Deflection and Gravitational lensing of null and timelike signals in the Kiselev black hole spacetime in the weak field limit

In this work we study the deflection and gravitational lensing of null and timelike signals in the Kiselev spacetime in the weak field limit, to investigate the effects of the equation of state parameter $\omega$ and the matter amount parameter $\alpha$. In doing this, we extend a perturbative method previously developed for asymptotically flat spacetimes whose metric functions have integer-power asymptotic expansions to the case that may or may not be asymptotically flat but with non-integer power expansions. It is found that in the asymptotically flat case ($-1/3<\omega<0$) the deflection angles are expressable as quasi-power series of the dimensionless quantities $M/b,~b/r_{s,d}$ and $\alpha/M^{1+3\omega}$ where $M,~b,~r_{s,d}$ are respectively the lens mass, impact parameter and source/detector radius. A similar series exists for the non-asymptotically flat case of ($-1<\omega<-1/3$), but with the closest radius $r_0$ replacing $b$. In the asymptotically flat (or non-flat) case, the increase of $\alpha$ or decrease of $\omega$ will increase (or increase) the deflection angle. Since the obtained deflection angles naturally take into account the finite distance effect of the source and the detector, we can establish an exact gravitational lensing equation, from which the apparent angles of the images and their magnifications are solved. It is found that generally for the asymptotically flat case, increasing $\alpha$ or decreasing $\omega$ will increase the apparent angles of the images. While for the non-asymptotically flat case, increasing $\alpha$ or $\omega$ will both lead to smaller apparent angles.