The above paper "Creation of Universes from Nothing" was published in 1982, which was subsequently followed up in 1984 by a paper titled "Quantum Creation of Universes". I decided it would be a good idea to talk about these proposals, since last time I talked about the Hartle-Hawking model which was, as it turns out, inspired by the above work.
Alexander Vilenkin also explains in a non-technical way the essential idea in his book; Many World's in One – one of the best books I've ever read – it mostly covers cosmic inflationary theory but the 17th chapter covers how inflation may have begun. In fact Vilenkin is one of the main preponderant who helped develop inflation along with Steinhardt, Guth, Hawking, Starobinsky, Linde and others.
Although I won't talk about it here, Vilenkin also discovered a way of doing cosmology by using something called "topological defects" and he has been known for work he's done on cosmic strings, too.
In explaining Vilenkin's 'tunneling from nothing' model its helpful to first reconsider the Hartle-Hawking no boundary proposal (even though this came later), Hartle-Hawking describe a universe which contracts from large size, it tunnels through a "nonphysical" region and at t = 0 begins to expand into our classical Big Bang universe. Vilenkin's model is similar except that he avoids the existence of this "prior" universe by setting some boundary conditions at beginning of time* and interpreting the nonphysical region as lacking any classical description, not just a quantum vacuum but a state devoid of all space, time, matter and energy.**
This here is a screenshot I've taken of a diagram that appears on pg. 180 of Vilenkin's book "Many World's in One" from chapter 17.
Try to imagine a rubber ball 'tunneling' through a barrier in classical physics, the limit of the balls motion will be when its total energy equals its potential energy, E = V(x). So if the ball's energy is less than the threshold to overcome the barrier, there will exist a "forbidden region" on the other side where you have a zero probability of finding the ball. Now imagine the ball is a particle described by a wave function, (which obeys a time independent Schrodinger equation).
This time you will find a solution on either side of the barrier including in the classically forbidden region, on one the incoming and reflected wave and on the other side, a transmitted wave. The probability of finding the particle between x and x + dx is given by the standard Born rule:
The origin of the universe is a lot like this, in Vilenkin's proposal. In fact the Wheeler-DeWitt equation is identical to a time independent Schrodinger equation if the particle has one-half unit mass, zero energy and subject to a certain potential. In this case, the region smaller than the universe's minimum radius is the "classically forbidden" region and that in affect will tell us how to select the right boundary*** condition; we don't want a universe with a contracting phase like Hartle-Hawking but one with just an outgoing region (refer back to the diagram above if this is confusing).
Just like Hartle-Hawking the solution will give you two different components of the wave function of the universe in the minisuperspace**** model. One is an exponential component and the other is an oscillating component, our universe at a = 0 what Vilenkin interprets as literally nothing, may simply tunnel through the "classically forbidden region" and begin to inflate.
A lot more maths is required to fully grasp the model but based on how physicists obtain a probability for ordinary quantum tunneling events and based on the potential selected earlier you can fairly easily show the probability amplitude for a small FRW universe filled with false vacuum and some bits of matter is
Of course this model has been simplified an awful lot.***** As you've probably by now worked out, the probability is an exponential decay, which means tunneling to large size in incredibly unlikely but Vilenkin is able to explain this away by appealing to the Many World's Interpretation of quantum mechanics.******
* Hartle-Hawking (1983) does not set any boundary conditions, hence the name "no boundary proposal".
** The laws of physics would still have some kind of Platonic existence.
*** We have to be very careful when we use the term "boundary" in the context of quantum cosmology, in the tunneling model it does not represent a physical boundary or a singularity. Instead its a boundary on the space of all physically possible universes.
**** A minisuperspace is an abstract mathematical space, like Hilbert spaces or phase space but one which has the number of degrees of freedom of gravitational and matter fields restricted. Each point in the minisuperspace is a 3-geometry and a path through that minsuperspace represents a possible history of a (3+1 dimensional) FRW universe.
***** Christopher Isham has written an excellent non-technical description of the model in his article "Quantum Theories of the Creation of the Universe".
****** The MWI to me at least, is unappealing and this may be one line of criticism the model is open to.