If we start analyzing the properties of the accelerated scaling solution given by the Point I of Table 3.1, we immediately realize that on this solution the Universe is dominated by the energy density of the dark energy scalar field φ, and that Ωφ = 1 along the solution I.
This means that if we want to build a cosmological model which could possibly describe the observed Universe, we need to assume that this stable attractor has not yet been reached at the present stage of cosmic evolution, that the Universe is in a transient phase approaching it, but that the expansion is nevertheless already accelerated. This means that if we want to obtain the observed energy ratio between dark energy and cold dark matter, we need to tune the initial conditions for the dynamics of the scalar field.
This sort of tuning is, however, of a different type than the fine tuning that has to be invoked to address the Fine Tuning Problem. In this case, what has to be tuned is not the total amount of dark energy present in the early Universe, but the time of the transition between a decelerated matter dominated regime and an accelerated dark energy dominated attractor. Therefore this type of tuning is more related to the Coincidence Problem than to the Fine Tuning problem. As we have already stressed in Chapter 1, for the case of a cosmological constant Λ the two problems coincide, and they reduce to
3.4 The weak coupling regime and the φMDE phase 63
Stability regions in the parameter space
-4
-2
0
2
4
!
0
2
4
6
8
10
µ
I
II
III
VI
V
Figure 3.1: The parameter plane (β, µ) for the autonomous system of Eqs. 3.20-3.22 as presented in Figure 1 of Amendola (2000), replotted here with the definitions and the conventions adopted in this work. Each region is labelled with the number corresponding to the only critical point that is stable in that region. The shaded areas correspond to the regions of the parameter space where the stable solution is also accelerated.
the extreme fine tuning of a single number. However, for dynamic dark energy models in general, and therefore also for the case of dynamic coupled dark energy, the additional degree of freedom introduced by the dynamic nature of the scalar field splits the problem into the two different issues described above.
In any case, the stable attractor solution given by Point I could not be a viable description of our Universe without the initial conditions selection required to drive the system to the present-day coincidence of comparable Ωc and Ωφ.
The positive feature of the solution I is that it can be accelerated also for small values of the coupling β, and can therefore be treated in terms of deviations from the standard ΛCDM cosmological model.
coupling (β = 0) and by setting the equation of state of the scalar field to the cosmological constant value of −1, which can be realized by ensuring that the kinetic energy of the scalar field φ always vanishes: x2 = const. = 0. This can be done by setting the initial
velocity of the scalar field to zero ( ˙φi = 0) and by choosing a constant self-interaction
potential (V(φ) = const. →µ≡0).
Therefore, the point of parameter space (β, µ) = (0,0) corresponds to the usual ΛCDM cosmology, and as can be seen from Fig. 3.1, in this case the only accelerated stable solution is the Λ-dominated stage given by Point I which drives a De Sitter exponential expansion.
For a ΛCDM cosmological model, the Universe starts in a radiation dominated epoch at very early times, then it moves to a matter dominated epoch, and eventually reaches the final Λ-dominated stage. This means that a ΛCDM cosmic trajectory would start with
z ∼1 at early times (radiation domination), would then move to a stage wherez ∼0 and
x , y ∼0 (matter domination), and would finally reach the solution I where ΩΛ = 1. The
trajectory of a ΛCDM cosmology is therefore given by the sequence IV→VI→I.
If we now move to dynamic and coupled dark energy models, we can explore the region of parameter space close to the ΛCDM case, but with non-vanishing values of β and µ. While in the ΛCDM case the saddle point corresponding to the matter dominated epoch is given by the point (0,0,0) in the three-dimensional phase-space (x, y, z), for the coupled dark energy case this saddle will move to (q2
3β,0,0), with a non-vanishing dark energy
density given by Ωφ = 23β2.
This is the new type of tracking scaling solution already introduced in the previous section, which is sustained by the continuous net flux of energy flowing from the cold dark matter fluid to the dark energy scalar field φ. This saddle point is not anymore completely dominated by cold dark matter, but presents a non-negligible fraction of dark energy, in a very similar fashion as what happens for the usual trackingsolutions already discussed for the uncoupled dynamic dark energy case in Chapter 2 (and corresponding to the last point of Table 2.1 and to the limit β = 0 for the Point III of Table 3.1). However, although giving rise to a subdominant fraction of Early Dark Energy during matter domination in a very similar fashion, the two tracking solutions are inherently different. In fact, while the usual tracking solution seen in Chapter 2 is driven by a dynamic balance of the kinetic and the potential energy of the scalar field that sticks the