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

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]

Overview of EMRI sources I: Masses: m~1-100 M , Mbh ~ 106 -108 M , q=m/Mbh ~10-4 -10-8 captured from stellar cusps into elliptical orbits Orbit decays via GW emission; in LISA band ~105-106 orbits during last year of inspiral (v/c ~ 1) (Finn & Thorne 2000) Orbits circularize, but expect substantial eccentricity at last stable orbit

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

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

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)

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

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

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

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 ¸ ¾ ¡ ] Z d G e z ; s n Difficult to compute in practice

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

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

The back-reaction problem III: Adiabatic radiation reaction Implementing Mino’s prescription: h _ E i = X l m n r µ 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

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.

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 

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 2 [ ; ´ ] 4 5 : o w h e r » v p , b m q ¹ 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 2 [ ; ^ O ] 4 5 : o solve equation of motion numerically.

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 ¼ 1 2 d ¡ ª w + Z µ ¶

Estimating adiabatic phase error II: u l t i ª ( f ) = 2 ¼ c + Á 3 1 8 ´ » 5 F w h r M a n d F ( » Á ; ´ ) = 1 + 2 3 Ã 7 5 6 9 ! " ¡ ¼ a º # 4 8 l n ° : circular piece (3.5PN order)

Estimating adiabatic phase error II: Compute eccentric corrections to F (1PN order, small eccentricities): (extends previous computation by Krolak, Kokkotas, Schafer 1995) Result: ª = 2 ¼ f t c + Á 3 1 8 ´ » 5 ½ µ 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 ª / R 1 ¡ [Damour, Gopakumar, Iyer 2004]

Estimating adiabatic phase error III: Can now compute error due to post-adiabatic phase corrections: ª = 2 ¼ f t c + Á 3 1 8 ´ ( M ) 5 F ; a d q b h ! ¢ ¡

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

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

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

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

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

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 © [ ] ¡ ª ½ ¯ ¾

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