1. Influence of dark energy on gravitational lensing Kabita Sarkar 1, Arunava Bhadra 2 1 Salesian College, Siliguri Campus, India 734001 2 High Energy Cosmic Ray Research Centre, University of North Bengal, Siliguri, India 734013

2. Introduction Recent cosmological observations suggest a considerable part of the Universe consists of the dark energy which may be in the form of a vacuum energy ( Cosmological Constant ) or a dynamically evolving scalar field with a negative pressure. It is natural that dark energy must have some influence on local gravitational phenomena. 8/17/2011232nd ICRC, Beijing

3. A number of recent cosmological observations indicate the presence of a cosmological constant with a value of [E.Komatsu et al.,2010] For its very small value the magnitude of such effects is also expected to be very small, but for certain situations may be detectable by the experiments [Ishak & Rindler 2010]. In the present work we explored such effects due to a cosmological constant in lensing phenomenon, both in the weak and strong field regime. 8/17/2011332nd ICRC, Beijing

4. Gravitational deflection of light in the Schwarzschild–de Sitter (SDS) metric in the weak field regime Phy.Rev D 82, 063003(2010) 8/17/201132nd ICRC, Beijing4

5. Geometrical configuration for the phenomenon of gravitational bending of light 8/17/201132nd ICRC, Beijing5

6. 8/17/201132nd ICRC, Beijing6

7. 8/17/201132nd ICRC, Beijing7

8. In presence of cosmological constant the exterior space-time due to a static spherically symmetric mass distribution is the SDS space- time which is with and (1) where m being the mass of the lens object. 8/17/2011832nd ICRC, Beijing

9. r  (3/  ) 1/2 gives the de Sitter horizon  r  does not make any sense in SDS space time. The source and the observer cannot be taken as at infinite distance away. The angle that the tangent to the light trajectory made with a coordinate direction at a given point may be obtained for the metric (1) is 8/17/2011932nd ICRC, Beijing

10. 8/17/201132nd ICRC, Beijing10

11. For the null geodesic the equation (2) reduces to r o being the coordinate distance of closest approach. For r>>r o, the angle between the tangent to the light trajectory at point r,  and the polar axis to the leading order in m,  and r o /r, is 8/17/20111132nd ICRC, Beijing

12. The reference object and the source are far away from the lens in comparison to the lens-observer distance the relative deflection angle becomes b S = impact parameter of the light rays from the source.  b = difference in impact parameter at two positions d LR = distance between lens and reference d LS = distance between lens and source 8/17/20111232nd ICRC, Beijing

13. d LR - d LS = 10 kpc &  b = earth-sun distance the contribution of the bending angle due to  will be about 10 -16 of the total bending angle. The expected angular precision of the planned astrometric missions using optical interferometry is at the level of microarcseconds, at least 10 orders lower than the  contribution to the bending angle when the lens system is within the galaxy. 8/17/20111332nd ICRC, Beijing

14. Gravitational lensing of light in the SDS metric in the strong field regime A general static and spherically symmetric spacetime of the form (6) 8/17/20111432nd ICRC, Beijing

15. The deflection angle as a function of closest approach is (7) (8) 8/17/20111532nd ICRC, Beijing

16. The divergent integral is first splitted into two parts to separate out the divergent and the regular parts. (9) so that(10) 8/17/20111632nd ICRC, Beijing

17. The function is regular for all values of z and (11) diverges as i.e. as one approaches to the photon sphere. (12) 8/17/20111732nd ICRC, Beijing

18. The integral (10) is splitted into two parts (13) (14) (15) (16) 8/17/20111832nd ICRC, Beijing

19. (17) (18) 8/17/20111932nd ICRC, Beijing

20. The function is simply the difference of the original integrand and divergent integrand (19) As,, the integral (11) diverges logarithmically 8/17/20112032nd ICRC, Beijing

21. Expanding both the integral around and approximating by the leading terms, the analytical expression of the deflection angle close to the divergence is (20) (21) (22) (23) 8/17/20112132nd ICRC, Beijing

22. Estimation of deflection angle from Schwarzschild-de Sitter Metric The radius of the photon sphere for the SDS metric is (24) (25) (26) 8/17/20112232nd ICRC, Beijing

23. (27) (28) (29) The equations (20) & (24)-(29) finally give the expression for the bending angle in the strong field regime. 8/17/20112332nd ICRC, Beijing

24. CONCLUSION The cosmological constant affects the gravitational bending angle. The contribution of cosmological constant is quite small. In the weak field expression for bending angle there are two leading order terms involving cosmological constant: one of them is purely local in the sense that it does not contain any information about the location of the observer/source. 8/17/20112432nd ICRC, Beijing

25. This term has the same signature to that of the classical expression of general relativistic bending (4m/b) i.e. this term will cause an increase of the bending angle. The other term, which is the dominating one, involves the radial distances of the source and the observer and it bears the repulsive characteristics of the positive cosmological constant. 8/17/20112532nd ICRC, Beijing

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27. The effect of cosmological constant will be prominent for sources of large distances. The strong field expression for bending also involves cosmological constant. This effect mainly occurs through the expression of impact parameter. For definite numerical value of the effect, a massive black hole will be considered. This part of the work is under progress. 8/17/20112732nd ICRC, Beijing

28. Thank You for your ATTENTION 8/17/201132nd ICRC, Beijing28