(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

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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