claim that one is bigger than the other.
On the other hand, we saw in the case of gravity with a cosmological constant but no
matter fields that the no boundary condition could lead to a universe that was predictable
within the limits of quantum theory. This particular model did not describe the universe
we live in, which is full of matter and has zero or very small cosmological constant. However one can get a more realistic model by dropping the cosmological constant and including
matter fields. In particular, one seems to need a scalar field φ with potential V ( φ). I shall assume that V has a minimum value of zero at φ = 0. A simple example would be a massive scalar field V = 1 m 2 φ 2.
2
V(φ)
φ
Energy – Momentum Tensor of a Scalar Field
Tab = φ,aφ,b − 1 gabφ,cφ,c − gabV ( φ)
2
52
One can see from the energy momentum tensor that if the gradient of φ is small V ( φ) acts like an effective cosmological constant.
The wave function will now depend on the value φ 0 of φ on Σ, as well as on the induced metric hij. One can solve the field equations for small round three sphere metrics and large values of φ 0. The solution with that boundary is approximately part of a four sphere and a nearly constant φ field. This is like the de Sitter case with the potential V ( φ 0) playing the role of the cosmological constant. Similarly, if the radius a of the three sphere is a bit bigger than the radius of the Euclidean four sphere there will be two complex conjugate
solutions. These will be like half of the Euclidean four sphere joined onto a Lorentzian-
de Sitter solution with almost constant φ. Thus the no boundary proposal predicts the spontaneous creation of an exponentially expanding universe in this model as well as in
the de Sitter case.
One can now consider the evolution of this model. Unlike the de Sitter case, it will not
continue indefinitely with exponential expansion. The scalar field will run down the hill of
the potential V to the minimum at φ = 0. However, if the initial value of φ is larger than the Planck value, the rate of roll down will be slow compared to the expansion time scale.
Thus the universe will expand almost exponentially by a large factor. When the scalar
field gets down to order one, it will start to oscillate about φ = 0. For most potentials V , the oscillations will be rapid compared to the expansion time. It is normally assumed that
the energy in these scalar field oscillations will be converted into pairs of other particles and will heat up the universe. This, however, depends on an assumption about the arrow
of time. I shall come back to this shortly.
The exponential expansion by a large factor would have left the universe with almost
exactly the critical rate of expansion. Thus the no boundary proposal can explain why
the universe is still so close to the critical rate of expansion. To see what it predicts
for the homogeneity and isotropy of the universe, one has to consider three metrics hij
which are perturbations of the round three sphere metric. One can expand these in terms
of spherical harmonics. There are three kinds: scalar harmonics, vector harmonics and
tensor harmonics. The vector harmonics just correspond to changes of the coordinates xi
on successive three spheres and play no dynamical role. The tensor harmonics correspond
to gravitational waves in the expanding universe, while the scalar harmonics correspond
partly to coordinate freedom and partly to density perturbations.
One can write the wave function Ψ as a product of a wave function Ψ0 for a round
three sphere metric of radius a times wave functions for the coefficients of the harmonics.
Ψ[ hij, φ 0] = Ψ0( a, ¯
φ)Ψ a( an)Ψ b( bn)Ψ c( cn)Ψ d( dn) 53
Tensor harmonics – Gravitational waves
Vector harmonics – Gauge
Scalar harmonics – Density perturbations
One can then expand the Wheeler-DeWitt equation for the wave function to all orders in