The Cosmic Background Radiation — Question 10

Question 10

Cosmic variance limits precision even with a noiseless sky map

For a Gaussian statistically isotropic full sky, use $\sigma(C_\ell)/C_\ell=\sqrt{2/(2\ell+1)}$. For observed sky fraction f, approximate the uncertainty by an additional factor $f^{-1/2}$.

1. Calculate full-sky fractional uncertainty for ell=2 and ell=200.

2. Calculate the ell=200 uncertainty for f=.25 and explain why longer observing time cannot remove this term.

Question 10 - Solution

Cosmic variance limits precision even with a noiseless sky map

(a) Method and calculation

$$\boxed{[\sigma/C]_{\ell=2}=\sqrt{2/5}=.632456}.$$$$\boxed{[\sigma/C]_{\ell=200}=\sqrt{2/401}=.0706225}.$$

(b) Interpretation and further calculation

$$\boxed{[\sigma/C]_{\ell=200,f=.25}=.0706225/\sqrt{.25}=.141245}.$$$$N_{\mathrm{modes}}\simeq(2\ell+1)f,\quad\boxed{\sigma/C\simeq\sqrt{2/N_{\mathrm{modes}}}}.$$

Physical check

Cosmic variance reflects a finite set of sky modes. The sky-fraction approximation ignores mode coupling and mask details.