On the Problem of Initial Conditions for Inflation

Abstract

I review the present status of the problem of initial conditions for inflation and describe several ways to solve this problem for many popular inflationary models, including the recent generation of the models with plateau potentials favored by cosmological observations.

Appendix: Quantum Creation of Universes with Non-trivial Topology

In the main body of the paper, I was using simple intuitive arguments not requiring familiarity with the tools of quantum cosmology. However, quantum cosmology [36] and the theory of quantum creation of the universe “from nothing” [37,38,39,40] allows to look at the problem of initial conditions from a different perspective.

Most of the related investigations described creation of a closed universe. But an initial size of a closed inflationary universe studied in [37,38,39,40] should be greater than $H^{-1}$, which is five orders of magnitude greater than the Planck length in the models considered above. In some cases, this may lead to exponential suppression of the probability of quantum creation of such universes, but this problem can be solved using anthropic considerations [40]. There is no exponential suppression of the probability of quantum creation of an open or flat compact universe [20, 28, 35]. I will briefly remind the corresponding results, following [20].

Consider a flat compact universe having the topology of a torus, $S_1^3$,

$$\begin{aligned} ds^2 = dt^2 -a_i^2(t)\,dx_i^2 \end{aligned}$$

(21)

with identification $x_i+1 = x_i$ for each of the three dimensions. We will assume for simplicity that $a_1 = a_2 = a_3 = a(t)$. In this case the curvature of the universe and the Einstein equations written in terms of a(t) will be the same as in the infinite flat Friedmann universe with metric $ds^2 = dt^2 -a^2(t)\,d\mathbf{x^2}$. In our notation, the scale factor a(t) is equal to the size of the universe in Planck units $M_{p}^{{-1}} = 1$.

In order to derive the Wheeler–DeWitt equation [36] for the compact flat toroidal universe, one should first consider the gravitational action

$$\begin{aligned} S = \int dt\, d^3 x\,\sqrt{-g}\,\left( -{1\over 2}{R} +{1\over 2}\partial _{\mu }\phi \,\partial ^{\mu }\phi -V(\phi )\right) \end{aligned}$$

(22)

and take into account that the volume of the 3D box is equal to $a^{3}$. Let us assume for a moment that $\phi$ is constant, which is the case if the field stays at the top of the potential, or at the dS plateau, as in the models which we discussed here. In this case one can represent the effective Lagrangian for the scale factor as a function of a and $\dot{a}$,

$$\begin{aligned} L(a) = -3\dot{a}^{2} a - a^{3}V\ . \end{aligned}$$

(23)

Finding the corresponding Hamiltonian and using the Hamiltonian constraint $H \Psi (a) =0$ yields the Wheeler–DeWitt equation

$$\begin{aligned} \left[ {d^2\over da^2} +12 a^4 V\right] \Psi (a) = 0 \end{aligned}$$

(24)

For large a, the solution of Eq. (24) can be easily obtained in the WKB (semiclassical) approximation, $\Psi \sim a^{{-1}}\exp \left[ \pm i {2a^3\sqrt{V}\over \sqrt{3}}\right]$; positive sign corresponds to an expanding universe. This approximation breaks down at $a \lesssim V^{-1/6}$. At that time the size of the universe is much greater than the Planck scale, but much smaller than the Hubble scale $H^{-1}\sim V^{-1/2}$. The meaning of this result, to be discussed below in a more detailed way, is that at $a \gg V^{-1/6}$ the effective action corresponding to the expanding universe is very large, and the universe can be described in terms of classical space and time. Meanwhile at $a \lesssim V^{-1/6}$, the effective action becomes small, the classical description breaks down, and quantum uncertainty becomes large. In other words, contrary to the usual expectations, at $a \lesssim V^{-1/6}$ one cannot describe the universe in terms of a classical space-time even though the size of the universe at $a \sim V^{-1/6}$ is much greater than the Planck size, and the density of matter as well as the curvature scalar in this regime remains small, $R = 4 V \ll 1$.

The general solution for Eq. (24) can be represented as a sum of two Bessel functions:

$$\begin{aligned} \Psi (a) =\beta \sqrt{a} \left( \mathrm{J}_{-{1\over 6}}\left( {2\sqrt{V} a^3\over \sqrt{3}}\right) + \gamma \, \mathrm{J}_{{1\over 6}}\left( {2\sqrt{V} a^3\over \sqrt{3}}\right) \right) , \end{aligned}$$

(25)

where $\beta$ and $\gamma$ are some complex constants, see Fig. 5.

Fig. 5

Two eigenmodes of the Wheeler–DeWitt equation for the wave function of the flat compact toroidal universe. The scale factor a is shown in units of $V^{-1/6}$

Figure 5 confirms that the WKB approximation is not valid at small a, and the “cosmic clock” starts ticking only at $a>V^{-1/6}$.

One can provide an alternative interpretation of this result, without invoking the Wheeler–DeWitt equation. By substituting the classical solution $a = e^{Ht}$ into the effective Lagrangian (23), one finds that the total action of the universe is proportional to $\sqrt{V} a^{3}(t) \sim \sqrt{V} a^{3}(t) e^{{3Ht}}.$

For $a < V^{-1/6}$, the action is smaller than 1, so the wave function does not oscillate. Not surprisingly, the total energy of the universe at the critical time when a becomes equal to $V^{-1/6}$ is of the same order as Hawking temperature $T_{H} =O(H)$, which corresponds to the typical energy of a single quantum fluctuation in dS universe. Thus the universe gradually emerges from nothing, and its wave function does not oscillate until its total energy reaches O(H).

Once the universe grows larger than $a \sim V^{-1/6}$, its action rapidly becomes exponentially large, classical description of the new-born universe becomes possible, and its topology becomes irrelevant due to the magic of inflation.

Of course, if it is so easy to create a homogeneous universe, it may not be too difficult to create an inhomogeneous universe as well. The most important conclusion of the investigation performed in [20, 28, 35] is that the probability of quantum creation of a compact homogeneous inflationary universe with non-trivial topology is not exponentially suppressed. Meanwhile the main result obtained recently in [22, 23] is that even if the new-born universe is grossly inhomogeneous, it typically becomes homogeneous at later stages of the cosmological evolution.