2. Gravitational potential
Under the assumptions that the gravitational potential fluctuations are small, parametrized by , and their peculiar accelerations are small (thus ensuring that peculiar motions remain much less than the speed of light) for characteristic inhomogeneity scales with L the background curvature or horizon length scale, the following metric provides a consistent description:
(Futamase 1989; JLW1). Here a is the FRW expansion factor (to the order required), the conformal time, and the spatial part of the conformally stationary Robertson-Walker metric. The conditions are expected to hold cosmologically everywhere far outside the radii of black holes and neutron stars. We call the full time dependence and presence of in the spatial part of the metric the post-Newtonian corrections.
The Einstein field equations produce the following relation between the scalar harmonic modes of and the density contrast ,
(JLW1; JLW2). Here k is the trichotomic FRW curvature parameter, the unperturbed density, and is the mode variable conjugate to position. No restrictions are made on the size of , i.e. the density field could be nonlinear.
Solution of this equation gives (JLW2)
with the Green function in the case (used here throughout)
Transformation of the time integration variable to reveals
In the Newtonian limit of late times () and subhorizon scales (), and it is found that
(JLW2), the usual result. [We have neglected the second, initial conditions term from (3) due to its exponential die off far from the initial hypersurface, but see Sect. 3.3.]
Going beyond JLW2 we adopt a dust background ( where is the Hubble constant) and inhomogeneity behavior to yield
In the Newtonian limit this reduces to . Some particular cases of interest are (density unevolving in physical coordinates) and (as for the growth of linear density fluctuations ):
with erfc the complementary error function.
Since the geodesic equations determining light propagation involve derivatives of the potential we calculate in comoving coordinates and .
with special cases
The tidal field is , necessary for calculations involving shear of a light ray bundle and the resulting image distortions. Because of its length we do not show the expression for it here but it is interesting to consider the Laplacian . For
The second term thus illustrates the (not linearized) general relativistic correction to the Poisson equation (in the fully specified longitudinal gauge), involving a gaussian weighting over the extent of the inhomogeneity. In the "Newtonian" limit the fluctuation is restricted to regions much smaller than the dispersion so its contrast is averaged to zero, leaving the Poisson result, while in the opposite ("superhorizon" fluctuation) limit the gaussian becomes a delta function, giving the Laplace equation appropriate for a uniform density field.
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998