  Calculations of the evolution of cosmological perturbations generally involve
solution of a large number of coupled differential equations to describe the
evolution of the multipole moments of the distribution of photon intensities
and polarization. However, this "Boltzmann hierarchy" communicates with the
rest of the system of equations for the other perturbation variables only
through the photon-intensity quadrupole moment. Here I develop an alternative
formulation wherein this photon-intensity quadrupole is obtained via solution
of two coupled integral equations -- one for the intensity quadrupole and
another for the linear-polarization quadrupole -- rather than the full
Boltzmann hierarchy. This alternative method of calculation provides some
physical insight and a cross-check for the traditional approach. I describe a
simple and efficient iterative numerical solution that converges fairly
quickly. I surmise that this may allow current state-of-the-art
cosmological-perturbation codes to be accelerated.
