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

## 3. Solving the horizon problem

An inspection of Eqs. (1) and (4), as done in CS, shows that the shell-crossing singularity surface is null: it cannot be crossed by any ingoing null geodesic.

The solution of the horizon problem, for an off-center observer in a DBB model, can thus proceed from its representation using a Penrose-Carter conformal diagram (see Fig. 32.

 Fig. 3. Penrose-Carter diagram sketching the permanent solution of the horizon problem by the DBB model. The current observer O sees, on the last-scattering surface, a causally connected (DE) region, included in the forward light cone of P. The same will hold when the point is reached, and from any other time in the observer's past or future history.

The light-like character of the shell-crossing surface forces all matter to be causally connected at , and any finite region to have been in causal contact at some .

The horizon problem is thus solved permanently in this model, a priori , for any location of the observer 3.

It is worth emphasizing that if the inflationary assumption also solves the horizon problem, it does so only temporarily. In effect, if one considers the horizon problem in a standard FLRW universe, as sketched in Fig. 4, the Big-Bang surface is space-like. It thus implies the existence of a limiting point L, in the history of the observer O, beyond which the observer sees, on the last-scattering surface, some no causally connected points. The current observer, being located above L, is confronted with this horizon problem.

 Fig. 4. Penrose-Carter diagram showing the horizon problem in a FLRW universe. The thin lines represent the light-cones. The CMBR, as seen by the observer located on the vertical axis, corresponds to the intersecting point of the observer backward light-cone and the last-scattering line. For a complete causal connection between every point seen in the CMBR, the backward light-cone issued from this intersecting point must reach the vertical axis before the Big-Bang curve. L is thus the limiting time beyond which the observer O experiences the horizon problem.

The solution proposed by the inflationary assumption is presented in Fig. 5. Adding an inflationary phase in the early history of the universe amounts to adding a slice of de Sitter space-time betwen the Big-Bang and the last-scattering, thus postponing the limit L when the observer can see non-causally connected points in the CMBR. Inflation thus solves the horizon problem, but only temporarily.

 Fig. 5. Penrose-Carter diagram showing the temporary solution of the horizon problem by inflation. The slice of de Sitter space-time corresponding to an inflationary phase, and added to Fig. 4, is shown between the Big-Bang and dashed lines. L is thus postponed, allowing the current observer O to see a causally connected CMBR. When time elapses and the observer reaches the above L region, the horizon problem returns.

© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000