When we were describing the properties of the scalar field in Section 5, we emphasized its versatility. It can produce p = + for large ^{2} and p = - for small ^{2}.
Belinskii and Khalatnikov (1972), who studied the singularity in a space with a massless scalar field, obtained the stiff behavior p = + (with a corresponding a(t) (t_{0} - t)^{1/3} cosmological singularity; see the original paper for details). The inflationary universe is based on the p = - or p - situation created by the scalar field. How can these different approaches be reconciled? Or, more appropriately, what is the rule for choosing one extreme case over the other?
The physical essence of the answer can be obtained by solving a simplified problem.
Assume a given constant rate of expansion H - this does not violate any general principle, since one can imagine fields other than the scalar field that also contribute to the energy density in the right-hand side of the cosmological equation. These fields ensure that = const and H = const over the entire time under consideration. On the other hand, we assume that these additional fields do not interact directly with the scalar field . The (inevitable) gravitational interaction between the fields is transmitted via the expansion rate; this interaction has already been included in the equation for , which we rewrite here:
For constant H, this is a linear equation. The general solution can easily be obtained by a college freshman:
where
and the constants _{1} and _{2} are determined by the initial conditions.
We shall study the case in which the exponent is real (a complex would mean an oscillatory solution).
To do this, we assume that H is large: 9H^{2}/4 >> µ^{2}. We shall discuss two cases:
(1) H > 0: Expansion
In this case, the general solution is a sum of two decreasing solutions, the first of which, _{1} exp (-_{2}t), is slowly decreasing, and the second of which, _{2} exp (-_{2}t), is rapidly decreasing. It is clear that in general, when both _{1} 0 and _{2} 0 and _{1} and _{2} are of the same order, it is the slowly decreasing solution, _{1} exp (-_{1}t), that survives beyond the time span required to eliminate the rapidly decreasing solution _{2} exp (-_{2}t).
But the very fact that the solution is slowly decreasing obviously means that ^{2} is small. So we obtain the qualitative result that the solution which survives the expansion has a slow time dependence, which means that ^{2} is small and, consequently, that p -.
We shall now write down the formula for p / in the limiting case H^{2} > µ^{2} for the "slow" solution. Carrying out a series expansion, we obtain
Therefore,
As µ^{2} / H^{2} -> 0, p -> -.
It is instructive to see how the ratio p / varies during the expansion (see Fig. 6, where the upper and lower horizontal lines are the extreme values p / = +1 and p / = -1; the upper dotted line describes p / for the pure rapidly decreasing _{2} exp (-_{2}t) solution (p / = 1 - 2/9 µ^{2} / H^{2}); the lower dotted line is p / for the pure slowly decreasing solution _{1} exp (-_{1}t), which approaches - 1 at large H; and the mixed solutions are shown by the dot-dash lines.
Figure 6. |
Since the mixed solutions are of two types, depending on whether _{1} and _{2} have the same or opposite sign, it is sufficient to calculate one solution of each type; all others of a given type can be obtained by simple translation along the t axis. The entire band between p / = -1 and p / = 1 is therefore filled, with two solutions going through every point.
What is the salient characteristic of Fig. 6? The general solutions diverge from the upper dotted line and approach the lower dotted line. One concludes that, in general, p - in the rapid expansion case.
(2) H < 0: Contraction
It is now easy to study the case of contraction, i.e., H < 0, in the same way. We have two solutions in this case as well: a rapidly growing solution and a slowly growing solution. The rapidly growing solution dominates in the general case. We shall not carry out all of the trivial calculations: Fig. 6 is valid with the time axis reversed!
So we conclude that p + for compression. That is the end of the story; the differences between the inflationary models and Belinskii and Khalatnikov's work have been resolved.
There is a moral to be learned from these simple calculations. The result by and large conforms to Braun and Le Chatelier's principle, which also holds true in human relations:
Every system resists outside forces.
The scalar field expands, and a negative pressure (or tension) builds up. If the expansion were created by the motion of a piston in a cylinder containing , the tension would decelerate the piston. On the other hand, if the field is being compressed, a positive pressure builds up, producing a force opposing the motion of the piston. The above remarks are very crude. Belinskii, Grishchuk, Khalatnikov, and the author (1985) have analyzed the exact self-consistent equations of the field and the universe in a detailed paper. This is not the right place to give all of the results. The important result is that the general tendency - the build-up of negative pressure due to expansion - is conformed.