String Theory and Cosmology
(2003, NS 127)
Ulf Danielsson, Ariel Goobar, Bengt Nilsson
August 14-19
Sigtuna, Sweden
Proceedings: to be published in Physica Scripta, The Royal Swedish Academy of Sciences in 2004.

 

A Non Singular Universe
by S. Hawking

The last chapter of my PhD thesis, contained my first singularity theorem. This showed that under certain reasonable conditions, any cosmological solution of the field equations, would have a big bang singularity. At this singularity, classical general relativity would break down, so one could not use it to predict how the universe began. It was therefore necessary to develop a quantum theory of gravity, in order to understand the origin of the universe.

We are still working on this problem, and haven't yet reached a definitive conclusion. But in this talk, I want to explore if the origin of the universe can be semi classical, and non singular. This is possible despite the singularity theorems, because like so many other no go theorems, they have a get out clause. In this case, the get out is the strong energy condition.

The strong energy condition, is the inequality on the energy momentum tensor, that ensures gravity is attractive in any frame. It is satisfied by the Maxwell and Yang Mills fields, and by the gauge field of eleven dimensional supergravity. Thus it is not possible to avoid cosmological singularities, in purely classical M theory.

On the other hand, a scalar field, or quantum corrections, can violate the strong energy condition. I shall therefore investigate models with a scalar field with an effective potential that includes quantum corrections. I shall take the effective gravitational theory to be four dimensional, because we have no observational evidence for extra dimensions back to the period of inflation. At this time, any extra dimensions must either be compactified smaller than the Hubble radius, or be part of an anti de Sitter space, on which the universe lives on a three brane. In either case, the effective theory is four dimensional.

An example would be trace anomaly inflation. This can be regarded as four dimensional gravity, coupled to the quantum effective action, of a large number of light matter fields. The trace anomaly, will generate an effective cosmological constant, which can give de Sitter space, as a self consistent solution. By ADS, CFT, the quantum effective action of the matter fields, in the four dimensional spacetime, can be calculated by taking the spacetime to be the boundary of ADS5. This means trace anomaly inflation, is more or less the world sheet theory of a stack of D3 branes.

There can be other four dimensional effective theories, that do not obey the strong energy condition. I shall therefore discuss theories with a scalar, phi, and a potential, V. in order to end up in nearly flat space, I shall assume that the scalar potential, has a minimum at, or very near, zero. I shall say that the potential belongs to class A, if it has a local or global maximum, which is not too sharply peaked, that is the second derivative is below a certain bound. Potentials with a maximum that is more sharply peaked, belong to class B. Potentials without a maximum, belong to class C.

Historically, potentials of class B were used in the old inflation scenario. The potentials were of class A in the new inflation scenario, and of class C in chaotic inflation.