Between z=1000 and today, photons free-stream to the observer assuming that reionization did not take place too early. However, gravitational effects still play a role in modifying the anisotropy spectrum. The simplest example is the gravitational redshift effects from potential perturbations. The gravitational redshift felt by the photons climbing out of the potential well at last scattering has already been discussed and encorporated in the effective temperature of the acoustic oscillations. However, if the depth of the potential wells between the last scattering surface and the observer change as the photons cross it, the differential redshifts will induce further anisotropies in the CMB.
There are two pieces to the gravitational redshift that leads to a doubling of the effect. If the depth of the potential well changes as the photon crosses it, the blueshift from falling in and the redshift from climbing out no longer cancel leading to a residual temperature shift. The second effect is general-relativistic in nature. A potential fluctuation represents a fluctuation in the space-time curvature. Heuristically, the wavelength of the photon is "stretched" along with the fabric of space-time. If the potential well decays, the photons will be blueshifted as the space-time fabric "contracts"
Fig. 3: Dilation Effect
The wavelength of a photon stretches with the space-time fabric in a potential well leading to redshifts and blueshifts as the potential varies.
Now let us consider several common manifestations of this basic ISW effect.
