(c)2013 van Putten
Last time
(c)2013 van Putten
1967
L
S
Swift, HETE II: X-
ray afterglows also
to short GRBs
050509B,
050709,…
BeppoSax GRB 970228: Z-0.695
GRBs
CC-SNe GRB-SNe are rare
making up < 1% of all
SN Ib/c and < 0.2% of
all CC-SNe
Swift Era LGRBs with no
association to massive
stars: GRB 059820A,
050911, 060418, 060505,
060614, 070125
Hyper energetic events
GRB 031203/SN2003lw
GRB030329/SN2003dh
defy max Erot of NS
CC-SNe are
diverse and
produce NS and
BH remnants
1/50 yr in MW
Thumbnail overview…
(c)2013 van Putten
PN
SN
S
( )2
52
%1
1.5 spin, maximal erg103
Mc
M Sun×
BH
( )2
54
%30
10 spin, maximal erg106
Mc
M Sun×
NS
NS
BH
NS
low mass
rapidly
spinning BH
high mass
BHs with a
diversity in
spin
Orphan
LGRB
Orphan LGRB
Astronomical origin of GRBs
(c)2013 van Putten
Detailed derivation of quadrupole emission formula
Lie derivative
Linearized Ricci tensor
Quadrupole GW emission
GW sources in the Local Transient Universe
(c)2013 van Putten
Independent of Christoffel symbols
( ) ( ) ( ) ( ) fufufufufu
uw
b
b
c
c
b
b
c
c
b
b
c
c
b
b
c
c
b
b
bb
∂∂−∂∂=∂∂−∂∂=∂
=
ξξξξξ
ξ
],[
],[
b
c
cb
c
cb uuu ξξξ ∂−∂=],[
Vector fields from vector fields
fufu b
bb ∂→:
Vector field
Lie derivative (I)
Tp
ub
ξb
(c)2013 van Putten
( )
c
a
aa
c
c
cbccb
b
b
bbb
b
bb
b
b
b
b
bb
b
b
bb
b
b
bb
b
bbb
uu
xuxux
x
uuuL
xx
x
x
xA
xx
xA
xx
ξξ
εξε
ε
ξεξ
ε
ξεξ
ε
εξ
ε
εξ
∂−∂=
−+
∂∂
=
−=
∂+=∂∂
+=
∂∂
=∂∂
=
∂−=∂∂
−=∂∂
=
−→
−
→
−
)()(
lim
11
11
'
'
1
*
0
''
1
''
'
'''
'
e.g. Stephani, H.
Taking along one’s own coordinate system along a vector field:
Simultaneous coordinate
transformation
Lagrangian displacement
Lie derivative (II)
(c)2013 van Putten
c
e
e
cce
ec
c
c
c
cc
e
e
c
c
e
e
cc
cc
cc
c
c
c
e
ec
e
e
cce
ec
c
c
uvvuvLu
vLuuvuvvLuuLvvuL
vuuvvuvuL
ξξ
ξξ
ξξξ
ξ
ξξξξ
ξ
∂+∂=
+∂−∂=+≡
∂=∂+∂≡
c
acac
c
a vvvL ξξξ ∂+∂=
Lie derivative (III)
Independent of Christoffel symbols
A tensor vb is dual to a vector ub if ucvc is a scalar, e.g., vb=φ,b
(c)2013 van Putten
c
abc
c
bacabc
c
c
abc
c
bacabc
c
ab
ggg
ggggL
ξξξ
ξξξξ
∇+∇+∇=
∂+∂+∂=
abbaabgL ξξξ ∇+∇=
0=∇+∇ abba ξξ
bξWe say is a Killing vector whenever
If so, it represents a symmetry of the metric tensor
Lie derivative and symmetries
(c)2013 van Putten
If are the vector fields associated with two independent
coordinates (s,t):
( ) ( )0
],[
≡−=∂−=
∂∂−∂∂=∂
sttssa
a
t
a
c
c
a
a
c
c
a
a
b
b
fffvfu
fuvfvufvu
),( bb vu
0],[ =bvu
Lie derivative and coordinates
(c)2013 van Putten
( )
[ ] ( )
hhhR
OhggggR
g
ggg
OR
hhhg
caecad
de
ace
e
ac
caeaccaeaec
de
b
d
adc
ccde
ded
dc
d
bca
d
acb
d
abc
ab
ab
ababab
∂∂−∂∂+∂∂−=
+∂∂−
−+∂=
∂==Γ
+Γ∂−Γ∂=
=+=
2
1
2
1
2
1
2
1
2
1
,
)(
2
,,,
,
2
η
ε
ε
ηη
Linearized Riemann and Ricci tensor
(c)2013 van Putten
hhhh
hhhh
accaae
e
ace
e
a
accaaea
e
cea
e
∂∂−∂∂−∂∂+∂∂
∂∂−∂∂−∂∂+∂∂
4
1
4
1
2
1
2
1
4
1
4
1
2
1
2
1
hhh
hhR
abab
ea
e
cace
e
ac
η2
1
2
1)(
−=
∂∂+∂∂−=
:2
1)( hh caecad
de ∂∂−∂∂η
Linearized Ricci tensor
(c)2013 van Putten
hhhhhR ababea
e
cace
e
ac η2
1,
2
1)( −=∂∂+∂∂−=
Gravitational radiation is in transverse waves. With
Choose
In transverse traceless gauge
abbaabgL ξξξ ∇+∇=
.2
1
0,under invariant ,
h
h
hhh
a
a
a
a
baaaab
a
b
abbaababbaabab
−=∂+∂
=∆+→−∂=∆
∂+∂+=∇+∇+→
ξµ
µµξξξ
ξξξξ
0,2
1=∂∂−= RhR ace
e
ac
linearized Ricci tensor - simplified
(c)2013 van Putten
∫∫∫ −∂−=−==−= xdghhxdghRSxdgRS ab
abab
ab
4244
32
1
16
1
16
1
ππδ
π
About a solution of the Einstein equations, we have
[ ] ( )∫ −∂= xdghhS abcab
42
2
1
16
1
π
The metric fluctuation hab in Coulomb gauge :
[ ] ( )∫ −∂=↔ xdgS c
42
2
1ϕϕ
××
++
+×
×+ +=
−= ababab eheh
hh
hhh
0
0
00 =ah
[ ] ( ) ( )[ ] :16
1 422
∫ −∂+∂= ×+ xdghhhS ccab π ( ) ( )22000
16
1×+ ∂+∂=== hhttt tt
zzz
π
Time-average/orbital-average
Stress-energy tensor of GWs
(c)2013 van Putten
0
02
2
2
2
==→==×=
====→=
∑∑
∑∑∑
Jdt
d
dt
dJjpx
Pdt
dp
dt
dmx
dt
dm
dt
ddxmd
AAA
A
A
A
A
A
A
µµ
No electric or magnetic dipole radiation in GWs
baabababab jAThTG πππ 482
1:8 22 −=∂↔=∂−=
Lienard-Wiechert potential for the radiation field produced by a source
ki
k
jkj
k
ij
k
iki
k
kjij TxTxxTxTxdT ∂+∂−=∂+∂= ∫∫∫ 2
1
2
13
( ) ijjiijjiij Idt
dxdxxTxdTxTxxdT
2
23002
0
300
0
3
2
1
2
1
2
1≡∂=+∂= ∫∫∫
|'|,'),'(4
),( 3 xxrxdrtxTr
trhV
TT
ij
TT
ij −=−= ∫
Making GWs
(c)2013 van Putten
TT
ij
TT
ij Idt
d
rh
2
22= III ijij
T
ij δ3
1−=
TT
jk
TT
jk
TT
jk
TT
jk
z
Idt
dI
dt
d
dtd
Ed
ht
ht
T
3
3
3
32
0
8
1
counting) double avoid to1/2 (extra 32
1
π
π
=Ω
∂∂
∂∂
=
In a given direction, there are 2 polarization modes. They represent
2/5 times the contributions of all 5 (=6-1) components in traceless
part of T
ijI
’t Hooft 2002
T
jk
T
jk
T
jk
T
jkGW Idt
dI
dt
dI
dt
dI
dt
dL
3
3
3
3
3
3
3
3
5
1
8
14
5
2=××=→
ππ
Counting degrees of freedom
(c)2013 van Putten
( ) ( ) ( ) 24622
2
26
3
3
3
3
3
3
3
3
322cos22cos22
12 µϕϕµ aaI
dt
dI
dt
d
Idt
dI
dt
d
T
jk
T
jk
ij
T
ij
Ω=+
Ω=
=
GW emission in binary motion
( ) ( )
( ) const.const.const. const.2cos2
1
const.2cos2
1cos
,,,,5
1
2
222
22
2
11
21
21213
212
3
3
3
3
===+−=
+=+=
+=+=
+=ΩΩ==
yxxyzzyy
xx
T
jk
T
jkGW
IIIaI
aamamI
mm
mmaaa
a
mmtI
dt
dI
dt
dL
ϕµ
ϕµϕ
µϕ
trace of moment of inertia reduces to a constant
(c)2013 van Putten
( )5
23
21246
3
3
3
3
5
32
5
32
5
1
a
mmaI
dt
dI
dt
dL T
jk
T
jkGW
µµ
+=Ω==
1-595
s erg106.3 ×=G
c
( )25
3/10
5
32~
5
32
≅Ω=M
m
a
MMLGW
δ
( )2/32/1
5/1
21
5/3
2
5/3
1 ,~ −≅Ω
+= aM
mm
mmM
Quadrupole emission from an inhomogeneity
Appreciable luminosities arise ONLY for
high-density matter formed in extreme
transient events, like CC-SNe and binary
mergers. The disk-mass (and its
inhomogeneities) in generic accretion
systems (e.g. GRS1915+105 are too small.
Making a wave in a high density disk or torus
Chirp mass
(c)2013 van Putten
LF:
Inspiral of NS/NS
binaries, stellar mass
BHs
HF:
Emissions
from extreme
transient
emissions?
VHF:
Quasi-normal
mode ringing
of stellar
mass BH
burst-sources in
the local universe
HF Shot-noise
2/11 −−∝ lasern PLh
Present and Advanced sensitivities
(c)2013 van Puttenvan Putten 2002
At a critical slenderness: m=1,2,... instability
Ma
mδb
mδ
m=2
Formation of mass-inhomogeneitiesVan Putten, 2002, ApJ, 575, L71
(c)2013 van Putten
Negative chirp during black hole spindownVan Putten, 2008, ApJ, 684, L91
(c)2013 van Putten
+
+
GW-modes
LIGO detector
Unraveling the
mysterious explosion
mechanism of CC-SNe
Observational strategy: CC-SN triggers from optical-radio surveys
(c)2013 van Putten
+= −
M
Mmetf SunTtGW
chirp
10
2]2971.27029.0[)( 90/5.7
Chirp diagramVan Putten, 2009, MNRAS,
396, L81
(c)2013 van Putten
Nearby CC-SNe –어디있어, 얼마나자주?
(c)2013 van Putten
Challenge: initial sensitivity range ~ tens of Mpc
D = 200 Mpc, N ~ 10 million
D = 10 Mpc, N ~ few hundred
CfA redshift survey(SAO 1998)
GW detectors are “near sighted”
(c)2013 van Putten
Van Putten, 2004, ApJ, 611, L81
Guetta & Della Valle 2007, ApJ, 657,
L73
In rare instances, CC-SNe produce GRBs
Branching ratio of SN Ib/c:
~ 0.2-4 %
Relative supernova rates:
SN Ia :SN II : SN Ib/c ~ 50:50:10
(depends on survey, e.g., 68:22:7 in PTF)
Remnants of CC-SNe: bubbles and shells
of the LMC (HI-Halpha)
Kim, S., et al. (1999,2003)
(c)2013 van Putten
Local rates: Milky WayDiehl, R., et al., 2006,
Nature, 439, 45
CC-SNe rate: 1.9(+/- 1.1) events / century
1.809 MeV line
(c)2013 van Putten
M51: farmland of CC-SNe at 8 Mpc
(c)2013 van Putten
CC-SNe are potentially powerful broad band sources of GWs,
especially those associated with long GRBs. Nearby events are
relevant to LIGO-Virgo and KAGRA
Believed to be aspherical and radio-loud. Program for combined
optical-radio monitoring of nearby (e.g. KKT) galaxies.
Efficient surveys (maximal use of telescope time) would exploit correlations of CC-SN rates and galaxy properties, i.e., identify galaxies
with potentially high yields like M51 or M82
EM counterparts to GW searches Van Putten, 2001, Phys.
Rev. Lett., 87, 091101;
2004, JKAS, 45, S77; Van
Putten, et al., 2004, Phys.
Rev. D, 69, 044007; van
Putten, et al., 2011, Phys.
Rev. D, 83, 044046
(c)2013 van Putten
Famaey, B., &
McGaugh, S.S., 2012,
Living ReviewsMissing mass problem
(c)2013 van Putten
Galaxy rotation curves Famaey, B., &
McGaugh, S.S., 2012,
Living Reviews
(c)2013 van Putten
Galaxy rotation curves Famaey, B., &
McGaugh, S.S., 2012,
Living Reviews
(c)2013 van Putten
Characteristic acceleration in weak gravity Famaey, B., &
McGaugh, S.S., 2012,
Living Reviews
(c)2013 van Putten
Scales in gravity Famaey, B., &
McGaugh, S.S., 2012,
Living Reviews
(c)2013 van Putten
Famaey, B., &
McGaugh, S.S., 2012,
Living ReviewsCosmological baryon distribution
(c)2013 van Putten
Lensing in curved spacetime
(c)2013 van Putten
)sin(2
12
1 22222
1
22 ϕθθ ddrdrr
Mdt
r
Mds ++
−+
−−=−
gH RR 2=
ba
ab
ba
ab
a
a
a
a dxdxgdsuuguud
dx
d
dxL ===
= 22 ,ττ
Lensing in Schwarzschild line element
(c)2013 van Putten
Aging is slower nearby the surface of the earth
than far out in space: gravitational redshift
Orbital precession (Mercury, PSR1913+16, 4U1820-30)
Bending of light by gravitation: gravitational lensing
Inner most stable circular orbit: critical inner radius of accretion disks
Circular photon-orbits
Event horizon
periastron
aphelion
Some phenomenology
(c)2013 van Putten
Chandra X-ray image (false colors) with
radio (contour) (Chartas et al. 2001)
Discovery: Dennis
Walsh, Bob Carswell,and
Ray Weymann (1979)
Rhee, G., et al. (NASA)
Lensed AGN
(c)2013 van Putten
4 images
(c)2013 van Putten
( )
b
Mr
rOrrb
v
v
44][
)()(2~)2cos(2
1
2
3sin
)2cos(2
1
2
3
sin
serieson perturbatiRegular
2
1
0
==±∞=∆⇒
∞→++
++=⇒
+=
=
εϕ
ϕεϕϕεϕ
ϕ
ϕ
ϕb
Photon trajectories
10
2
2
)(3''
3''
1,
vvv
buvvvv
M
rb
M
εε
κκκ
κε
+=
==+
=+
==
ϕ∆
(c)2013 van Putten
Young et al. 1980
Turner, Ostriker & Gott (1984)
Blandford & Narayan (1986,1992)
ξ
∧
αθ
αβ
ξα
M=
∧
Source-image plane Observer’s planeLensing object
ξ
dsDdD
sDLens equation: βαθ +=
αα sds DD =∧
Image (“mirage”)
source
observer
Lensed trajectory
Unlensed trajectory
Photon trajectories
(c)2013 van Putten
arcsec-milliGpc110
32
)(
)(,~
411
2
1
,4
2/12/1
6
2
2
2
−
==
<<±
>>−
+±=
=
==−
D
M
M
D
R
D
DDD
DD
DM
sun
g
E
EE
EE
E
ds
sd
d
s
ds
θ
θβθ
θββθ
ββθ
βθ
θξβξ
θ (all vectors are are aligned)
one image outside,
one image insideEθ
Einstein ring
(source near axis)
Point mass
(c)2013 van Putten
==
−−=
=−
ds
sd
ds
ds
D
DDD
D
M
etravel tim
DD
DM
θθψ
θψβθθτ
βθ
θ
ln4
)(
)()(2
1)( theis
4 of integral The
2
Find images in terms of local extrema of photon travel times
Fermat’s principle
(c)2013 van Putten
Numerical example
(c)2013 van Putten
αβ ire=Source positions
quadrupole mass-moment
(a binary of galaxies or a
model for an elliptical galaxy)
observer
Quadrupole lens
(c)2013 van Putten
Multiple images
(not to scale)
(c)2013 van Putten
0 1 2 3 4 5 6-16
-15.5
-15
-14.5
-14
-13.5
-13
-12.5
-12Merger and annihilation of a minimum and saddle point
τ(θ)
S1
L1
S2
L2
S1
angle along Einstein ring
β=0
β=0.1*eiπ/4
β=0.2*eiπ/4
β=0.3*eiπ/4
Merger of S and L in quadrupole lensing
(c)2013 van Putten
Cre
ate
extrem
um
+ s
addle
poin
t
Split
an e
xtrem
um
into
tw
oS
X
X
X XS
Always create two new extra extrema (leaving a total of 2n+1)
On the boundary
Boundary reconnection
Burke 1981, Blandford &
Narayan 1986
ϕcos:Limacon bar += 2||||:Lemniscate ccpcp =+−
X (high,low), S out of change in topology
(c)2013 van Putten
( )
inside) (const.
outside) (||ln2)(
inside) (0
outside) (2
''
1
'
1
'
'
'
1)('
)(Re'lnRe'ln)(
,2'
const.''ln4
''ln)'(4
)(
2
0
2
0
0
''
0
∴
=⇒
=
−−
=−
=
≡−=−=
====
+−=−=
∫∫
∫∫
∫∫
∫∫
==
z
zzzI
z
zzdz
zzzz
i
iz
dz
zzzf
zfdzzdzzzI
D
DDDDdDdsM
dzzD
Dd
D
D
ds
sdd
RD
d
Rz
s
ds
s
ds
d
ππ
ϕϕ
µπθµµ
ϕµ
θθθθµθψ
ππ
θ
)(
''
Rd
d
DD
dDRd
≡
= θϕ observer
=ring) inside(0
ring) outside(ln4
)(θθψ D
M
Lensing by a ring
(c)2013 van Putten
>
<=
=
)()ln(4
)()ln(4
)(
22
22
d
dd
dd
D
MD
D
DM
θθθ
θθθθπµ
θψ
θπµ
'' θϕ dDRd d=
observer
Lensing by a uniform disk
(c)2013 van Putten
Gpc10kpc5
25.1
disk of radius nalgravitatio4
1
4
1)4(
observer)-emitter(symmetry 1414
later) on this more-distancesangular (,4
,44
1
11
2
,
1**
222
−
−
=
=
===
−=⇔
−=
+==
===
sun
dd
dd
d
gd
ddsd
ds
dsd
d
dds
ddssd
ds
s
ds
sdEdenclE
M
MR
RR
RR
RDD
D
DD
D
DD
DDDDD
D
D
DDD
D
D
D
M
µπ
µπµπ
πµ
θπµθ
*
dD*
dsD
Minimal focusing distances
(c)2013 van Putten
Elliptical galaxy modeled by two disks of uniform density
Oops! – too close cq. insuffient lensing mass
Einstein radius
M=0.1
Lensing by two elliptical galaxies
(c)2013 van Putten
M=0.5 M=1
Lensing by two elliptical galaxies
(c)2013 van Putten
H
M=4
L,S
LS are no more
Lensing by two elliptical galaxies
(c)2013 van Putten
νθ ∝d
[ ]
[ ] .0det
:i.e.
thatprovided on,continuati l)l/numerica(analyticaby uniquely )( ofbranch aTrack
0)),((
Then
.0)),((
distance) source:..(parameter control aConsider
locations. source toresponsein extremum isolatedan ofy trajectorThe
0::),( of points Extremal
1
≠
−=
=+=
=
=∂=
−
θθ
θλθθ
θλθθθ
θ
θ
τ
ττλθ
λθ
τλθ
τλλθτλ
λλθτλ
τθτ
d
d
d
d
d
d
ge
GΛ
Confluence of images (isolated extrema)
(c)2013 van Putten
[ ]
same) theremains imagescomplex ofnumber total(The
2."by jumps images real ofnumber thesurfaces, caustic a across"
:Precisely 2.by jumps produced images ofnumber thesurface, caustic a Across
sky. in the positions image possible for ),( positions source Solve
0det,0
equations 3 thesolving space source-3Din preimages are caustics The
θβ
ττ θθθ
allDds
==
One real root
Two complex
rootsThree real rootsTwo real roots
(one double)
Caustics: changing multiplicity of images
(c)2013 van Putten
[ ]
.spanKer and spanIm definesIt
leavesreduction one-rank a andconfluent are images twoHere, .point a is
0det :confluent are images When two
⊥=∈=
=
=
ξτθξτ
ξξτ
τ
θθθθ
θθ
θθ
d
nbifurcatio
T
Trajectory of confluent images
Trajectory of source:
generator of a caustic surface
)(),( λλβ sdD
)(λθ
Trajectory of confluent images
(c)2013 van Putten
( )
0:0
:0
)(2
1
1
2
=−=
−==
−−=
−λψτ
λψθβτ
θλψβθτ
θθθθ
θθ
HHH det)(tr4
1tr
2
1 21 −±=−λ
Travel time
Fermat’s principle
Eigenvalue problem
Hessian of potential
Away from lens
Apex
µµ
ψ θθ
2tr, == HH
HH
D
M
d
det4)(tr
0
2
2
=
=
=
µ
πµ
D
M
M
dM
4
'ln)(
=
−= ∫
λ
θθθψ
Hdet1 −±=−±λ
Generators of a caustic
(c)2013 van Putten
.)det of isocurves are theseregion, source (Outside
.)( seigenvalue theof isocurves
: fixedat caustic a ofsection -Cross
0
H
Dds
λθλ =+
d
ddsd
DM
DDD
−=+=
+
−
λψλθβ θ
/4,1
:),( sDβθ →
Image of a constant distance cross section of a caustic
(c)2013 van Putten
Caustic surfaceImages of constant Dds-caustics
Caustic surface of a quadrupole lens at (+/-p,0,0)
(c)2013 van Putten
[ ] [ ]
[ ][ ]θθ
θθ
θβθθθβθθ
θβθθ
θ
ττ
βθ
τδβτδβττδθ
δβτδθτ
τ
det
1det
)1(
:0
:0
1
11
==∂∂
≡−=−=
=+
=
−
−−
Fermat’s principle
introduces the linearized relation
and hence an expression for the amplification (of brightness, expressed per unit
of opening angle) in terms of the determinant of the associated Jacobian
Amplification
