheoremwhichstatesthatthe
probability distribution density (pdf) of a sum of independent variables is always more Gaussian
than the pdf of one single component. The inverse application of that theorem implies that a
single component can be extracted from the mixture by maximizing the non-Gaussianity of the
pdf. Thisisturnimpliesthat FASTICA cannotbeappliedtoGaussianvariables. Wecanformulate
the problem in the linear equation x=As, where x is the signal, A the mixing matrix and s the
independentcomponents(IC).FASTICAblindlysolvestheinvertedproblem
s=Wx (3.3)
where the weighing matrix W defined as the inverse of A is unknown. As a measure of the non-
GaussianityoftheICs,FASTICAmaximizesaproxyforthenegentropy.
FASTICAhasbeenemployedtosubtractforegroundsinintensitymappingsimulationsinWolz
et al. (2014). Since the foregrounds are highly correlated between frequencies, FASTICA incorpo-
rates them into the ICs. The 21cm signal has a very low correlation which is close to Gaussian.
Hence, FASTICA blindly reconstructs the foregrounds and the residuals of this analysis are cos-
mological signal plus the receiver noise. In Wolz et al. (2014), it has been shown that foreground
subtraction with FASTICA does not affect cosmological distance measures such as the BAO scale.
An residual analysis of the SKA simulations with FASTICA is demonstrated in the following sec-
tions.
8
ForegroundSubtractioninIntensityMapping LauraWolz
4. ResidualErrorsandBAOrecovery
Inthissection,wepresenttheforegroundresidualsandsystematicerrorswhenapplying FASTICA
tothedescribedSKA1-MIDsimulation.
4.1 ExperimentalSet-up
The SKA1-MID band 1 simulation is based on the 21cm and Galactic foregrounds simulation
presented in Alonso et al. (2014), where the frequency range is 406MHz<ν <945MHz with a
frequencywidthofδν=0.7MHzperchannel. ThebeamisapproximatedbyaGaussianbeamwith
constant solid angle Ω =1.6deg2. The instrumental noise is simulated according to Equ. 2.4
beam
foranSKA-MIDband1likeobservation. Wemaskouttheskyabove63deglatitudeinequatorial
coordinateswhichresultsinacoverageof30,000deg2. Theintensitymapsarestackedintoconstant
redshift bins with width δz 0.05 after the foreground removal to increase the signal-to-noise in
≈
thesystematicsanalysis.
4.2 Results
WeevaluatetheforegroundsubtractionbyestimatingtheangularpowerspectrumC((cid:96))oftheorig-
inal21cmintensitymapsandthereconstructedmaps. Wecorrectforthepartialskycoveragewith
the Peebles approximation, as described in Wolz et al. (2014). In Fig. 2(a), we show the relative
errorCICA((cid:96))/Corig((cid:96)) of the FASTICA-cleaned maps for different number of ICs. The black error
barsindicatethestatisticalvarianceofthemeasurementsgivenbysamplingerrorandinstrumental
noise. It can be seen that for a number of ICs higher than 4 the reconstruction converges towards
the original inputC((cid:96)) for scales (cid:96)>100. For smaller scales, the broadband spectrum is slightly
distortedandweobservedeviationsofC ((cid:96))outsidethestatisticalerrors.
ICA
In Fig. 2(b), the relative systematic error δ((cid:96),z) is depicted for all redshift bins z on the
i i
x-axis and for multipole bins δ(cid:96) = 20 in the y-direction. The systematic error is defined as
∆ = C ((cid:96)) C ((cid:96)) . Wedividethaterrorbythestatisticaluncertaintyonthemeasurements
sys orig ICA
(cid:112) | − |
σ(C((cid:96)))composedofsamplingvarianceandinstrumentalnoise. Therelativeerrorsarepictured
in a logarithmic scale such that values larger than zero imply insignificant systematic errors and
smallerthanzeroindicatescaleswheresystematicsdominatethestatisticalerrors. Weseethatthe
foregrounds at the edges of the redshift bins are poorly subtracted due to incomplete frequency
informationandthoseregionsshouldbeneglectedinsubsequentanalysis.
Furthermore, we present BAO scale measurements of the SKA1 simulation in Fig. 3, the
details of the methods can be found in Wolz et al. (2014). In Fig. 3(a), the BAO wiggles in the
powerspectrumproducedbydividingbyaBAO-featurelesspowerspectrummodelareshownfor
4 different redshift bins. The lines represent the fitted models to the estimated data points. The
blackmeasurementsmarkthesimulationwithoutanyforegroundcontaminationsandtheredlines
include systematic errors. We see that there is no significant shift of the BAO features induced
by the foreground subtraction. Fig. 3(b) presents the scale distortion parameter α as a function
of redshift where α = D (z)/D (z). The BAO recovery is not significantly affected by the
A i A,fid i
foregroundsubtractionformostredshiftsz.
i
9
ForegroundSubtractioninIntensityMapping LauraWolz
0
5. Variance+Noise
ICA= 4
ICA= 5 292
ICA= 6 272 4
252
0 232 2
2. 212
(l)/C(l)Aorig 1.0 l 111135792222 −02
CIC 112 −4
92
0.5 5722 −6
32
12 −8
1 3 5 7 9 111315171921232527293133353739
0.2 zbin
2 5 10 20 50 100 200
l
(a) RelativeerroronC((cid:96))for1.45<z<1.50 (b) Relativesystematicerrors;FASTICAwithIC=5
Figure2: TherelativeerrorsoftheforegroundsubtractionwithFASTICAontheangularpowerspectrumas
afunctionofscale. Ontheleftside,wepresentthefractionC ((cid:96))/C ((cid:96))fordifferentnumbersofICA
ICA orig
foranexampleredshiftbin. Therelativesystematicerrorsδ((cid:96),z)aregivenonanaturallogarithmicscales
i
intherightpanel.
(a) BAOscaleinangularpowerspectrum;FASTICAwithIC=5 (b) BAOfit;FASTICAwithIC=5
Figure3: TheBAOmeasurementsoftheoriginalsimulationswithoutforegrounds(black)arecomparedto
themeasurementsincludingforegrounds(red). Intheleftpanel,wepresenttheBAOwigglesintheangular
power spectrum divided by a featureless fiducial model. The lines are produced by fitting a model to the
measurements. TherightpanelshowstheBAOscaledistortionparameterα withoutandwithforegrounds
asafunctionofredshift.
10