A Perturbative Analysis of Synchrotron Spectral Index Variation over Microwave Sky

In this paper, we implement a perturbative approach, first proposed by Bouchet & Gispert (1999), to estimate variation of spectral index of galactic polarized synchrotron emission, using 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 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, = -2.84 \pm 0.01, which matches very closely with position of a peak at \beta_s(p) = -2.85, of empirical probability density function of input synchrotron indices, obtained from the same sky region. For thermal dust, mean recovered spectral index, = 2.00 \pm 0.004, from simulations, matches very well with spatially fixed input thermal dust spectral index \beta_d = 2.00. As accompanying results of the method we also reconstruct CMB, thermal dust and a synchrotron template component with fixed spectral indices over the {\it entire} sky region. We use full pixel-pixel noise covariance matrices of all frequency bands, estimated from the sky region being analyzed, to obtain reference spectral indices for synchrotron and thermal dust, spectral index variation map, CMB map, thermal dust and synchrotron template components. The perturbative technique as implemented in this work has the interesting property that it can build a model to describe the data with an arbitrary but enough degree of accuracy (and precession) as allowed by the data. We argue that, our method of reference spectral index determination, CMB map, thermal dust and synchrotron template component reconstruction is a maximum likelihood method.