No Rescue for the No Boundary Proposal

TL;DR

The paper scrutinizes the Hartle-Hawking no-boundary proposal by formulating the Lorentzian gravitational path integral and evaluating it semiclassically with Picard-Lefschetz theory. It finds that perturbations are governed by an inverse Gaussian distribution and that no choice of complex lapse contour can avoid unsuppressed fluctuations, even when backreaction is considered nonperturbatively. The analysis argues that topology change via smooth no-boundary processes is ill-defined and that viable quantum cosmology likely requires singular or nontrivial pre-inflationary phases. These results cast doubt on the no-boundary scenario for quantum de Sitter and inflation, while highlighting the need for alternative formulations of quantum cosmology.

Abstract

In recent work, we introduced Picard-Lefschetz theory as a tool for defining the Lorentzian path integral for quantum gravity in a systematic semiclassical expansion. This formulation avoids several pitfalls occurring in the Euclidean approach. Our method provides, in particular, a more precise formulation of the Hartle-Hawking no boundary proposal, as a sum over real Lorentzian four-geometries interpolating between an initial three-geometry of zero size, {\it i.e}, a point, and a final three-geometry. With this definition, we calculated the no boundary amplitude for a closed universe with a cosmological constant, assuming cosmological symmetry for the background and including linear perturbations. We found the opposite semiclassical exponent to that obtained by Hartle and Hawking for the creation of a de Sitter spacetime "from nothing". Furthermore, we found the linearized perturbations to be governed by an {\it inverse} Gaussian distribution, meaning they are unsuppressed and out of control. Recently, Diaz Dorronsoro {\it et al.} followed our methods but attempted to rescue the no boundary proposal by integrating the lapse over a different, intrinsically complex contour. Here, we show that, in addition to the desired Hartle-Hawking saddle point contribution, their contour yields extra, non-perturbative corrections which again render the perturbations unsuppressed. We prove there is {\it no} choice of complex contour for the lapse which avoids this problem. We extend our discussion to include backreaction in the leading semiclassical approximation, fully nonlinearly for the lowest tensor harmonic and to second order for all higher modes. Implications for quantum de Sitter spacetime and for cosmic inflation are briefly discussed.

Figures (13)

• Figure 1: The Morse function for the background is plotted in the complex $N$-plane, for a closed, homogeneous and isotropic $\Lambda$-dominated cosmology. The solid orange line is the defining integration contour ${\cal C}$, and the dashed orange line is the corresponding deformed contour, passing along Lefschetz thimbles. As $N$ tends to infinity in the complex $N$-plane, the real part of the exponent in the integrand (the Morse function) tends to $+\infty$ in the red regions or $-\infty$ in the green regions. It is constant along the blue contours. Upper panel: the real Lorentzian contour $0^+<N<\infty$ used for the causal Lorentzian propagator. Lower left panel: the contour for $N$ running from $-\infty$ to $+\infty$below the origin, as proposed by Diaz Dorronsoro et al.Dorronsoro:2017. Lower right panel: the real part of the causal propagator, equivalent to a continuous contour for $N$ running from $-\infty$ to $+\infty$above the origin.
• Figure 2: The classical background geometries appearing in the no boundary path integral. Left: The regular, complex saddle point geometry. Middle: A real Lorentzian off-shell background geometry appearing at $N_-^2\leq N^2\leq N_\star^2$, possessing one strong singularity. Right: The real Lorentzian geometry appearing at $N^2> N_\star^2$, possessing two strong singularities.
• Figure 3: The branch cuts (in red) on the real $N$-axis, for $-N_+< N < -N_-$ and $N_-< N < N_+$, form impenetrable barriers for Picard-Lefschetz theory. The classical scale factor squared $q$ crosses zero for a second time (as in the right panel of Fig. \ref{['fig:onoffshell']}) on the blue lines. The Hartle-Hawking and Lorentzian-Picard-Lefschetz saddles are indicated HH and L-PL respectively. The gray lines are the lines of steepest ascent and descent emanating from the four saddle points, with the arrows indicating directions of descent.
• Figure 4: Our results imply that a smooth, $\Lambda$-mediated topology changing transition is ill-defined. Thus topology change should not be thought of as illustrated in the left panel (where the physical regions of spacetime are blue, space is horizontal and time vertical). Rather, topology change most likely requires passage through a singularity, where massless degrees of freedom play a crucial role in enabling the transition and extensions of the semi-classical methods employed in the present paper are needed.
• Figure 5: The Morse function $h = Re(iS/\hbar)$ around a branch cut, in units where $\hbar=1$ and for the parameters $\Lambda = 3, q_1 = 101, l=10, \phi_1=1.$ At the cut, the Morse function reaches its maximum at $N_\star = 10$ coming from the upper half plane, and its minimum also at $N_\star,$ though approaching the cut from below.