Joint Bayesian component separation and model estimation

The main goal of Commander is to perform joint Bayesian component separation and model estimation of the CMB data (see http://arxiv.org/abs/0709.1058). Here we describe everything that can go into a Commander analysis of CMB data; of course, one is free to sample only a subset of the parameters in a given run.

Input

Data

The input is a number of maps, indexed by \(i\), each containing the following information:

Temperature and polarization maps \(\mathbf{d}_{i}\):
In the HEALPix pixelization.
Noise properties \(\mathbf{N}_i\):
Currently supported: RMS maps (independent noise between pixels)
Mask \(\mathbf{m}_i\):
Internally the mask is treated as part of \(\mathbf{N}^{-1}_i\)
Beams \(\mathbf{B}_i\):
Currently supported: Symmetric beams; input the spherical harmonic transfer function. Todo: FEBeCOP beam
Frequency band information \(f(\nu)\):
Either \([\nu_\text{min}, \nu_{max}]\) or \(F_i(\nu)\)

Priors

Output

\(\xi^2\) map:
...

Instrument parameters

Monopole, dipole:
One per detector map \(i\). Choose to arbitrarily fix monopole at some frequencies.

Bandpass shift:

Gain \(g_i\):

Noise calibration factor \(\tau_i\):
Multiplicative factor with \(\tau_i\)

Foreground parameters

Dust \(A^d_p f(\nu; T, e) g_\nu\):
Priors on all parameters.

Synchrotron, \(A^s_p (\frac{\nu}{\nu_\text{ref}}^{\beta + C \log(\frac{\nu}{\nu_\text{ref})}\):

Free-free, \(A^f_p (\frac{\nu}{\nu_\text{ref}}^{\beta_f}\)
Prior: \(N(-2.15, 0.02^2)\)
CO \(A^c_p \alpha_\nu\)
Specifically, \(\alpha^{217}, \alpha^{353}\)

CMB model parameters

Currently only power spectrum estimation is supported:

CMB power spectrum \(C_\ell\), \(\sigma_\ell\) :
Standard inverse-Wishart