The Cosmic Background Radiation — Question 9

Question 9

Two frequency channels can separate a simple foreground from a common signal

In thermodynamic-temperature units assume $d_1=s+f$ and $d_2=s+2f$, where s is the same CMB fluctuation in both bands. Measured values are 80 and 110 microkelvin. Each channel has independent noise standard deviation 6 microkelvin.

1. Solve for s and f.

2. Derive the uncertainty in s and compare it with either single-channel noise.

Question 9 - Solution

Two frequency channels can separate a simple foreground from a common signal

(a) Method and calculation

$$f=d_2-d_1\Rightarrow\boxed{f=30\ \mu{\mathrm{K}}}.$$$$s=2d_1-d_2\Rightarrow\boxed{s=50\ \mu{\mathrm{K}}}.$$

(b) Interpretation and further calculation

$$\sigma_s^2=4\sigma_1^2+\sigma_2^2=5(6)^2.$$$$\boxed{\sigma_s=6\sqrt5=13.4164\ \mu{\mathrm{K}}}.$$

Physical check

Foreground removal increases noise in this simple combination. Incorrect frequency scaling or correlated calibration errors would add further uncertainty.