Effect of the cosmological constant on the bending of light and the cosmological lens equation

Abstract

We revisit the effect of cosmological constant à on the light deflection and its role in the cosmological lens equation. First, we reexamine the motion of photon in the Schwarzschild spacetime, and explicitly describe the trajectory of photon and deflection angle up to the second order in G. Then the discussion is extended to the contribution of the cosmological constant à in the Schwarzschild-de Sitter or Kottler spacetime. Contrary to the previous arguments, we emphasize the following points: (a) the cosmological constant à does appear in the orbital equation of light, (b) nevertheless the bending angle of light does not change its form even if Ã Þ 0 since the contribution of à is thoroughly absorbed into the definition of the impact parameter, and (c) the effect of à is completely involved in the angular diameter distance D A .

Effect of the Cosmological Constant on Light Deflection: Time Transfer Function Approach

“…We list in Table 1 the expressions of bending angle due to the cosmological constant previously obtained [14][15][16][17][18][19][20][21][22][23][24][25], and estimate the numerical value using c = 3.0 × 10 8 …”

“…Nevertheless, presently, it seems that a conclusion has not yet been reached, for instance, on whether the leading order effect due to Λ is coupled with the mass of the central body M or not. See [14] for a review and also [15][16][17][18][19][20][21][22][23][24][25]. In addition, for the cosmological constant and cosmological lensing equation, see, e.g., [17,18,26,27].…”

Bending of light in a universe filled with quintessential dark energy

“…Thus, at small enough ϕ angle, in which r can be compared with the horizon value, the cosmological constant can have a significant effect on the ψ angle and thus on the deflection angle of photons. One can easily show that replacing b and r g withB andr g in relation (25)(see relation (13) in [1]) and then expanding it, the resulted expression for deflection angle is different from what can be obtaind by (30).…”

“…The first three equations are exactly those are obtained from the Kottler space-time in [1] and their solution yields (28). Substituting (35) into (39-41) and assuming the minimum of r occurs at ϕ = π/2 at any order of expansion, we find the photon trajectory as following 1 : Although in this paper we have assumed that the dark energy equation of state is very close to the cosmological constant, it is known that there are several candidate for dark energy in addition to the cosmological constant and the quintessence such as the chaplygin gas, phantom, k-essence and so on [2].…”

“…In deriving the above relations, the integration constants are chosen such that the radial distance is minimized where ϕ = π/2 at any order of expansion. The effect of cosmological constant Λ on the photon trajectory and the bending of light is also examined in [1]. In Kottler space-time (21), the geodesic equation in equatorial space-time can be written as…”