5.4 Window Functions 127
transform of the beam image is the set of Zernike polynomials that form an orthonormal basis on the unit disk (Born and Wolf, 1999). The Zernike polynomials, expressed in polar coordinatesρ andϕon the aperture plane, are:
Vnm(ρ, ϕ) =Rmn(ρ)eimϕ, (5.4.21) wherem and n are integers such that n ≥ 0, n > |m| andn − |m|is even. In the case of an azimuthally symmetric beam, we need only consider the m = 0radial polynomials, which can be expressed in terms of Legendre polynomials,Pn(x)as:
R2n0 (ρ) = Pn(2ρ2−1). (5.4.22) The radial Zernike polynomials have a convenient analytic form for their Fourier transform:
R˜02n(θ) = Z
ρdρ e−iρθR02n(ρ) = (−1)nJ2n+1(θ)/θ, (5.4.23) whereJnis a Bessel function of the first kind. Motivated by this, we adopt
bn(θ) = µθ
θ0
¶−1 J2n+1
µθ θ0
¶
(5.4.24) as our set of basis functions to fit the radial beam profile.4 Here, we have introduced a fitting parameter,θ0,to control the scale of the basis functions.
5.4.2 Fitting Basis Functions to the Beam Profile
Below θ1 (c.f. Table 5.1), we fit the bases bn of Eq. (5.4.24) to the measured beam profile, and beyond θ1, we use the power law defined in Eq. (5.3.14) with the parametersθw listed in Table 5.1. We assume vanishing covariance between the power law and the basis functions as they are fitted to independent sets of data points.
We employ a nonlinear, least-squares method to solve for the coefficientsanand their covari- ance matrix Cmnaa0. The algorithm uses a singular value decomposition to determine if the basis
4It may be asked why the Airy pattern, which was somewhat suitable for the high-θfit in §5.3.2, is not used here.
We find that at lowθ, it is a poor fit since the optics are more complicated than the perfect-aperture model assumed by the Airy pattern.
5.4 Window Functions 128
functions accurately characterise the data and also computes a goodness-of-fit statistic (Press et al., 1992, §15.4). As inputs to the fitting procedure we are required to specify the scale pa- rameter, θ0, and polynomial order, nmax. We searched the {θ0, nmax} parameter space until a reasonable fit was obtained that kept nmax as small as possible. For all three bands,nmax = 13 gives a reducedχ2 ≈ 1. No singular values is found for any of the fits. The parametersθ0 we use for each frequency band are listed in Table 5.1.
5.4.3 Window Functions and their Covariances
Given the amplitudesanof the radial beam profile fitted to the basis functions and the covariance matrixCmnaa0 between the amplitudesam andan, the beam Legendre transform is:
b` =
nXmax
n=0
anb` n, (5.4.25)
and the covariance matrix of the beam Legendre transformsb` andb`0 is:
Σb``0 =
nXmax
m,n=0
∂b`
∂amCmnaa0∂b`0
∂an (5.4.26)
and therefore the covariance for the window function is:
Σw``0 = 4w`w`0Σb``0. (5.4.27) In Fig. 5.5 we show the window functions for each of the three frequency bands with diagonal error bars taken from the covariance matrix,Σw``0. We observe that the window function for each of the frequency bands has fallen to less than 15% of its maximum value at ` = 10000. The statistical diagonal errors are at the1.5%, 1.5%, and6% levels for the 148 GHz, 218 GHz, and 277 GHz bands respectively, as shown in Fig. 5.5. They are computed following Eq. (17) of Page et al. (2003). The off-diagonal terms in the beam covariance matrix are comparable in magnitude to the diagonal terms. Singular value decompositions ofΣb``0 yield only a handful of modes with singular values larger than10−3 of the maximum values: 5 modes for 148 GHz, 4 for 218 GHz, and 2 for 277 GHz. Thus, the window function covariances can be expressed in a compact form which will be convenient for power spectrum analyses.
5.4 Window Functions 129
Table5.2:SelectionofSZClustersDetectedbyACT ACTDescriptorCatalogueNameJ2000Coordinatesa rmsb tintc SNR(θ)d ∆TSZe 1010 ×Y(θ)f [µK][min][µK] RADec.θ≤20 θ≤40 θ≤60 (±0.2)(±0.6)(±1.2) ACT-CLJ0245−5301AbellS029502h 45m 28s −53◦ 010 3600 4410.115.2(6.80 )−2500.892.363.91 ACT-CLJ0330−5228Abell3128(NE)03h 30m 50s −52◦ 280 3800 4910.312.8(4.30 )−2600.942.694.34 ACT-CLJ0509−5345SPT-CL0509−534205h 09m 20s −53◦ 450 0000 4710.17.7(5.20 )−700.331.071.50 ACT-CLJ0516−5432AbellS052005h 16m 31s −54◦ 320 4200 556.84.2(4.10 )−1100.19-0.11-0.55 ACT-CLJ0546−5346SPT-CL0547−534505h 46m 35s −53◦ 460 0400 469.513.9(5.80 )−2500.912.363.67 ACT-CLJ0638−5358AbellS059206h 38m 46s −53◦ 580 4000 557.58.1(3.10 )−2300.701.402.07 ACT-CLJ0645−5413Abell340406h 45m 29s −54◦ 130 5200 599.32.8(2.00 )−1200.12-0.18-0.69 ACT-CLJ0658−55561E0657−56(Bullet)06h 58m 33s −55◦ 560 4900 803.412.1(2.70 )−5101.602.953.56 aPositionofthedeepestpointin20FWHMGaussiansmoothedmap,exceptforACT-CLJ0509−5345whichhasapositionwhichgivesamaximalSNR(seetext). bMaprmsmeasuredoutsidea60maskandreportedforaonesquarearcminutearea. cIntegrationtime,definedastheapproximatetotaltime(inminutes)thatthetelescopewaspointedinthemapregion. dMaximumsignal-to-noiseratio(Eq.(5.5.29))andtheradiusθatwhichitwasobtained. eClusterdepth,asmeasuredina20FWHMGaussiansmoothedmapatthelistedcoordinates;intendedasaguidetothemagnitudeofthedecrement. fSeeEq.(5.5.32)andfollowingdiscussion.
5.4 Window Functions 130
0 0.25 0.5 0.75 1
w`
0 0.025 0.05
2000 4000 6000 8000 10000
∆w`/w`
Multipole`
148 GHz 218 GHz 277 GHz
Figure 5.5: Normalised window functions (top) and diagonal errors (bottom) computed from the basis functions for each of the three frequency bands. The window functions have been normalised to unity at ` = 0. In practice, the normalisation will take place over the range of multipoles corresponding to the best calibration. Only statistical errors are shown.
For 277 GHz we estimate a10% systematic uncertainty from destriping. Another source of systematic error in the window functions arises from the beams not being perfectly symmetric.
The symmetrised beam window function generally underestimates the power in the beam (e.g., Fig. 23, Hinshaw et al., 2007). In practice, the scans in our survey field are cross-linked (see
§5.5.1) so that the quantity of relevance to the power spectrum calculation is the cross-linked window function i.e.,
w`= 1 2π
Z 2π
0
b(`)b∗(`)dφ`, (5.4.28)
where φ` is the polar angle in spherical harmonic space andb(`)is the transform of the beam profile. To study the magnitude of this effect on the window function we computed the fractional difference between the window function derived from the Legendre transform of the symmetrised beam and the cross-linked window function, derived for a typical cross-linking angle of60◦. The difference between the two was found to be at the1%,level for the 148 GHz and 218 GHz arrays, and at the4%level for the 277 GHz array.