MASTER Of The CMB Anisotropy Power Spectrum: A Fast
MASTER of the CMB Anisotropy Power Spectrum: A Fast Methodfor Statistical Analysis of Large and Complex CMB Data SetsEric Hivon1,2 , Krzysztof M. Górski3,4 , C. Barth Netterfield5 ,Brendan P. Crill1 , Simon Prunet6 , Frode Hansen7arXiv:astro-ph/0105302 v1 17 May 200111.5Observational Cosmology, MS 59-33, Caltech, Pasadena, CA 911252IPAC, MS 100-22, Caltech, Pasadena, CA 911253European Southern Observatory, Garching bei München, Germany4Warsaw University Observatory, Warsaw, PolandDept. of Phys. and Astron., U. of Toronto, 60 St George St, Toronto, Ontario, M5S 3H8, Canada6CITA, University of Toronto, 60 St George St, Toronto, Ontario, M5S 3H8, Canada7MPA, Garching, GermanyABSTRACTWe describe a fast and accurate method for estimation of the cosmic microwave background(CMB) anisotropy angular power spectrum — Monte Carlo Apodised Spherical TransformEstimatoR. Originally devised for use in the interpretation of the Boomerang experimental data,MASTER is both a computationally efficient method suitable for use with the currently availableCMB data sets (already large in size, despite covering small fractions of the sky, and affected byinhomogeneous and correlated noise), and a very promising application for the analysis of very largefuture CMB satellite mission products.IntroductionDuring the past decade since the ground-breakingdiscovery of the cosmic microwave background radiation anisotropy by the COBE satellite (Smootet al. 1992), numerous successful measurements ofmicrowave sky structures have provided us with thedata for powerful tests of the current cosmologicalparadigm, and created an unprecedented opportunity to estimate key parameters of the candidatetheoretical models of the Universe.Recent ground-based and balloon-borne experiments with improved sky coverage, angular resolution, and noise performance (see de Bernardis et al,2000, Hanany et al, 2000, Padin et al, 2000, Jaffeet al, 2001, Lee et al, 2001, Halverson et al, 2001,Pryke et al, 2001 and references therein for someof the most recent experiments and their interpretation) have both given us a taste of what futuresatellite missions MAP1 and Planck2 should accomplish, and revealed the growing challenges that wewill have to meet in the analysis of the forthcomingCMB data sets.In the currently favoured structure formationmodel of inflation induced, Gaussian distributed,1 MicrowaveAnisotropy Probe, http://map.gsfc.nasa.gov/2 http://astro.estec.esa.nl/Planck/curvature perturbations all the statistical information contained in a CMB map can be summarisedin its angular power spectrum C . General maximum likelihood methods for extracting C from aNpix -pixel map with non uniform coverage and correlated noise (Górski 1994, Bond 1995, Tegmark& Bunn 1995, Górski 1997, Bond, Jaffe, & Knox1998, Borrill 1999b) involve computations of complexity Npix 3 , and become prohibitively CPU expensive for the Npix 104 maps produced by current experiments. With the presently anticipatedcomputer performance such methods appear totallyimpractical for application to the Npix 106 mapsexpected from the future space missions (Borrill1999a). Hence, there is a well recognised need forfaster, more economical, and accurate C extractionmethods, which should enable a correct comologicalinterpretion of the CMB anisotropy observations.In this paper we introduce and discuss a newmethod for fast estimation of the CMB anisotropyangular power spectrum from fluctuations observedon a limited area of the sky. This method is based ona direct spherical harmonic transform (SHT) of theavailable map and allows one to incorporate a description of the particular properties of a given CMBexperiment, including the survey geometry, scanning strategy, instrumental noise behaviour, and
possible non-gaussian and/or non-stationary eventswhich can occur during the data acquisition. Theestimated power spectrum is affected by the unwanted contribution of the instrumental noise andthe effects of any necessary alteration of either therecorded data stream (such as high pass filtering) orthe raw map of the observed region of the sky, whichare introduced during the data analysis. These effects are calibrated in Monte Carlo (MC) simulations of the modeled observation and analysis stageof the experiment and can then be removed, or corrected for in the estimated power spectrum. Theharmonic mode-mode coupling induced by the incomplete sky coverage is described analytically bythe SHT of the sky window and corrected for in order to obtain an unbiased estimate of the C . Hereafter we refer to this method with an acronym MASTER ( Monte Carlo Apodised Spherical TransformEstimatoR).Netterfield et al. (2001) described an applicationof this method in the extraction of the CMB angular power spectrum C , for 75 1025, fromthe sky map (analysed region comprised 1.8% ofthe sky covered with 57000 pixels of 70 size) made bycoadding four frequency channel data of the 1998/99Antarctic long duration flight of the Boomerangexperiment (Boom-LDB). The first derivation ofthe CMB anisotropy spectrum from the same data(de Bernardis et al. 2000) involved the MADCAPmethod (Borrill 1999b) applied to a smaller subset of the data (one frequency channel, 1% of thesky covered with 8000 pixels of 140 size). TheMADCAP approach is too CPU intensive for repeated applications to the new, enlarged subset ofthe Boomerang data, and, hence, the MASTERapproach was the method of choice for extractionof the high- angular power spectrum of the CMBanisotropy.Other fast methods have recently been proposedfor estimation of the angular spectrum of the CMBanistropy. Szapudi et al. (2001) advocate the useof the 2-point correlation function for extraction ofthe angular power spectrum from the CMB maps.The computational demands of this method scalequadratically, Npix 2 , with the size of data set(that may be improved to Npix log Npix ). Inthe same way as in the case of MASTER, the effects of the noise and correlations of the derivedC -s are quantified by Monte Carlo simulations (although the demonstrated applications involved onlythe case of a uniform white noise). Doré, Knox &Peel (2001b) proposed a hierarchical implementation of the usual quadratic C estimator with a computational scaling proportional to Npix 2 , that maybe reduced to Npix (with a large prefactor) at theprice of additional approximations.Experiment specific techniques have also beenproposed : Oh, Spergel, and Hinshaw (1999) described a fast power spectrum extraction techniquedesigned for usage with the MAP satellite data.Their method scales like Npix 2 with the size ofthe pixellised map, and takes advantage of uncorrelated pixel noise with approximate axisymmetricdistribution on the sky. Wandelt (2000) advocatesthe use of the set of rings as a compressed form of thePlanck data set from which to extract optimally theC -s in the presence of correlated noise. The applicability of this approach is limited by its assumptionof the the symmetry of the scanning strategy.This paper is organised as follows: In section 2we describe how a data stream of observations is reduced to a CMB fluctuation map, and how the ane is extracted fromgular pseudo power spectrum Csuch a map by SHT. In section 3 we show how an unbiased estimate of the true underlying power spece -s with the aidtrum can be recovered from the Cof the Monte Carlo simulations. The tests of themethod on simulated Boom-LDB observations aredescribed in section 4, and the application of themethod is discussed in section 5.2.From Time Ordered Data to PseudoPower SpectrumSingle dish CMB experiments produce for eachdetector a data stream, or the time ordered data(TOD), of the direction of observation and the skytemperature as measured through the instrumentalbeam. We assume that the beam is known, that itis close to isotropic in the main lobe, that the sidelobes are negligible, and and that the pointing ateach time is known to an accuracy better than thesize of the main lobe of the beam. Exceptions tothese assumptions will be addressed in section 3.6.We will also assume that all the TOD samples affected by transient events, such as cosmic ray hits,have been removed and that in order to preserve theTOD continuity the resulting gaps are filled withfake data having the same statistical properties asthe genuine observations (eg, Prunet et al, 2000,Stompor et al, 2000).2.1.From TOD to Sky MapThe data produced by each detector at a time tcan be modeled asdt Ptp p nt ,(1)where p is the sky temperature, that we assumeto be pixelised and smoothed with the instrumentbeam, Ptp is the pointing matrix, p is the pixel indexand nt is the instrumental noise.
If the TOD noise is Gaussian distributed with aknown correlation function Ntt0 hn(t)n(t0 )i, theoptimal solution for the sky map† 1 †Ppt Ntt 1mp (PptNtt 10 dt00 Pt0 p0 )(2)minimises the residual noise in the pixellised map, p mp (Lupton 1993, Wright 1996, Tegmark1997). While being completely general this procedure is impractical for very long TOD streamsbecause of the required inversion of the large matrix Ntt0 . A simplification is possible under the assumption of the TOD noise being piece-wise stationary, and its correlation matrix representable ascirculant, Ntt0 N (t t0 ). Eq. (2) can then besolved either directly (with a computational scalingof Npix 3 ), or by using iterative methods as discussed by Wright (1996), or Natoli et al. (2001) (inwhich case the computation time is dominated byFourier space convolutions of the TOD corresponding to the product Ntt 10 dt0 in Eq. 2). Iterative approaches scale like Niter Nτ log Nτ , where Nτ is thenumber of time samples, and Niter is the number ofiterations. Niter depends on the required accuracyof the final map, and it is of the order of a few tensin the case of a conjugate gradient method of linearsystem solution (Natoli et al. 2001).If the TOD noise properties are not known beforehand, however, as is generally the case, theEqs (1) and (2) can be solved iteratively together.This returns at each time step an estimate of thenoise stream, n(t), and, hence, of the noise timepower spectrum. The required computational scaling involves a somewhat larger Niter (Ferreira andJaffe 2000, Prunet et al. 2000, Stompor et al. 2000,and Doré et al. 2001a).Since the MASTER method requires repetitiveTOD simulations, processing, and map making, theiterative solution of Eq. (2) can be too time consuming for practical applications. Therefore, to avoidthe necessity to iterate, we use a suboptimal, fastmap making method involving the high pass filtering of the TOD stream, which improves the longtime scale behaviour of the noise, and reduces thestriping of the resulting map (see section 3.2). Themap solution is nowX †(3)Ppt f (t t0 )dt0 ,mp (Nobs (p)) 1tt0†where Nobs (p) PptPtp is the number of observations in the pixel p, and f denotes the high passfilter. The computational scaling is now reducedto Nτ log Nτ . Clearly, Eq. (3) is only equivalentto Eq. (2) if the TOD noise is white, i.e. Ntt0 N0 δDirac (t t0 ) (in which case the filter would be reduced to f δDirac (t), i.e. no filtering would be applied). While the application of the high pass filterreduces the long term noise correlations, it degradesthe CMB signal at low frequencies (see Fig. 1) andaffects the resulting angular power spectrum derivedfrom the filtered map solution Eq. (3). This effect isquantified and corrected for with the Monte Carlosimulations and analysis involving the filtered mapmaking technique applied to the simulated TODs ofthe pure CMB signal. This procedure will be discussed in detail later on.2.2.From Sky Map to Pseudo Power SpectrumA scalar field T (n) defined over the full sky canbe decomposed in spherical harmonic coefficientsZ a m dn T (n)Y m(n),(4)with T (n) X X̀a m Y m (n).(5) 0 m If the CMB temperature fluctuation T is assumed to be Gaussian distributed, each a m is anindependent Gaussian deviate withha m i 0,(6)ha m a 0 m0 i δ 0 δmm0 hC i,(7)andwhere hC i C th is specified by the theory ofprimordial perturbations, and parametrised accordingly, and δ is the Kronecker symbol. An unbiasedestimator of C th is given byC X̀12 a m .2 1(8)m C -s are χ2ν -distributed with the mean equal to C th ,ν 2 1 degrees of freedom (dof), and a variance2of 2C th /ν.In the case of CMB measurements the temperature fluctuations can not be measured over the fullsky, either because of ground obscuration or galactic contamination for example, and a position dependent weighting W (n) can also be applied to themeasured data, for instance to reduce the edge effects. If fsky represent the sky fraction over whichthe weighting applied is non zero, thenZ1dnW i (n)(9)fsky wi 4π 4πis the i-th moment of the arbitrary weightingscheme. The window function can also be expanded in spherical harmonics with the coefficients
w m trumR dnW (n)Y m(n), and with a power spec-W 1 X2 w m ,2 1 m(10)2 2forP which W( 0) 4πfsky w1 and 0 W( )(2 1) 4πfsky w2 .A sky temperature fluctuation map T (n) onwhich a window W (n) is applied can be decomposedin spherical harmonics coefficientsZ ã m dn T (n)W (n)Y m(n)(11)X T (p)W (p)Y m(p),(12) Ωppwhere the integral over the sky is approximated bya discrete sum over the pixels that make the map,with an individual surface area Ωp .e can be defined asThe pseudo power spectrum Ce CX̀1 ã m 2 .2 1(13)m The computation of Eq. (12) for each ( , m) upto max performed on an arbitrary pixelisationof the sphere would scale as Npix max 2 . However,if one uses an adequate lay out of the pixels to exploit the symmetries of Spherical Harmonics, suchas for example the ECP (Muciaccia et al. 1997),HEALPix (Górski et al. 1998), or Igloo (Crittendenand Turok 1998) this computation actually scaleslike Npix 1/2 max 2 . In our implementation of theMASTER method, after application of the windowfunction on the map, the program anafast from thepackage HEALPix was used to compute the pseudopower spectrum.Wandelt, Hivon & Górski (2000) showed that thee C th , N ) of the pseudomarginalised likelihood P (Ce for a given underlying theory C th and a givenCnoise covariance N can be computed analyticallyin O(Npix 1/2 max 2 ) operations, under conditions ofaxisymmetric sky window function and (non necessarily uniform) white noise. Under these assumpe C th ) can be used to perform a maximumtions P (Clikelihood fit of the cosmological parameters to theobserved data set. Hansen et al (2001) extend thise covariance maapproach by using the full pseudo Ctrix. We will now build, starting from the measurede , and under more general conditions on the noiseCproperties and shape of the observing window, anew estimator of the full sky power spectrum thatcan be compared directly to C th .3.From Pseudo Power Spectrum to FullSky Power Spectrum Estimatore rendered by theThe pseudo power spectrum Cdirect spherical harmonics transform of a partial skymap, Eq. (13), is clearly different from the full skyangular spectrum C , but their ensemble averagescan be related byXe i hCM 0 hC 0 i,(14) 0where M 0 describes the mode-mode coupling resulting from the cut sky. As described in the appendix, this kernel depends only on the geometry ofthe cut sky and can be expressed simply in terms ofthe power spectrum W of the spatial window applied to the survey (see Eq. (A31) and Eq. (A14) forthe spherical and planar geometry, respectively).The effect of the instrumental beam, experimental noise, and filtering of the TOD stream can beincluded as followsXe i,e i (15)M 0 F 0 B 20 hC 0 i hNhC 0where B is a window function describing the combined smoothing effects of the beam and finite pixele i is the average noise power spectrum, andsize, hNF is a transfer function which models the effect ofthe filtering applied to the data stream or to themaps. The determination of each of these termswill be described below.It is often assumed that the χ2ν 2 1 distribution of C -s on the full sky can be generalised tocut sky observations by scaling ν to the number ofdof effectively available. Given the large value of νthe central limit theorem is also invoked to furthersimplify this to a Gaussian of the same mean andvariance. From these successive (and excessive) simplifications we will only retain, as a rule of thumb,that the rms of C averaged over a range is approximately r 2N ( ),(16) C C 2B ( )ν withν (2 1) fskyw22,w4(17)where the factor w22 /w4 accounts for the loss ofmodes induced by the pixel weighting. We willshow in section 4 how this compares to the resultsof Monte Carlo simulations.3.1.Mode-mode Coupling KernelThe resolution in of the measured power spectrum is ultimately determined by the extent of the
Fig. 1.— Simulation of the Boom-LDB experiment and application of MASTER to extract the CMB angularpower spectrum. The oval contour on the maps shows the ellipse (distorted by projection) within which thepower spectrum is estimated (fsky 1.8% of the sky). Top left panel: A random realisation of the CMB skyfrom the theoretical model described by the power spectrum shown in the top right panel (red line). Middle leftpanel: A noiseless map of the same region of the sky made from the TOD with actual Boom-LDB pointing andprocessed with the 100 mHz high-pass filter (see text). Bottom left panel: the difference between the two CMBsky maps shown above, which shows the component of the sky signal lost due to the combination of Boomerangscanning and data processing. Middle right panel: A simulation of the same Boomerang CMB sky map withthe instrumental noise included. Bottom right panel: Integration time per pixel for the actual scanning of theBoom-LDB channel B150A; the average integration time is about 500s/Deg2. Top right panel: The input powerspectrum smoothed by the beam and pixel window function (red line); The average angular power spectrumof the instrumental noise (black line); The pseudo C -s directly measured on the sky map shown in the middleright panel and divided by fsky (orange line); The binned MASTER estimate of the full sky power spectrumafter removal of noise contribution and correction of the effect of the high pass filtering and mode-mode coupling(blue histogram).
oberved area of the sky, its geometrical shape, andthe pixel weighting W (n) applied to the survey (seeHobson & Magueijo 1996, and Tegmark 1996). Although we only tested numerically the method on acircular or elliptically shaped window, nothing prevents the use of a more complex window, speciallyfor a pixel starved experiment with a nonconvexsurvey area, for which a well designed apodisationcould help improving the achievable spectral resolution. We will show in section (4) how the choiceof window changes the estimated C spectrum.3.2.TOD Filter Transfer FunctionThe transfer function F introduced in Eq. (15)describes the effect of any filtering applied to theTOD stream or to the map on the angular powerspectrum. A specific example of the latter is theremoval of parallel stripes extending along a direction different from the scanning direction observedin some channels of the Boom-LDB data (Netterfield et al. 2001, Contaldi et al. 2001). The filteringof the TOD has broader applications, however, andcan take the form of a high pass filtering that servesseveral purposes, as follows: reduce the contribution of the low frequencynoise (1/f noise) to the map, specially if thescanning strategy and/or the map makingtechnique used do not optimize the removalof these modes, reduce the scan or spin synchronous noise,which may appear at the scan frequency andits harmonics, remove from the signal the contribution fromthe large scale anisotropies, which are poorlyconstrained on an incomplete sky survey, andare likely to contaminate the estimated powerspectrum at all the smaller scales.It should be noted that the validity of Eq. (14)fo
inhomogeneous and correlated noise), and a very promising application for the analysis of very large future CMB satellite mission products. 1. Introduction Duringthe pastdecadesincethe ground-breaking discovery of the cosmic microwave background ra-diation anisotropy by the COBE satellite
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