Figure 4.3: Input maps for several Ninkasi runs in Stokes I with clean maps on the left, signal added maps in the middle and the difference between them on the right. The temperature scale is in Kelvin.
The aim of our investigation was to determine the degree to which the final Ninkasi map estimate depends on the input guess. This was done by investigating the difference between the resulting sky maps of the two series of Ninkasi runs after each run. Merely investigating the differences between the maps provides no information on the angular scale of potential discrep- ancies. This is particularly important, for example, when considering the relevance of our maps
to foreground removal inB-mode searches. Thus an investigation of the angular power spectrum is required. We took the auto correlated spectrum of the residual signal in the differenced maps and compared this to the auto spectrum of the injected artificial signal. If the auto spectrum of the residual was of similar order to that of the artificial signal, then the injected artificial signal (or at least a large part of it) still remained, and the maps had not converged. Conversely, if the auto spectrum of the residual was small relative to that of the artificial signal, then the injected artifi- cial signal (or at least most of it) had been removed and the maps had converged. This was tested further by taking the cross correlation of the residual signal and the injected artificial signal. A high cross correlation between these two signals would also indicate that the injected artificial signal was still influencing the final Ninkasi map, resulting in an unconverged map, while a low cross correlation would indicate a converged map.
To investigate this mathematically, we look at the fractional error in the power spectrum that comes from lack of convergence, or the degree to which the input guess leaks into the final calculated result. Let us say thatqrepresents the injected artificial signal map, andprepresents the residual map (which can be thought of as the sum of systematic and statistical errors). The fractional power spectrum error is then given by
<(p+q)2>−<q2>
<q2> = <(p+q)2>
<q2> −1. (4.9)
If we expand the terms, then the fractional power spectrum error is
<p2+ 2qp+q2>
<q2> −1 = <p2+ 2qp>
<q2> , (4.10)
which reduces to
<2qp>
<q2> (4.11)
in the limit thatpis much smaller thanq. This expression provides us with a reasonable estima- tion of the fractional error in the power spectrum arising from map making systematics. We can also investigate the worst case scenario for the fractional power spectrum error by replacing the cross spectrum<qp>with an upper bound from the triangle of inequality
|<qp>| ≤p
<q2><p2>. (4.12)
4.1. MAP MAKING 79 The right hand side of this equation relies only on the auto spectra<q2>and<p2>, and so the worst case estimate of the fractional power spectrum error is then
2p
<q2><p2>
<q2> (4.13)
which simplifies to
2 s
<p2>
<q2>. (4.14)
Thus, we have two estimates of the fractional error in the power spectrum that result from map making systematic errors: the cross spectrum expression (Equation 4.11), which represents our best guess, and the auto spectrum expression (Equation 4.14), which represents the estimate for the worst case scenario. Note that in the results presented in this chapter, I have ignored the factor of 2 present in Equations 4.11 and 4.14.
The maps in Figure 4.3 show clear evidence of a trend towards convergence, with a large residual signal in the first runs becoming very small in later runs. This trend is confirmed when looking at the binned power spectra of the residuals as shown in Figure 4.4 (a), which shows the auto spectra of the residuals as well as that of the injected artificial signal. The spectra have been grouped into`= 30bins to smooth the plots and improve clarity. The requirement that the residuals are small relative to the added signal is confirmed in Figure 4.4 (b) which shows the square root of the ratio between the auto spectra of the residuals and the auto spectrum of the artificial signal, as shown in Equation 4.14. These plots show that while the residuals of the first runs were large, the residuals of later runs are only a few percent of the artificial signal, with the 12th run showing residuals of about 6%of the artificial signal. It is important to check that this residual signal is not a result of the injected artificial signal. This is confirmed by looking at the cross correlation between the residual and the artificial signal.
(a)Auto spectra of the residuals after selected Ninkasi runs and the auto spectrum of the injected artificial signal in Stokes I.
(b)Square root of the ratio of the auto spectra of the residuals after selected Ninkasi runs, and the auto spectrum of the injected artificial signal in Stokes I.
Figure 4.4: Auto spectra of the residuals after selected Ninkasi runs and the injected artificial signal (denoted as ‘noise’) in Stokes I, binned by` = 30.
The cross correlation between the injected artificial signal and the residual decreased with each successive run as the maps began to converge, ending with a cross correlation of order 2% by the 12th run. This is low enough when compared to other instrumental systematic errors that we can conclude that the intensity maps converged sufficiently within twelve Ninkasi runs.
The cross correlation is shown in Figure 4.5 (a), while Figure 4.5 (b) shows the ratio between the cross correlated signal and the auto spectrum of the injected artificial signal, as shown in Equation 4.11.
(a) Cross correlation between the residuals and the injected artificial signal.
(b)The ratio between the cross correlated signal and the auto spectrum of the injected artificial signal.
Figure 4.5: Cross correlation between the residuals and the injected artificial signal in Stokes I, binned by`= 30.
4.1. MAP MAKING 81