Abstract
We study the geometry of Euclidean instantons in loop quantum cosmology (LQC) such as those relevant for the no-boundary proposal. Confining ourselves to the simplest case of a cosmological constant in minisuperspace cosmologies, we analyze solutions of the semiclassical (Euclidean) path integral in LQC. We find that the geometry of LQC instantons have the peculiar feature of an infinite tail which distinguishes them from Einstein gravity. Moreover, due to quantum-geometry corrections, the small-a behaviour of these instantons seem to naturally favor a closing-off of the geometry in a regular fashion, as was originally proposed for the no-boundary wavefunction.
Highlights
The introduction of the ‘no-boundary’ proposal in loop quantum cosmology (LQC), for minisuperspace models, has unveiled a lot of interesting physical possibilities [1]
Universe 2019, 5, 22 in its phase space realization, retains all the crucial aspects of the quantum geometry which appear in the canonical LQC.) Other than this, our formalism shall be exactly the same as in the original no-boundary proposal: We shall look only at the saddle-point approximation of the Euclidean path integral and consider the wavefunction to be a functional of the value of the scale factor only at the final boundary
The interesting fact is the quantum-geometry regularization is such that this tail closes the geometry in a regular way without requiring any additional fine-tuning even though the field equations are heavily modified in LQC. This is suggestive of the fact that the no-boundary proposal is robust and, if anything, such a necessary tail-off of LQC instantons to zero points towards it being more natural in the presence of quantum-geometry corrections
Summary
The introduction of the ‘no-boundary’ proposal in loop quantum cosmology (LQC), for minisuperspace models, has unveiled a lot of interesting physical possibilities [1]. Universe 2019, 5, 22 in its phase space realization, retains all the crucial aspects of the quantum geometry which appear in the canonical LQC.) Other than this, our formalism shall be exactly the same as in the original no-boundary proposal: We shall look only at the saddle-point approximation of the Euclidean path integral and consider the wavefunction to be a functional of the value of the scale factor only at the final (spatial) boundary. One gets rid of the singular behaviour of any function which contains some inverse power of a due to the replacement by these aforementioned inverse-triad corrections Once again, their form for minisuperspace cosmologies is rather simple, as shall be explicitly demonstrated later. This is suggestive of the fact that the no-boundary proposal is robust and, if anything, such a necessary tail-off of LQC instantons to zero points towards it being more natural in the presence of quantum-geometry corrections
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