The ordinary Sachs-Wolfe effect gives a ΔT = φ/3c2 which is three times smaller than a naive expectation. This factor of three comes because the Universe is expanding as the 2/3 power of the time at the time of recombination, and a high φ makes clocks, including the "expansion clock" of the Universe, run faster, so 2/3 of the normal gravitational redshift is cancelled.
The integrated Sachs-Wolfe effect gives a ΔT of 2Δφ/c2 where Δφ is the change in the potential while a photon traversed the potential well. This is twice the naive expectation and the factor of 2 is the same factor of 2 that occurs in the deflection of starlight by the Sun.
The early integrated Sachs-Wolfe effect is caused by the change in potential when the photons, which were trapped in structures prior to recombination, suddenly leave those structures.
The late integrated Sachs-Wolfe effect is caused by the switch-over to an accelerating expansion of the Universe in dark energy dominated models. This causes the depth of potential wells to decrease.
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© 2007 Edward L. Wright. Last modified 26 Aug 2007