In this paper , we implement a perturbative approach , first proposed by ( 5 ) , 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 } _ { 0 s } , over the sky portion being analyzed .
Using Monte Carlo simulations we find that , \langle \hat { \beta } _ { 0 s } \rangle = -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 , \langle \hat { \beta } _ { d } \rangle = 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 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 .