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Astron. Astrophys. 362, 840-844 (2000)

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2. The inhomogeneous delayed big-bang model

The main features and properties of the model are here briefly mentioned. For a more detailed presentation and a discussion of the assumptions retained, the reader is referred to CS and SC.

The proposed DBB model is a subclass of the LTB flat model. Its line-element, in comoving coordinates ([FORMULA]) and proper time t, is

[EQUATION]

An appropriate choice of the radial coordinate r yields

[EQUATION]

The Big-Bang function, [FORMULA], verifies 1

[EQUATION]

The physical singularity of the model - i.e. the first surface, encountered on a backward path from "now", where the energy density and the invariant scalar curvature go to infinity - is the shell-crossing surface, represented in the [FORMULA] plane by the curve:

[EQUATION]

As the energy density increases approaching this surface, radiation becomes the dominant component in the universe, pressure can no longer be neglected, and the LTB model no longer holds. The region between the Big-Bang surface [FORMULA] and the shell-crossing one is thus excluded from the part of the model retained to describe the matter dominated region of the universe, which we discuss here.

The optical depth of the universe to Thomson scattering is approximated by a step function (see SC). The last-scattering surface is thus defined, in the local thermodynamical equilibrium approximation (see CS), by its temperature, [FORMULA], as is the now-surface, [FORMULA] K, where the observer is located. The equal temperature surfaces verify

[EQUATION]

The value of the entropy function [FORMULA] is assumed to be constant with r. The shell-crossing surface is thus asymptotic to every (monotonically increasing with r) [FORMULA] curve.

An observer located at a distance from the center sees an axially symmetrical universe in the center direction. In the geometrical optics approximation, the light travelling from the last-scattering surface to this observer follows null-geodesics, which is thus legitimate to consider in the meridional plane. The photon path is uniquely defined by the observer's position [FORMULA] and the angle [FORMULA] between the direction from which the light ray comes and the direction of the center of the universe, C.

The microwave radiation observed in the direction of this symmetry center follows a light-cone issued from point D on the last-scattering surface, then passes through the center [FORMULA] and reaches the earth at point O. Observed in the opposite direction, it starts from point E, and travels on the EO null-geodesic to the observer (see Fig. 2).

The radial null-geodesic equation, as established in CS, is

[EQUATION]

It is easy to see that, with a function [FORMULA] verifying conditions (3), for a fixed t, [FORMULA] decreases with increasing r, and thus: [FORMULA].

If observed with an angle [FORMULA] in the inward direction, a light beam issued from point A on the last-scattering surface approaches C to a comoving distance [FORMULA], then proceeds toward O. In the outward (opposite) direction, it follows the BO geodesic (see Fig. 1 and Fig. 2).

[FIGURE] Fig. 1. The CMBR observed from O with an angle [FORMULA]. Schematic illustration of the trajectory of two CMBR light beams received by the observer O and making an angle [FORMULA] with the direction of the symmetry center C of the universe. The inner circle is the 2-sphere on the last-scattering surface from which the beam issued from point A is emitted (the observer looks inward). The outer circle is the one from which the beam starting from B is emitted (the observer looks outward).

[FIGURE] Fig. 2. The CMBR observed from O in the [FORMULA] plane. The dotted curve represents the Big-Bang surface [FORMULA]. The broken line with dots represents the shell-crossing singularity and the last-scattering surfaces that cannot be resolved at the scale of the figure. The broken curve is the now-surface [FORMULA]. The solid lines are the light-cones.

The corresponding null-geodesics are solutions of the system of differential equations established in SC (in units [FORMULA]):

[EQUATION]

[EQUATION]

[EQUATION]

with a plus sign in Eq. (6) from O to [FORMULA] ([FORMULA]), and a minus sign from [FORMULA] to A ([FORMULA]). The equations corresponding to the observer looking outward (OB curve) require a minus sign.

If one considers Eq. (6), for a fixed t and for a same variation [FORMULA] of the [FORMULA] affine parameter, the absolute value of the radial coordinate variation dr is smaller with [FORMULA] or [FORMULA] than with [FORMULA] or [FORMULA]. It follows that [FORMULA].

With [FORMULA] taking every value between [FORMULA] and [FORMULA], and considering the two oppposite directions of sight, the CMBR is observed from O as coming from a set of points each located on a 2-sphere belonging to the the subset [FORMULA] on the last-scattering surface [FORMULA].

To prove that this set of points can be causally connected, it is sufficient to show that there is at least one forward radial light-cone, i.e. issued from a point [FORMULA] including the DE subset.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000
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