Analysing the polarisation of the CMB with spin scale-discretised wavelets

0.84 MB·English

by  Boris Leistedt

Analysing the polarisation of the CMB
with spin scale-discretised wavelets
Boris Leistedt∗, Jason D. McEwen†, Martin Bu¨ttner∗, Hiranya V. Peiris∗, Pierre Vandergheynst‡, and Yves Wiaux§.
∗
Department of Physics and Astronomy, University College London (UCL), London WC1E 6BT, UK
†
Mullard Space Science Laboratory (MSSL), University College London (UCL), Surrey RH5 6NT, UK
‡
Institute of Electrical Engineering, Ecole Polytechnique Fe´de´rale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
§
Institute of Sensors, Signals & Systems, Heriot-Watt University, Edinburgh EH14 4AS, UK
Abstract—Wediscussanewscale-discretiseddirectionalwavelettrans- II. APPLICATIONTOTHECMB
form to analyse spin signals defined on the sphere, in particular the
The polarisation of the CMB is currently an intense avenue of
polarisationofthecosmicmicrowavebackground(CMB).
research, since it may reveal signatures of primordial gravitational
5 I. SPINSCALE-DISCRETISEDWAVELETSONTHESPHERE waves and a glimpse on the initial conditions of the universe.
1 We design a directional scale-discretised wavelet transform to However,thephysicalquantitiesnecessaryfortheseinvestigations–
0 analyse the directional features of signals of arbitrary spin on the namely the E- and B-modes of the polarisation, or gradient and curl
2 sphere. Following Refs. [1]–[4], the spin-discretised wavelets Ψj modes – are not directly available. They are obtained by reducing
s
n on the sphere are defined in harmonic space as Ψj ≡ κj((cid:96))ξ , and transforming frame-dependent observables: the Q and U “local”
s (cid:96)m (cid:96)m
a whereκj((cid:96))characterisestheangularlocalisationofthewavelets(for polarisation measured on the sky. This Q-U to E-B transformation
J the jth scale), while ξ controls their directionality. Fig. 1 shows is unambiguous when dealing with data covering the entire sky.
(cid:96)m
0 examples of wavelets obtained with this construction. However,whenCMBobservationscoverfractionsoftheskyonly,E-
3 B reconstruction is imperfect near the boundaries of the observation
mask, causing leakage and potential biases in the recovered E- and

] B-modemaps.Thisissueistraditionallyaddressedbysmoothingand
M
extendingthemasktoremoveboundaryregionswheretheleakageis
I important.Wedevelopamethodexploitingthenoveldirectionalspin
.
h wavelet transform to decrease the leakage near the mask boundaries
p and obtain a more accurate E-B reconstruction.
- First,wecomputethewaveletcoefficientsoftheobservableQ+iU
o
(a)Spin-0scale-discretised(directional)wavelets usingourspinwavelettransform.Second,wedealwiththepartialsky
r
t coveragebymaskingthedatainspinwaveletspace.Third,weapply
s
a totherealandimaginarypartsofthesewaveletcoefficientsascalar
[ inverse wavelet transform. It can be shown that this yields estimates

oftheEandBsignalsprovidedthewaveletsofthisinversetransform
1
arespin-loweredversionsofthoseusedinthefirsttransformRef.[4].
v
This new method to estimate the E- and B-mode contributions from
0
2 Q and U observations can significantly reduce the E-B leakage by
1 exploiting improved masking in spin wavelet space.
3 We anticipate our novel spin wavelet transform will be of general
0 use in analysing CMB polarisation data, beyond this first applica-
. tion to E-B separation. A more detailed description of spin scale-
2
0 discretised wavelets, fast algorithms, and a rigorous evaluation of
5 their performance will be given in a series of forthcoming articles.
1
REFERENCES
: (b)Spin-2scale-discretised(directional)wavelets
v
[1] Y. Wiaux, J. D. McEwen, P. Vandergheynst, and O. Blanc, “Exact
i
X Fig.1. Spin0and2waveletsconstructedwithangularband-limitL=512, reconstructionwithdirectionalwaveletsonthesphere,”MNRAS,vol.388,
azimuthalband-limitN =15,andtilingparametersB=2andJmin=2. no.2,pp.770–788,2008.
r [2] B.Leistedt,J.D.McEwen,P.Vandergheynst,andY.Wiaux,“S2LET:A
a
codetoperformfastwaveletanalysisonthesphere,”A&A,vol.558,no.
The wavelet transform of a spin signal f on the sphere is
s A128,pp.1–9,2013.
given by the directional convolution with the wavelets. Provided the [3] J.D.McEwen,P.Vandergheynst,andY.Wiaux,“Onthecomputationof
wavelets satisfy an admissibility property, the original signal can be directional scale-discretized wavelet transforms on the sphere,” in SPIE
synthesised exactly from its wavelet coefficients. More details about WaveletsandSparsityXV,2013.
[4] J. D. McEwen, M. Bu¨ttner, B. Leistedt, H. V. Peiris, P. Vandergheynst,
this transform can be found in Ref. [4].
andY.Wiaux,“Onspinscale-discretisedwaveletsonthesphereforthe
analysisofCMBpolarisation,”inProceedingsIAUSymposium306,2014,
pp.119–126.