Quasilocal horizons in inhomogeneous cosmological models

We investigate quasilocal horizons in inhomogeneous cosmological models, specifically concentrating on the notion of a trapping horizon defined by Hayward as a hypersurface foliated by marginally trapped surfaces. We calculate and analyse these quasilocally defined horizons in two dynamical spacetimes used as inhomogeneous cosmological models with perfect fluid source of non-zero pressure. In the spherically symmetric Lema\^{i}tre spacetime we discover that the horizons (future and past) are both null hypersurfaces provided that the Misner-Sharp mass is constant along the horizons. Under the same assumption we come to the conclusion that the matter on the horizons is of special characte - a perfect fluid with negative pressure. We also find out that they have locally the same geometry as the horizons in the Lema\^{i}tre-Tolman-Bondi spacetime. We then study the Szekeres-Szafron spacetime with no symmetries, particularly its subfamily with $\beta_{,z}\neq 0$, and we find conditions on the horizon existence in a general spacetime as well as in certain special cases.