Gravitational waves as a source for the polarization of the cosmic microwave background

The polarization of the Cosmic Microwave Background (CMB) induced by gravitational waves (GWs) is studied by solving in a semi-analytical way the Chandrasekhar radiative transfer equation; following the Polnarev approach, the equation is written as a second-kind Volterra integral equation and its kernel is handled by performing a series expansion of the trigonometric functions it contains. In this way, a recursive calculation of the Volterra equation gets possible and the polarizing effect of the gravitational waves can be brought out. The polarization degree of the CMB coming from this analysis shows a peak for a wavenumber corresponding to GWs re-entering the horizon at the end of the recombination epoch: the position and the size of the maximum are in agreement with the results of other works, based on a totally numerical calculation. However, a difference quite relevant can be remarked when one looks at the shape of the polarization plot: a semi-analytical calculation of the solution of the Volterra integral equation gives a sharp peak due to the fact that the contribution of each packet of GWs of fixed wavenumber Ic is strongly singled out when one substitutes the integrals with series and sums. As a consequence, this solution method may have some usefulness when one wants to point out the contributions really dominating in producing a polarization for the CMB. From this analysis one can also infer that the best angular scales to test in order to detect a polarization for the CMB are theta similar to 20 degrees - 30 degrees, smaller than those investigated by COBE.