2. QUINTESSENCE

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

(1)

where H is the expansion rate

(2)

Here is the energy density of the scalar field

(3)

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

(4)

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

(5)

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 10⁹ °K to 10¹⁰ °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_Planck⁴ (or m_W⁴, or m_e⁴) 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

(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.