A Perturbative Analysis of Synchrotron Spectral Index Variation over the Microwave Sky

Abstract

In this paper, we implement a perturbative approach, first proposed by Bouchet & 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{β }}_0s, over the sky portion being analyzed. Using Monte-Carlo simulations, we find that < {\hat{β }}_0s> =-2.84+/- 0.01, which matches very closely with the position of a peak at {β }_s(p)=-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{β }}_d> =2.00+/- 0.004 from simulations, matches very well with the spatially fixed input thermal dust spectral index {β }_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.