Starting inflation in asymptotically flat spacetimes

Technical Summary: Starting Inflation in Asymptotically Flat Spacetimes

Problem Statement
A central question in early universe cosmology is whether inflation can be robustly seeded by a generic, localized fluctuation in the inflaton field that probes the inflationary plateau of its potential. Previous numerical relativity (NR) studies have primarily utilized periodic spacetime topologies (e.g., $T^3$ or $S^3$) to model initial conditions. While these setups allow for the study of non-perturbative dynamics, they impose restrictive integrability conditions on the initial data. Specifically, periodic boundaries often enforce a volume-averaged intrinsic curvature that vanishes or is spatially constant, and tie the average expansion rate directly to the average energy density. These constraints may bias results toward successful inflation by artificially suppressing the collapse of small, high-energy fluctuations. The authors aim to revisit the robustness of inflation by studying localized fluctuations in asymptotically flat spacetimes, which allow for more general profiles of intrinsic and extrinsic curvature and avoid the artificial homogeneity imposed by periodic boundaries.

Methodology
The authors employ numerical relativity simulations within the framework of General Relativity, utilizing the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation in spherical symmetry.

1. Initial Data Construction:

   • The initial data must satisfy the Hamiltonian and momentum constraints. The authors adopt a conformal transverse-traceless decomposition where the conformal metric is flat ($\bar{\gamma}_{ij} = \delta_{ij}$).
   • The inflaton field $\phi$ is initialized with a Gaussian profile $\phi = \phi_0 e^{-r^2/\sigma^2}$ and zero conjugate momentum ($\Pi=0$). The amplitude $\phi_0$ is fixed such that a homogeneous FLRW universe with this value would undergo 60 e-folds of inflation.
   • A key parameter, $\epsilon$, is introduced to parametrize the split of the potential energy density between the source for intrinsic curvature and the source for extrinsic curvature. The Hamiltonian constraint is solved to determine the conformal factor $\psi$, and the momentum constraint is solved iteratively to determine the trace of the extrinsic curvature $K$.
   • This approach reveals two distinct branches of solutions for a given fluctuation width $\sigma$ and parameter $\epsilon$: a weak-field branch (smaller Misner-Sharp mass) and a strong-field branch (larger Misner-Sharp mass, potentially containing a "throat" or "bag of gold" geometry). Solutions only exist up to a critical value $\epsilon_{crit}$.

2. Numerical Evolution:

   • The system is evolved using a fourth-order finite-difference scheme with a fourth-order Runge-Kutta time integrator.
   • Gauge conditions include a modified $1+log$ slicing for the lapse function and zero shift.
   • The authors track the formation of cosmological horizons, the local acceleration of the scale factor, and the relative contributions of various terms (potential, kinetic, gradient, and curvature) to the Hamiltonian constraint.

3. Criteria for Inflation:

   • Inflation is deemed to have started in a region if the potential energy contribution dominates the Hamiltonian constraint (at least 90% of the $L_1$ norm of all contributions) and the local equivalent of $\ddot{a}$ is positive.
   • The number of e-folds is extrapolated based on the central values of the field and momentum at the moment these criteria are met.

Key Results
The study maps the parameter space defined by the fluctuation width $\sigma$ and the curvature split $\epsilon$:

• Existence of Solution Branches: For a fixed fluctuation width, solutions to the constraint equations exist only up to a maximum $\epsilon_{crit}$. Below this limit, two solution branches exist. The strong-field branch supports larger spatial volumes and is more robust against collapse.
• Minimum Fluctuation Size:
  • On the weak-field branch (specifically the $\epsilon=0$ limit), inflation fails if the proper size of the fluctuation is too small. The authors find a minimum proper radius of approximately $\sigma_{prop} \approx 2 L_{infl}$ (where $L_{infl}$ is the inflationary length scale). Below this size, the region collapses to form a black hole.
  • On the strong-field branch, the minimum size required for inflation is slightly smaller in coordinate terms, but the physical requirement remains consistent when measured in proper volume.
• Universality of Proper Volume: When the results are plotted against the proper radius (or proper volume) of the initial fluctuation rather than the coordinate radius, the data from different $\epsilon$ values and solution branches collapse onto a single curve. This indicates that the primary driver for successful inflation is the physical volume of the fluctuation, not the specific geometric split between intrinsic and extrinsic curvature.
• Comparison to Previous Work: The results confirm and generalize the findings of Goldwirth and Piran (GP), who established that a "Hubble-sized" patch is required for inflation. However, this work demonstrates that this threshold is robust across a much wider class of initial data, including those with non-uniform intrinsic and extrinsic curvatures in asymptotically flat spacetimes.
• Failure in Small Fluctuations: Unlike some periodic simulations where inflation was found to occur somewhere in the domain even if black holes formed, this study shows that for sufficiently small fluctuations in asymptotically flat spacetimes, the entire central region collapses into a black hole, and inflation fails to start.

Significance and Claims
The paper claims to provide a more rigorous test of inflation's robustness by removing the artificial constraints of periodic boundary conditions. The authors argue that:

1. Generalization of GP: The requirement for a fluctuation size of several times the inflationary scale ($1/\sqrt{G\rho_{infl}}$) is confirmed to be a generic feature, valid even when the initial data allows for arbitrary (within constraints) distributions of intrinsic and extrinsic curvature.
2. Bias in Periodic Simulations: The authors suggest that periodic simulations with scales close to the inflationary scale may bias results toward finding successful inflation. In such setups, the "regularity" imposed by the periodicity and the requirement to satisfy global constraints (like vanishing average curvature) may artificially stabilize small fluctuations that would otherwise collapse in an asymptotically flat setting.
3. Physical Volume as the Key Metric: The study establishes that the proper volume of the initial fluctuation is the decisive factor for the onset of inflation, superseding the specific choice of gauge or the split between curvature terms.
4. Inherent Restrictions: The work highlights that the balance between intrinsic and extrinsic curvature is not a free choice; geometric constraints limit the maximum intrinsic curvature that can balance a given energy density, often necessitating non-zero extrinsic curvature (expansion) for solutions to exist.

The authors conclude that while their study is limited to spherical symmetry and a quadratic potential, it provides a foundational understanding that successful inflation requires a physically large, localized region, and that smaller fluctuations in generic asymptotically flat spacetimes are prone to gravitational collapse rather than inflation.