We have investigated a spatially homogeneous and anisotropic Bianchi type II universe filled with anisotropic dark energy within the framework of Lyra geometry with time-varying displacement field vector. The exact solutions of field equations are obtained by using Berman's law of variation for the Hubble parameter that yields a negative constant deceleration parameter. Some physical and kinematical behaviors of the cosmological model are discussed. This model may play relevant and significant roles to study spatially homogeneous cosmologies with anisotropic dark energy and astrophysics.
The relationships between the redshifts, the luminosity distance and the true distances at the times of light emission and reception, are formally calculated in Euclidean spaces whose expansion is describable using simple functions: proportional to cosmic time, power law or exponential, and for the different assumptions about the origin of the redshift: kinematic, relativistic and wave stretching effects. None of these combinations gives the same results in terms of redshift, of horizon, of distances and of deviation from the Hubble law. This systematic analysis provides the limit cases of expansion scenarios.
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