and some unexpected effects from the sources I discovered in the process of doing the mosaic analysis.
In Section 4.5 I describe the data that went into the first-year CBI papers. Finally, in Section 4.6, I describe the final power spectrum from the first-year mosaics and cosmological results from the spectrum.
more than just a few (3-4) 8-minute scans. Fortunately, FFT’s are the magic bullet we need. This is because the distribution of the sum of two random variables is the convolution of their individual distribution functions. So, all we have to do is take the FFT of a distribution, and raise it to the power of however many samples we want to combine. The two quantities we need to understand are
*
w1
w1+P
i>1wi
2+
(4.1)
and
X wi
−2
(4.2)
For the first term, we can convolve all of the wi fori > 1 to get a new variable, sayq. Then the desired quantity is
* w1
w1+q 2+
(4.3)
So, we have reduced the problem to a two-dimensional integral, which is quite feasible computation- ally. The other term becomes even simpler—all thewi can be combined, to get a one-dimensional integral. The main subtlety is that since the FFT implicitly assumes periodic boundary conditions, the length of region of real-space to be transformed must be large enough so that only one period of the function contributes. Since eachwi is peaked around one, the convolution ofn of them will be peaked aroundn, and the real-space coverage of the distribution that gets transformed must be substantially larger thann. Once one does that, then the answers are quite good. For instance, I checked the expectation value of the first term for ν = 50 and two scans. The theoretical value is 1 + ν+11 = 1.019607843 and the value I get from the FFT integral is 1.019607855.
We expect the first-order calculations to be close for the CBI. The CBI typically has of order 50 points per scan, with both real and imaginary points used in estimating the noise, for a total of 100 degrees of freedom in the PDF. Figure 4.1 shows the correction factor calculated using FFT’s to convolve the PDF of single weights. If the correction factor required to scale the variance is expressed as 1 +d.o.f.x , then Figure 4.1 showxfor varying numbers of scans, with 100 d.o.f. per scan
100 101 102 1.5
2 2.5 3 3.5 4 4.5
Number of Scans Combined
Correction Factor
Correction to Scatter−Based Noise for 100 d.o.f.
Figure 4.1 Plot of numerical estimates of the correction factor that needs to be applied to scatter- based estimates of the variance. If multiple scans whose noises are estimated from internal scatter are combined with optimal weighting, then there is a systematic underestimate of the true variance of the final, averaged data point. The values plotted arexwhere the correction factor to be applied to the variance is of the form 1 + d.o.f.x where d.o.f. is the number of samples in the scan minus the degrees of freedom we may have removed in subtracting off means. First-order calculations predictx= 2 for 2 scans, andx= 4 for infinitely many scans. The first-order calculations can have substantial corrections to xif there are few d.o.f., but with the CBI’s typical value of 100 d.o.f., the first-order prediction is close. Note that few scans are needed to approach the limiting value of the correction. All data points going into the scans have identical variances and are Gaussian distributed.
and each individual point an identically distributed Gaussian. The first-order predictions are 2 for 2 scans and 4 for infinitely many scans. The FFT values are 1.95 for 2 scans, and 4.21 for 100 scans.
At 10 scans, the correction factor is 3.8, or about 90% of its limiting value, so the correction factor approaches its limiting value with relatively few scans. Because of roundoff issues in the FFT’s, it is difficult to push the numerical integrals to much higher accuracy or to many more scans combined.
4.1.2 Noise Correction Using Monte Carlo
We use Monte Carlo simulations of the noise to estimate the final noise correction factor. There are multiple factors that can break the assumptions in the theoretical calculations that are better
treated by Monte Carlo. Not all scans necessarily have the same number of points, due to outlier tossing or un-matched lead/trail points. Also, the noise on baselines at the same UV point from different receivers will be different, as each receiver has its own system temperature. These effects are difficult to treat theoretically but can be simulated without undue effort. I calculated the final noise correction factor from a set of 50 simulations created using the program MOCKCBI, written by Tim Pearson. MOCKCBI takes a set of visibilities and a map, then replaces the visibilities by the value they would have from the map, and adds Gaussian noise. By forcing MOCKCBI to use the undifferenced estimates of the noise rather than the scatter-based weights of the differenced data, the final combined data points will have the proper noise behavior. The data set can then be combined andχ2 calculated. To avoid confusion caused by the presence of CMB signals, the maps were simulated with the CMB set to zero. Once simulated, the data were run through the standard pipeline to combine them into scans, and then combine the scans with scatter-based weights into final UV values for each antenna pair. I then calculated theχ2 values for antenna pairs at identical UV points to estimate the final noise correction. Using the 20 hour deep field as a visibility template, the final noise correction value is 1.057±0.002. The answer has been skewed somewhat by a minor bug in our pipeline program that mis-estimated the degrees of freedom by 1, leading to an error in the noise estimate of about 1%. So, the true value of the noise correction is probably more like 1.047, which is in excellent agreement with the predicted first-order theoretical value of 1.04, and the Fourier integral value of 1.042. The difference is likely due to the fact that some scans have fewer than 100 d.o.f., which will skew the correction to a larger value. The noise correction value that should be used is in actuality probably a bit higher. The reason is that individual UV points are not independent, but rather are correlated because of the primary beam. As such, maximum likelihood is combining, with weights, several different UV points to create independent estimators of the CMB. Those independent estimators will have contributions from many more scans than a single UV point in the final data set, which will have approximately 50 nights’ worth of data at each point (since that’s how many nights went into the 20 hour deep field). It is for this reason that the result from the Fourier integral calculations that the excess noise converges to its final
value in relatively few scans is critically important. Because of that, the true final value can be only marginally different from the Monte Carlo value for individual UV points.