Matching collapse and expansion across Matter Trapping surfaces in inhomogeneous $\mathrm{\Lambda }$CDM models

In the present work we examine the MTS, for the restriction to spherical dust plus $\mathrm{\Lambda }$, 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 $\mathrm{\Lambda }$ ($\mathrm{\Lambda }$CDM, Schwarzschild-de,Sitter, Lemaître-Tolman-Bondi-de,Sitter: LTBdS – i.e. the inhomogeneous, spherically symmetric $\mathrm{\Lambda }$CDM). The LTBdS model presents a static, stable MTS for the first time.