Next Contents Previous

C. Inflation and dark energy

The negative active gravitational mass density associated with a positive cosmological constant is an early precursor of the inflation picture of the early universe; inflation in turn is one precursor of the idea that Lambda might generalize into evolving dark energy.

To begin, we review some aspects of causal relations between events in spacetime. Neglecting space curvature, a light ray moves proper distance dl = a(t) dx = dt in time interval dt, so the integrated coordinate displacement is

Equation 26 (26)

If OmegaLambda0 = 0 this integral converges in the past -- we see distant galaxies that at the time of observation cannot have seen us since the singular start of expansion at a = 0. This "particle horizon problem" is curious: how could distant galaxies in different directions in the sky know to look so similar? The inflation idea is that in the early universe the expansion history approximates that of de Sitter's (1917) solution to Einstein's field equation for Lambda > 0 and Tµnu = 0 in Eq. (20). We can choose the coordinate labels in this de Sitter spacetime so space curvature vanishes. Then Eqs. (11) and (12) say the expansion parameter is

Equation 27 (27)

where HLambda is a constant. As one sees by working the integral in Eq. (26), here everyone can have seen everyone else in the past. The details need not concern us; for the following discussion two concepts are important. First, the early universe acts like an approximation to de Sitter's solution because it is dominated by a large effective cosmological "constant", or dark energy density. Second, the dark energy is modeled as that of a near homogeneous field, Phi.

In this scalar field model, motivated by grand unified models of very high energy particle physics, the action of the real scalar field, Phi (in units chosen so Planck's constant hbar is unity) is

Equation 28 (28)

The potential energy density V is a function of the field Phi, and g is the determinant of the metric tensor. When the field is spatially homogeneous (in the line element of Eq. [15]), and space curvature may be neglected, the field equation is

Equation 29 (29)

The stress-energy tensor of this homogeneous field is diagonal (in the rest frame of an observer moving so the universe is seen to be isotropic), with time and space parts along the diagonal

Equation 30 (30)

If the scalar field varies slowly in time, so that dot{Phi}2 << V, the field energy approximates the effect of Einstein's cosmological constant, with pPhi appeq - rhoPhi.

The inflation picture assumes the near exponential expansion of Eq. (27) in the early universe lasts long enough that every bit of the present observable universe has seen every other bit, and presumably has discovered how to relax to almost exact homogeneity. The field Phi may then start varying rapidly enough to produce the entropy of our universe, and the field or the entropy may produce the baryons, leaving rhoPhi small or zero. But one can imagine the late time evolution of rhoPhi is slow. If slower than the evolution in the mass density in matter, there comes a time when rhoPhi again dominates, and the universe appears to have a cosmological constant.

A model for this late time evolution assumes a potential of the form

Equation 31 (31)

where the constant kappa has dimensions of mass raised to the power alpha + 4. For simplicity let us suppose the universe after inflation but at high redshift is dominated by matter or radiation, with mass density rho, that drives power law expansion, a propto tn. Then the power law solution to the field equation (29) with the potential in Eq. (31) is

Equation 32 (32)

and the ratio of the mass densities in the scalar field and in matter or radiation is

Equation 33 (33)

In the limit where the parameter alpha approaches zero, rhoPhi is constant, and this model is equivalent to Einstein's Lambda.

When alpha > 0 the field Phi in this model grows arbitrarily large at large time, so rhoPhi -> 0, and the universe approaches the Minkowskian spacetime of special relativity. This is within a simple model, of course. It is easy to imagine that in other models rhoPhi approaches a constant positive value at large time, and spacetime approaches the de Sitter solution, or rhoPhi passes through zero and becomes negative, causing spacetime to collapse to a Big Crunch.

The power law model with alpha > 0 has two properties that seem desirable. First, the solution in Eq. (32) is said to be an attractor (Ratra and Peebles, 1988) or tracker (Steinhardt, Wang, and Zlatev, 1999), meaning it is the asymptotic solution for a broad range of initial conditions at high redshift. That includes relaxation to a near homogeneous energy distribution even when gravity has collected the other matter into nonrelativistic clumps. Second, the energy density in the attractor solution decreases less rapidly than that of matter and radiation. This allows us to realize the scenario: after inflation but at high redshift the field energy density rhoPhi is small so it does not disturb the standard model for the origin of the light elements, but eventually rhoPhi dominates and the universe acts as if it had a cosmological constant, but one that varies slowly with position and time. We comment on details of this model in Sec III.E.

Next Contents Previous