CMB component separation by parameter estimation

Abstract

We propose a solution to the CMB component separation problem based on standard parameter estimation techniques. We assume a parametric spectral model for each signal component, and fit the corresponding parameters pixel by pixel in a two-stage process. First we fit for the full parameter set (e.g., component amplitudes and spectral indices) in low-resolution and high signal-to-noise ratio maps using MCMC, obtaining both best-fit values for each parameter, and the associated uncertainty. The goodness-of-fit is evaluated by a chi^2 statistic. Then we fix all non-linear parameters at their low-resolution best-fit values, and solve analytically for high-resolution component amplitude maps. This likelihood approach has many advantages: The fitted model may be chosen freely, and the method is therefore completely general; all assumptions are transparent; no restrictions on spatial variations of foreground properties are imposed; the results may be rigorously monitored by goodness-of-fit tests; and, most importantly, we obtain reliable error estimates on all estimated quantities. We apply the method to simulated Planck and six-year WMAP data based on realistic models, and show that separation at the muK level is indeed possible in these cases. We also outline how the foreground uncertainties may be rigorously propagated through to the CMB power spectrum and cosmological parameters using a Gibbs sampling technique.

Figures (10)

• Figure 1: Component separation using multi-frequency measurements (linear units in left panel, logarithmic units in right panel). Most signal components has a well-defined frequency spectrum that may be parametrized by one or a few parameters, and component separation may therefore be viewed as a standard parameter estimation problem. The example shown here is based on one single pixel in a simulated data set corresponding to the six-year WMAP and the Planck experiments, as discussed in § \ref{['sec:planck']}. The error bars on the data points are multiplied by a factor of 50 in order to make them visible on this scale. (Due to modeling errors, this particular fit has a $\chi^2$ of 44, and with five degrees of freedom, it is rejected at the 99.9999% confidence level.)
• Figure 2: The "high-resolution" simulations used in this paper. Shown are the 23 GHz channel from the WMAP experiment, the 70 GHz channel from the LFI experiment, and the 217 GHz channel from the HFI experiment. All maps are smoothed to a common resolution of $1^\circ$ FWHM.
• Figure 3: Marginalized parameter probability distributions for an arbitrarily chosen pixel inside the Galactic plane, generated by MCMC as described in the text. The vertical lines show the true input value for the pixel. (The true value is not well defined for the dust spectral index, since dust is modeled by a two-component spectrum, while a one-component model is fitted. However, the stronger of the two dust components has an index parameter of $\alpha_1 = 1.67$.)
• Figure 4: Marginalized two-dimensional probability distributions for the same pixel as in Figure \ref{['fig:marg_1d_dist']}, computed by MCMC. Boxes indicate the true input values, and the contours mark the peak and the 68, 95 and 99.7% confidence levels.
• Figure 5: Low-resolution parameter maps reconstructed by MCMC. The left column shows the parameter estimates, the middle column shows the difference between reconstructed and input maps, and the right column shows the rms errors estimated by MCMC. From top to bottom we show: 1) the thermodynamic CMB temperature; 2) the synchrotron emission amplitude relative to 23 GHz; 3) the synchrotron spectral index; 4) the free-free emission amplitude relative to 33 GHz; 5) the thermal dust emission amplitude relative to 90 GHz; and 6) the thermal dust spectral index.