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

Alan Maciel, M. Le Delliou, José P. Mimoso

TL;DR

This work shows that Matter Trapping Surfaces (MTS) in spherically symmetric spacetimes with dust and a cosmological constant are characteristic Cauchy surfaces, which makes the interior and exterior evolutions decoupled and allows infinite compatible solutions across the MTS. It develops the 1+1+2 formalism and the GPG coordinate form to recast Einstein’s equations in terms of the Misner–Sharp mass $M_{ms}$ and energy $E$, enabling a precise Cauchy-problem treatment and the identification of CCs along fluid flow. The authors illustrate the principle with three canonical cases (LTB with Λ, Schwarzschild–de Sitter, and Einstein–de Sitter) and then construct a composite LTBdS model featuring a central NFW-like core and a smooth cosmological transition, demonstrating a static, stable MTS in LTBdS. They also explore how MTSs can serve as matching surfaces between different spacetimes, offering a flexible framework for modeling bound structures in cosmological backgrounds with controlled discontinuities in global quantities. The findings have implications for modeling matter collapse, cosmic structure formation, and the interface between local dynamics and global expansion, while inviting extensions beyond spherical symmetry and into nonzero-pressure regimes.

Abstract

In the present work we examine the MTS, for the restriction to spherical dust plus $Λ$, proving that it actually is a characteristic surface of the Cauchy problem (generated by its characteristic curves), which opens the possibility for infinite solutions. This translate as the MTS being a boundary between arbitrarily independent solutions, reminiscent of the Birkhoff theorem effects. This property is illustrated with combinations of 3 examples containing MTSs and $Λ$ ($Λ$CDM, Schwarzschild-de\,Sitter, Lemaître-Tolman-Bondi-de\,Sitter: LTBdS -- i.e. the inhomogeneous, spherically symmetric $Λ$CDM). The LTBdS model presents a static, stable MTS for the first time.

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

TL;DR

This work shows that Matter Trapping Surfaces (MTS) in spherically symmetric spacetimes with dust and a cosmological constant are characteristic Cauchy surfaces, which makes the interior and exterior evolutions decoupled and allows infinite compatible solutions across the MTS. It develops the 1+1+2 formalism and the GPG coordinate form to recast Einstein’s equations in terms of the Misner–Sharp mass and energy , enabling a precise Cauchy-problem treatment and the identification of CCs along fluid flow. The authors illustrate the principle with three canonical cases (LTB with Λ, Schwarzschild–de Sitter, and Einstein–de Sitter) and then construct a composite LTBdS model featuring a central NFW-like core and a smooth cosmological transition, demonstrating a static, stable MTS in LTBdS. They also explore how MTSs can serve as matching surfaces between different spacetimes, offering a flexible framework for modeling bound structures in cosmological backgrounds with controlled discontinuities in global quantities. The findings have implications for modeling matter collapse, cosmic structure formation, and the interface between local dynamics and global expansion, while inviting extensions beyond spherical symmetry and into nonzero-pressure regimes.

Abstract

In the present work we examine the MTS, for the restriction to spherical dust plus , proving that it actually is a characteristic surface of the Cauchy problem (generated by its characteristic curves), which opens the possibility for infinite solutions. This translate as the MTS being a boundary between arbitrarily independent solutions, reminiscent of the Birkhoff theorem effects. This property is illustrated with combinations of 3 examples containing MTSs and (CDM, Schwarzschild-de\,Sitter, Lemaître-Tolman-Bondi-de\,Sitter: LTBdS -- i.e. the inhomogeneous, spherically symmetric CDM). The LTBdS model presents a static, stable MTS for the first time.
Paper Structure (22 sections, 67 equations, 3 figures, 3 tables)

This paper contains 22 sections, 67 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The MTS as a Cauchy problem for dust with $\Lambda$
  3. From 1+1+2 to GPG
  4. Dynamical Equations
  5. The dust and $\Lambda$ model
  6. The MTS as the characteristic surface for the dust with $\Lambda$ model
  7. Illustrative cases with dust and a cosmological constant
  8. MTS in LTB with $\Lambda$
  9. MTS in Schwarzschild-de Sitter
  10. MTS in Einstein-de Sitter
  11. Building an MTS in LTB example
  12. The Heaviside MTS jump
  13. Smooth cosmological transition
  14. Induced initial time limit
  15. Smoother function
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Mass determination with smooth transition
  • Figure 2: Adjusting inner Hubble parameter to the no-shell-crossing condition.
  • Figure 3: GLTB model with NFW inner halo and FLRW outer behaviour containing a stable MTS. The velocity jump is restricted to a factor 5 for illustration.