Can someone explain probability flux in the tunneling boundary condition of Vilenkin? This is what's leading to the notion of a quantum universe tunneling from nothing into existence, right? The idea is that probability flux flows out of superspace (configuration space) at singularities and into superspace at non-singular geometries at past timelike infinity if I have it correct.
If I were to picture it, it's a 2 dimensional Minkowski space (coordinates (a, phi)) where there could be, arbitrarily, a ring representing the boundary composed of singularities at which probability flux goes out towards past timelike infinity until it reaches the non-singularity areas in the center (a=0, phi=finite) where it then turns "ingoing" and gets injected into superspace... and eventually back to the singular boundary and then outgoing again.
The outgoing wave should be present in the classically allowed region. 
So this means probability flux gets injected into superspace, universe tunnels into existence and as universe collapses back into singularity, probability flux goes back out of superspace?
 A: Thanks - that looks like a very useful reference.  A brief summary of my (superficial) understanding of what the paper is talking about:
The context is wave function(al)s on superspace, which is the space of three metrics $h_{ij}$ (and matter fields $\phi$) modulo diffeomorphisms.  Dirac quantization is applied, which results in the operator relations
$$H_i\Psi = 0$$ and
$$\mathcal{H}\Psi = 0$$
The latter is the Wheeler DeWitt equation, and is second order hyperbolic.
Minisuperspace is a vastly simplified version of superspace, in which only highly symmetric metrics are allowed.  The paper uses, for many of its examples, a minisuperspace with homogeneous and isotropic metrics for which the only parameter is the scaling factor (as in FRW metrics).
Having defined the Wheeler DeWitt equation for the wave functions on minisuperspace, the question of a probability measure arises.  We'd like such a thing in order, for example, to compare probabilities of universes having certain parameter sets.  The equation is a bit like the Klein Gordon equation in relativistic QM, so the question arises of defining a probability measure that doesn't produce negative probabilities.  
The minisuperspace itself has a metric (De Witt metric), which for the simple minisuperspace with coordinates $(a,\phi)$, makes it into a two dimensional space with a Minkowski signature metric.  Although this space-with-metric obviously has no direct interpretation in terms of space-time itself, the usual space-time techniques can be applied, for example a boundary can be attached as it is in the process of constructing Penrose diagrams.  This boundary consists of three-metrics-plus-matterfields which are in some way singular.  
Haliwell shows how the probability measure is defined in terms of the current derived from the wavefunction and a three-surface $\Sigma$ cutting across the trajectories.  The three-surfaces can't simply be defined in the obvious way as surfaces of constant timelike minisuperspace coordinate, since for a given trajectory, $a$ may change direction from increasing to decreasing (a universe can expand then contract).
For the simple $(a,\phi)$ minisuperspace case, the part of the boundary represented by $a=0; \phi$ finite is considered as non singular.  The other parts are singular.  In the terminology of minisuperspace as a Penrose diagram, the non singular part is past timelike infinity $i^-$.
Vilenkin's ansatz is to consider wavefunctions which consist of "outgoing modes" at the singular part of the boundary.  What this means is that, since the Wheeler De-Witt equation on minisuperspace with Minkowski signature metric is analogous to the Klein Gordon equation, we can look for solutions which are the analogs of positive and negative frequency KG modes.  The wavefunctions which Vilenkin selects have probability currents which originate at $i^-$ and terminate somewhere else on the boundary.  These are the outgoing modes which start off in the exponential region $a^2V(\phi)<1$, where the wavefunction does not correspond to a classical geometry, and enter the oscillatory region $a^2V(\phi)>1$, where it does represent classical geometry. The exit point on the singular part of the boundary represents a final singularity of the 4 geometry of the classical universe. 
So to answer your first question, yes, the application of the WKB approximation here is formally similar to its application in tunneling problems, except that here, the tunneling isn't really taking place "from" anything.
