JHEP11(2016)090
{
ǎ = ∞}
{ǎ = 0}
{â
= 0}
{â = ∞}
(a)
{
ǎ
=
∞
}
{ǎ = 0}
{â = 0}
{â = ∞}
(b)
Figure 2. Causal diagrams for dual universes filled with fluids with constant equation of state
parameters. The blue diagrams represent decelerated expansion ( ˇw > −
1
3
) and have only a particle
horizon. The dual red diagrams are both accelerating ( ˆw < −
1
3
). (a) Expansion/Expansion: the
dual red diagram has accelerating expansion and only an event horizon. (b) Expansion/Contraction:
the red diagram has accelerated contraction and only a particle horizon.
For a pair of C/E dual universes we do not have the τ 7→ −τ mapping, and thus
there is no “flipping” of the conformal diagram. We may, however, as has been noted
above, choose the origin of conformal times such that conformal infinity and the dual big-
bang coincide. The scale factor inversion, nevertheless, still exchanges the roles of these
asymptotic surfaces. The overall effect for a universe with a finite conformal lifetime can
be seen in figure 1(b). As to what regards the horizons, we have now: η
p
=
R
a
∞
da(a
2
H)
−1
and η
h
=
R
0
a
da(a
2
H)
−1
, which leads us to conclude that ˇη
p
= ˆη
p
. A similar calculation is
valid for event horizons ˇη
h
= ˆη
h
. Therefore the particle horizons of the expanding and the
contracting universes are mapped into each other, and so are the respective event horizons.
We shall be concerned in this paper with universes like the ones appearing in figure 1,
in which there are two different phases of acceleration (such as in the ΛCDM model),
thus possessing both kinds of horizon and marked by a finite total conformal time duration
η
f
=
R
∞
0
da (a
2
H)
−1
= η
p
+ η
h
. This may, of course, not be always the case. For example,
universes filled with a single perfect fluid with constant equation of state parameter ˇw >
−
1
3
, and their duals, with ˆw < −
1
3
, have, each, only one kind of horizon as shown in figure 2.
Then either η
p
or η
h
are infinite, resulting in an infinite η
f
so eq. (2.5), for example, has
to be modified. The crucial point here is this: when both boundary surfaces {a = 0}
and {a = ∞} are space-like, as in figure 1, we may identify the coinciding boundaries of
consecutive aeons as one single space-like hypersurface X, which we shall call a ‘crossover
surface’. It is conformally equivalent to the Euclidean 3-dimensional space (that is, a
constant-time-surface in Minkowski space-time), and with our choice of conformal time,
X = {η = 0}. Now in the case of figure 2, one cannot define properly a transition
surface because this would amount to identify hypersurfaces with different signatures. For
example, in a C/E pair the contracting universe ends in a null hypersurface while the
– 7 –
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