frequency (kHz)

Sources of Gravitational Waves

2

talk

Astrophysical Sources of GWs

z

Binary systems (NS/NS, NS/BH, BH/BH)

z

Supernova

z

Bounce

z

Fall back

z

Oscillations & Instabilities

z

Old and Isolated NS

z

Cosmological origin

GW sources in ground-based detectors

Supernovae, BH/NS formation

Young Neutron Stars BH and NS Binaries

Spinning neutron stars in X-ray binaries

Sources in LISA

Galaxy mergers Galactic Binaries

Binary systems

(NS/NS, NS/BH, BH/BH)

The best candidates and most reliable

sources for broad band detectors

Coalescence of Compact Binaries

z During the frequency change from 100-200Hz GWs carry away5x10^-3M_~c².

z In LIGOs band

z NS/NS (~16000 cycles)

z NS/BH(~3500 cycles)

z BH/BH(~600 cycles)

z The GW amplitude is:

z Larger total mass improves detection probability.

2 / 3 2 /

3

3

2 100

7.5 10 2.8 0

.7 100

M f Mpc

h M M Hz r

− ⎛ ⎞ ⎛ µ ⎞⎛ ⎞ ⎛ ⎞

≈ × ⎜⎝ ⎟ ⎜⎠ ⎝ ⎟⎜⎠⎝ ⎟ ⎜⎠ ⎝ ⎟⎠

: :

events/y ear

LIGO-I LIGO-II

NS/NS ~0.05 ~60-500

BH/NS ~0.02 ~80

BH/BH ~0.8 ~2000

Total 0.8 22000

•Phase effects are important, if the signal and the template get out of phase their cross correlation will be reduced.

Gravitational Waves from Binaries

Generically, there are 3 regimes in which black holes radiate:

z Orbital in-spiral: PN- approximations ^or point-particle orbits.

z Plunge/merger after the last stable orbit: numerical simulations or point- particle orbits^.

z Ring-down of the disturbed black hole as it settles down to a Kerr hole: perturbation theory of black holes.

BH/BH coalescence

z The inspiral, merger, and ringdown waves from 50M_: BH binaries as observed by initial and advanced LIGO.

z The energy spectra are

coming from crude estimates (10% of the total mass energy is radiated in merger waves and 3% in ringdown waves).

z We observe that the inspiral phase is not visible with initial LIGO, for this case Numerical Relativity is important.

Possible First Source:

Binary Black Hole Coalescence

z 10M_~ + 10 M_~ BH/BH binary

z Event rates based on population synthesis,

z mostly globular cluster binaries.

z Totally quiet!!

First gen

Second gen

100 M

pc inspiralRate: 1 / 300yrs to 1 / yr

merger

z=0.4

inspiral Rate: 2/mo to 10/day

merger

NS/BH Binaries

43 Mpc

NS-BH inspiral and NS Tidal Disruption

650 Mpc

NS-BH Event rates

z Based on Population Synthesis

z Initial interferometers

z Range: 43 Mpc

z 1/1000 yrs to 1per yr

z Advanced interferometers

z Range: 650 Mpc

z 2 per yr to several per day

Merging phase: NS/NS & BH/NS

z Tidal disruption of a NS by a BH

(Vallisneri)

z GWs could carry information about the EOS of NS eg. estimation of NS radius (15% error).

z The disruption waves lie in the band 300-1000Hz

z A few events per yearat 140Mpc (LIGO-II)

z Merging of NS-NS(Rasio et al)

z Imprint of the NS radii just before merging (ƒ11kHz)

z During the merging we could get important information about the EOS (ƒ21kHz)

Core-collapse Supernova

The most spectacular astronomical event

with exciting physics

Supernovae/gravitational collapse

Rate 1/30yr in typical galaxy Detection would provide unique insight into SN physics:

– optical signal hours after collapse – neutrinos after several seconds – GWs emitted during collapse

Supernova core collapse was the primary source of GW detectors.

GW amplitude uncertain by factors of 1,000’s?

Simulations suggest low level of radiation (<10^-6 M_:c²?), but

– rotational instabilities possible

– observational evidence for asymmetry from speeding final neutron stars (release of 10^-6 M_:c² could explain 1000 km/s?)

– convective “boiling” observable to LMC

Core-Collapse Supernovae I

z Stars more massive than ~8M_:end in core collapse (~90%are stars with masses ~8-20M_:).

z Most of the material is ejected

z If M>20M_: more than 10% falls back and pushes the PNS above the maximum NS mass leading to the formation of BHs (type II collapsars).

z If M>40M_: no supernova is launched and the star collapses to form a BH (type I collapsars)

z Formation rate:

z 1-2per century in the Galaxy (Cappellaro & Turatto)

z 5-40%of them produce BHs through the fall back material

z Limited knowledge of the rotation rate! Initial periods probably <20ms.

z Chernoff & Cordesfit the initial spin with a Gaussian distribution peaked at 7ms. This means that 10%of pulsars are born spinning with millisecond periods.

Core-Collapse Supernovae II

Dimmelmeir, Font & Muller 2002 z Rotation increases strongly during the collapse.

•Multiple bounces are possible for a few models.

The ringing phase might last much longer

Core-Collapse Supernovae III

z GW amplitude

z Signals from Galactic supernova detectable.

z Frequencies ~1 kHz

z The numerical estimates are not conclusive. A number of effects (GR, secular evolution, non-axisymmetric instabilities)have been neglected!

(Axisymmetric collapse, Mathews-Wilson approximation…)

z Kicks suggest that a fraction of newly born NSs (and BHs) may be strongly asymmetric.

z Polarization of the light spectra in SN indication of asymmetry.

23 10

TT 10 Mpc

h d

 −

Black-Hole Ringing I

z The newly formed BH is ringing till settles down to the stationary Kerr state (QNMs).

z The ringing due to the excitation by the fallback material might last for secs

z Typical frequencies: ~1-3kHz

z The amplitude of the ringdown

waves and their energy depends on the distortion of the BH.

z Energy emitted in GWs by the

falling material: ∆E>0.01µc²(µ/M)

3/10 2

9 0

20 1 1

/

[1 0.63(1 / ) ] 2(

3.2 Hz

) k

1

m M a

f

Q f

M π τ a

=

−

− − −

= ≈ −



d 1

ε µ

−

− ⎛ ⎞

⎛ ⎞

⎛ ⎞

≈ ×

Oscillations & Instabilities

The end product of gravitational collapse

Neutron Stars

z Suggested: 1932

z Discovered: 1967

z Known: 1070+

z Mass: ~ 1.3-1.8 Μ_:

z Radius: ~ 8-14 Km

z Density: ~10¹⁵gr/cm³

Stellar pulsation primer

For spherical stars we can (in the Cowling approximation) write the Euler equations as ²

1

2 ( )

i

i i j

j

p

p A

t

∂ ξ δ ξ

∂ ρ ρ

⎛ ⎞ Γ

= −∇ ⎜ ⎟+ ∇

⎝ ⎠

Two main restoring forces, the pressure and the buoyancy associated with

internal composition /temperature gradients, lead to:

p-modes g-modes

2 2

2

( 1) _s l l c

ω +r

≈

2

i

Ai p

ω gA ∇

≈− =

(δ p∝Y_lm( , ))θ ϕ

NS ringing : Stellar Modes

z P-modes: main restoring force is the pressure

z G-modes: main restoring force is the buoyancy force

z F-mode: has an inter-mediate character of p- and g-mode

z W-modes: pure space-time modes (only in GR) (KK & Schutz) z Inertial modes (r-modes) :main

restoring force is the Coriolis force

z Superfluid modes: Deviation from

chemical equilibrium provides the main restoring agent

0.6 0.8 1.0 1.2 1.4 1.6 1.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

w₁-modes

p₁-modes

f-modes

A B C D E F G I L G₂₄₀ G₃₀₀ WWF

M (in 1.4 solar mass units)

frequency (kHz)

Each type of mode is sensitive to the physical conditions where the amplitude of the mode is greatest.

Grav. Wave Asteroseismology

⎛ ⎞1/ 2 ⎡ ⎤

0.15 0.18 0.21 0.24 0.27 0.30 50

60 70 80 90 100 110 120 130 140 150

A B C D E F G I L G₂₄₀ G₃₀₀ WFF Q_1 Q_2 Q_3

M/R R ω w-mode

0.03 0.04 0.05 0.06 0.07 0.08 1.2

1.6 2.0 2.4 2.8 3.2 3.6 4.0

A B C D E F G I L G₂₄₀ G₃₀₀ WFF Q_1 Q_2 Q_3

(M/R³)^1/2

ω f-mode

Grav. Wave Asteroseismology

⎤

⎡

1 M3 M

6 8 10 12 14 16

1.0 1.5 2.0 2.5

A B C D E F G I L WFF G₂₄₀ G₃₀₀

Radius (Km)

M (Solar Masses)

0.15 0.18 0.21 0.24 0.27 0.30 0.01

0.02 0.03 0.04 0.05

A B C D E F G I L G₂₄₀ G₃₀₀ WFF

M/R R4 / M3 τ f-mode

Unique estimation of Mass and Radius

Stability of Rotating Stars

Non-Axisymmetric Perturbations

Dynamical Instabilities

z Driven by hydrodynamical forces (bar-mode instability)

z Develop at a time scale of about one rotation period

Secular Instabilities

z Driven by dissipative forces

(viscosity, gravitational radiation)

z Develop at a time scale of several rotation periods.

z Viscosity driven instability causes a Maclaurin spheroid to evolve into a non-axisymmetric Jacobi ellipsoid.

z Gravitational radiation driven instability causes a Maclaurin

spheroid to evolve into a stationary but non-axisymmetricDedekind ellipsoid.

A general criterion is:

W

= T β

T : rot. kinetic energy W : grav. binding energy

27 .

≥ 0

β

The bar-mode instability I

For rapidly (differentially!) rotating stars with:

the “bar-mode” grows on a dynamical timescale.

If the bar persists for many (~10-100) rotation periods, the signal will be easily detectable from at least Virgo cluster.

–A considerable number of events per year in Virgo: ≤10^-2 /yr/Galaxy

–Frequencies ~1.5-3.5kHz

dyn 0.27

T

β = W >β ≈

2

23 2

1.4 10

15 Mpc

9 10 0.2 3 kHz d

h _{− ⎛} ε _⎞⎛ f _{⎞ ⎛ ⎞}M R

≈ × ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠

Remember mini-Grail: f₀~3.2kHz

The pattern speed

d

dt m

ϕ ω

σ

= − =

z The pattern speed ^σ of a mode is:

z If a star rotates very fast, a backward moving mode, might change to move forward, according to an

inert rot

inert rot

m

σ

ω ω

σ

= + Ω

= +Ω

Chandrasekhar 1969: Gravitational waves lead to a secular instability

Friedman & Schutz 1978: The instability is generic, modes with sufficiently large m are unstable.

A neutral mode of oscillation signals the onset of CFS instability.

The CFS instability

•Radiation drives a mode unstable if the mode pattern moves backwards

according to an observer on the star (J_rot<0), but forwards according to someone far away (J_rot>0).

•They radiate positive angular

momentum, thus in the rotating frame the angular momentum of the mode increases leading to an increase in mode’s amplitude.

in rot

m m

ω _{= −}ω _{+ Ω}

F-mode-(I)

z F-mode is the

fundamental pressure mode of the star

z It corresponds to polar perturbations

z Frequency for uniform density stars

z For ℓ=2 is ~2-4kHz

z Rotation breaks the symmetry:

the various -ℓ≤m≤ℓ decouple

z There is coupling between the polar and axial modes

z The frequency shifts:

2

3

1 3 4

2 ( 1)

2 1

growth time(if unstable)

( ) ~ 0.07 1.4 sec

10

l GW

l l GM

l R

R M R

t f l R

M M km

ω

+

= −

+

⎛ ⎞

⎛ ⎞⎟ ⎜ ⎟ ⎛ ⎞⎟

⎜ ⎜

≈ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠

:

inert( ) ( 0) m

ω Ω = ω Ω = +κ Ω

1 0 1 1 1 2 1 3 1 4 1 5 1 6

σ (kHz)

m = - 2 m = - 1 m = 0 m = + 1 m = + 2

The r-mode-(I)

z A non-rotating star has only trivial axial modes

z Rotation provides a restoring force (Coriolis) and leads in the appearance of the inertial modes.

z The l=m=2 inertial mode is called r- mode

z In a frame rotating with the star, the r- modes have frequency

z Meanwhile in the inertial frame

z r-modes are appear retrograde in the rotating system while in the inertial frame the prograde at all rotation rates!

rot

2 ( 1)

m ω = l l Ω

+

inertial rot 2

1 ( 1)

m m l l

ω = −ω + Ω = Ω −⎛⎜⎝ + ⎞⎟⎠

2

~

, , ~

r

u

u u

ϕ

θ

δ

δ δ δρ Ω

Ω

R-modes have:

Growth ^vs Damping

z Viscosity tends to suppress a GW instability.

z An instability is only relevant if it grows sufficiently fast that is not completely damped by viscosity

z Bulk viscosity: arises because the pressure and density variations associated with the mode oscillation drive the fluid away from beta

equilibrium. It corresponds to an estimate of the extent to which energy is dissipated (via neutrino emission) from the fluid motion as the weak interaction tries to re-establish equilibrium.

z Shear viscosity: in matter hotter than superfluid transition temperature T~10⁹ K, due to neutron-neutron scattering, and for superfluids, due to electron-electron scattering

2

1 1 1

2

1

1

BV SV

GW

dE E dt

E dV

τ τ

ρ ξ

−τ

= + +

=

∫

Timescales

z Dissipation due to bulk viscosity

z Dissipation due to shear viscosity

z Dissipation/growth due to gravitational radiation

( )

SV

*

2 9

( )

2 2

,

0

2 ~

1

ab

ab

c

a b b a ab c

i m

dE dV

d

g T t

K

δσ ω ξ ξ ξ η

ηδσ δσ + Ω −

⎛ ⎞⎟

⎜ ⎟ = −

⎜ ⎟

⎜

⎛ ⎞⎟

⎜ ⎟

⎜ ⎟

⎜⎝ ⎠

= − ∇ ∇

⎠

∇ + −

⎝

∫

BV

6 9

2

, ( ) , ~

10 dE

d p K

T m p

t ζ σδ δσ i ω ζ

⎛ ⎞⎟

⎜ ⎟ =

⎜ ⎟

⎛ ⎞⎟

⎜ ⎟

⎜ ⎟

⎜⎝

= − + Ω ∆

Γ ⎠

⎜⎝ ⎠

∫

( )

( )

2 2

2 1 G

*

2

1/ 2

* W

( )

2

, 1

l

l B

l

l lm

l

m l

lm l

lm lm m

dE m N D

D r Y dV

d J

J l r Y dV

l t

δ δρ

ω ω δ

δ ρδυ υ

δ

δρ

∞ +

=

⎛ ⎞⎟

⎛ ⎞⎟

⎜ = − + Ω +

= ⎜ +

=

⎟

⎜ ⎟⎟

⎜⎝ + ⎠

⎜ ⎟

⎜⎝ ⎠

∑

∫ ∫

R-mode: Instability window

z For the r-mode (ℓ=2) we get:

z Instability window

z Many astrophysical

applications both on newly

5 9 6 2

BV

5 / 4 23/ 4 2

SV 9

4 0

8

W

1

G

1.4 10

10km 1ms sec

1.4 sec

10km 10

1 2.4 10

1.2 10

22 .4

10km 1ms

M R K P

M T

M R T

M K

M R P

M τ

τ τ

−

⎛ ⎞

⎛ ⎞⎛⎟ ⎞⎟ ⎜ ⎟ ⎛ ⎞⎟

⎜ ⎜ ⎟ ⎜

≈ ⎜⎜⎝ ⎟⎟⎟⎠⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠

⎛ ⎞ ⎛⎟ ⎞⎟ ⎛ ⎞⎟

⎜ ⎜ ⎜

≈ ⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠

⎛ ⎞⎛⎟ ⎞ ⎛⎟

⎜ ⎜ ⎜

≈ ⎜⎜ ⎟⎟⎟⎜⎜ ⎟⎟ ⎜⎜

×

⎝ ⎠ ⎝

⎝ ⎠

×

−

:

:

:

6

⎞⎟ sec

⎟⎟⎠

R-modes

z GW amplitude depends on α (the saturation amplitude).

z Mode coupling might not allow the growth of instability to high amplitudes (Schenk etal)

z The existense of crust, hyperons in the core, magnetic fields,

affect the efficiency of the instability.

z For newly born neutron stars might be quite weak; unless we have the creation of a strange star

z Old accreting neutron (or

strange) stars, probably the best source! (400-600Hz)

22 1Mpc

( ) 10 Hz

h t 1 k

α d

− ⎛ ⎛ ⎞

≈ ⎜⎝ ⎟

Ω ⎞

⎜ ⎟

⎠ ⎠

⎝

2 3

10 10

α 

−

Lindblom-Vallisneri-Tohline

F-mode (astrophysics)

z F-mode is naturally excited in any process.

• In GR the m=2 mode becomes unstable for Ω>0.85^Ω_Kepleror β>0.06-0.08

• The instability window significantly smaller than the r-mode

• Detectable from as far as 15Mpc (LIGO-I), 100Mpc (LIGO-II) (depending on the

saturation amplitude).

• Differential rotation affects the onset of the instability

• Recent non-linear calculations by Shibata

& Karino (2004) suggest that:

• Up to 10% of energy and angular

momentum will be dissipated by GWs.

• Amplitude (ar ~500Hz): ⎛ ⎞ ⎛^{1/ 4} ⎞ ⎛^{3/ 4} ⎞

Isolated & Old NS

Isolated NS

z Wobbling or Deformed NS (many

interesting features but highly uncertain the degree of deformation)

z LMXBs : if accretion spin-up torque on NS is counterbalanced by GW emission then Sco X-1 and a few more might be detectable around 500-700 Hz.

⎟⎟⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

× ⎛

≥ ⁻

kpc r f

kHz

10 10 1

2

2

ε 8

LMXBs might be as robust source of GWs

Slowdown from pulsar

zUpper limits on amplitudes from known pulsars, set by assuming spindown due to the emission of gw energy. The points represent all pulsars with gravitational wave frequencies above 7 Hz and amplitudes above 10^-27.

zExpected sensitivities of three first-generation interferometers in a one-year

The Wagoner mechanism (1984) Papaloizou &Pringle (1978)

Possible GW mechanisms:

– accretion induced asymmetry

– unstable r-modes: strong bulk viscosity may shift instability window to lower temperatures;

accreting stars can reach quasi-equilibrium state Key idea: Emission of GW balances accretion torque.

Strength of waves can be inferred from X-ray flux.

Requires deformation:

Observational evidence (?):

clustering of spin-frequencies in LMXB (250-590 Hz)

1/ 2 5 / 2

8

9

300 Hz 4.5 10

10 / _s

M

M yr

ε ν

−

−

⎛ ⎞ ⎛ ⎞

= × ⎜ ⎟ ⎜ ⎟

⎝ ⎠

⎝ ^: ⎠



LMXBs & r-modes

UNSTABLE

5ms

Period

clustering of ms pulsars

STABLE Limiting Period 1.5ms

Fastest known pulsar 1.56ms

LIGO narrow banding

•LIGO-I phase

•The only detectable source is BBHs (10M:)

•LIGO-II phase (2006)

•Many sources…

Narrow banding for LMXBs

Stochastic Background

GW from the Big Bang

2

3H0

ρ =

gw gw

c

f d df

ρ Ω = ρ

d gw

f ρ

Ω = Stochastic background reflecting

fundamental physics in the early universe;

- Phase transitions - Inflation

- Topological defects

- String-inspired cosmology - Higher dimensions

After the Big Bang, photons

decoupled after 10⁵ years, neutrinos after 1s, GWs before10^-24 s!

Strength expressed as fraction of closure energy density;

14 5

10⁻ ≈ Ω <10⁻

Detection: Requires cross-correlation of detectors.

Best window, free of “local” GW sources, is around 0.1-1 Hz.

Need LISA follow-on mission?

18 1Hz 2

10 ( )

h ⁻ ⎛ ⎞ h f

⎜ ⎟ Ω



GW from Inflation

The Dark Side of the Universe

-- Kip Thorne

z Our present understanding of the Universe is based almost entirely on electromagnetic radiation.

z Black holes can emit only gravitational radiation.

z More than 90% of the Universe is dark, but it still interacts by gravity.

z There are 5-10 times as many dark baryons as luminous ones.

z If part of the dark matter forms compact clumps, then

gravitational wave detectors will be the only way to see it directly.

The Rewards Are Huge

(Very) Early Universe

Gravitational Wave Observations

Stellar interiors Cosmology

Quantum theory

Astrophysics

Fundamental physics Extreme Gravity

The END