Analysis of "No smooth beginning for spacetime"
In "No smooth beginning for spacetime," Feldbrugge, Lehners, and Turok present an examination of two prominent semiclassical proposals regarding the universe's origin: the Hartle-Hawking no boundary proposal and Vilenkin’s tunneling proposal. Both rely on integrating over 4-geometries bounded by a final 3-dimensional geometry. The authors introduce Picard-Lefschetz theory to address the path integral formulation for gravity, which is pivotal for discerning the meaningfulness of the Lorentzian version over the Euclidean alternative.
Overview of Semiclassical Proposals
The Hartle-Hawking proposal involves a compact Euclidean 4-geometry, while Vilenkin advocates for a Lorentzian geometry transitioning from zero initial geometry to the final state. Specifically, the paper addresses a significant challenge inherent to such theories: primordial tensor fluctuations—essentially gravitational wave perturbations—are not suppressed in the predicted framework, notwithstanding the Euclidean path integral’s divergence due to the errant sign in the kinetic term and cosmological constant. Moreover, a theorem proved within the paper generalizes this insight across a range of theories.
Introduction of Picard-Lefschetz Theory
The Picard-Lefschetz theory serves here as an indispensable mathematical tool for refining the path integral approach. By deforming the usual real-time integration contour into the complex plane, it renders the Lorentzian path integral conditionally and absolutely convergent. Importantly, the authors highlight that while the resulting predictions from this method are unique, they are theoretically untenable due to the unsuppressed perturbations.
Implications and Findings
The use of Picard-Lefschetz theory reverses the semiclassical weightings, aligning them closer to Vilenkin’s proposal rather than Hartle and Hawking’s. Through this, large perturbations become favored, a result that contradicts observational and theoretical expectations for a smooth early universe—thus indicating a breakdown of the theory. The paper analytically extends this condition to models inclusive of a slowly rolling inflaton, finding similar weightings that favor small rather than large initial universes.
Technical Insights
When engaging with perturbations on this cosmological background, the path integral accentuates an unexpected preference for inverse Gaussian perturbations. This result arises from utilizing the Lorentzian path integral which, in the context of Picard-Lefschetz, diverges from expected positive frequency mode function results that align with ground state principles—leading to theoretical instability.
Future Directions
The unsuppressed fluctuations issue prompts a reconsideration of how these semiclassical concepts are formulated. In future work, resolving these foundational discrepancies will be crucial for any robust theory on the universe's inception. Moreover, further exploration using Picard-Lefschetz theory might identify new pathways in complex manifold geometries to redefine the convergence and weighting of path integrals.
In sum, Feldbrugge et al. offer a critical examination of established theories on spacetime origins, challenging assertions with robust mathematical techniques. For practitioners and theorists, the paper underscores a pivotal consideration in cosmology: that interpretations of semiclassical path integrals must be refined, and potentially redefined, in pursuit of viable cosmological models reflecting both observational constrains and theoretical consistency.