5.2. Scaling solutions and trackers

One reason for having optimism that quintessence can at least address the coincidence problem comes from an interesting class of solutions known as scaling solutions or trackers. These arise because the scalar field does not have a unique equation of state p p(). Although we can usefully define an effective equation of state

(50)

in which terminology cosmological constant behaviour corresponds to w = - 1, the scalar field velocity depends on the Hubble expansion, which in turn depends not only on the scalar field itself but on the properties of any other matter that happens to be around. That is to say, the scalar field responds to the presence of other matter.

A particularly interesting case is the exponential potential

(51)

which we already saw in the early Universe context as Eq. (30). If there is only a scalar field present, this model has inflationary solutions a t^2/2 for ² < 2, and non-inflationary power-law solutions otherwise. However, if we add conventional matter with equation of state p = w, a new class of solutions can arise, which turn out to be attractors whenever they exist [28, 29, 30]. These solutions take the form of scaling solutions, where the scalar field energy density (indeed both its potential and kinetic energy density separately) exhibit the same scaling with redshift as the conventional matter. That is to say, the scalar field mimics whatever happens to be the dominant matter in the Universe. So, for example, in a matter-dominated Universe, we would find 1/a³. If the matter era were preceded by a radiation era, at that time the scalar field would redshift as 1/a⁴, and it would make a smooth transition between these behaviours at equality. The ratio of densities is decided only by the fundamental parameters and w. So, at any epoch one expects the scalar field energy density to be comparable to the conventional matter.

Unfortunately this is not good enough. We don't want the scalar field to be behaving like matter at the present, since it is supposed to be driving an acceleration, and we need it to be negligible in the fairly recent past. This requires us to consider alternatives to the exponential potential, a common example being the inverse power-law potential [28]

(52)

where < 0. In fact, Liddle and Scherrer [31] gave a complete classification of Einstein gravity models with scaling solutions, defined as models where the scalar field potential and kinetic energies stay in fixed proportion. The exponential potential is a particular case of that, but in general the scaling of the components of the scalar field energy density need not be the same as the scaling of the conventional matter, and indeed the inverse power-law potential is an example of that; if the conventional matter is scaling as 1/a^m where m = 3(1 + w), there is an attractor solution in which the scalar field densities will scale as

(53)

With negative , the scalar field energy density is redshifting more slowly and eventually overcomes the conventional matter, at which point the Universe starts to accelerate.

This type of scenario can give a model capable of explaining the observational data, though it turns out that quite a shallow power-law is required in order to get the field to be behaving sufficiently like a cosmological constant (current limits require w < - 0.6 at the present epoch, where 0.7 [32]). Also, the epoch at which the field takes over and drives an acceleration is still more or less being put in by hand; it turns out that the acceleration takes over when m_Pl, and so 10^-120 is required to ensure this epoch is delayed until the present.

Various other forms of the potential have been experimented with, and many possibilities are known to give viable evolution histories [33]. While such models do give a framework for interpretting the type Ia supernova results, in many cases with the possibility ultimately of being distinguished from a pure cosmological constant, I believe it is fair to say that so far no very convincing resolution of either the cosmological constant problem or the coincidence problem has yet appeared. However, quintessence is currently the only approach which has any prospect of addressing these issues.