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

Equation 1 (1)

where H is the expansion rate

Equation 2 (2)

Here rhophi is the energy density of the scalar field

Equation 3 (3)

while rhoM is the energy density of matter and radiation, which decreases as

Equation 4 (4)

with pM the pressure of matter and radiation.

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

Equation 5 (5)

where alpha > 0, and M is an adjustable constant. If the scalar field begins at a value much less than the Planck mass and with V(phi) and phidot2 much less than rhoM, then the field phi(t) initially increases as t2/(2+alpha), so that rhophi decreases as t-2alpha/(2+alpha), while rhoM 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 rhoM at temperatures of order 109 °K to 1010 °K.) Eventually a time is reached when rhoM becomes as small as rhophi, after which the character of the solution changes. Now rhophi becomes larger than rhoM, and rhophi decreases more slowly, as t-2/(4+alpha). The expansion rate H now goes as H propto sqrt[V(phi)] propto t-alpha/(4+alpha), so the Robertson-Walker scale factor R(t) grows almost exponentially, with log R(t) propto t4/(4+alpha). In this approach, the transition from rhoM-dominance to rhophi-dominance is supposed to take place near the present time, so that both rhoM and rhophi 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 rhoM-dominated expansion to a later rhophi-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 rhophi at late times would be spoiled if we added a constant of order mPlanck4 (or mW4, or me4) 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 rhophi at which rhophi approx rhoM close to the present critical density rhoc0. The value of the field phi(t) at this crossover can easily be seen to be of the order of the Planck mass, so in order for rhophi to be comparable to rhoM at the present time we need

Equation 6 (6)

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.

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