Deflection and gravitational lensing of null and timelike signals in general asymptotically (anti-)de Sitter spacetimes

The deflection and gravitational lensing of light and massive particles in arbitrary static, spherically symmetric and asymptotically (anti-)de Sitter spacetimes are considered in this work. We first proved that for spacetimes whose metric satisfying certain conditions, the deflection of null rays with fixed closest distance will not depend on the cosmological constant $\Lambda$, while that of timelike signals and the apparent angle in gravitational lensing still depend on $\Lambda$. A two-step perturbative method is then developed to compute the change of the angular coordinate and total travel time in the weak field limit. The results are quasi-series of two small quantities, with the finite distance effect of the source/detector naturally taken into account. These results are verified by applying to some known de Sitter spacetimes. Using an exact gravitational lensing equation, we solved the apparent angles of the images and time delays between them and studied the effect of $\Lambda$ on them. It is found that generally, a small positive $\Lambda$ will decrease the apparent angle of images from both sides of the lens and increase the time delay between them. The time delay between signals from the same side of the lens but with different energy however, will be decreased by $\Lambda$.