AN ANALYTICAL SOLUTION OF A SECOND KIND VOLTERRA INTEGRAL EQUATION REPRESENTING THE CONTRIBUTION OF GRAVITATIONAL WAVES TO THE POLARIZATION OF THE COSMIC MICROWAVE BACKGROUND

Abstract

Volterra integral equations of the second kind have a remarkable importance in many physical problems. In this paper, I shall discuss the particular case of a Volterra equation whose solution represents the polarization of the cosmic microwave background (CMB) induced by cosmological gravitational waves. The analytical study of the radiative transfer equation, written in the Chandrasekhar formalism, is performed after a change of variables that follows the formalism developed by Polnarev in 1980. Then, the problem turns into the solution of a second kind Volterra equation, containing the gravitational wave spectrum in its source term. An analytical solution for this Volterra equation can be found if its kernel has a polynomial or an exponential form: in these cases, respectively, the methods of the resolvent and of the iterated kernels can be successfully applied; the solution thus determined represents the polarization of the CMB photons induced by cosmological gravitational waves. The calculated polarization degree for the CMB is plotted versus the wave number k of the gravitational waves inducing the polarization. In the case of an exponential kernel, i.e. for a Universe fast reionized at small redshift (z≤20), I obtain a remarkable polarization peak at small k (k≤4); such a degree of polarization for the CMB should be observable at angular scales near 9°, thus providing also a test for many reionization models. On the contrary, for a standard ionization history, a CMB polarization comparable in magnitude to the one predicted above could only be observed at small angular scales (θ≤2°).