Cosmological-perturbation equations from the total-angular-momentum formalism

We derive the equations for the linear evolution of cosmological perturbations using total-angular-momentum (TAM) waves. The cosmic microwave background temperature and polarization transfer functions are first derived as integral equations and from these the usual Boltzmann hierarchy of differential equations are then obtained for both scalar and tensor perturbations. We include the effects of isotropic cosmic birefringence and discuss anisotropic cosmic birefringence. We note a subtle effect of cosmic birefringence on temperature fluctuations. The TAM derivation of the equations provides physical insights to results obtained previously.