The Real No-Boundary Wave Function in Lorentzian Quantum Cosmology

TL;DR

The authors provide a Lorentzian formulation of the no-boundary wave function by evaluating a lapse-integrated path integral with a real-axis contour via Picard-Lefschetz theory. In a de Sitter minisuperspace, the method yields a real solution to the Wheeler-DeWitt equation that describes two identical, time-reversed ensembles of inflationary universes with nearly Gaussian fluctuations. Extending to a scalar field with a cosh potential, they identify the classical regime that produces regular, slow-roll histories and demonstrate Gaussian perturbations consistent with no-boundary regularity. Comparing to prior Lorentzian approaches, the work argues that the chosen contour leads to physically sensible predictions and motivates holographic and future-boundary perspectives on cosmology.

Abstract

It is shown that the standard no-boundary wave function has a natural expression in terms of a Lorentzian path integral with its contour defined by Picard-Lefschetz theory. The wave function is real, satisfies the Wheeler-DeWitt equation and predicts an ensemble of asymptotically classical, inflationary universes with nearly-Gaussian fluctuations and with a smooth semiclassical origin.

Figures (2)

• Figure 1: The four saddle points \ref{['dSMSPsaddles']} of the contour integral \ref{['wavefunction']} in the complex $N$-plane together with their steepest ascent and descent curves. In the shaded region $\text{Re}(i S_0) > 0$, suggesting divergent behaviour of an integral along a contour running to complex infinity or to the essential singularity at the origin in this domain. The Lorentzian contour $\mathcal{C}=(-\infty,+\infty)^{\downarrow}$ avoids the origin by passing along a parametrically small circle of radius $\varepsilon$ below that point. Analyticity away from $N=0$ ensures that the value of $\varepsilon>0$ does not affect the outcome of the integral. The parameter values $\Lambda = 3$ and $q_1 = 10$ were taken and to lift the degeneracy we considered the perturbation $S_0 \rightarrow S_0 + i \, .02 N^2$.
• Figure 2: The continuous deformation ${\cal C}'$ implied by Picard-Lefschetz theory of the original contour $\mathcal{C}=(-\infty,+\infty)^{\downarrow}$ that passes through the two saddle points in the lower half complex $N$-plane. The two Lefschetz thimbles both tend to the negative imaginary axis at complex infinity, where $\text{Re}(i S_0) \rightarrow -\infty$. The contribution coming from the 'arc at infinity' connecting the Lefschetz thimbles and the positive and negative real $N$-axes vanishes.