Bayesian CMB foreground separation
with a correlated log-normal model
N. Oppermann & T.A. Enßlin
Separation problem
for diffuse foregrounds
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Prior model
1) Log-normal prior for foreground components:
(α)
Several physical components: s
( ν)
Observations at different frequencies: d
( ν , α)
Components have different frequency spectra: R
(ν)
Observational noise: n
P (log(s)) Gaussian
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d =R s+n
s always positive
Fluctuations varying over orders of magnitude (Galactic
plane and halo)
Bayesian reconstruction of all signal components:
P (d∣s) P (s)
P (s∣d )=
P (d )
2) Allow for correlations, both within one component and
across components
State-of-the art: Use Gaussian prior for CMB, flat prior else.
3) Gaussian CMB prior
〈s
(α) (β)
x
y ( s)
s 〉 ≠0
Test cases
log-normal components:
reality
frequency spectra:
dust-like
free-free-like
synchrotron-like
frequency
CMB-like
max.-likel.
reconstr.
Bayesian
reconstr.
data
reality
based on
Planck Coll. 2013 XII
max.-likel. Bayesian
reconstr. reconstr.
5 frequency
bands
data
4 components:
dust-like
CMB-like
CMB-like synchrotron-like free-free-like
realistic components:
synchrotron-like
based on
Haslam et al., 1982
free-free-like
based on
WMAP 2009
dust-like
based on
Planck Coll. 2013 XII
Bottom line:
The isotropic correlated
log-normal model leads
to more accurate foreground reconstructions
than a model with flat
priors, even if the model
assumptions are only
approximately fulfilled.