Transcript Презентация PowerPoint - Louisiana State University

Gravitational lensing by
cosmic string loops



Lagrangian of scalar field:
L          14     2
Euler-Lagrange equation:
      12     2  0

2


2
Particular solution:
Energy-momentum tensor:
  e in f (r )
T    ( x) ( y)diag (1,0,0,1)
R
  22 ln  
 
Space-time metric for the case of straight string:
ds 2  c 2 dt 2  dz 2  (1  8G ln( r / r0 ))( dr 2  r 2 d 2 )
G  r 
ln  
c 2  r0 
G 

,     1  4 2 
G
c 

1 8 2
c
1 8
r  r
ds 2  c 2 dt 2  dz 2  dr  2  r  2 d  2 ,
Energy-momentum tensor:
1

T  (r, t )  c 2  d  2 f  f   f  f    3 (r  f ( , t )),
c

Equations of motion:
f  f   0,
f 2  f 2  1,
f - f   0,
Lens equations:
 (1  f3 ) 2  f 32

xi  f i
q
yi  xi 
d 
, i  1,2

2 L 
( x1  f1 ) 2  ( x2  f 2 ) 2  t t t
1  f3
0 1
1
Gμ  v  Dl Dls
q  8 2 1  
,
c  c  Ds R
t0  f 3 ( , t0 ).
Magnification of a point source:
2


 (1  f3 ) 2  f 32 ( x  f ) 2  ( x  f ) 2 
q2 

1
1
2
2
m  1  2   d 

 

2
2 2
4 
1

f
( x1  f1 )  ( x2  f 2 )  t t t

3
0 1 



 (1  f ) 2  f 2
2( x1  f1 )( x2  f 2 )
q2 
3
3
 2   d 
4 
1  f3
( x1  f1 ) 2  ( x2  f 2 ) 2









2

 t t0 t1 

2 1
.
Lens equation for circular loop:
x1

x

q
, if
 1
2
2
y1  
x1  x2
 x1 ,
if

x2

, if
 x2  q 2
2
y2  
x1  x2
 x2 ,
if

x | cos t |,
x | cos t |,
x | cos t |,
x | cos t |,
Magnification of a point source:
1


q2
1 
m
2
2
x

x
1
2

1,

, if
x  cos t ,
if
x  cos t .

2
Examples of brightness curves for lensing by circular loop.
Lens equations for asymmetric loop:
(1  x1  x2 ) 2  4 x1  1  x1  x2
2
y1  x1 
qx1
y 2  x2 
qx2
2 x1
2
2
(1  x1  x2 )  4 x1
2
2
2 x2
2
2 2
2
2
(1  x1  x2 ) 2  4 x1  1  x1  x2
2
2
2
2
(1  x1  x2 )  4 x1
2
2
2 2
2
Magnification of a point source:
1
m
1
q ( x1  x2 )
2
2
2
((1  x1  x2 ) 2  4 x1 )3 / 2
2
2
2
.
,
2
,
Critical curves (dark) and loop
(red).
Caustics (dark), lines of doubling
(blue) and loop (red).
Observable data and theoretical curves.
Parameters of the loop:   8 1021g/cm,
L  0.1pc, 2θR  3 ( Dl  3 kpc)
Position of asymmetric loop at different time.
Lens equations for binary system:
q
yi  xi  b
2
qb 
a
b


xi  xi (t )
xi  xi (t )


a
a
b
b
2
2
2
2
(
x

x
(
t
))

(
x

x
(
t
))
(
x

x
(
t
))

(
x

x
(
t
))
2
2
1
1
2
2
 1 1

8Gmc Dl Dls
c 2 Ds r 2
x1 (t )  cos t
x2 (t )  sin  t
a
a
xi (t )   xi (t )
b
a
Magnification of a point source:
1
m
qb ( x1  x2 )
2
2
2
((1  x1  x2 ) 2  4 x1 )3 / 2
2
1
2
2
.
Necessary values of system parameters: mc  78 MSun , Dl  1.2 pc, r  1.8 a.u.
1 / 2
Magnification of stars near ends of caustics:
5  Rs 

m  3  10 
 RSun 
Magnification of stars on cusp of caustics:
 R 
m  1.1  10 9  s 
 R Sun 
2 / 3
Loop distribution:
n(l , t ) 
vm
, l   ct
2 2
2
c t (l   c t )
vm  0.5,  ~ 10 4
Source distribution:
dN s 4 N tot 2  z 2

z e
dz

Lensing probability
N g N tot  1.5  10 4 mmin 
For
N tot  10 6
mmin  0.04
1 / 3
N g  440