3.2 Power Spectrum Estimation Using Scanning Rings
3.2.2 An Example
Figure 3.5: The noise power spectrum used in the simulation. The vertical line shows the knee frequency. The frequency k is shown in units of the Nyquist frequency NT OD/2.
To test the above method for power spectrum estimation I simulated an ex- ample whereNr=NR = 243 meaning that there are 243 scanning rings with 243
pixels in each. Each ring is only scanned once Nc = 1 so that the total number
of pixels NT OD = 59049, a similar number of pixels to the BOOMERANG ex-
periment. The opening angle θE = 5◦ and the radius of each ring θ = 5◦. All
the rings intersect at C (figure (3.1)). For simplicity a symmetric Gaussian beam with 180 FWHM was used. This is however not necessary as the method can deal
with any beam pattern. The noise power spectrum was of the form (2.8) with α= 1. WritingP(k) in terms of wavenumber k one has
P(k) =σ2 1 + kknee k , (3.46)
withσ = 9µK and kknee= 4.3×10−3 in units of the Nyquist frequency NT OD/2.
3.2 Power Spectrum Estimation Using Scanning Rings 87
to noise ratio was one at a multipole of about`= 600. The simulation was carried out with the following steps:
1. First a standard CDM power spectrum was selected and a timestream was simulated using the Wandelt-G´orski method described in section (3.1). A set of a`ms were created using a random number generator and the CDM
power spectrum up to a multipole L = 1024. Then with equation (3.12) the Fourier transformed time stream was created and finally equation (3.14) (FFT) was used to create the time stream in pixel space. The ringset is shown in figure (3.6).
Figure 3.6: The ringset from a scan on a simulated sky without noise. On the plot a line from the bottom to the top describes one ring. In this way the rings are put next to each other from the left to the right. The central pointC where all rings intersect corresponds to the horizontal line of constant value in the middle of the plot.
3.2 Power Spectrum Estimation Using Scanning Rings 88
In the figure, each ring goes from the bottom to the top along theφ axis. The rings are put next to each other from the left to the right along theφE
axis. At φ = π each ring intersects the central point C and therefore this value is the same for all the rings giving the horizontal line of constant value in the middle. The lines of constant value going from upper left to lower right are the other points of intersection between rings which are close to each other.
2. Noise was generated using the power spectrum above. In Fourier space Gaussian random numbers were created with the power spectrum P(k). The noisestream was then Fourier transformed to pixel space (with FFT). The noise ringset is shown in figure (3.7).
Figure 3.7: Same as figure (3.6) for the noise.
3.2 Power Spectrum Estimation Using Scanning Rings 89
experiment scanning on rings. The ringset is shown in figure (3.8). This is the input data from which the power spectrum was to be estimated. From the figure the vertical striping caused by the noise which was discussed in section (2.2.1) is clearly visible.
Figure 3.8: Same as figure (3.6) for signal plus noise. This is the result of adding the ringsets in figure (3.6) and (3.7)
4. Now the simulated data set was used for likelihood estimation. A conju- gate gradient solver using only first derivative to find the minimum of the log-likelihood was used (Press, Teukolsky, Vetterling, and Flannery 1992). Since the solver can easier handle a solution in which all the parameters have roughly the same order of magnitude, the following change of variables was made: C` = Dbe−σ 2 b`(`+1) `(`+ 1) , (3.47)
3.2 Power Spectrum Estimation Using Scanning Rings 90
where Db is the binned power spectrum (b is bin number) for which the
log-likelihood was minimized. The Gaussian beam has σ = σb. For this
minimization 20 bins with 50 multipoles in each were used. The starting guess for the minimizer wasDb = 104(µK)2 for all b= [0,19]. For this test
the correlation matrix was calculated in pixel space (equation 3.22) and then Fourier transformed using FFT. With the binning, equation (3.22) takes the form
Cpr,pS 0r0 = X b Db X `b e−σ2 b`(`+1) `(`+ 1) 2`+ 1 4π P`(cosδθ), (3.48) and its derivative is
∂CS pr,p0r0 ∂Db =X `b e−σ2b`(`+1) `(`+ 1) 2`+ 1 4π P`(cosδθ), (3.49) where the sum over `b is the sum over all multipoles ` in the bin b. The anglesδθbetween all possible pairs of pointsprandp0r0 were precalculated.
5. After about 30 likelihood and derivative evaluations taking about 2 hours each on a single processor on a 500MHz DEC Alpha Work Station the minimum was found. If the initial guess was better (it could have been taken from a faster approximate power spectrum estimation algorithm as discussed in section (2.3)), the number of likelihood evaluations could have been significantly reduced. The result is shown in figure (3.9). The solid line is the input average power spectrum and the crosses are the estimated bins. The crosses are plotted in the middle of each bin.
3.2 Power Spectrum Estimation Using Scanning Rings 91
Figure 3.9: The figure shows the result of the likelihood maximisation using the Fourier transformed ringset as input data. The solid line shows the input average power spectrum and the crosses show the estimated bins with error bars. The error bars are approximate asymmetric 2σ confidence regions showing where the log-likelihood drops by 2 compared to the maximum.
6. Finally the error bars were found. First the 2σ (95%) confidence regions were found by finding where the log-likelihood had been reduces by 2 com- pared to the maximum. These asymmetric error bars are plotted on the estimates in figure (3.9). Then these error bars were compared to the 2σ error bars found from approximating the Fisher matrix by taking the numer- ical second derivative of the log-likelihood. These error bars were consistent with the ones found from the likelihood contours.