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, ds2 = −dt2 + dr2 + (r2 + a2)(dθ2 + sin2 θdφ2), where a is a positive constant. Using ρ2 ≡ r2 + a2, the line element is given by, ds2 = −dt2 +

(

1 − a2 ρ2

)−1

dρ2 + ρ2(dθ2 + sin2 θdφ2),

• ρ = ±a corresponds to the wormhole throat.
• E ≡ −gµνkµtν and L ≡ gµνkµφν are constant.

tµ∂µ = ∂t, φµ∂µ = ∂φ:Killing vectors, kµ:photon wave number.

• b ≡ L/E：impact parameter.

3

Impact parameter b and photon orbit.

From kµkµ = 0, a photon trajec- tory is described by,

(dr

dλ

)2

+ Veff = 1, where the effective potential is de- fined as, Veff ≡ b2 r2 + a2. 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,

(

du dφ

)2

= 1 b2(1 − a2u2)(1 − b2u2). The deflection anlge α is given by, α = 2

∫ b−1

du

√

G(u) − π, G(u) ≡ a2(a−2 − u2)(b−2 − u2).

• α

increases with the de- creasing b.

• In the strong field limit |b| → a,

α → log ∞.

• ρ = a is the photon shpere.

❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❖

• ✒

③ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❖ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✗

L

α 2 + π 2 α 2 α 2

φ u(φ)

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

du

√

G(u) =

∫ π/2

dθ

√

1 −

(a

b

)2 sin2 θ

= K

(a

b

)

The deflection angle α is expressed by, α = 2K

(a

b

)

− π. In the weak field limit |b| ≫ a, α ≃ π 4

(a

b

)2

. In the strong field limit |b| → a, α → log ∞.

6

Lens configuration.

・|¯

α|, |θ|, |φ| ≪ 1

・Effective deflection anlge ¯

α : α = ¯ α + 2πn

・Winding number of the light ray n (non-negative integer)

Lens equation Dls¯ α = Ds(θ − φ). We sets φ = 0, then, 2K

( a

bn

)

− Ds DlDls bn = (2n + 1) π This equation gives a unique (rel- ativistic) Einstein ring anlge θn = bn/Dl. for each n.

・θn monotonically decreases with

respect to n and approaches a/Dl.

✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ❄ ✻ ❄ ✻ ❄ ✻ ✛ ✲ ❥ ③ ③ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆✆ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔✔ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈

Ds Dl Dls O L b S I φ θ ¯ α

7

(Relativistic) Einstein ring.

・In the weak field limit |b| ≫ a (=

⇒ n = 0), Using α ≃ π

4

(a

b

)2,

θ0 ≃

(

π 4 Dls DsD2

l

a2

)1

3

≃ 2.0 arcsecond

(

Dls 10Mpc

)1

3 (20Mpc

Ds

)1

3

(

10Mpc Dl

)2

3 (

a 0.5pc

)2

3 .

・In the strong field limit a ∼ bn (⇐

⇒ n ≥ 1), θn≥1 ≃ a Dl ≃ 1.0 × 10−2 arcsecond

(

10Mpc Dl

) (

a 0.5pc

)

. Measuring the Relativistic rings ∼ measuring the photon shpere. The relation between θ0 and θn≥1 is obtained by, θn≥1 ≃

(

4 π Ds Dls

)1

2

θ

3 2

0.

8

Schwarzschild BH.

• K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D62, 084003 (2000).
• V. Bozza et al., Gen. Relativ. Gravit. 33, 1535 (2001).

・The critical impact parameter is b = (3

√ 3/2)rg.

・In the weak field limit b ≫ rg, using α ≃ 2rg/b,

θ0 ≃

√

2Dls DlDs rg ≃ 2.0 arcsecond

(

Dls 10Mpc

)1

2

(

M 1010M⊙

)1

2 (

10Mpc Dl

)1

2 (20Mpc

Ds

)1

2 .

・In the strong field limit,

θn≥1 ≃ 3 √ 3 2 rg Dl ≃ 5.1 × 10−5 arcsecond

(

M 1010M⊙

) (

10Mpc Dl

)

. The relation between θ0 and θn≥1 is obtained by, θn≥1 ≃ 3 √ 3 4 Ds Dls θ2

9

We can distinguish between BH and WH.

・Given Ds/Dls. ・The lens parameter is a or rg ・

The relation between θ0 and θn≥1 is obtained by, θn≥1 ≃

(

4 π Ds Dls

)1

2

θ

3 2

• r

θn≥1 ≃ 3 √ 3 4 Ds Dls θ2

0.

• n1

• [ar se ond℄

• rmhole

10

We summarize the points so far.

• Measuring the Relativistic rings ∼ measuring the photon shpere.
• Given Ds/Dl, θ0 and θn≥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 Ds/Dl, θ0 and θn≥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

α =

Cθ Dn

l

• θn+1
• ,

C:positive constant, n:non-negative integer

• n = 0 and C = 4πσ2: the singular isothermal sphere lens

σ : velocity dispersion of particles

• n = 1 and C = 4M: the Schwarzschild lens

M : lens mass

• n = 2 and C = πa2/4: the Ellis wormhole lens
• n ≥ 3: some exotic lens objects and the gravitational lens effect
• f modified gravitational theories.

13

The light curves.

• 2
• 1

• 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 ˆ θn+1 − ˆ φˆ θn ∓ 1 = 0, where ˆ θ ≡ θ/θ0, ˆ φ ≡ φ/θ0, the Einstein ring angle θ0 ≡

(

DlsC/Ds/Dn

l

)

1 n+1.

The solutions ˆ θ1, ˆ θ2, · · ·, ˆ θn+1 satisfies

n+1

∏

i=1

(ˆ θ − ˆ θi) = 0. ˆ φ2 =

 

n+1

∑

i=1

ˆ θi

 

2

=

n+1

∑

i=1

ˆ θ2

i − 2δ1n.

where δ1n = 0 for n ≥ 2 and δ1n = 1 for n = 1. This implies

n+1

∑

i=1

ˆ θi ˆ φ dˆ θi dˆ φ = 1.

15

The number of the real solutions.

We can express the lens equation as follows ˆ θ/

• ˆ

θn+1

• = ˆ

θ − ˆ φ.

• The intersections are the real

solutions of the lens equation.

• The number of the real solu-

tions is always two regardless

• f ˆ

φ.

• ˆ

θ+:positive solution

• ˆ

θ−:negative solution

• 3
• 2
• 1

• 4
• 2

y = x

• xn+1
• and

y = x − ˆ φ

16

Signed magnification in the directly-aligned limit

The signed magnification is given by µ0± ≡ ˆ θ± ˆ φ dˆ θ± dˆ φ . The directly-aligned limit is lim

ˆ φ→0

µ0± = lim

ˆ φ→0

1 1 + n 1 ± ˆ φ ±ˆ φ . The total magnification is given by lim

ˆ φ→0

• µ0+
• + |µ0−| = lim

ˆ φ→0

2 1 + n 1 ˆ φ. The signed magnification sum is obtained by lim

ˆ φ→0

(

µ0+ + µ0−

)

= 2 1 + n.

17

The signed magnification sum.

• 0+

• ^
• n=1

18

Summary of this talk.

• Given Ds/Dls, 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