| MadSci Network: Physics |
Hi Patrick! I am not really an expert on inflation. But I guess, you have messed a few things up a bit. I have found a nice reference on inflation, however.
In standard cosmology time evolution of the (model) universe is given by a
solution to the socalled Friedmann equation, which is also stated
in the reference above. The time derivative ot the scale, i.e., the size
of the universe, is determined by the curvature of space-time and by
energy density. This leads to solutions showing a polynomial dependence on
time. So there is no inflation. Inflation was invented to explain the
local inhomogeneity of our universe. Quantum fluctuations are pushed up to
galaxies and star clusters.
As I understand it, quantum tunneling does not allow the universe to
inflate, but it ends the inflation period! The usual approach to inflation
is introducing an additional term in the Friedmann equation. This
additional term results from a scalar particle (phi) one has to introduce.
This particle is called inflaton, and it is assumed to be that heavy that
it cannot be observed by today's accelerators. The contribution of the
inflaton consists of a kinetic term (derivatives) and a potential V(phi).
Let us assume that we can neglect the contribution of the kinetic term
with respect to the contribution of the potential. The solution of the
modified Friedmann equation we get, when we also consider an inflaton are
exponentials,
a = A Exp(Ht)a is the since of the universe, A is a positive constant, t is the time, and H is the Hubble constant. Therefore the introduction of a scalar particle results in an inflationary behaviour of the universe. Now the potential of the inflaton has to be chosen such that the inflation may end after a suitable time, since we do not have any inflation nowadays. One way is to assume that we start in a local minimum of the potential (i.e., a classical stable state). The global minimum of the potential has the value V=0. That means, there is no contribution of the inflaton to the evolution of the universe at the global minimum, and therefore no inflation. Quantum mechanically the local minimum we start off is not stable, but this system will tunnel to the global minimum state. The tunneling time (duration of the inflationary state) is given by the hight and width of the potential barrier seperating both minima. These values have to be adjusted to match experimental data. If the inflaton tunnels to its potential minimum, inflation vanishes.
However, one can also explain the ending of the inflationary phase of the universe differently, without using quantum tunneling. In fact this method is more widely used. It is also discussed in the reference given above.
If one talks about the wave function of the universe, it is an attempt to introduce quantum mechanical ideas in the theory of gravity. The whole universe is not considered as a particle, but as a quantum mechanical system, described by a wave function which obeys Schrödinger's equation. So the wave function is a superposition of posible eigenfunctions of the Schrödinger equation. An eigenfunction is a possible universe. Observation determines in which eigenfunction we live, and the wave function "collapses". Time evolution is given by the Schrödinger equation. The probability of observing a specific eigenstate (universe) is given by the square of the absolute value of the coefficient of that eigenstate, not by its frequency (energy eigenvalue). E.g. consider the following wave function
psi = a . psi1 + b . psi2,where psi1 and psi2 are the possible eigenstates (universes). The probability that we live in universe psi1 is given by |a|^2.
I hope you could help you on
greetings,
Michael.
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