Abstract
We extend the concept of Hawking-Moss, or up-tunnelling, transitions in the early universe to include black hole seeds. The black hole greatly enhances the decay amplitude, however, order to have physically consistent results, we need to impose a new condition (automatically satisfied for the original Hawking-Moss instanton) that the cosmological horizon area should not increase during tunnelling. We motivate this conjecture physically in two ways. First, we look at the energetics of the process, using the formalism of extended black hole thermodynamics; secondly, we extend the stochastic inflationary formalism to include primordial black holes. Both of these methods give a physical substantiation of our conjecture.
Highlights
At φ = φF and the true vacuum is at φ = φV
We extend the concept of Hawking-Moss, or up-tunnelling, transitions in the early universe to include black hole seeds
Note that the BHHM transition between black hole spacetimes with differing cosmological constants before and after the transition suggests that we explore the extended black hole thermodynamical description [21,22,23,24], in which the cosmological ‘constant’ determines a thermodynamic pressure P = −Λ/(8π)
Summary
The HM instanton represents a simultaneous up-tunnelling event from a false vacuum φF , to the top of a potential barrier φT. A fluctuation that was larger than the event horizon could not arise in a causal process Once we impose this constraint, we find that there is a natural cut-off in parameter space that keeps the instanton solutions in the range consistent with the semi-classical approximation. As the seed mass increases, the action decreases until we reach the red curve boundary This is the equal area curve, where the area of the cosmological horizon is the same for initial and final states. Before moving on to examine the physics of our conjecture, note that the line mT = mNT in figure 2 does not close up with the equal area curve at mF √= mNF This is because the cosmological horizons in the Nariai limit are rcF,T = F,T/ 3, which clearly do not coincide for F = T
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