  We establish in a cosmological model based on Lyra's geometry a relationship
between a displacement vector field $ \phi_{\mu} $, the Hubble parameter and
the matter energy density $ \rho_{m} $ (with a constant equation of state $0
\le \omega_{m} < 1$) via an arbitrary function $\alpha(t)$. For a pressureless
matter ($ \omega_{m}=0 $) the effective equation of state $ \omega_{eff} $ is
completely determined by $ \alpha(t) $. We subsequently investigate and find
exact solutions in models yielding the $ \Lambda $CDM and a power-law
expansion.
