Research of the celestial objects by gravitational lensing Naoki - PowerPoint PPT Presentation

Research of the celestial objects by gravitational lensing Naoki Tsukamoto (Rikkyo U.) Collaboration with Tomohiro Harada and Kohji Yajima (Rikkyo U.) arXiv:1207.0047 (accepted by PRD) arXiv:1211.0380 (accepted by PRD) 11/12/2012 JGRG22 @Tokyo Univ. 1

Our Purpose. The behaviors of the light ray on Schwarzschild space- time and Ellis spacetime are very similar. （ Photon sphere, asymptotic flatness ） V. Perlick, Phys. Rev. D 69 , 064017 (2004). = ⇒ Can we distinguish between black holes and wormholes by their Einstein ring systems? 2

Ellis WH. H. G. Ellis, J. Math. Phys. 14 , 104 (1973). One of the static spherically symmetric solutions which are asymptotic flat, ds 2 = − dt 2 + dr 2 + ( r 2 + a 2 )( dθ 2 + sin 2 θdφ 2 ) , where a is a positive constant. Using ρ 2 ≡ r 2 + a 2 , the line element is given by, ) − 1 1 − a 2 ( ds 2 = − dt 2 + dρ 2 + ρ 2 ( dθ 2 + sin 2 θdφ 2 ) , ρ 2 • ρ = ± a corresponds to the wormhole throat. • E ≡ − g µν k µ t ν and L ≡ g µν k µ φ ν are constant. t µ ∂ µ = ∂ t , φ µ ∂ µ = ∂ φ :Killing vectors, k µ :photon wave number. • b ≡ L/E ： impact parameter. 3

Impact parameter b and photon orbit. From k µ k µ = 0, a photon trajec- tory is described by, ) 2 ( dr + V eff = 1 , dλ where the effective potential is de- fined as, b 2 V eff ≡ r 2 + a 2 . λ :affine parameter. | b | > a :The photon is scattered. | b | = a :Unstable circular orbit. | b | < a :The photon reaches the throat ρ = a . We only consider a scattering case, or | b | > a . 4

Deflection angle α . Using u ≡ 1 /ρ , 2 + π α 2 ) 2 ( du = 1 ❖ ❈ ❈ ❖ b 2 (1 − a 2 u 2 )(1 − b 2 u 2 ) . ❈ ❈ ❈ ❈ dφ ❈ ❈ α ❈ ❈ u ( φ ) ❈ 2 ❈ The deflection anlge α is given by, ❈ ❈ ✒ � ❈ ❈ � ❈ � ∫ b − 1 ❈ � du ❈ � α = 2 − π, ❈ � L ❈ � √ 0 φ G ( u ) ❈ � ❈ � ③ 0 G ( u ) ≡ a 2 ( a − 2 − u 2 )( b − 2 − u 2 ) . • α increases with the de- ✄ ✗ ✄ α ✄ creasing b . ✄ 2 ✄ ✄ • In the strong field limit | b | → a , ✄ ✄ α → log ∞ . • ρ = a is the photon shpere. 5

α is expressed by the complete elliptic inte- gral of the first kind K ( a/b ) . L. Chetouani and G. Cl´ ement, Gen. Relativ. Gravit. 16 , 111 (1984). K. Nakajima and H. Asada, Phys. Rev. D 85 , 107501 (2012). Using sin θ ≡ bu , ∫ b − 1 ∫ π/ 2 du dθ ( a ) = = K ) 2 sin 2 θ √ √ b 0 0 G ( u ) ( a 1 − b The deflection angle α is expressed by, ( a ) α = 2 K − π. b In the weak field limit | b | ≫ a , ) 2 α ≃ π ( a . 4 b In the strong field limit | b | → a , α → log ∞ . 6

Lens configuration. ・ | ¯ α | , | θ | , | φ | ≪ 1 ・ Effective deflection anlge ¯ α : α = ¯ α + 2 πn ・ Winding number of the light ray n (non-negative integer) Lens equation S I ③ ❥ ❈ ✻ ✻ ✆✆ ✔ ❈ ✔ D ls ¯ α = D s ( θ − φ ) . ✆ ❈ ✔ ✆ ❈ ✔ ✆ ❈ ✔ ✆ ❈ ✔ We sets φ = 0, then, ✆ ❈ ✔ D ls ✆ ❈ ✔ ( a ✆ ❈ ✔ D s α ¯ ) ✆ ❈ ✔ 2 K b n = (2 n + 1) π ✆ − ❈ ✔ ✆ ❈ ✔ b n D l D ls ✆ ❈ ✔ ✆ ❈ ✔ D s ✆ L ❄ ✛ ✲ ❈ ✔ This equation gives a unique (rel- ③ ✆ ✻ b ✔✔ ✆ ✆ ✔ ativistic) Einstein ring anlge ✆ ✔ φ ✆ ✔ ✆ ✔ ✆ ✔ D l θ n = b n /D l . ✆ ✔ ✆ ✔ θ ✆ ✔ ✆ ✔ for each n . ✆ ✔ ✆ ✔ ・ θ n monotonically decreases with ✆ ✔ ❄ ❄ ✔ ✆ O respect to n and approaches a/D l . 7

(Relativistic) Einstein ring. ・ In the weak field limit | b | ≫ a (= ⇒ n = 0), ) 2 , Using α ≃ π ( a 4 b ) 1 ( π D ls 3 a 2 θ 0 ≃ D s D 2 4 l ) 2 ) 1 ) 1 ) 2 ( D ls 3 ( 20Mpc 10Mpc 3 ( a 3 . ( 3 ≃ 2 . 0 arcsecond 10Mpc 0 . 5pc D s D l ・ In the strong field limit a ∼ b n ( ⇐ ⇒ n ≥ 1), ( ) ( θ n ≥ 1 ≃ a 10Mpc a ) ≃ 1 . 0 × 10 − 2 arcsecond . 0 . 5pc D l D l Measuring the Relativistic rings ∼ measuring the photon shpere. The relation between θ 0 and θ n ≥ 1 is obtained by, ) 1 ( 3 4 D s 2 2 θ n ≥ 1 ≃ θ 0 . π D ls 8

Schwarzschild BH. K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62 , 084003 (2000). V. Bozza et al. , Gen. Relativ. Gravit. 33 , 1535 (2001). √ ・ The critical impact parameter is b = (3 3 / 2) r g . ・ In the weak field limit b ≫ r g , using α ≃ 2 r g /b , √ 2 D ls θ 0 ≃ r g D l D s ) 1 ) 1 ) 1 ) 1 ( 2 ( 2 . D ls M 10 Mpc 2 ( 20 Mpc ( 2 ≃ 2 . 0 arcsecond 10 10 M ⊙ 10 Mpc D l D s ・ In the strong field limit, √ ( ) ( ) 3 3 10 Mpc r g M ≃ 5 . 1 × 10 − 5 arcsecond θ n ≥ 1 ≃ . 10 10 M ⊙ 2 D l D l The relation between θ 0 and θ n ≥ 1 is obtained by, √ θ n ≥ 1 ≃ 3 3 D s θ 2 0 4 D ls 9

� 2 10 � 4 10 � 6 10 � 8 10 We can distinguish between BH and WH. � 10 10 � 12 ・ Given D s /D ls . 10 ・ The lens parameter is a or r g � 14 W ormhole 10 [ar se ond℄ ・ Bla k hole The relation between θ 0 and θ n ≥ 1 � 16 10 � 5 � 4 � 3 � 2 � 1 0 1 is obtained by, 10 10 10 10 10 10 10 � [ar se ond℄ 0 ) 1 n � 1 ( 3 4 D s 2 2 θ n ≥ 1 ≃ θ � 0 π D ls or √ θ n ≥ 1 ≃ 3 3 D s θ 2 0 . 4 D ls 10

We summarize the points so far. • Measuring the Relativistic rings ∼ measuring the photon shpere. • Given D s /D l , θ 0 and θ n ≥ 1 , we can distingusih WH and BH But • The case where φ = 0 is very rare. • The relativistic rings are very small. = ⇒ Next we will consider the case where φ ̸ = 0 and use the signed magnification sums to determine the lens objects. 11

We summarize the points so far. • Measuring the Relativistic rings ∼ measuring the photon shpere. • Given D s /D l , θ 0 and θ n ≥ 1 , we can distingusih WH and BH But • The case where φ = 0 is very rare. • The relativistic rings are very small. = ⇒ Next we will consider the case where φ ̸ = 0 and use the signed magnification sums to determine the lens objects. 12

General spherical lens model in the weak field. The deflection angle C θ α = , � � D n � θ n +1 � � l � C :positive constant, n :non-negative integer • n = 0 and C = 4 πσ 2 : the singular isothermal sphere lens σ : velocity dispersion of particles • n = 1 and C = 4 M : the Schwarzschild lens M : lens mass • n = 2 and C = πa 2 / 4: the Ellis wormhole lens • n ≥ 3: some exotic lens objects and the gravitational lens effect of modified gravitational theories. 13

5 n=1 n=2 n=3 4 n=4 The light curves. 3 2 1 j 0 � j � -2 -1 0 1 2 + j time 0+ j � F. Abe, Astrophys. J. 725, 787 (2010). T. Kitamura, K. Nakajima and H. Asada, arXiv:1211.0379 14

The lens equation. The lens equation is given by θ n +1 − ˆ θ n ∓ 1 = 0 , ˆ φ ˆ 1 ( ) D ls C/D s /D n where ˆ θ ≡ θ/θ 0 , ˆ n +1 . φ ≡ φ/θ 0 , the Einstein ring angle θ 0 ≡ l The solutions ˆ θ 1 , ˆ θ 2 , · · · , ˆ θ n +1 satisfies n +1 (ˆ θ − ˆ ∏ θ i ) = 0 . i =1 2  n +1  n +1 φ 2 = θ 2 ˆ ˆ ˆ ∑ ∑ θ i = i − 2 δ 1 n .   i =1 i =1 where δ 1 n = 0 for n ≥ 2 and δ 1 n = 1 for n = 1. This implies n +1 ˆ d ˆ θ i θ i ∑ φ = 1 . ˆ d ˆ φ i =1 15

3 2 The number of the real solutions. 1 0 -1 We can express the lens equation -2 as follows -3 -4 -2 0 2 4 y � θ n +1 � ˆ � ˆ � = ˆ θ − ˆ x θ/ φ. � � • The intersections are the real solutions of the lens equation. • The number of the real solu- tions is always two regardless x y = of ˆ � � φ . � x n +1 � � � • ˆ θ + :positive solution and • ˆ θ − :negative solution y = x − ˆ φ 16

Signed magnification in the directly-aligned limit The signed magnification is given by ˆ d ˆ θ ± θ ± µ 0 ± ≡ φ . ˆ d ˆ φ The directly-aligned limit is 1 ± ˆ 1 φ lim µ 0 ± = lim φ . ± ˆ 1 + n ˆ ˆ φ → 0 φ → 0 The total magnification is given by 2 1 � � lim � µ 0+ � + | µ 0 − | = lim φ. � � ˆ 1 + n ˆ ˆ φ → 0 φ → 0 The signed magnification sum is obtained by 2 ( ) lim µ 0+ + µ 0 − = 1 + n. ˆ φ → 0 17

2 n=1 n=2 n=3 1.5 n=4 The signed magnification sum. 1 0.5 0 � 0 � 0 1 2 3 4 5 6 + ^ � 0+ � 18

Summary of this talk. • Given D s /D ls , we can distinguish between WH and BH by their Einstein ring systems. • The signed magnification sums is a powerful tool to find exotic lens objects because it only depends on the deduced source angle ˆ φ and n . • The method to distinguish the lens objects by using the signed magnification sums can be used in both the magni- fication and demagnification phase. Thus, we not have to rely on only the demagnification to detect the Ellis worm- holes. 19