  The Aether Scalar Tensor (AeST) theory is an extension of General Relativity
(GR), proposed for addressing galactic and cosmological observations without
dark matter.The action for the theory includes a function that can currently
only be constrained by phenomenological considerations. In antecedent work,
forms of this function were considered that led to an effective fluid
contribution to the cosmological evolution equations that approximated that of
dust more and more closely at late cosmic times. In this work we consider an
alternative set of functions that most closely approximate dust at the earliest
cosmic times and where deviations from dust-like behaviour gradually emerge
with time. We use the dynamical system formalism to analyze example models from
both possible sets of functions, introducing a complete set of dynamical
variables describing the spacetime curvature, energy density parameters of
different matter components, and AeST scalar field, and obtain the dynamical
equations describing cosmological evolution. The cosmological phase space is
found to feature invariant submanifolds associated to the absence of the matter
components, as well as equilibrium states associated with well-known
cosmological behaviors e.g. matter, radiation, and cosmological constant
dominated epochs. A full numerical integration of the dynamical system is
performed for the models and it is shown that each can closely approximate the
$\Lambda \mathrm{CDM}$ model at the level of the cosmic background.
Generalizations of the models are considered and it is shown that the new
models likely can simultaneously replicate the cosmological successes of cold
dark matter whilst satisfying constraints on the theory from the weak-field
quasistatic regime.
