[Paper Review] Slow Contraction and the Weyl Curvature Hypothesis

Anna Ijjas|arXiv (Cornell University)|Apr 20, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper demonstrates via numerical relativity that a period of slow contraction driven by a canonical scalar field with a steep negative potential dynamically drives spacetime toward a flat, homogeneous, and isotropic state with vanishingly small Weyl curvature, even from generic, inhomogeneous initial conditions. The mechanism achieves this by inducing ultralocal behavior, where spatial gradients decay rapidly, ensuring robust satisfaction of the Weyl Curvature Hypothesis across the entire simulation domain.

ABSTRACT

Using the power of numerical relativity, we show that, beginning from generic initial conditions that are far from flat, homogeneous and isotropic and have a large Weyl curvature, a period of slow contraction rapidly drives spacetime towards vanishingly small Weyl curvature as the total energy density grows, thus providing a dynamical mechanism that satisfies the Weyl Curvature Hypothesis. We also demonstrate a tight correlation between the Weyl Curvature Hypothesis and ultralocal behavior for canonical scalar fields with a sufficiently steep negative potential energy density.

Motivation & Objective

• To resolve the cosmic initial conditions problem by demonstrating a dynamical mechanism that satisfies the Weyl Curvature Hypothesis.
• To show that slow contraction can achieve universal smoothing and vanishing Weyl curvature from generic, non-perturbative initial conditions far from flat FRW geometry.
• To establish frame/gauge-invariant evidence that curvature invariants (Weyl and Chern-Pontryagin) shrink to negligible values during contraction.
• To link the robustness of slow contraction to ultralocality and the attractor properties of the scalar field's negative potential.

Proposed method

• Numerical relativity simulations of the full nonlinear Einstein-scalar field equations with a minimally coupled canonical scalar field.
• Use of a negative exponential potential $ V(\phi) = -V_0 e^{-\phi/M} $, yielding $ \varepsilon = 1/(2M^2) \gg 3 $, to drive slow contraction.
• Frame/gauge-invariant diagnostics using curvature invariants: conformal Weyl tensor $ \mathcal{C} $ and Chern-Pontryagin invariant $ \mathcal{P} $.
• Tracking the evolution of dimensionless geometric variables and their spatial gradients in comoving time $ N \approx \ln a $.
• Identification of ultralocal behavior via the condition that maximum spatial gradients fall below the minimum of geometric variables.
• Analytic scaling laws in the ultralocal limit: $ \bar{\mathcal{C}} \propto \Theta^{2(1-3/\varepsilon)} $, $ \bar{\mathcal{P}} \propto \Theta^{2(1-2/\varepsilon)} $.

Experimental results

Research questions

• RQ1 Can slow contraction dynamically achieve vanishing Weyl curvature from generic, inhomogeneous initial conditions?
• RQ2 Does the Weyl Curvature Hypothesis hold in a frame- and gauge-invariant manner during slow contraction?
• RQ3 What is the role of ultralocality in enabling robust smoothing and curvature suppression?
• RQ4 How do curvature invariants scale in the ultralocal regime of slow contraction?

Key findings

• The Weyl curvature invariant $ |\bar{\mathcal{C}}| $ and Chern-Pontryagin invariant $ |\bar{\mathcal{P}}| $ both decrease rapidly, reaching $ \mathcal{O}(10^{-6}) $ by $ N \approx 11 $.
• Ultralocal behavior emerges at $ N \approx 7.5 $, marked by the maximum of all spatial gradients falling below the minimum of all geometric variables.
• The ratio of spatial gradients to geometric variables drops below $ 10^{-6} $ by $ N \approx 22 $, confirming deep ultralocality.
• In the ultralocal limit, curvature invariants scale as $ \bar{\mathcal{C}} \propto \Theta^{2(1-3/\varepsilon)} $ and $ \bar{\mathcal{P}} \propto \Theta^{2(1-2/\varepsilon)} $, consistent with analytic predictions.
• The attractor solution for slow contraction has a large basin of attraction, ensuring robustness even from highly inhomogeneous initial data.
• The results confirm that slow contraction satisfies the Weyl Curvature Hypothesis universally, providing a dynamical resolution to the initial conditions problem.