Geodesic "curve"-of-sight formulae for the cosmic microwave background:
  a unified treatment of redshift, time delay, and lensing
  In this paper, we introduce a new approach to a treatment of the
gravitational effects (redshift, time delay and lensing) on the observed cosmic
microwave background (CMB) anisotropies based on the Boltzmann equation. From
the Liouville's theorem in curved spacetime, the intensity of photons is
conserved along a photon geodesic when non-gravitational scatterings are
absent. Motivated by this fact, we derive a second-order line-of-sight formula
by integrating the Boltzmann equation along a perturbed geodesic (curve)
instead of a background geodesic (line). In this approach, the separation of
the gravitational and intrinsic effects are manifest. This approach can be
considered as a generalization of the remapping approach of CMB lensing, where
all the gravitational effects can be treated on the same footing.
