The idea of quintessence
^{(4)}
is that the
cosmological constant is small because the universe is old. One imagines a
uniform scalar field (*t*)
that rolls down a potential
*V*(), at a rate governed by
the field equation

where *H* is the expansion rate

Here _{} is the energy density of the
scalar field

while _{M}
is the energy density of matter and radiation, which decreases as

with *p*_{M} the pressure of matter and radiation.

If there is some
value of (typically,
infinite) where
*V'*() = 0, then it is
natural that should approach this
value, so that it eventually changes only slowly with time. Meanwhile
_{M} is
steadily decreasing, so that
eventually the universe starts an exponential expansion with a slowly varying
expansion rate *H*
sqrt[8 *G*
*V*() / 3]. The problem, of
course, is to
explain why *V*() is small or
zero at the value of where
*V'*() = 0.

Recently this approach has been studied in the context of so-called `tracker'
solutions. ^{(5)}
The simplest case arises for a potential of the form

where > 0, and *M* is an
adjustable constant. If the scalar field begins
at a value much less than the Planck mass and with
*V*() and
^{2} much less than
_{M}, then the field
(*t*) initially increases as
*t*^{2/(2+)}, so
that _{} decreases as
*t*^{-2/(2+)},
while _{M} is
decreasing faster, as *t*^{-2}. (The existence of this phase
is important, because the success of cosmic nucleosynthesis calculations would
be lost if the cosmic energy density were not dominated by
_{M} at
temperatures of order 10^{9} °K to 10^{10}
°K.) Eventually a time is reached when
_{M} becomes
as small as _{}, after which the
character of the solution changes. Now
_{} becomes larger than
_{M}, and
_{} decreases more slowly, as
*t*^{-2/(4+)}. The
expansion rate *H* now goes
as *H*
sqrt[*V*()]
*t*^{-/(4+)}, so the
Robertson-Walker scale factor *R(t)* grows almost exponentially,
with log *R(t)
*
*t*^{4/(4+)}. In
this approach, the transition from
_{M}-dominance to _{}-dominance is supposed to take place near the
present time, so that both
_{M} and
_{} are now both contributing
appreciably to the cosmic expansion rate.

The nice thing about these tracker solutions is that the
existence of a cross-over from an early
_{M}-dominated expansion to a later
_{}-dominated expansion does not
depend on any fine-tuning of the
initial conditions. But it should not be thought that *either* of the two
cosmological constant problems are solved in this way. Obviously, the decrease
of _{} at late times would be spoiled if
we added a constant of order
*m*_{Planck}^{4} (or
*m*_{W}^{4}, or *m*_{e}^{4})
to the potential (5). What is
perhaps less clear is that, even if we take the potential in the form (5)
without any such added constant, we still need a fine-tuning to make the value
of _{} at which
_{}
_{M} close to
the *present*
critical density
_{c0}. The
value of the field (*t*) at
this crossover can easily be seen to be of the order of the Planck mass, so
in order
for _{} to be comparable to
_{M} at the
present time we need

Theories of quintessence offer no explanation why this should be the case.
(An interesting suggestion has been made after Dark Matter
2000. ^{(6)})

^{4} P. J. E. Peebles and B. Ratra:
Astrophys. J. **325**, L17 (1988);

B. Ratra and P. J. E. Peebles:
Phys. Rev. **D 37**, 3406 (1988);

C. Wetterich: Nucl. Phys. **B302**, 668 (1988).
Back.

^{5} I. Zlatev, L. Wang, and
P. J. Steinhardt:
Phys. Rev. Lett. **82**,
896 (1999);

Phys. Rev. **D 59**, 123504 (1999).
Back.

^{6} C. Armendariz-Picon, V. Mukhanov, and
P. J. Steinhardt: astro-ph/0004134.
Back.