  The effects of several dark energy models on gravitational time delay of
particles with non-zero mass are investigated and analytical expressions for
the same are obtained at the first order accuracy. Also the expression for
gravitational time delay under the influence of conformal gravity potential
that well describes the flat rotation curve of spiral galaxies is derived. The
findings suggest that i) the conformal gravity description of dark matter
reduces the net time delay in contrast to the effect of normal dark matter and
therefore in principle the models can be discriminated using gravitational time
delay observations and ii)the effect of dark energy/flat rotation curve may be
revealed from high precision measurements of gravitational time delay of
particles involving megaparsec and beyond distance-scale.
