Inflationary cosmology 45towards use of the fanciful term ‘quintessence’. In any case, it is important to
appreciate that the idea uses exactly the same physical elements that we discussed
in the context of inflation: there is someV(φ), causing the expectation value of
φto obey the damped oscillator equation of motion, so the energy density and
pressure are
ρφ=φ ̇^2 / 2 +V
pφ=φ ̇^2 / 2 −V.This gives us two extreme equations of state:
(i) vacuum-dominated, withVφ ̇^2 /2, so thatp=−ρ;
(ii) kinetic-dominated, withVφ ̇^2 /2, so thatp=ρ.In the first case, we know thatρdoes not alter as the universe expands, so the
vacuum rapidly tends to dominate over normal matter. In the second case, the
equation of state is the unusual=2, so we get the rapid behaviourρ∝a−^6.
If a quintessence-dominated universe starts off with a large kinetic term relative
to the potential, it may seem that things should always evolve in the direction of
being potential-dominated. However, this ignores the detailed dynamics of the
situation: for a suitable choice of potential, it is possible to have atracker field,in
which the kinetic and potential terms remain in a constant proportion, so that we
can haveρ∝a−α,whereαcan be anything we choose.
Putting this condition in the equation of motion shows that the potential is
required to be exponential in form. More importantly, we can generalize to the
case where the universe contains scalar field and ordinary matter. Suppose the
latter dominates, and obeysρm∝a−α. It is then possible to have the scalar-field
density obeying the sameρ∝a−αlaw, provided
V(φ)=2
λ^2( 6 /α− 1 )exp[−λφ].The scalar-field density isρφ=(α/λ^2 )ρtotal(see, e.g., Liddle and Scherrer 1999).
The impressive thing about this solution is that the quintessence density stays a
fixed fraction of the total, whatever the overall equation of state: it automatically
scales asa−^4 at early times, switching toa−^3 after the matter–radiation equality.
This is not quite what we need, but it shows how the effect of the overall
equation of state can affect the rolling field. Because of the 3Hφ ̇term in the
equation of motion,φ‘knows’ whether or not the universe is matter dominated.
This suggests that a more complicated potential than the exponential may allow
the arrival of matter domination to trigger the desired-like behaviour. Zlatevet
alsuggest two potentials which might achieve this:
V(φ)=M^4 +βφ−β or V(φ)=M^4 [exp(mP/φ)− 1 ].The evolution in these potentials may be described byw(t),wherew=p/ρ.We
needw 1 /3 in the radiation era, changing tow−1 today. The evolution