No-Boundary and Tunneling Wavefunction Proposals

No-boundary and tunneling proposals select cosmological wavefunctions by different regularity, contour, and outgoing-branch conditions. Because the gravitational lapse and conformal directions require complex cycles, the declared thimble determines which saddles contribute. A semiclassical weight is not a normalized probability until a physical inner product and measure are supplied.

Required background. Wheeler–DeWitt Cosmology: Boundary Conditions, Inner Products, and Probabilities supplies interpretation. Conformal-Factor Problem, Integration Contours, and Resurgent-Completion Proposals supplies contour requirements.

Helpful background. Negative Modes, Determinants, and Decay Prefactors and Contours, iε, and Initial-Boundary Terms supply fluctuation and state data.

Lapse contours and regular saddles

For a closed de Sitter minisuperspace with final squared scale factor $q_1$, integrating out $q(t)$ gives a lapse integral of the generic form

$$\Psi(q_1)=\int_{\mathcal C_N}dN\,\mu(N) \exp\left[\frac{i}{\hbar} \left(\frac{A}{N}+BN-CN^3\right)\right],$$

where $A,B,C$ depend on $q_1$, curvature, and $\Lambda>0$. Saddle lapses obey

$$3CN^4-BN^2+A=0,$$

so

$$N^2=\frac{B\pm\sqrt{B^2-12AC}}{6C}.$$

The four square roots correspond to different complex geometries and WKB branches. Their mere existence does not determine intersection numbers with $\mathcal C_N$.

Application: compare branch selection

The Hartle–Hawking proposal sums compact regular Euclidean geometries with one final boundary Hartle and Hawking 1983. The tunneling proposal selects an outgoing expanding WKB branch at large scale factor Vilenkin 1986. In a thimble calculation, start from an explicit Lorentzian or complex contour, compute upward-flow intersections, and list the contributing $N_s$:

$$\Psi\simeq\sum_s n_s\, \mu_s\frac{e^{iS(N_s)/\hbar}} {\sqrt{S''(N_s)}}.$$

Different cycles can reverse exponential weightings or mix expanding and contracting branches. Feldbrugge, Lehners, and Turok found that a particular Lorentzian contour leads to problematic inverse-Gaussian perturbation weights Feldbrugge et al. 2017. This is a contour-specific obstruction and a demand to state the cycle, not a license to omit perturbations.

Adversarial contour and perturbation tests

Cross a Stokes line and verify that thimble coefficients jump so the original integral remains continuous. Choose a homologically inequivalent contour and show which prediction changes. For each background saddle, compute scalar and tensor quadratic actions with the same contour; a background-damped wavefunction with enhanced fluctuations is not a viable semiclassical state.

Scope

These proposals define candidate boundary states within minisuperspace and semiclassical saddle expansions. They do not settle the physical measure, full superspace, or singularity resolution. A distinct difference-dynamics model follows on Loop Quantum Cosmology and Effective Difference Dynamics.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Feldbrugge, J., J.-L. Lehners, and N. Turok. “Lorentzian Quantum Cosmology.” Physical Review D 95 (2017): 103508. DOI.
• Hartle, J. B., and S. W. Hawking. “Wave Function of the Universe.” Physical Review D 28 (1983): 2960–2975. DOI.
• Vilenkin, A. “Boundary Conditions in Quantum Cosmology.” Physical Review D 33 (1986): 3560–3569. DOI.