Matching collapse and expansion across Matter Trapping surfaces in inhomogeneous $Λ$CDM models

Technical Summary: Matching Collapse and Expansion Across Matter Trapping Surfaces in Inhomogeneous $\Lambda$CDM Models

Problem Statement
The paper addresses the formulation of the initial data problem in General Relativity (GR) within spherically symmetric spacetimes containing dust and a cosmological constant ($\Lambda$). Specifically, it investigates the nature of Matter Trapping Surfaces (MTS), previously defined as hypersurfaces separating cosmologically expanding regions from regions of gravitational collapse. The central question is whether the MTS can serve as a well-posed boundary that entirely determines the dynamics on either side, or if it allows for independent evolution of interior and exterior solutions. This inquiry is motivated by fundamental questions regarding the decoupling of overdense regions from global expansion and the transition between Newtonian and relativistic regimes.

Methodology
The authors employ the 1+1+2 formalism applied to spherically symmetric metrics, utilizing a timelike unit vector $n^a$ (matter flow) and a spacelike unit vector $e^a$ (orthogonal to spheres of symmetry). This framework is translated into Generalized Painlevé-Gullstrand (GPG) coordinates, characterized by a lapse function $\alpha(t,r)$ and a shift vector $\beta(t,r)$.

The analysis focuses on the Einstein Field Equations (EFEs) for a perfect fluid source reduced to dust and $\Lambda$. The dynamical equations are recast in terms of the Misner-Sharp mass $M_{ms}$ and the energy function $E$. The authors reformulate the system as a Cauchy problem to analyze the MTS. They define the MTS by two conditions:

1. Kinematic condition: The areal radial velocity vanishes ($L_n r = 0$, implying $\beta = 0$).
2. Dynamic condition: The radial acceleration vanishes ($L_n (L_n r) = 0$, implying $\dot{\beta} = 0$).

By treating the MTS as a boundary surface $r = r^*$ rather than an initial time slice $t = t_0$, the authors analyze the characteristic curves of the resulting partial differential equation (PDE) system.

Key Contributions and Results

1. MTS as a Characteristic Surface:
   The primary theoretical result is the proof that for dust plus $\Lambda$, the MTS is a characteristic surface of the Cauchy problem. In the context of the PDE system governing $E$ and $M$, the condition $\beta = 0$ at the MTS means the surface coincides with a characteristic curve (the flow line).

   • Consequence: Standard Cauchy data cannot propagate across a characteristic surface. Consequently, the MTS does not uniquely determine the solution on either side. Instead, the system admits infinite solutions: any interior solution can be matched with any exterior solution across the MTS, provided the boundary values (mass and energy function) are continuous. The MTS acts as a "shield," decoupling the detailed dynamics of the interior from the exterior.

2. Static, Stable MTS in LTBdS:
   The authors construct a specific inhomogeneous model (Lemaître-Tolman-Bondi-de Sitter, or LTBdS) featuring a central Navarro-Frenk-White (NFW) density profile transitioning to a flat FLRW background. They demonstrate that this model supports a static, stable MTS for the first time. This is achieved by carefully matching the kinematic and dynamic conditions, ensuring the areal velocity and acceleration vanish simultaneously and remain zero during evolution.

3. Matching Scenarios:
   The paper illustrates the decoupling property by constructing six distinct matching scenarios between three solution types:

   • Schwarzschild-de Sitter (SdS)
   • Einstein-static (ES) universe
   • Inhomogeneous LTBdS
     The authors show that these solutions can be joined at an MTS even if physical quantities such as density ($\rho$), total mass ($M$), curvature ($E$), and the cosmological constant ($\Lambda$) are discontinuous across the boundary. The only requirement is the continuity of the matching conditions at the surface itself.

Significance and Claims
The paper claims that the MTS acts as a boundary that "protects" each side from the detailed dynamics of the other, analogous to the effects of the Birkhoff theorem in vacuum solutions. Just as Birkhoff's theorem allows any interior Schwarzschild solution to be matched to an exterior vacuum solution given mass and radius continuity, the MTS allows for the arbitrary matching of expanding and collapsing regions in inhomogeneous $\Lambda$CDM models.

The authors emphasize that this result provides a rigorous mathematical foundation for treating MTSs as separators between bound and unbound regions. It suggests that for dust+$\Lambda$ models, the transition between local collapse and global expansion is not a smooth dynamical evolution determined by global initial data, but rather a characteristic boundary where independent solutions can be "glued" together. The work is presented as a step toward understanding how overdense regions dynamically decouple from the universe's expansion and offers a tool for constructing complex cosmological models with distinct inner and outer dynamics.

The authors note limitations, specifically that the analysis assumes strict spherical symmetry and zero pressure (dust), and they suggest that extending these results to models with non-zero pressure or angular momentum remains an open question.