Forecast B-modes detection at large scales in presence of noise and foregrounds

0.21 MB·English

by  A. Bonaldi

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed26January2011 (MNLATEXstylefilev2.2)
Forecast B-modes detection at large scales in presence of
noise and foregrounds
1
Bonaldi A. 1 and Ricciardi S. 2
1
0 1INAF, Osservatorio Astronomico di Padova, Vicolo dellOsservatorio 5, I-35122 padova, Italy
2 2INAF/IASF, Sezione di Bologna, Via Gobetti 101, I-40129 Bologna, Italy

n
a
J 26January2011

5
2
ABSTRACT

] We investigate the detectability of the primordial CMB polarization B-mode power
O
spectrumonlargescalesinthepresenceofinstrumentalnoiseandrealisticforeground
C contamination. We have worked out a method to estimate the errors on component
. separation and to propagate them up to the power spectrum estimation. The perfor-
h
mancesofour methodareillustratedby applyingit to the instrumentalspecifications
p
of the Planck satellite and to the proposed configuration for the next generation
-
o CMB polarization experiment COrE. We demonstrate that a proper component sep-
r arationstep is required in order achieve the detection of B-modes on large scales and
t
s thatthe finalsensitivitytoB-modesofagivenexperimentisdeterminedbyadelicate
a
balance between noise level and residual foregrounds, which depend on the set of fre-
[
quenciesexploitedintheCMBreconstruction,onthesignal-to-noiseofeachfrequency

1 map, and on our ability to correctly model the spectral behavior of the foreground
v components. We have produced a flexible software tool that allows the comparison
6 of performances on B-mode detection of different instrumental specifications (choice
7 of frequencies, noise level at each frequency, etc.) as well as of different proposed
8
approaches to component separation.
4
. 1 INTRODUCTION amplitudes while the CMB B-mode is far weaker than the
1
0 E-mode.
Primordial B-mode (curl component) polarization of the
1 The constraints in detecting primordial B-modes due
CMB provides a unique opportunity to detect the imprint
1 toforeground contamination and residuals from foreground
of the primordial gravitational waves predicted by the in-
: subtractionhavebeeninvestigatedbyseveralauthors.Many
v flationary paradigm. Measuring theamplitudeof theseten-
papersincludeamodelingoftheresidualsintheFisherma-
Xi sorperturbationswouldfixtheenergyscaleofinflationand trix (see. e.g., Tucci et al. 2005, Amarie et al. 2005, Verde
its potential and would provide a powerful constraint on a
r et al. 2006) but generally bypass any realistic component
a broad class of inflationary models. Moreover, the confirma- separation approach and thus cannot properly deal with
tion of inflation and the determination of the inflationary
component separation errors. Other analyses focussed on
potential would haveprofound implications, byproviding a
a certain component separation method and evaluated its
direct observational link with the physics of the early uni-
performances through simulations tailored for aspecific ex-
verse.
periment.Forexample,Betouleetal.(2009)andEfstathiou
The bulk of the statistical information on inflationary etal.(2009)proposedifferentstrategies(aspectralmatching
B-modes is concentrated in two features of the power spec- componentseparation method andaninternaltemplatefit-
trum: the reionization bump at multipoles ℓ 2 10 and tingmethod,respectively)targetedtoB-modedetectionfor
∼ −
themainbumpatmultipolesℓ 20 100.Giventhatmost thePlanckexperiment.Morerecently,Stivolietal.(2010)
∼ − ◦
of thesignal lies on large angular scales (larger than 1 ), assessed thecapabilities ofaparametric componentsepara-
∼
full-skysurveys,andthussatelliteexperiments,arerequired tion method to detect B-modes for suborbital experiments.
to limit cosmic variance. Thepurposeofourworkisdifferent:whileunderstand-
When designing a satellite experiment targeted to B- ingtheneedofperformingdetailedsimulationsoncetheex-
mode detection on large scales it is mandatory to consider, perimental configuration is established, we want to provide
togetherwiththesignal-to-noiseratio,alsotheissuesrelated afast,agileandflexibletool1toforecastthedetectabilityof
to foreground contamination and component separation. In B-modesallowingustoeasilychangeboththeinstrumental
fact,asdemonstratedbytheWMAPdata(Goldetal.2009), specifications and theCMB reconstruction strategy. Sucha
the cleaning of CMB E-mode from polarized foregrounds tool can be usually exploited to help optimizing the instru-
is still a manageable problem. But polarized foregrounds
are expected to dominate by a wide margin over the CMB
B-mode because foreground E- and B-modes have similar 1 thesoftwarecouldbeavailableuponrequest
2 Bonaldi A. and Ricciardi S.
mental design and the data reduction pipeline for future quency ν is the superposition of signals coming from Nc
CMB polarization experiments. different physical processes s˜j:
The different components are reconstructed as a suit-
Nc
able linear mixture of the data. Different choices for the
x˜(rˆ,ν)= s˜j(rˆ,ν). (4)
linear mixture operator are available, and we carry out a
Xj=1
comparison of their performances The forecast, targeted to
large scales, on the B-mode detectability is performed at Theinstrumentintegrates thesignal overfrequencyin each
the power spectrum level and includes the computation of ofitsNd channels,convolvesit withacertain kernel,repre-
thefinalpowerspectraofnoiseandforegroundresidualson senting the spatial response function, and adds noise. The
the CMB map. We explicitly account for errors in the fre- measurementyieldedbythegenericchannelcenteredatthe
quencyscalingsassumedfortheforegroundcomponents.On frequency ν,with beam axis in thedirection rˆis:
the other hand, we do not account for errors due to actual
Nc
power spectrum estimation (leakage effects) nor for instru- xν(rˆ)= B(rˆ rˆ′,ν′) tν(ν′)s˜j(r′,ˆν′)dr′dν′+nν(rˆ), (5)
mental systematics. The former should be subdominant for Z −
Xj=1
thealmostfull-skycoverageconsideredhere,especiallywhen
exploiting a power spectrum estimation method optimized where B(rˆ rˆ′,ν′) is the (azimuthally-symmetric) beam,
forlowmultipoles(e.g.Gruppusoetal.2009).Instrumental tν(ν′) is the−frequency response of the channeland nν(rˆ) is
systematics obviously depend on the particular instrument the noise map. The data model in eq. (5) can be simplified
and thuscannot beincluded in our general treatment. underthefollowing assumptions:
The outline of our paper is the following: in 2 we de-
scribeourmodelsforthedataandthecomponents§eparation (i) Each source signal, s˜j(rˆ,ν),is a separable function of
direction and frequencyat least within limited skypatches;
process. In 3wederiveanalyticalexpressionsfor thenoise
§ (ii) B(rˆ,ν) is independent of frequency within the pass-
and residual foreground contributions to the errors in the
band of each channel;
BBpowerspectrum.In 4weapplyourmethodtothespec-
ifications of the Planck§ satellite and of a proposed CMB (iii) B(rˆ,ν) is thesame for all thechannels.
polarizationexperiment(COrEwhitepaperinpreparation). The first two are reasonable approximations. To meet the
Ourresultsarepresentedin 5andourmainconclusionsare thirdoneweneedtosmooththedatatothelowestresolution
§
summarized in 6. we are dealing with. This is not a problem in the present
§
context,since we are interested in large angular scales.
Undertheseassumptions,thelinearmixturedatamodel
applies. In vector form, for each direction of the sky (each
2 STATEMENT OF THE PROBLEM
pixel) we may write:
2.1 Data model x=Hs+n. (6)
Themicrowaveskycontains,besidestheCMB,severalfore- where x and n are N -vectors containing data and in-
d
groundcomponents,bothdiffuseandcompact.Forouranal- strumental noise respectively, s is the Nc-vector containing
ysis, which is focused on large and intermediate scales, we sources,andHistheNd Nc mixingmatrix,containingthe
will consider only diffuseforegrounds. The main diffuse po- ×
frequencyscalingofthecomponents.Thespatialvariability
larized foregrounds are Galactic synchrotron and thermal
ofthesynchrotronanddustspectralindicesimpliesthatthe
dust (the free-free emission is unpolarized and the anoma- mixing matrix H is in general different for different pixels.
lous dust emission is also expected to be essentially unpo-
larized).
Thesynchrotroncomponentdominatesatthelowerfre- 2.2 CMB recovery
quencies. Its spectral behavior in antenna temperature can
bemodeled as a power law: If the linear mixture data model holds, the components
which are mixed in the channel maps x [eq. (6)] can be
TA,synch(ν) ν−βs , (1) reconstructed as:
∝
wherethesynchrotronspectralindexβs canvaryinthesky sˆ=Wx. (7)
ibnehthaveiroaunrgoef2t.h5e<rmβasl<du3st.5e(mGisoslidone,ttahl.at20t0a9k)e.sTovheersaptechtirgahl wheresˆisanestimateofthecomponentssandWisaNc
×
frequencies, follows approximately a grey-body law: Ndmatrixcalledreconstructionmatrix.Wechoosetorelyon
alinearestimatorbasically becauseitwillallowustoeasily
νβd+1 deal with the component separation process in the forecast
T (ν) . (2)
A,dust
∝ exp(hν/kTdust) 1 of B-mode detection, as we will see in the next section. In
−
additionthisapproachishandyforMonteCarlosimulations
Both β and T are spatially varying around β 1.7 and
d d d
∼ needed to accurately control error sources.
T 18K.ThepolarizedCMBsignalhasablackbodyspec-
d
∼ We adopt the so-called Generalized Least Square solu-
trum:
tion (GLS):
TA,CMB(ν)∝ (hν/(ekxTpC(MhBν/)2kTexCpM(Bh)ν/k1T)C2MB) (3) W=[HˆTN−1Hˆ]−1HˆTN−1. (8)
−
with TCMB =2.73. This requires the noise covariance N of the channel maps
The sky radiation, x˜, from the direction rˆ at the fre- and an estimate Hˆ of the mixing matrix H that can be ob-
Forecast B-modes detection at large scales 3
tained exploiting any of the successful component separa- simulation and,finally,allthenoisepowerspectraareaver-
tion methods discussed in the literature (see, e.g., Bonaldi aged.HerewemakethesimplifyingassumptionsofGaussian
et al. 2006, Eriksen et al. 2006, Stompor et al. 2009). We whitenoiseandspatiallyinvariantforegroundspectralprop-
stressthatthischoiceisnotcompletelygeneral,asotherre- erties. This allows us to proceed analytically. In the frame-
construction matrices could beused.Theadaptation ofour workofarealisticspatially-varyingspectralmodel,wehave
approach to other choices for W, still in the framework of verifiedthattheanalyticcomputationofthenoiseerrorwith
thelinear mixturedata model, is howeverstraightforward. mean spectral dependencies over the sky is still enough ac-
The fact that our reconstruction matrix explicitly con- curate for the purpose of the forecast here considered. In
tains the estimated mixing matrix will allow us to easily fact, due to the intrinsic uncorrelation between noise and
introduceinourforecast alsotheeffectoferrorsinthemix- foregrounds, the mean level of the noise bias is predicted
ing matrix estimation (see 3). with good accuracy.
Thenoisecovariancete§rm,N,ineq.(8)allowustoper- ThetheoreticalpowerspectrumofGaussianwhitenoise
formthecomponentseparationkeepingthenoiselevelunder at thefrequency ν is:
control.Nevertheless,everycomponentseparationleadstoa
4πfsky
nthoeisecoanmdpitliiofinciantgioonfbtyheanmaixminogunmtautlrtiixmHaˆt,elaysdweepewnidllinsghoown Nν = Npix σν2B2ν (10)
in 4. Despite this drawback, in the low multipole regime where fsky is the sky fraction considered, Npix the number
con§sideredwecannotdispensefromthecomponentsepara- of pixels, σν2 thenoise variance at frequency ν and B2ν(ℓ) is
tionstep(e.g.byresortingtoanaggressivemaskingoffore- the beam function applied to the channel maps to obtain a
ground contaminated regions) because the foreground con- commonresolution.Asinourcasewedealwithpolarization,
taminationismoretroublesomethanthenoiseamplification σν is the noise of the detector at frequency ν multiplied
and large areas are essential to restrain cosmic variance. by √2 which, in the Gaussian white noise hipothesis here
On the contrary, the use of a suitable mask could be considered, gives the RMS of the noise in Q and U maps.
enough to keep foreground contamination under control GiventhelinearityoftheCMBrecoveryprocess[eq.(7)]
on smaller scales, where the power spectra of diffuse fore- and under the assumption of a spatially invariant recon-
groundsdecreasemorerapidlythanthoseofCMBandnoise. struction matrix, the noise bias is obtained by combining
In this case we could completely change strategy and sim- thechannelnoise spectra Nν with the matrix W2:
pwliythrewceoivgehrtsthgeivCenMbByatsheaiwnveeigrshetendoimseeavnaroiafntchee. Tchhaisnncealns NCMB = wν2,CMBNν, (11)
Xν
beviewedagainasalinearcombinationasineq.(7),where
thereconstruction matrix W hasnull elementsfor thefore- wherewν2,CMB aretheelementsofthematrixW2 pertaining
grounds and inverse noise variance weights for the CMB. to theCMB component.
This approach is complementary to the one previously de- The error on the CMB power spectrum is due to the
scribed: it allows noise reduction but does not separate the fact that we do not know the actual noise realization, but
components; only one component is reconstructed and con- onlyameannoisebias.Inotherwords,∆Cℓ,noiseistheerror
sidered as CMB. dueto thesampling variance of thenoise bias NCMB:
In this work we will consider both approaches, which
2/(2ℓ+1)
will belabeled as “component separation” (CS) and “mini- ∆Cℓ,noise=rfskynbin(ℓ)NCMB, (12)
mum variance” (MV). The comparison of the results in 5
§
will give us an idea of the multipole ranges where each ap- where nbin is a function containing the number of multi-
proach works better. poles around any chosen ℓ to be averaged according to a
certain binningscheme.
The error ∆C [eq. (9)] due to the imperfect fore-
ℓ,foreg
groundsubtraction can becomputedfollowing Stivoliet al.
3 FORECAST OF B-MODE DETECTION (2010). The map of residuals, s sˆ, for a linear mixture
−
source reconstruction can be estimated as:
Our forecast is done at the power spectrum level and ac-
countsfor both noise errors and foreground residuals: s sˆ=(WH I)s˜, (13)
− −
∆C =∆C +∆C . (9) where I is the identity matrix and s˜ is a set of simulated
ℓ ℓ,noise ℓ,foreg
components. ∆C is the power spectrum of the residu-
ℓ,foreg
In this section we formalize the estimate of the two contri- als computed over a certain area of the sky. We note that
butions ∆Cℓ,noise and ∆Cℓ,foreg to the error on the power in the computation of the foregorund residuals we do not
spectrum, exploiting the model of the sky [eq. (6)] and of proceed analytically up to the power spectrum level. This
the component separation process [eq. (7)] described in the means that, in this case, we could easily exploit a realis-
previous section. tic spatially-varying foreground spectral model. Given the
To compute the noise error ∆C we have to es- lack of observational constraint at present,however, for the
ℓ,noise
timate the noise bias on the CMB power spectrum, say purpose of the paper we are relying again on the spatially-
NCMB, i.e. the power spectrum of the noise in the recon- invariant model.
structed CMB map. This can be done with “noise-only” According to eq. (13) the foreground residuals vanish
Monte Carlo simulations whereby several sets of realistic for W=H−1,which istheexact solution of thecomponent
noisemapsaregeneratedandeachofthemismixedwiththe separation problem in the absence of noise. For the recon-
matrix W. Then the power spectrum is computed for each structionmatrixofeq.(8),theforeground contamination is
4 Bonaldi A. and Ricciardi S.
Table1.InstrumentalcharacteristicsconsideredinthepresentstudyforthePlanckandCOrEexperiments.
Thermsperpixelisreferredtonside1024(size∼3.5arcmin)
Planckspecificationsa
ν(GHz) 30 44 70 100 143 217 353
FWHM(arcmin) 33 24 14 9.5 7.1 5.0 5.0
29mRMS∆T(µKRJ) 66 64 60 20 10 7.6 7.5
b
COrEspecifications
ν(GHz) 45 75 105 135 165 195 225 255 285 315 375 435 555 675 795
FWHM(arcmin) 23 14 10 7.8 6.4 5.4 4.7 4.1 3.7 3.3 2.8 2.4 1.9 1.6 1.3
24mRMS∆T(µKRJ)) 1.4 0.7 0.6 0.5 0.4 0.3 0.2 0.4 0.5 1.1 3.6 0.9 1.2 1.2 1.3
a
LFIspecifications asinMandolesietal.(2010), HFIspecifications asinLamarreetal.(2008)
b
ESAM3call(Dec2010)availableathttp://www.core-mission.org
non null (although very low) even for Hˆ = H. It obviously gravitationallensing.Wehaveaddedtensormodeswithten-
increaseswithincreasingerrorinthemixingmatrixestima- sor to scalar ratio r=[0.1,0.03,0.01,0.001] and re-ionization
tion. In the MV case the mismatch between W and H−1 is optical depth τ =0.1.
higher, and the foreground contamination increases.
It is clear that the estimate of ∆C through the
ℓ,foreg
previousrelationismodel-dependentsinceitreliesonsimu-
4.2 Minimum noise variance weighting
lationsofthedatas˜whicharehamperedbyourpoorknowl-
edge of polarized foregrounds. The situation will substan- As mentioned above, the MV approach assumes that, out-
tially improve in the near future as new polarization data, side a suitable mask, foregrounds are negligible so that we
aboveallthosefromthePlanckmission,willbecomeavail- needtodealonlywithCMBandnoise.Thereforetherecon-
able. struction matrix, W, contains only inverse noise variance
weights for the CMB and null elements for the foreground
components. For this assumption to be plausible we must
4 APPLICATION TO THE PLANCK AND restrict ourselves to a “CMB sensitive” set of channels and
adoptawideGalactic mask.TocomputeHandWwehave
CORE EXPERIMENTS
thus considered only the channels in the frequency range
The adopted experimental specifications for the Planck 70 6 ν < 200 GHz and computed the power spectrum of
and COrE experiments are reported in Table 1. Planck the foreground residual map [eq. (13)] only over the high
specifications are derived from Mandolesi et al. (2010) for Galactic latitude sky [b > 30◦, so that fsky = 0.5, see
| |
the Low Frequency Instrument (LFI) and from Lamarre et eq.(12)].
al. (2008) for theHigh FrequencyInstrument(HFI)assum- Given the different sensitivities of Planck and COrE
ing a mission duration of 29 months. The specification for we haveused different bin sizes. Bin centersfor ℓ6150 are
COrE are from the ESA M3 call (Dec 2010) available at ℓ =[4,9,14,20,28,39,53,80,140] for Planck and ℓ =
bin bin
http://www.core-mission.org. [3,6,10,19,31,49,74,106,150] for COrE.
For each experiment we computed the error on the
power spectrum using eqs. (9)–(13) for both the minimum
variance (MV) and component separation (CS) case.
4.3 Component separation weighting
For the component separation approach we generated the
4.1 Sky model
estimated mixing matrix assuming the two standard po-
larized foreground components (synchrotron and thermal
ThepolarizationQandUsynchrotronanddusttemplatesof
dust) with spatially invariant spectra, but allowing for er-
ourmodelwereobtainedat100GHzbyrunningthePlanck
SkyModel(PSM).2 Extrapolationstolowerandhigherfre- rorsintheestimateofsynchrotronanddustspectralindices.
Three error regimes, labeled as “conservative”, “intermedi-
quenciesweremadeusingthespectraofeq.(1)withβs =3
ate”, and “goal” were considered (see Table 2).
for synchrotron and of eq. (2) with β =1.7 and T =18K
d d
The “conservative” errors on spectral indices are those
for dust. Is has been necessary to slightly adjust the inten-
estimatedbyRicciardietal.(2010)applyingtheCCAcom-
sity of the synchrotron template to reproduce the WMAP
ponentseparationmethod(Bonaldietal.2006)toarealistic
7-yr K-band polarization map. The polarized CMB simu-
simulationofPlanckpolarizationdata,includingspatially-
lation is based on a standard ΛCDM model with WMAP
varying mixing matrix and realistic noise for 2 all-sky sur-
7-yrcosmological parameters(Larsonetal.2010),including
veys.ThustheyareslightlypessimisticfortherealPlanck
mission,nowextendedto4surveys.Moreoverthenextgen-
2 http://www.apc.univ-paris7.fr/APC CS/Recherche/Adamis/PSM/psekrya-tion experimentstargeted toB-modes detection will un-
en.php doubtedlyimprovetheaccuracyofthedeterminationoffore-
Forecast B-modes detection at large scales 5
nallyexploiting46% ofthesky.Inthiscase, theforeground
Table2.Adoptederrorsintheestimationofthesynchrotronand
dustspectralindicesforthreedifferentregimes contaminationlowersapproximatelyby30% atlargescales.
Withamaskoptimizedforthefrequencycoverageherecon-
βs βd sidered we could do even better; nonetheless, the residual
contaminationwouldbestillveryhighrespecttotheCMB.
conservative 0.05 0.01
Weconcludethat onlarge andintermediatescales aproper
intermediate 0.01 0.005
component separation is necessary.
goal 0.005 0.001
InthePlanckcase,thetotalerrorfortheCSapproach
is noise dominated even for a 85% sky coverage (b >10◦).
| |
Extending the frequency range from 50 < ν < 200 GHz to
ground spectral properties. That’s why we also considered
the40<ν <300GHzimprovessubstantiallythenoiselevel.
the“intermediate” and “goal” regimes.
Tobemorequantitative,let’scallA2thesumoftheelements
To propagate the errors on spectral indices to thefinal
ofthematrixW2 pertainingtotheCMBcomponent.These
result, we generated a set of 10 estimated mixing matrices
Hˆ,drawingtheactualspectralindicesfromGaussiandistri- elementsweightthenoisepowerspectrainthecomputation
butionswithσequaltothespectralindexerrors.ForeachHˆ of the noise bias NCMB. Thus A can be interpreted as the
wecomputedthereconstructionmatrixW,theresidualmap amplificationofthenoiserms.Inoursimulation,thesmaller
setofchannelsyieldsA 3,whilethelargeroneyieldsA
through eq. (13) and its power spectrum. Since in this case ∼ ∼
1.3.Oursimulationsindicatethat,usingthewiderfrequency
we minimize foreground residuals the usable sky fraction is
range, Planck can detect B-modes for r = 0.1 and, with
wider. We adopt fsky = 0.85, corresponding to a Galactic
lower significance, for r = 0.03. The fact that the error is
cut of 10 deg around the equator. The values of ∆C
ℓ,foreg
± dominated by the noise term implies that the improvement
shown in Fig. 1 are the mean for the10 simulations.
in themixing matrix estimation from the“conservative” to
Another issue is the choice of the set of channels to be
the“intermediate” case has a minor effect.
used for the source reconstruction. On one hand, using a
Although our conclusions on Planck’s capability for
wide frequency range brings in channels heavily foreground
detectingB-modesaresimilartothoseofEfstathiou&Grat-
contaminatedand,sincetheforegroundspectraarenotper-
ton (2009), our approach differs substantially from theirs.
fectly known, this increases the foreground residuals in the
Theseauthorsbasedtheiranalysisonaminimumnoisevari-
final map. On the other hand, using a too limited set of
ancecombination of the70, 100 and 143 GHzchannels and
channels in the linear combination leads to a higher noise
assumed that foregrounds can beremoved tohigh accuracy
level in the reconstructed CMB map, because the mixing
if the highest and lowest polarized Planck channels are
matrix is less well-conditioned.
used as foreground templates and the most contaminated
The best trade-off clearly depends on the instrument,
areas (37% of the sky) are masked. However, as shown by
and in particular on its sensitivity. To exemplify this situ-
our analysis, the foreground contamination corresponding
ation we considered two sets of channels: a restricted one,
to the minimum noise variance combination is very high
limited to the frequency range 50 < ν < 200 GHz, and a
even at high Galactic latitude. Therefore a sufficiently ac-
wider one (40 < ν < 300 GHz) for both the considered
curate foreground removal by template fitting is very chal-
experiments.
lenging.Conversely,amorestandardcomponentseparation
We stress that this analysis on the choice of channels
approach, as the one exploited for our forecast, keeps both
referstothesource reconstruction.Fortheestimationofthe
foreground contamination and noise undercontrol.
mixing matrix, i.e. of the foreground spectral properties, it
The bottom panels of Fig. 1 show our expectations for
isusefultoexploitafrequencyrangeaswideaspossible.To
the COrE mission. Due to the higher sensitivity of this ex-
this end the lowest and highest frequencies are particularly
periment, the total error for the CS case is dominated by
useful, as they map respectively polarized synchrotron and
noise only for very low errors in the mixing matrix esti-
dust emissions with high signal-to-noise and low contam-
mation (“goal”). For “conservative” errors in the spectral
ination from the other components. Even if not explicitly
indices,thebest resultsareobtained with therestricted set
addressed in this paper, the mixing matrix estimation has
of channels. B-mode polarization can be detected down to
a key role in the detection of B-modes, as will beshown by
r = 0.01 even with the ”conservative” errors in the spec-
the comparison of the results on the spectral index errors
tral indices. The minimum value of r that can be reached
for our different regimes.
dependson our ability to estimate themixing matrix.
5 RESULTS
6 CONCLUSIONS
In Fig. 1 we show the result of the forecast for Planck
(upper panels) and COrE (lower panel) for the two choices We presented a method to forecast the detectability of
of the frequency range (left: 50 < ν < 200 GHz, right: CMB B-mode polarization on large and intermediate angu-
40<ν <300 GHz) used for the CMB map reconstruction. lar scales given the experimental specifications of a full-sky
We can immediately note that the MV approach performs multi-frequencyexperiment.Ourforecast accountsfor both
worse: the noise level is low, but the total error is strongly noiseandresidualforegroundcontaminationaftertheCMB
dominated by foregrounds even with a Galactic mask cov- has been extracted from the data by means of a suitable
ering 50% of the sky (b 6 30◦). To verify wether this re- linear combination of the frequency maps. The computa-
| |
sult depends critically on the mask, we also combined the tion of foreground residuals explicitly includes errors in the
b 630◦ mask with the WMAP 7yrs polarization mask, fi- modelingofthedataandintheestimation ofthefrequency
| |
6 Bonaldi A. and Ricciardi S.
Figure 1. Forecast for B-modes detection by Planck (upper panels) and COrE (lower panel) for two choices of the frequency range
usedforthepolarizedCMBreconstruction:50<ν<200GHz(left-handpanels)and40<ν<300GHz(right-handpanels).Blacksolid
lines: fiducial models for tensor to scalar ratios, r, of 0.1, 0.03 0.01 and 0.001. Dotted lines: cosmic variance limits (cyan for 50% sky
coverage, grey for 85%). Coloured solid lines: total error for MV (red) and CS with conservative (blue), intermediate (green) and goal
(magenta)errorsinthemixingmatrixestimations(seeTable2).Thosetotalerrorsarethesumofthenoise(redtrianglesforMV,grey
trianglesforCS)andforeground(colouredsquares)errors.
scalings of the components. The forecast is quick and flexi- the linear combination leads to a higher noise level in the
ble, and allows the selection of different sets of frequencies reconstructed CMB map, because themixing matrix is less
and of different methods for the CMB reconstruction. In well-conditioned.Wefindthatifthetotalerrorisdominated
thispaperweconsideredtheminimumnoisevariance(MV) bynoise, like in thecase of Planck, it is convenient to use
and GLS component separation (CS) methods, but other a rather broad frequency range (40 < ν < 300 GHz) while
methods can be easily implemented. in the case of more sensitive experiments, like the planned
We haveinvestigated several issues: COrE mission, it is preferable to restrict ourselves to the
“cleaner” frequency range 50 < ν < 200 GHz. We stress,
How do the performances of minimum noise variance however, that this applies to the CMB reconstruction step.
•
methods compare with methods exploiting component sep- Forthepreliminary“mixingmatrixestimation”stepamuch
aration? We have shown that, although component separa- broader frequency coverage is essential.
tion leads toa noise amplification byan amount ultimately
dependingon theconditioning of the mixing matrix, in the
lowmultipoleregimeconsideredinthispaperwecannotdis- How critical is a better understandingof polarized dif-
•
pensefromthecomponentseparationstep,e.g.byresorting fuseforegroundsformeasurementsofthepowerspectrumof
to an aggressive masking of foreground contaminated re- primordialB-modepolarization?Wefindthatinthecaseof
gions, because theforeground contamination is moresevere Planckthemainlimitation comesfrom thedetectornoise.
than that of noise and the mask cannot be too extended Weestimatethat withfourall-sky surveysPlanckcan de-
becauselargeareasareessentialtorestraincosmicvariance. tect the re-ionization bump of the B-mode power spectrum
Which is the optimal frequency range for the recon- for values of the tensor to scalar ratio r down to < 0.1, in
•
struction of CMB polarization maps? Using a wide fre- agreementwiththeearlierconclusionbyEfstathiou&Grat-
quencyrangebringsinchannelsheavilyforegroundcontam- ton(2009).Ontheotherhand,tofullyexploitthesensitivity
inated and, since the foreground spectra are not perfectly of planned experiments specifically aimed at measuring the
known, increases the foreground residuals in the final map. primordial B-mode anisotropies, it is essential to substan-
On the other hand, using a too limited set of channels in tially improve the determination of spectral properties of
Forecast B-modes detection at large scales 7
polarized foregrounds. Planck maps will allow an impor-
tant step forward in this direction.
As soon as more accurate multi-frequency maps of po-
larized foreground emissions are available, our method will
allow us to make more quantitative predictions that may
helpthedesignofnextgeneration CMBpolarization exper-
iments and theoptimization of thedata analysis strategy.
7 ACKNOWLEDGEMENTS
Work supported by ASI through ASI/INAF Agreement
I/072/09/0 for the Planck LFI Activityof Phase E2.
ThisresearchusedresourcesoftheNationalEnergyRe-
search Scientific Computing Center, which is supported by
theOfficeofScienceoftheU.S.DepartmentofEnergyunder
Contract No. DE-AC02-05CH11231.
We thank C. Baccigalupi, P. Natoli and G. Polenta for
theircontributionstoourwork andthePlanck andCOrE
teams for useful discussions. We acknowledge the anony-
mous referee for useful comments and suggestions leading
to improvementsin thepaper.
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