**6. Conclusions**

It is an old expectation that *mathematical consistency* alone shall be sufficient to derive the boundary conditions in quantum cosmology [44]. The form of the LQC instantons sugges<sup>t</sup> that it might indeed be possible to identify *typical* smooth initial conditions due to quantum-geometry corrections. This demonstrates that one of the most fundamental proposal for the initial condition in quantum cosmology can appear naturally in another—LQC—due to putative corrections coming from quantum geometry. The introduction of the no-boundary proposal in LQC has also opened new physical possibilities for the latter. Instead of replacing the big bang singularity with a deterministic

bounce, as is predicted by semiclassical states in some restricted models of LQC, this opens up the opportunity to allow for Euclidean trajectories leading to a bubble nucleation of our universe. The semiclassical saddle-point approximation of the no-boundary proposal is distinct from the semiclassical sharply-peaked states in LQC and therefore the former acts as an example of how a new state can unveil novel features in a well-established theory. At this point, it is difficult to compare the probability of a bounce versus that of tunneling of the universe from nothing. However, the no-boundary proposal also provides a (set of) *natural* initial conditions for considering inhomogeneous perturbations in LQC leading to effects observable from early-universe cosmology without having to resort to *ad hoc* choices for the initial state.

Regarding the geometry of no-boundary instantons in LQC, we have demonstrated that the feature of having an infinite tail distinguishes these instantons from the original Hartle-Hawking ones in Einstein gravity. We end our discussion with a few caveats. Firstly, it has been pointed out recently that the Euclidean path integral in gravity is not a good approximation for the original Lorentzian path integral due to several conceptual reasons [21]. However, even if one works with the Lorentzian path integral and applies a different mathematical trick (Pecard-Lefshetz theory) to improve its convergence, the resulting theory typically has runaway perturbations due to the old conformal factor problem in gravity [22–24]. Interestingly, LQC can come to the rescue of the no-boundary proposal [45], written as a Lorentzian path integral, even in this case. However, the main physical effect from LQC responsible for this is 'dynamical signature-change' [46–48], something we have ignored in this work as a first pass. The second caveat is regarding the fact that our discussions were limited to the case of a pure cosmological constant in this paper. Indeed, the more interesting physical scenario is that of having a scalar field in some potential. However, preliminary investigations have already revealed that the solution space of the no-boundary wavefunction is greatly enhanced for such a system in the presence of LQG corrections. Finally, one can ask how physical is the fact that the tail of these instantons are stretched to infinity? As already mentioned, this is only true in Euclidean time and therefore not directly meaningful. However, it might even be possible that for some different gauge choice (i.e., *N* = 1), one can even avoid such an infinite stretching altogether. Nevertheless, all the interesting effects of having such a geometry as explained in this paper would still be valid in this case. Most importantly, no matter what the gauge choice, the remarkable conclusion that the modified Friedmann equation in LQC not only allows for the no-boundary condition to be imposed, but also somehow makes it more natural, seems to be robust and points towards a new paradigm in quantum cosmology merging these two mainstream approaches.

**Author Contributions:** All of the work was done equally by both the authors.

**Funding:** This research was supported in part by the Ministry of Science, ICT & Future Planning, Gyeongsangbuk-do and Pohang City and the National Research Foundation of Korea Grant No. 2018R1D1A1B07049126.

**Conflicts of Interest:** The authors declare no conflict of interest.
