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

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2. Gravitational potential

Under the assumptions that the gravitational potential fluctuations [FORMULA] are small, parametrized by [FORMULA], and their peculiar accelerations [FORMULA] are small (thus ensuring that peculiar motions remain much less than the speed of light) for characteristic inhomogeneity scales [FORMULA] with L the background curvature or horizon length scale, the following metric provides a consistent description:

[EQUATION]

(Futamase 1989; JLW1). Here a is the FRW expansion factor (to the order required), [FORMULA] the conformal time, and [FORMULA] 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 [FORMULA] 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 [FORMULA] and the density contrast [FORMULA],

[EQUATION]

(JLW1; JLW2). Here k is the trichotomic FRW curvature parameter, [FORMULA] the unperturbed density, and [FORMULA] is the mode variable conjugate to position. No restrictions are made on the size of [FORMULA], i.e. the density field could be nonlinear.

Solution of this equation gives (JLW2)

[EQUATION]

[EQUATION]

[EQUATION]

with the Green function in the [FORMULA] case (used here throughout)

[EQUATION]

[EQUATION]

[EQUATION]

Transformation of the time integration variable to [FORMULA] reveals

[EQUATION]

[EQUATION]

where [FORMULA].

In the Newtonian limit of late times ([FORMULA]) and subhorizon scales ([FORMULA]), [FORMULA] and it is found that

[EQUATION]

(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 ([FORMULA] where [FORMULA] is the Hubble constant) and inhomogeneity behavior [FORMULA] to yield

[EQUATION]

[EQUATION]

In the Newtonian limit this reduces to [FORMULA]. Some particular cases of interest are [FORMULA] (density unevolving in physical coordinates) and [FORMULA] (as for the growth of linear density fluctuations [FORMULA]):

[EQUATION]

[EQUATION]

[EQUATION]

with erfc the complementary error function.

Since the geodesic equations determining light propagation involve derivatives of the potential we calculate in comoving coordinates [FORMULA] and [FORMULA].

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

with special cases

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The tidal field is [FORMULA], 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 [FORMULA]. For [FORMULA]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA].

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 [FORMULA] 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.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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