Annu. Rev. Astron. Astrophys. 1994. 32: 319-70
Copyright © 1994 by . All rights reserved

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B. Sachs-Wolfe Effect

In this appendix we give a brief derivation of the Sachs-Wolfe effect in a "flat" cosmology with no cosmological constant, using the metric perturbation approach. Our starting point is the metric (following Sachs & Wolfe 1967)

Equation 42 (42)

where a(eta) is the scale factor, eta is the conformal time and gµnu(0) is the unperturbed metric (Minkowski space). To understand the temperature fluctuations induced by the perturbations we need to study the photon trajectories in ds2. For photon (null) geodesics ds2 = 0, so by (42) there is a 1-to-1 correspondence between photon paths in ds2 and ds bar2. This allows us to consider the problem first in ds bar2 and later translate our results in ds2.

We can solve for the geodesics by extremizing the Lagrangian gµnu dot xµ dot xnu; the geodesic (Euler-Lagrange) equations for ds bar2 are

Equation 43 (43)

where zeta is a parameter along the photon trajectory and the overdot represents differentiation w.r.t. zeta. The term in parenthesis is the 4-momentum kµ. Integrating, we find

Equation 44 (44)

where E is the unperturbed energy and x(0) = (const + zeta', zeta'e) is the unperturbed photon path.

The photon energy seen by an observer with 4-velocity u(|u2| = 1) is k . u. Using u = (1 - 1/2 h00, v) with |v| << 1 and (43)

Equation 45 (45)

where e and r refer to "emission" and "reception" respectively. The corresponding expression in ds2 comes from multiplying the whole expression by a(etar) / a(etae) to account for the cosmological redshift.

If we assume a uniform source and use the correspondence h00 = 2Phi between the metric perturbation and the Newtonian potential the temperature fluctuation induced is

Equation 46 (46)

The three terms can be identified as the gravitational potential redshift, the Doppler effect due to motion of the emitter and receiver, and an extra effect due to the time dependence of the metric (see also Stebbins 1993). In a flat Lambda = 0 universe Phi is constant in time in linear theory, so the last (integral) term vanishes and in the absence of Doppler shifts the potential change is known as the Sachs-Wolfe effect. In this limit the Sachs-Wolfe effect is simply the red-shifting of the photon as it climbs out of the potential on the surface of last scattering (assuming Phi = 0 at the time of observation). In some cases, such as with gravitational waves, non-flat or Lambda-dominated cosmologies, or nonlinear fluctuations, the integral term can also play a role.

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