10. LINDE INFLATION WITH MONOTONIC V()

The Higgs potential and the corresponding inflationary scenario were based on two ideas: (1) the Higgs potential is needed in other branches of particle physics (one does not have to shoulder the burden of inventing it) and (2) the preinflationary stage was assumed to be an extremely hot universe; this is a simple extrapolation of our current knowledge about the 3 K background, nucleosynthesis, etc.

However, both of these premises have been losing some of their appeal in recent years:

(1) There is a direct argument for the existence of scalar fields. In the supersymmetry theories, the existence of scalar fields is a consequence of existence of the experimentally verified spin- 1/2 fields - the electron quarks, etc.

Those scalar fields which have not yet been experimentally observed may have a V() without a maximum. The form V = (/4) ⁴ is appealing, because is dimensionless. The simple form 1/2 µ^{2 2} is also not excluded, although the question of what can be used to calculate µ arises in this case.

(2) "Complete" cosmological theories (see, for example, (Quantum Gravity 1984)) which intend to describe the spontaneous birth of the universe are now more fashionable than before. This is a case of feedback from the inflationary expansion theory - we shall explain this loose statement in the last section of the paper.

In the spontaneous creation scenario, one can imagine the birth of a very small closed universe with a large, already-imprinted scalar field. This field is coherent (or almost coherent) and uniform precisely because the newborn universe is very small.

The new approach proposed by Linde (1983) is to look at (8.1), (8.2), and (8.3) in Section 8 with a monotonic V() and the following initial conditions: large , small a, and an already-built-in expansion velocity H > 0. I will illustrate this approach for the case V = 1/2 µ^{2 2}, omitting the curvature term in (8.3). We assume that ² is much less than µ^{2 2} and can be neglected (we shall see later on what the domain of applicability of this assumption is) and use units in which = c = 1. We then introduce the Planck mass M = M_Pl = G^-1/2 in place of G. In the CGS system, G = 6.67 x 10^-8 cm³f^-1s^-2 and

(10.1)

The second equation, (8.2), can be written in the form

(10.2)

In the first equation, (8.1), we neglect compared to 3H:

(10.3)

(10.4)

where the initial conditions are t = 0, = ₀ and a = a₀.

The radius of the universe is now given by

(10.5)

where the upper limit of the integral is the time t₁ at which has fallen to zero; the approximations we made above fail somewhat earlier, but this does not change the final result. We obtain

(10.6)

It is well known that cosmology requires an inflationary expansion of at least a factor of 10³⁰ = e⁷⁰. Therefore, we require

(10.7)

We shall now repeat the sequence of the calculations in simple words. We take the case of a scalar field that has no exact solution: = const and V() = 0, i.e., there is no "false vacuum". It turns out that one can still construct an expanding cosmological solution. The scalar field decreases more slowly than in the flat Minkowski case: the expansion plays the role of a brake or friction. Given a high enough initial ₀, the expansion that occurs during the slow decrease in is large enough.

If the mass of the scalar particles µ is much smaller than the Planck mass, M = 10^-5 g (= 10¹⁹ GeV), then the initial ₀ is much greater than µ, but only on the order of 2.5-4 times larger than the Planck mass M.

The ideal de Sitter solution, exp (H₀t), is now changed to exp ( H dt). The beautiful symmetry of the de Sitter solution is lost. However, this is an achievement of the theory; not a deficiency. This makes the transformation from the inflationary de Sitter H₀t or quasi-de Sitter H(t) dt solution to the plasma solution easier.

The regime changes simultaneously at all points in space. When becomes of order µ, the universe goes into an oscillatory regime in which cos µ t; the scalar particles then decay into plasma. However, this is beyond the scope of our review. One difficult (and as yet unsolved) question is, to what extent was the spatial uniformity of ₀ created during the birth of the universe and to what extent is it required as an initial condition? What would happen if ₀ were nonuniform? Linde argues that the region with the largest initial ₀ expands more rapidly; its final volume will be much larger than that of the regions that start out with smaller ₀, so that those regions will be unimportant. His idea is that one can begin with random values of ₀ in different regions of space; the inflationary process will choose the region with maximum ₀ by itself and smooth out the effects of nonuniformity. This is why this is called the "chaotic theory". We shall not discuss this theory any further, but discuss some simpler points instead.