A dynamical system study of the inhomogeneous Λ-CDM model

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We consider spherically symmetric inhomogeneous dust models with a positive cosmological constant, Λ, given by the Lemaître–Tolman–Bondi metric. These configurations provide a simple but useful generalization of the Λ-CDM model describing cold dark matter (CDM) and a Λ term, which seems to fit current cosmological observations. The dynamics of these models can be fully described by scalar evolution equations that can be given in the form of a proper dynamical system associated with a four-dimensional phase space whose critical points and invariant subspaces are examined and classified. The phase space evolution of various configurations is studied in detail by means of two two-dimensional subspaces: a projection into the invariant homogeneous subspace associated with Λ-CDM solutions with FLRW metric, and a projection into a subspace generated by suitably defined fluctuations that convey the effects of inhomogeneity. We look at cases with perpetual expansion, bouncing and loitering behavior, as well as configurations with ‘mixed’ kinematic patters, such as a collapsing region in an expanding background. In all cases, phase space trajectories emerge from and converge to stable past and future attractors in a qualitatively analogous way as in the case of the FLRW limit. However, we can identify in both projections of the phase space various qualitative features absent in the FLRW limit that can be useful in the construction of toy models of astrophysical and cosmological inhomogeneities.