A perturbative analysis of synchrotron spectral index variation over the microwave sky

In this paper, we implement a perturbative approach, first proposed by Bouchet and Gispert, to estimate the variation of the spectral index of galactic polarized synchrotron emission, using a linear combination of simulated Stokes Q polarization maps of selected frequency bands from WMAP and Planck observations on a region of sky dominated by the synchrotron Stokes Q signal. We find that a first order perturbative analysis recovers the input spectral index map well. Along with the spectral index variation map, our method provides a fixed reference index, ${\hat{\beta }}_{0s}$, over the sky portion being analyzed. Using Monte-Carlo simulations, we find that $⟨{\hat{\beta }}_{0s}⟩=-2.84\pm 0.01$, which matches very closely with the position of a peak at ${\beta }_s\left(p\right)=-2.85$ of the empirical probability density function of input synchrotron indices obtained from the same sky region. For thermal dust, the mean recovered spectral index $⟨{\hat{\beta }}_d⟩=2.00\pm 0.004$ from simulations, matches very well with the spatially fixed input thermal dust spectral index ${\beta }_d=2.00$. As accompanying results of the method, we reconstruct cosmic microwave background, thermal dust, and a synchrotron template components with fixed spectral indices over the entire sky region. We use, in our analysis, full pixel–pixel noise covariance matrices of all frequency bands, estimated from the sky region being analyzed. The perturbative technique of this work (1) can build a model with an arbitrary but sufficient degree of accuracy (and precession) as allowed by the data and (2) can produce maximum likelihood estimators for reference indices and templates asymptotically.