Effects of the dark energy and flat rotation curve on the gravitational time delay of particle with non-zero mass

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 the megaparsec and beyond distance scale.