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Astron. Astrophys. 326, 23-28 (1997)

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3. Sachs-Wolfe effect

Given the metric (1) and the results (7), (9), (11) one can investigate light propagation behavior by writing down the geodesic equation, geodesic deviation, and the beam or Raychaudhuri equation. Properties of the photon bundle, such as convergence and shear, and applications of the geodesic deviation equation to astrophysical problems such as correlations of observables in terms of the density power spectrum are in ongoing research. Here we concentrate on the geodesic equation for individual photon four momentum, in particular the Sachs-Wolfe effect on the redshift.

In longitudinal gauge ([FORMULA] and [FORMULA] proportional to [FORMULA] ; see Bardeen 1980 and Kodama & Sasaki 1984 for gauges and gauge invariance) the expression for the frequency of a photon emitted at [FORMULA] becomes

[EQUATION]

(Note the more familiar time derivative of [FORMULA] occurs in the unfully specified synchronous gauge expression, although one could convert the gradient into a time derivative and a surface term.) The ratio of expansion factors is the background cosmological redshift and [FORMULA] is the photon propagation unit vector. The inhomogeneity induced redshift, i.e. the Sachs-Wolfe effect, is then

[EQUATION]

Upon adopting a density field [FORMULA] one can compute the gravitational potential gradient by (9) and hence obtain the redshift, or equivalently temperature anisotropy in the CMB. This, being an observable, is gauge independent. We consider compact density distributions, both static and time dependent, and then a field of inhomogeneities, along with the Fourier representation.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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