Before Inflation

The inflation theory is a nice way in which to explain some of the mysteries which have bothered people about our world view of the Universe, in particular, the horizon and flatness problems. However, inflation does not address the ultimate question of the creation of the Universe. Here, we consider some recent speculations on this interesting and important (and probably currently intractable) problem.

The treatment of the origin of the Universe probably requires that we understand the nature of things during the Planck era. The notion of a smooth space-time in this era may break down and we may ultimately need to develop a consistent theory which weds the quantum theory and the theory of relativity (theory of gravity). At the present time, this feat has not been accomplished.

QUANTUM THEORY

Off and on throughout this year we have brought up aspects of the Quantum Theory (as needed). This in itself is interesting as it means that it is necessary to consider physics on the smallest and lowest energy scales when we try to understand the properties and nature of the Univere which is physics on the largest possible and highest energy scales imaginable. Let us review some things:

• Heisenberg Uncertainty Principle (HUP). As commonly stated, the HUP is
  dx dp > h/(2 pi)
  that is, one can not determine both the position and momentum of a particle to infinite precision. There will always be an uncertainty in such measurements. This leads to indeterminism in the quantum mechanical world in that one cannot predict trajectories (paths) individual particles follow with 100 % certainty. What happens is that one calculates many possible states with different probabilities of realization. When the motions of a bunch of particles are measured, we find that the paths they follow are apparently distributed as predicted by quantum mechanics. The HUP is a consequence of the wave-like nature of particles.

  The predictions of one form of quantum mechanics encodes the information on the motion of particles in something referred to as the Wave function and uses the Schroedinger Equation to calculate wave functions.

• An odd feature of the quantum mechanical world is the process known as Tunneling. We met this process in our discussion of nuclear fusion. In the Sun, the particles are not energetic enough to overcome (classically) the electrical repulsion between the nuclei. What happens is that sometimes particles can appear closer together than allowed by classical arguments. We say that the particles tunnel through the barrier which separates them. The tunneling does not occur in real time. It occurs in imaginary time, that is, time mulitplied by sqrt(-1). This is a strange world as time does not behave like the time we know; time behaves more like a spatial coordinate in this type of place.

QUANTUM FLUCTUATIONS (Tryon 1973)

Recall that the HUP can also be stated as

dE dt > h/(2 pi)

that is, one cannot measure both the size of an energy fluctuation and the time over which the fluctuation persists. (This has the consequence that no object will always have zero energy; quantum fluctuations will always be around). Well, let's look at the Universe for a second

-k = 1/2 [dR(t)/dt]**2 - (4 pi G/3) rho(t) R(t)**2 - (CC/3) R(t)**2

If inflation is correct, k = 0 and the energy of the Universe can be though of as zero (in the Newtonian way). Well, ignore this for a second and ask the question, If the Universe is a quantum mechanical energy fluctuation, how small must the energy fluctuation be in order for the Universe to have lasted for 10 - 20 billion years?

energy ~ (h/2 pi) / (10 - 20 billion years)

====> energy ~ 10**(-27) / (3-6) x 10**17 seconds ~ (2-3) x 10**(-45) ergs

or an amount of mass ~ 10**(-41) m(proton)!!

NO BOUNDARY CONDITION CONDITION (Hartle and Hawking)

• In a normal world, we can represent the evolution of the Universe as in (a) and (b). The radii of the rings represent the size of the Universe and time increases upward. As time moves backward (downward) the Universe gets smaller and eventually compresses into a singularity (a) or ends abruptly (b). This is unavoidable in standard (classical) theories. In both situations, since things emanate from well-defined conditions, one needs to specify initial conditions (twiddle some knobs) in order to describe how the rest of the evolution proceeds.

• Hartle and Hawking proposed something different. In their nonclassical (quantum mechanical) world, another type of solution becomes possible. They proposed that the tubes did not end abruptly but rather, proposed tubes which ended in a smooth and regular fashion (c). Just as for spheres, these smooth tubes have no boundary and thus, in a sense, Hartle and Hawking have escaped from the need to specify an initial condition for the Universe.

  The smooth end is not allowed in classical theory; it is a quantum mechanical beast and takes place in imaginary time (just as does tunneling). In the smooth end, time behaves like a spatial dimension; it does not behave like time as we know it. What this means is that it does not make sense to talk about the time before our Universe began as time does not exist in the sense in which we understand time.

TUNNELING UNIVERSES

Noting that imaginary time is also found in tunneling processes, Vilenkin and Linde proposed that perhaps the Universe tunneled through from nothing,

"Quantum Foam" ----- "tunnels" -----> Universe

The physics of the tunneling process are purported to select the most likely universe out of the plethora of possibilities (according to some people, inflationary universes are selected). The quantum foam is an era when space-time was not smooth; it was jumbled and jumps in space and time were allowed.