# Bias on r from Band Center Errors

Steve Palladino, Justin Willmert, Colin Bischoff -- 2017-09-13

In this posting, I looked at the systematic bias on r from band center errors using the data challenge 02.00 which has nine observing frequencies. The simulated maps were processed through to BB bandpowers by Justin. First, I applied a quadratic estimator for r to these bandpowers with 0% band center error. I then held each of the band center errors at 0% except for allowing one observing frequency band center to vary from -3% to 3% (Figure 1) while subtracting off estimate of r from the 0% case (let’s call this r0). As can be seen, the r bias appears to be quadratic and asymmetric about the 0% band center error. It is also interesting to note that the largest biases on r come from errors in the 20 GHz and 270 GHz band centers.

Figure 1

I next looked at how the r bias changed with allowing all the band center errors to vary. I looked at two cases of band center error (uncorrelated and correlated). Again, I constructed a quadratic estimator for r using the case with 0% band center error which gave an estimate for r (r0) and applied this estimator to the bandpowers. For the uncorrelated case, I used an uncorrelated nine-dimensional Gaussian distribution centered about 0 with a σ of 1% for the band center errors with 10,000 realizations. From these sets of 9 Gaussian random band centers, I calculated the bandpower expectation values, applied the quadratic estimator, subtracted off r0 and took the magnitude. This gave a distribution, Figure 2, for the r bias from which I calculated <r> and <|r|>.

Figure 2

Lensing Mean r Std r Mean mag(r) Std mag(r)
AL=1.0 1.49e-4 2.05e-4 1.81e-4 1.77e-4
AL=0.3 1.35e-4 1.61e-4 1.48e-4 1.50e-4
AL=0.1 1.18e-4 1.34e-4 1.24e-4 1.28e-4
AL=0.03 1.05e-4 1.19e-4 1.10e-4 1.14e-4

I repeated this process using a band center σ of 0.5%, 1.5%, and 2%. The mean of the magnitude of these distributions are shown in Figure 3 below. Also seen in Figure 3 is a quadratic fit to the band center errors.

Figure 3

Band Center Error Mean (AL=1.0) Mean (AL=0.3) Mean (AL=0.1) Mean (AL=0.03)
0.5% 0.64e-4 0.45e-4 0.36e-4 0.31e-4
1% 1.81e-4 1.48e-4 1.24e-4 1.10e-4
1.5% 3.70e-4 3.18e-4 2.73e-4 2.41e-4
2% 6.35e-4 5.58e-4 4.82e-4 4.27e-4
Benchmark ( r = 1e-4 ) 0.70% 0.81% 0.89% 0.95%

For the correlated band center error case, I varied the band center errors for all of nine observing frequencies simultaneously and by the same amount. From here, it was a similar procedure of calculating the bias on r; I calculated the bandpower expectation values, applied the quadratic estimator, subtracted off r0 and took the magnitude. The <|r|> bias can be seen in Figure 4 below with a quadratic fit.

Figure 4

Band Center Error Mean (AL=1.0) Mean (AL=0.3) Mean (AL=0.1) Mean (AL=0.03)
0.5% 0.15e-4 0.11e-4 0.09e-4 0.08e-4
1% 0.44e-4 0.35e-4 0.29e-4 0.25e-4
1.5% 0.87e-4 0.72e-4 0.61e-4 0.54e-4
2% 1.46e-4 1.23e-4 1.05e-4 0.94e-4
Benchmark ( r = 1e-4 ) -2.61%, 1.55% -2.58%, 1.72% -2.65%, 1.89% -2.74%, 2.03%

The bias on r for the correlated case also seems to scale as a quadratic and is asymmetric about the 0% band center error. However, these biases are less significant when compared to the uncorrelated case.