## Gravitational lensing computation

Gravitational lensing is a phenomenon that occurs when the path of light is bent by the gravitational field. It was first proposed by Albert Einstein in his theory of general relativity in 1915. However, it was not until the 1970s that the first gravitational lens was discovered. Gravitational lensing can be divided into three categories: strong lensing, weak lensing and micro lensing. It has allowed astronomers to study the distribution of dark matter in galaxies and clusters of galaxies, which cannot be directly observed. Gravitational lensing has also been used to measure the expansion rate of the universe and to study the properties of dark energy. However, the analytics for gravitational lensing is hard to get with complicated metrics. One solution is using some perturbative methods.

In my work, I used a perturbative method proposed before to get analytics like deflection angle, apparent angle, time delay, etc, for Kiselev black hole spacetime, which is important for cosmology. First, I calculated the first integral to get deflection angle for two cases (asymptotically flat case and non-asymptotically flat case) and analyze their relations to parameters in the spacetime, which can seen in Fig.1 and Fig.2. Then I solved the lensing equation to get impact parameters and used it to get apparent angles and time delay. I found that the relations to parameters are different for two cases.