#
waynehu

## Professor,
Department of Astronomy and Astrophysics

University of Chicago

# ISW Effect

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.