1. Marc Favata Cornell University
The validity of the adiabatic approximation for extreme mass ratio inspirals
Marc Favata
Cornell University

2. Outline:
Overview of extreme mass ratio inspiral (EMRI) sources for LISA
Scientific payoffs
Theoretical challenge: computing strong-field orbital evolutions and waveforms in the regime v ~ c, m << M
Computing back-reaction effects via conservation laws
Computing back-reaction effects via self-force
Adiabatic evolution of the Carter constant
Preliminary results:
Justifying the adiabatic approx. [Tanja Hinderer & E. Flanagan]
Estimating error from post-adiabatic terms. [M.F. & E. Flanagan]

3. Overview of EMRI sources I:
Masses: m~1-100 M , Mbh ~ M , q=m/Mbh ~
captured from stellar cusps into elliptical orbits
Orbit decays via GW emission; in LISA band ~ orbits during last year of inspiral (v/c ~ 1) (Finn & Thorne 2000)
Orbits circularize, but expect substantial eccentricity at last stable orbit

4. Overview of EMRI sources II: Event Rates
Gair et. al 2004

5. Scientific Payoffs:
Measure BH mass and spin with accuracy dM/M, da ~ 10-4
(Barack & Cutler 2004)
Constrain growth history (merger vs. accretion of BHs) (Hughes & Blandford 2003)
Census of central parsec of galactic nuclei (from event rate and distribution of inspiralling object masses
Holiodesy (space-time “geodesy”; aka “bothrodesy”): measure multipole moments of central object; test “no-hair” theorem

6. Scientific Problems:
Need to accurately model dynamics for fully-generic orbits about rapidly spinning holes
For data analysis, need phase errors D~1 over entire signal of ~ 1yr;
D/  ~ 1/Ncycles ~ m/Mbh ~ 10-6
For detection, only need D ~1 over ~ 3 weeks (Gair et. al ’04) (limited by computational resources)

7. Methods of computing templates:
post-Newtonian techniques:
expand in (v/c)2 ~ (M+m) / r << 1
gives crude waveforms useful for estimating event rates (Gair et. al 2004) and LISA’s ability to measure binary parameters (Barack & Cutler 2004)
Extreme-mass ratio limit
v ~ c , but expand in m << Mbh
use techniques of black hole perturbation theory, combined with some scheme to compute back-reaction on orbit

8. The back-reaction problem I:
How to compute corrections to the orbit that depend on the particle’s mass m
Conservation law approach
1. Describe geodesic orbits in terms of E, Lz, Q
2. For sequence of geodesics, solve Teukolsky eqn:
 4  dE/dt (E,Lz), dLz/dt (E,Lz)  dp/dt, de/dt, d/dt
3. generate p(t), e(t), (t)  h+(t), h(t)
Limitations:
1. can only evolve equatorial or circular orbits dQ/dt = F [dE/dt, dLz/dt]
2. ignores non-dissipative (conservative) effects

9. The back-reaction problem I:
A brief history of conservation-law calculations:
Freq. Domain
1. Cutler, Kennefick, Poisson 1994
2. Shibata 1994
3. Finn & Thorne 2000
4. Hughes 2000
5. Glampedakis & Kennefick 2002
Time Domain
1. Martel, Poisson 2003
2. Lopez-Aleman, Khanna, Pullin 2003
3. Khanna 2004
4. Laguna, Soperta 2005

10. The back-reaction problem II:
Compute the self-force acting on point-particle f = m a + O ( 2 ) 3 f ( 1 ) = 2 m g + u [ r h t a i l ] Z d G e z ; s n
Difficult to compute in practice

11. The back-reaction problem III:
Adiabatic radiation reaction
Expand orbital phase:
Adiabatic waveforms use only leading order orbital phase0 . Results in cumulative phase errors ~ O(1)
Orbits described by:
Adiabatic evolution requires only:
Rates of change of constants: ( t ) = 1
" [ + O 2 ] E = ( t ) u ; L z Á Q K h _ E i ; L z Q d E = ( t ) a s e l f L z Á Q 2 K u

12. The back-reaction problem III:
Adiabatic radiation reaction d E À = D ( t ) a s e l f [ h r ] L z Á Q 2 K u d E = ( t ) a s e l f [ h i ] L z Á Q 2 K u
Mino(’03)
hrad much easier to evaluate than htail

13. The back-reaction problem III:
Adiabatic radiation reaction
Implementing Mino’s prescription: h _ E i = X l m n r F 1 [ Z H ; ! ] L z 2 Q 3
Drasco, Flanagan, Hughes 2005 (scalar case)
Sago, Tanaka, Hikida, Ganz, Nakano (gravitational case)
Allows computation of adiabatic waveforms for generic orbits
<dE/dt>, <dLz/dt> implemented in Drasco-Hughes code; <dQ/dt> in progress

14. Justification of adiabatic waveforms:
Expand the self-acceleration
Two-timescale expansion of phase: a s e l f = 1 m
" [ ; d i + c o n ] 2 O ( 3 ) ( t ) = 1
" [ ; + O 2 ]
Tanja Hinderer & Eanna Flanagan (in prep) showed:
Leading order (adiabatic) phase depends only on leading-order dissipative-piece of self-force
Post-1-adiabatic corrections to the phase depend on leading-order conservative self-force and on the 2nd-order dissipative self-force.

15. Justification of adiabatic waveforms:
Sketch of derivation: [Hinderer & Flanagan]:
1. Use action-angle formulation of equations of motion
2. add small perturbing forces ~ 
3. perform two-timescale expansion, fast time T, slow time T
4. split perturbing forces into conservative and dissipative pieces
5. expand and solve order by order in 

16. Estimating adiabatic phase error I:
would need correction to self-force formula  ???
PN theory is our remaining tool (although not reliable for v/c~1) a = ( ) M t n 1 + 2 [ ; ] 4 5 : o w h e r v p , b m q i s d u c
schematic 2.5PN Eqs
Expand in small mass ratio q, and keep leading order terms (post-adiabatic corrections) a = ( ) M b h n 1 + q 2 [ ; ^ O ] 4 5 : o
solve equation of motion numerically.

17. Estimating adiabatic phase error II:
… or… compute phase errors analytically  do-able for small eccentricities [M.F. & E. Flanagan (in progress)] F o r n e a l y c i u b t s h ( ) = A Á ~ f 1 2 d w + Z

18. Estimating adiabatic phase error II:u l t i ( f ) = 2 c + Á 3 1 8 5 F w h r M a n d F ( Á ; ) = 1 + 2 3 Ã 7 5 6 9 !
" a # 4 8 l n :
circular piece (3.5PN order)

19. Estimating adiabatic phase error II:
Compute eccentric corrections to F (1PN order, small eccentricities):
(extends previous computation by Krolak, Kokkotas, Schafer 1995)
Result: = 2 f t c + Á 3 1 8 5 7 6 9 4 e i r u l a m s p o . P N : ( )
Sketch of derivation: d f t = K 1 [ ; e ] 2 F e x p a n d s o l v f r m ) = ( ; c u t h / R 1
[Damour, Gopakumar, Iyer 2004]

20. Estimating adiabatic phase error III:
Can now compute error due to post-adiabatic phase corrections: = 2 f t c + Á 3 1 8 ( M ) 5 F ; a d q b h !

21. Estimating adiabatic phase error III:
Error in Ncycles
Phase error zeroed at initial observation freq. M b h = 1 6 m a e f ; : 2 g n l i s c o t y r

22. Estimating adiabatic phase error III:
Error in Ncycles
Choose masses and coalescence time and phase to minimize phase error. M b h = 1 6 m a e f ; : 2 g n l i s c o t y r

23. Estimating adiabatic phase error III:
Error in Ncycles
Phase error zeroed at initial observation freq. M b h = 1 6 m a e f ; : 2 g n l i s c o t 3 w k r

24. Estimating adiabatic phase error III:
Error in Ncycles
Choose masses and coalescence time and phase to minimize phase error. M b h = 1 6 m a e f ; : 2 g n l i s c o t 3 w k r

25. Role of conservative terms in adiabatic approx:
Pound, Possion, Nickel (2005): look at electric charge + Newtonian gravity
Orbital elements evolve due to electro-magnetic self-force
Magnetic field gets out of phase if conservative self-force terms are neglected N o t i m p r a n f E M R I s ( ) c v d g 3

26. Role of conservative terms in adiabatic approx:y p f Ä x + V =
" ( ) w h . I , E _ 2 u 1 O T : k m g À Z [ ]

27. Conclusions: produces adiabatic waveforms
Carter constant evolution now possible --- can evolve generic orbits
produces adiabatic waveforms
adiabatic waveforms determined by leading order dissipative piece of self-force
post-adiabatic correction requires leading-order conservative self-force AND dissipative piece of 2nd order self-force.
With current tools, not yet possible to go beyond adiabatic waveforms
But adiabatic waveforms will be good enough for detection purposes