## 4. Static point massTo derive analytic estimates of the dependence of the redshift on
density parameters we begin by adopting the model of a simple point
inhomogeneity of mass For analytic computation rewrite (13) with (10a) as where the photon is emitted at ,
, , and
corresponds to closest approach at impact
parameter . We have suppressed the subscript 0
on Another interval giving a potentially large contribution is when or . Here the integral varies as . Physically this corresponds to the early epoch when the universe was smaller and so the mass was nearer the source, or equivalently the mass per physical volume was greater. The final order of magnitude estimate is therefore . For small angles . In terms of our original parametrization and so the "Newtonian" result is . Recall that , radians so , and the microwave background, for example, originates from so . In the post-Newtonian case, near the mass the integrand is
approximately the same, the correction factor being
, while far away the Green function cuts off the
contribution exponentially so only. The top
half of Table 1 gives
## 4.1. Time varying point massWe now examine , corresponding to an accreting (or evanescing) mass. In density perturbation theory inhomogeneities grow due to gravitational instability as in a dust background. Here of course we are not restricted to linear fluctuations ; the parametrization, as can be seen from (2) or as discussed in JLW2, is , which can be large. The analog of (14) in the Newtonian limit is Again splitting up the interval we find near that . As before, the other possibly significant contribution arises near (for ) and is found to vary as . Thus the total is . This of course agrees when with the static result, since . Inclusion of the post-Newtonian correction again cuts out the contribution. Note that for the logarithm dominates. Also note that for , as in linear theory, we find so if , i.e. the observer and emitter are situated symmetrically with respect to the inhomogeneity and so at the same value of gravitational potential, then there is zero redshift. Classically this corresponds to a ball gaining kinetic energy as it rolls into a dip then losing it coming out. If it rises to the same height on the far side and if the dip is time independent then its final velocity equals its initial. We see that for the Newtonian potential is time independent and only the endpoints contribute - the usual result. However, even for and symmetric
observer-emitter geometry there is a redshift (actually a blueshift)
in the post-Newtonian case due to symmetry breaking by the distance
dependence of the correction, i.e. roughly
which is neither symmetric nor time independent. The shift is of order
, just as for the asymmetric endpoint case. The
Sachs-Wolfe effect for point masses varying with time as The time dependent effect for a dynamically evolving isolated
Newtonian density fluctuation, e.g. a cluster decoupled from the
universal expansion, is sometimes known as the Rees-Sciama effect
(Rees and Sciama 1968). A gravitational dynamical time scale is
so we expect ,
parametrized as . Alternately consider an
effective mass within a fixed comoving radius at the times a light ray
enters and leaves the potential well: with
## 4.2. Fourier decompositionFor a more diffuse density distribution, e.g. a linear density
field, it is convenient to Fourier decompose the gravitational
potential into modes corresponding to a characteristic fluctuation
wavelength, or density inhomogeneity length or mass scale. In fact,
working in Fourier space allows simplification of many expressions.
For example, to obtain the Laplacian for
arbitrary Having seen that the variation of the gravitational potential over the photon propagation gives an insufficiently large effect to be readily observable, we are led to consider the effect of the potential at the endpoints of the path, i.e. at emitter and observer, on the photon energy. This gravitational redshift is simply . Whereas in the isolated mass cases of Sect. 3.1, 3.2 we could ignore the initial condition term in (3), either by saying it provided a term in the redshift that just added independently to the propagation effect, or by realizing that its contribution was exponentially suppressed far from the mass, now when dealing with a density field and an endpoint effect we must treat it more carefully. Either we can ignore it by pushing , or include it in the calculation. We choose the latter. We concentrate on the last scattering surface of the microwave
background radiation and the density perturbations
there, writing the wavenumber from now on as
This quantity can be written in terms of the density perturbation power spectrum by taking the Fourier transform of (8b), but first we must evaluate the initial condition term of (3) and add it in. It is Note the exponential suppression as grows larger (later) than . At this point we see that we really have two inputs to specify in (3), the initial density field and the initial potential . Because of the form of the full relativistic equation (2), both must be given. This is not unexpected though, because in JLW2 it was pointed out that (2) was essentially analogous to an inhomogeneous diffusion equation with time dependent parameters. It was found that the potential corresponded to the temperature in that sort of problem, and the density contrast to a heat source term. From that physical situation, however, we know that we must generally specify the initial distribution of both the temperature and the heat sources. It is only in the late time limit that the source completely determines the diffusive variable; in our case this corresponds (if compactness holds as well) to the Newtonian limit and the Poisson-Newton equation. For the initial condition constant we use
the ansatz , so where is the density power spectrum and as given by (4). Note that gives the overall correction factor to the usual Poisson relation (in Fourier space) between the gravitational potential entering the metric (1) and the energy density; in particular, at . It can also be obtained directly from transforming the version of the Laplacian (11). Thus can be viewed alternately as giving the post-Newtonian adaptation of the endpoint Sachs-Wolfe effect, or as an effective alteration to the density power spectrum , i.e. an effective transfer function. It is included in any (e.g. numerical) treatment using the full equation (2). Depending on the initial conditions it has the possibility of either enhancing or suppressing the low Fourier modes, i.e. increasing or decreasing the large scale power in the intrinsic power spectrum . This, for example, would cause a smaller (larger) overall normalization factor to be needed to match the large angle COBE microwave background anisotropy measurements, and hence also decrease (increase) the resultant predicted small scale power, thus ameliorating (exacerbating) the difficulties of the cold dark matter model. The correction factor is plotted in Fig. 1.
In the late time limit, , the results are
independent of and with the present day wavelength and the redshift of the last scattering surface, we see this corresponds to scales or angular scales , applicable to the COBE regime.
© European Southern Observatory (ESO) 1997 Online publication: April 20, 1998 |