What I did in grad school

Marc FavataB-exam

June 1, 2006

Kicking Black Holes

Crushing Neutron Stars

and the adiabatic approximation in extreme-mass-ratio inspirals

How black holes get their kicks:The Gravitational Radiation Rocket Effect

recoil

Work based on:

Favata, Hughes, & Holz, ApJL 607, L5, astro-ph/0402056

Merritt, Milosavljevic, Favata, Hughes, & Holz, ApJL 607, L9, astro-ph/0402057

Astrophysical Motivations

Galaxies:

• Ejection of black holes from galaxies: [700 - few ´ 1000km/s for large galaxies,

5 - 200 km/s for dwarf galaxies]

• “Wandering” of BHs not ejected.

• “Smearing” of central density cusp.

• Formation of SMBHs through hierarchical mergers

Globular clusters:

1. Ejection from cluster: Vesc~ (3-100) km/s

2. Formation of IMBHs: seed holes with M d 50 M susceptible to ejection.[Gültekin, Miller, Hamilton, astro-ph/0402532]

Gravitational wave recoil: a GR application for astrophysics, NOT gravitational wave detection.

Understanding Gravitational Radiation Recoil:

GW momentum flux:

[Wiseman, PRD 46, 1517]

GW momentum flux:

[fig. from Wiseman, PRD 46, 1517]

4 3 33

GW

4 3 3 3

2 16

63 45

jjab pa qaab

jpq

d d ddP d

dt dt dt dt dtε= + +⋯

I I SI

Understanding Gravitational Radiation Recoil:

Recoil relies on “symmetry breaking”…

42

1 2kick

max term.

( ) 2 ( ) /1480 km/s

f q G m m cV

f r

+=

Lowest order quasi-Newtonian calculation gives (circular orbits) [Fitchett (1983)]:

5

2

)1(

)1()(

q

qqqf

+−=

018.0)38.0( maxmax ≈== qffIf system is symmetric (m1=m2), recoil is zero (for non-spinning holes).

2

1

m

mq =

Recoil relies on “symmetry breaking”…

42

1 2kick

max term.

( ) 2 ( ) /1480 km/s

f q G m m cV

f r

+=

Lowest order quasi-Newtonian calculation gives (circular orbits) [Fitchett (1983)]:

5

2

)1(

)1()(

q

qqqf

+−=

018.0)38.0( maxmax ≈== qffIf system is symmetric (m1=m2), recoil is zero (for non-spinning holes).

2

1

m

mq =

Spin-orbit corrections to Fitchett’s formula (circular binary) [Kidder 1995]: [symmetry broken even for q=1]

4 9 / 2

SO 1 2kick

max term. SO,max term.

( , , )( ) 2 21480 km/s 883 km/s

f q a af q M MV

f r f r

= +

ɶ ɶ

2 5

SO 2 1( ) /(1 )f q a a q q= − +ɶ ɶ[valid for non-precessing binary, spins aligned/anti-aligned]

Accumulated

recoil for

a/M=0.8, η=0.1

Why isn’ the kick zero for circular orbits?

1. radiation reaction

means orbits are not

exactly circular.

2. final orbit before

horizon is not closed,

so momentum can’t

cancel.

orbit momentum vector

center of mass accumulated recoil

Famous Moments in

Recoil History

Foundations:

• Bonnor & Rotenberg (1961); Papapetrou (1962); Peres (1962): expressions for dP/dt

• Campbell & Morgan (1971); Dionysiou (1974); Booth (1974); Thorne (1980): generalizations

Recoil from gravitational collapse:

• Bekenstein (1973): [upper limit of 300 km/s]

• Moncrief (1979): [recoil ~ 25 km/s ]

Recoil from binaries:

Post-Newtonian:

• Fitchett (1983): quasi-Newtonian calculation; highly uncertain—as high as 20,000 km/s

• Pietila et.al (1995): includes 2.5PN rad. reaction in Fitchett’s calc; slightly larger values

• Wiseman (1992): full extends Fitchett’s calc. to 1PN order in dP/dt [2.5PN Eqs. of motion]

• Kidder (1995): includes spin-orbit contribution to dP/dt

Perturbation Theory

• Fitchett & Detweiler (1984): BH perturbation (a/M=0, no radiation reaction, circular orbits)

• Oohara & Nakamura (1983): plunge from infinity into Schwarzschild [~275 km/s]

• Kojima & Nakamura (1984): extension to Kerr

Numerical Relativity and the Head-on Collision:

• Nakamura & Haugan (1983): Kerr radial in-fall along symmetry axis [~ 5 km/s]

• Andrade & Price (1997): head-on; close-limit approx.; highly uncertain [1- several 100 km/s]

• Anninos & Brandt (1998): head-on; full numerical [~9 km/s]

• Lousto & Price (2004): BH perturbation theory; [~ 5 km/s]

• Brandt & Anninos (1999): BH distorted by axisymmetric Brill waves [~2 - 500 km/s]

History of Recoil Calculations

What we did:• Extended BH perturbation theory work of

Fitchett & Detweiler to BHs that are:

– spinning (but point-mass is nonspinning)

– inspiralling due to radiation reaction

• Estimated recoil due to final “plunge” from the last stable orbit

– (used more realistic orbits than Nakamura et. al, but

neglecting some relativistic effects)

• Extended Fitchett’s analytical computations to spinning holes

– (but neglected radiation reaction and all other post-

Newtonian effects)

Our approach: circular, equatorial Kerr orbits

Adiabatic inspiral:

• use BH perturbation theory to compute momentum flux up to the ISCO.

• this is an exact computation in the test mass limit.

Plunge into the horizon:

• use Kerr geodesic that plunges from the ISCO

• use two different approximations for the momentum flux dP/dt

Ringdown:

• ignore; its contribution to the total recoil is small; this was confirmed by Damour & Gopakumar (2006) / 0, =0.1a M η=

Split the coalescence into 3 phases:

mass ratio of BH binaries vs. redshift

[fig. from Volonteri, Haardt,& Madau; ApJ 582, 559 (2003) ]

The only way to correctly compute the recoil is with numerical relativity…

…but perturbation theorycan be more useful than you might think.

Main Approximation: small mass ratio, q = m1/m2

Physical motivation for small mass ratio assumption:

• effective-one-body treatment:

(m1,m2) ® (M,µ) 1S

2S

2m

1m

1 2M m m= +

µ

2| | aM=S ɶ

1

2

mq

m=

M

µη =

2

2

In q 1 limit, .

For head-on collision of two (non-spinning) BHs, the scaling law

, produces agreement with full numerical relativity.

dEq

dt

dEq

dtη η

∝

→ ∝

≪

[Smarr (1978)]

• perturbative calculations of the

head-on collision agree with

numerical relativity when scaled

to higher mass-ratios:

Physical motivation for small mass ratio assumption:

1S

2S

2m

1m

1 2M m m= +

µ

2| | aM=S ɶ

1

2

mq

m=

M

µη =

• perturbative calculations of the

head-on collision agree with

numerical relativity when scaled

to higher mass-ratios:

2

2

When 1, . To scale-up to large , we

use ( ). ( ) for small , ( 0) 0.

j

j

dPq q q

dt

dPf q f q q q f q

dt

∝

∝ → = =

≪

A similar scaling holds for the momentum

flux (Fitchett & Detweiler):

• Post-Newtonian studies

have recently shown that

this scaling is accurate

(Blanchet, Qusailah, & Will;

Damour & Gopakumar)(the scaling is more complicated for spinning bodies)

• effective-one-body treatment:

(m1,m2) ® (M,µ)

Calculations I: BH perturbation theory

Solve Teukolsky equation for Y4 to get momentum flux up to the ISCO:

[using code developed by Hughes(2000)]

*

4 24

( )

2

1( ) ( ; )

( cos )

1 1( ; ) ( ) (as )

2mk

im i t

lm lm

lm

i t rH im

lmk lm mk

lmk

d R r S a e er ia

Z S a e e h ih rr

φ ωω

ωφ

ω θ ωθ

θ ω

+∞−

−−∞

− −− + ×

Ψ =−

= = − → ∞

∑∫

∑ ɺɺ ɺɺ

1. pick a geodesic orbit with E, Lz

2. Solve Teukolsky equation for this geodesic.

3. Compute GW fluxes dE/dt and dLz/dt to infinity and down the horizon.

4. Update E, Lz for the orbit and generate an inspiral trajectory up to the ISCO

5. Use calculated quantities to compute dPj/dt along the orbit.

GW [ ( ), ( )]j

H

lmk mk

dPF Z t t

dtω=

Results I:

Center of mass velocity

for circular, equatorial

orbit up to ISCO.

[Schwarzschild, a/M=0]

[reduced mass ratio=0.1]

VMAX = 4.7 km/s

Agrees well with Fitchett

Results II:

Center of mass velocity

for circular, equatorial

orbit up to ISCO.

[Kerr, a/M=0.99]

[ reduced mass ratio=0.1 ]

VMAX = 257 km/s

Kick reduced by

• gravitational redshift

• wave scattering

Results III: final recoil up to ISCO

isco2.63 0.06( / )

kick,isco

max isco

( ) 2422 km/s

r M

f q MV

f r

+

=

0.1η =

( large spins should be excluded

due to finite-size effects. )

[a convenient fitting function]

Recoil depends strongly on the ISCO radius

Calculations II: Recoil from plunge (lower limit):

Use a ``semi-relativistic’’ or ``hybrid’’ method to compute recoil from ISCO to plunge into the horizon (circular-equatorial Kerr orbits)

• Match plunging geodesic onto adiabatic inspiral just before ISCO.

• Use orbit [x(t), y(t)] to compute Newtonian-order multipole moments:

• Plug into lowest order multipole expansion of momentum flux:

• Truncate when:

4 3 33

GW

4 3 3 3

2 16

63 45

jjab pa qaab

jpq

d d ddP d

dt dt dt dt dtε= + +⋯

I I SI

STF STF

STF

[ ( ) ( )] , [ ( ) ( ) ( )] ,

[ ( )[ ( ) ( )] ]

jk j k jki j k i

jk k j

x t x t x t x t x t

x t t t

µ µ

µ

= =

= ×x v

I I

S

horizon 2r r µ= +

Perform a “quick and dirty” over-estimate of the momentum flux

• Again, match plunging geodesic onto adiabatic inspiral near ISCO

• From BH perturbation code, the momentum flux follows a power-law in radius up to the ISCO:

• Extrapolate power-law into plunge region; stop power-law at 3M and use:

• Integrate dPGW/dt to get kick

• Truncate when:

( ) 1[ ]i t x yGW GW GWd

e P iPdt r

ϕα

−= + ∝ɺ ɺP

GW const, 3ddt

r Md dtτ

= ≤ P

horizon 2r r µ= +

Calculations II: Recoil from plunge (upper limit):

Limits on final recoil

• Large uncertainties, especially for retrograde

orbits

• rapid, prograde case is

more certain (dominated

by inspiral recoil)

• Vkick = 120 km/s bisects

shaded region.

• For a/M=0, scaling to

q=0.38 gives:

Vup,max= 465 km/s

Vlow,max= 54 km/s

0.1η =

upper and lower limits on total kick velocity:

(Large effective spins excluded because of finite-size effects)

summary of main results:

• first group to examine recoil from realistic orbits into spinning BHs

• made clear the importance of the plunge in determining the final kick

• performed BH perturbation calculation of recoil up to ISCO

- recoil reduced relative to Newtonian estimates → strong-field effects important

- kick of ~ few km/s for large ISCO radius; up to ~ 200 km/s for moderately large prograde inspiral

• final kick still uncertain due to modeling of plunge phase

• Summary of kick values:

Vkick d 100 km/s likely; Vkick ~ few 100 km/s not unexpected;

largest possible kicks have Vkick d 500 km/s.

Ejection of BHs from large galaxies is very unlikely !

recent recoil calculations:Since our paper was published, there has been much recent progress in kick

computations from post-Newtonian theory and numerical relativity:

Post-Newtonian Recoil Calculations (all for non-spinning holes):

• Blanchet, Qusailah, & Will (2005):

1. Computed momentum flux to 2PN order (for circular orbits)

2. Used this flux formula to perform a calculation analogous to our “lower-limit”

calculation

3. Most of the finite-mass ratio effects are contained in Fitchett’s function f(q),

justifying our scaling-up procedure

4. Their ISCO recoil (22 km/s) is a bit higher than our “exact” result (16 km/s)

[neglect of 3PN effects?]

5. They find maximum kicks of 250 ± 50 km/s

6. Although they use 2PN fluxes, they assume circular orbits and the wrong

“Kepler’s law” during the plunge to simply their expressions. (Our lower-limit

calculation does not). This overestimates the recoil.

recent recoil calculations:Since our paper was published, there has been much recent progress in kick

computations from post-Newtonian theory and numerical relativity:

Post-Newtonian Recoil Calculations (all for non-spinning holes):

• Damour & Gopakumar (2006):

1. Used “effective-one-body’’ (EOB) approach: models dynamics on a “deformed’’

Schwarzschild metric

2. “Corrected’’ the 2PN momentum flux of Blanchet et. al.

3. Their ISCO recoil (using Pade approximants) agrees more closely with our

“exact” result.

4. Also confirm that additional finite-mass ratio effects are small (< 8%)

5. Show analytically that the kick is dominated by the “peak” in |dPj/dt| that occurs

during the plunge

6. Show that the ringdown makes a relatively small contribution ( < 15% of total)

7. Their “best-bet” estimate is 74 km/s (but acknowledge that uncertainty remains)

8. Their quasi-Newtonian estimate (throwing away PN corrections to momentum

flux) agrees very well (< 7%) with our lower-limit calculation (as it should).

recent recoil calculations:Since our paper was published, there has been much recent progress in kick

computations from post-Newtonian theory and numerical relativity:

Numerical Relativity (all for initially non-spinning holes):

• UTB group [Campenlli (2005)]:

1. Use “Lazarus” approach (full GR plus close-limit approximation)

2. Recoils are highly uncertain. examples: 277 ± 160 km/s (rescaled from q=0.5); 172 ± 95 km/s (rescaled from q=0.83);

• Penn State group [Herrmann, Shoemaker, & Laguna (2006)]

1. full GR using “moving-puncture” method

2. max recoil of 118 km/s (rescaled from q=0.85)

• Goddard group [Baker et. al (2006)]

1. full GR using “moving-puncture method; most accurate simulations available.

2. max recoil of 163 km/s (rescaled from q=0.67)

summary of recent recoil calculations:

Nakamura Favata UTB Blanchet PennState Damour, Goddard

Recoil calculations from selected groups (non-spinning holes)

• all estimates remain within our upper and

lower bounds

• estimates seem to be converging to a range

of ~70 – 200 km/s