- #1
ChristianS
- 2
- 0
My general question is:
What is the angular power spectrum C_{l,N,ω} of N weighted (weight ω_i for event i) events from a full sky map with distribution C_l?
I'm interested in:
Due to energy dependent detector effects it is often important to weight each event i by the observed Energy ω_i(E). Maybe this problem is solved for the CMB-Powerspectrum, but I couldn't find anything :(.
For simplification I would like to start with the special case of a pure isotropic sky map. If we neglect the weights, we know from Poisson noise/shot noise (we observe N events at random positions):
What is the angular power spectrum C_{l,N,ω} of N weighted (weight ω_i for event i) events from a full sky map with distribution C_l?
I'm interested in:
- Mean of C_{l,N,ω}: <C_{l,N,ω}>
- Variance of C_{l,N,ω}: Var(C_{l,N,ω})
Due to energy dependent detector effects it is often important to weight each event i by the observed Energy ω_i(E). Maybe this problem is solved for the CMB-Powerspectrum, but I couldn't find anything :(.
For simplification I would like to start with the special case of a pure isotropic sky map. If we neglect the weights, we know from Poisson noise/shot noise (we observe N events at random positions):
- <C_{l,N}>=4π/N
- Var(C_{l,N})= (2/(2l+1)) (4π/N)^2