4. Static point mass
To 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 m at comoving position . The x -axis is aligned with the light ray (perturbations to the photon path produce effects of smaller order than we consider) and where is the cosmological redshift of the mass. The density contrast is , corresponding to . Such a static case is intended as a toy model only and is not realistic over long times or large scales.
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 E. First consider the Newtonian case, where the bracketed quantity is unity. Around the integrand is nearly odd so the dominant term from the symmetric part of the interval requires expanding to obtain an overall term even in s. One readily sees that this gives a logarithmic integral . (Although the mass is static, the background is not so an energy shift is expected; but see the next subsection for the case.)
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 z in units of for the static case, showing that the analytic dependences hold well.
Table 1. Point mass Sachs-Wolfe effect
4.1. Time varying point mass
We 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 a is given in the bottom half of Table 1. Because of the vanishing of the logarithmic term the results are angle independent (for small angles). For the logarithmic term reappears and the results regain the angle dependence of the case. The generalization to an extended matter distribution, e.g. a spherical density inhomogeneity, does not significantly affect the results (Kendall 1993).
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 v the typical matter velocity, due to infall or peculiar motions. With an impulse approximation (15) becomes since for bound systems. From this order of magnitude parametrization it is clear that to obtain a frequency shift (or equivalently temperature shift in the CMB) large enough to be observable (say ) one needs a rapid variation in the potential, more rapid than the gravitational time scale, such as from relativistic cosmic strings or black holes (but recall that the metric (1) was derived for nonrelativistic, weak field perturbations). Thus, within the context of a dynamical Sachs-Wolfe effect, the post-Newtonian formalism provides a significant but unobservable difference.
4.2. Fourier decomposition
For 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 p, just transform (3), multiply by , and perform the inverse transform. One finds that for general p, (11) is altered by a term multiplying and a sum inside the integral of polynomials up to order p involving and , multiplying the exponential.
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 k instead of q to agree with the usual notation in the literature. In the linear density perturbation regime grows linearly with scale factor a, corresponding to our case . The mean square gravitational redshift is proportional to the sum over Fourier modes of the square of the gravitational potential, assuming random phases between modes: . The magnitude of the temperature anisotropies is usually characterized by the zero separation correlation function but our case is slightly different. For lines of sight separated by some large angle the two point correlation function does not vanish because of the coherence introduced by the term. Rather than have for large, one has , i.e. plays the role of the zero lag correlation in setting the magnitude. Alternately one can say that offers only a constant, isotropic shift so one can ignore it and consider only .
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 A gives the deviation from the Poisson equation () in the initial conditions. Adding (16) to the transform of (8b) gives in terms of the density perturbation power spectrum
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 A. From the expression in (17) or the figure, three regimes in perturbation wavelength can be identified. When , the factor is close to one, i.e. this is the late time, compact inhomogeneity Newtonian limit. For , the factor begins to decline predominately due to the influence of the exponential, and when , the term becomes important as well, leading to . From Fig. 1, the deviation from the usual behavior of the Sachs-Wolfe effect becomes noticeable below , depending on the value of . Converting to wavelengths by
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