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)

 (42)

where a() is the scale factor, is the conformal time and gµ(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 d2. This allows us to consider the problem first in d2 and later translate our results in ds2.

We can solve for the geodesics by extremizing the Lagrangian gµ µ ; the geodesic (Euler-Lagrange) equations for d2 are

 (43)

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

 (44)

where E is the unperturbed energy and x(0) = (const + ', '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)

 (45)

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

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

 (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 = 0 universe 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 = 0 at the time of observation). In some cases, such as with gravitational waves, non-flat or -dominated cosmologies, or nonlinear fluctuations, the integral term can also play a role.