On the modeling of Intermediate- and Extreme-Mass-Ratio Inspirals

On the modeling of Intermediate- and 

Extreme-Mass-Ratio Inspirals 

Carlos F. Sopuerta 

Institute of Space Sciences (CSIC-IEEC) 

2nd Iberian Gravitational Wave Meeting [February 16th, 2012] 

1

Outline 

Introduction to IMRIs and EMRIs 

Review of modeling techniques and key 

Results 

A new Method: The Chimera Scheme 

Some Conclusions and Future Work 

2nd Iberian Gravitational Wave Meeting [February 16th, 2012] 

2


Intermediate- and Extreme mass ratio inspirals (IMRIs and 

EMRIs) are binary systems whose evolution is driven by 

Gravitational-Wave (GW) emission and such the mass ratio 

between their components is in the range: 

q = m ∗ 

M • 

= 

∼ 10 −7 − 10 −4 

∼ 10 −4 − 10 −2 

for EMRIs 

for IMRIs 

2nd Iberian Gravitational Wave Meeting [February 16th, 2012] 3


Intermediate- and Extreme mass ratio inspirals (IMRIs and 

EMRIs) are binary systems whose evolution is driven by 

Gravitational-Wave (GW) emission and such the mass ratio 

between their components is in the range: 

q = m ∗ 

M • 

= 

∼ 10 −7 − 10 −4 

∼ 10 −4 − 10 −2 

for EMRIs 

for IMRIs 

The main MOTIVATION to study IMRI/EMRIs is that they 

are important sources of GWs for space-based observatories, 

like eLISA/NGO, and also for advanced (LIGO, VIRGO, 

KAGRA) and third-generation ground-based detectors (ET). 



These systems, given that they are defined by mass ratios 

beyond 1:100, necessarily must involve Black Holes (BHs). 





More specifically: The systems we have in mind are: 

* EMRIs : Stellar Compact Object (SCO) + Massive BH (MBH) 

(Space-Based Detectors) 

* IMRIs : SCO + Intermediate-Mass BH (IMBH) 

(Ground-Based Detectors) 

IMBH + MBH (Space-Based Detectors) 





More specifically: The systems we have in mind are: 

* EMRIs : Stellar Compact Object (SCO) + Massive BH (MBH) 

(Space-Based Detectors) 

* IMRIs : SCO + Intermediate-Mass BH (IMBH) 

(Ground-Based Detectors) 

IMBH + MBH (Space-Based Detectors) 

Number of GW Cycles ∼ q −1 



After many years of observations the existence of stellarmass 

black holes (~4-50 solar masses) and (super)massive 

black holes (~100000 to billion solar masses ) has been 

established with a high degree of confidence. 







There is a gap in the BH mass range, between ~100 to several 

10000 solar masses, where there is not yet a strong 

observational evidence. The BHs in this mass range cannot 

have neither stellar origin nor primordial (early universe). 

These are the Intermediate-Mass BHs (IMBHs). 







There is a gap in the BH mass range, between ~100 to several 

10000 solar masses, where there is not yet a strong 

observational evidence. The BHs in this mass range cannot 

have neither stellar origin nor primordial (early universe). 

These are the Intermediate-Mass BHs (IMBHs). 

Therefore, there is no observational evidence that supports the 

existence of IMRIs, whereas the possible existence of EMRIs is 

sound and the uncertainties are related to formation 

mechanisms and their associated event rates. 



Why is this Mainly because it is extremely difficult to 

find dynamical evidence for IMBHs by looking at the most 

probable host scenarios: globular clusters and Ultra 

Luminous X-ray sources (ULXs). 

Cygnus X-1 

Sgr A* 



However, there are several claims in the literature about the 

presence of IMBHs in several systems, e.g.: 

S.A. Farrell, N.A. Webb, D. Barret, O. Godet & J.M. Rodrigues An intermediate-mass black 

hole of over 500 solar masses in the galaxy ESO 243-49, Nature 460, 73-75 (2 July 2009) 

Discovery of a variable ULX in the edge-on spiral galaxy ESO 

243–49. The extreme luminosity of the source (HLX-1) is 

consistent with the presence of an IMBH of more than 500 

solar masses 



Astrophysical mechanisms to produce EMRIs: The best 

studied one is the “Single Capture” mechanism: 

Scattering of a Stellar-mass 

Compact Object (through 2-body or 

multi-body encounters) to highly 

eccentric orbits around a MBH: 

M • ∼ 10 5 − 10 7 M ⊙ , 1 − e ∼ 10 −3 − 10 −6 . 

They expend the last year before 

plunging inside the eLISA band: 

e ∼ 0.5 − 0.9 , no. cycles ∼ 10 5 . 



Other Mechanisms: 

Stellar-Mass Compact Binaries passing close to the MBH 

can be tidally separated. One component gets bound to the 

MBH and the other one escapes to infinity. 







Capture of cores of giant stars close to the MBH by tidal 

stresses. 







Capture of cores of giant stars close to the MBH by tidal 

stresses. 

Inspiral of black holes produced in an accretion disc 

around the MBH (at distances ~ 0.1-1 pc). 



The orbits probe the strong-field region of the central MBH: 



Astrophysical Mechanisms for IMRIs: 

Dense clusters of stars -> Runaway Collisions and IMBH 

formation. IMRIs come from capture of stellar objects. 



Astrophysical Mechanisms for IMRIs: 

Dense clusters of stars -> Runaway Collisions and IMBH 

formation. IMRIs come from capture of stellar objects. 

Gravitational collapse of Population III stars at the early 

universe to form IMBHs. 


Detectability: 


IMRI signals are stronger than EMRI ones but sweep in the 

band faster too. 






The SNR of IMRIs (MBH+IMBH with M. = 10^5) for 

eLISA, in a given frequency bin, may be around 10 or more. 

Could they be followed without full templates 









IMBH + BH may be detected by Advanced ground 

detectors (LIGO/VIRGO/KAGRA) up to z = 2. 









IMBH + BH may be detected by Advanced ground 

detectors (LIGO/VIRGO/KAGRA) up to z = 2. 

IMBH + MBH with eLISA : everywhere... 


Modeling of IMRI/EMRIs 

Taking into account the mass ratios involved, both in the 

case of IMRIs and EMRIs, it is unkely that Numerical 

Relativity (NR) can produce full waveforms (at least 

extrapolating from the present status). However, NR can 

provide with valuable information about the dynamics of 

these systems. 



Taking into account the mass ratios involved, both in the 

case of IMRIs and EMRIs, it is unkely that Numerical 

Relativity (NR) can produce full waveforms (at least 

extrapolating from the present status). However, NR can 

provide with valuable information about the dynamics of 

these systems. 

In the case of EMRIs, we can resort to BH Perturbation 

Theory: The idea is that the spacetime geometry can be well 

aproximated as that of the MBH plus perturbations induced 

by the SCO. 





When we treat the SCO as a point-like object the deviations 

from geodesic motion can be described by the action of a local 

force, the self-force. The equation of motion for the SCO is the 

so-called the MiSaTaQuWa equation [Mino, Sasaki & 

Tanaka (1997); Quinn & Wald (1997)]: 

BH 

SCO 

F µ 








BH 

SCO 

F µ 

Needs 

Regularization 

D 2 z µ 

Dτ 2 = −1 2 

g µν + dzµ 

dτ 

dz ν 

dτ 

2∇ρ dz ρ 

h tail 

νσ −∇ ν h tail z(τ) 

ρσ 

dτ 

dz σ 

dτ 








BH 

SCO 

F µ 

Needs 


D 2 z µ 

Dτ 2 = −1 2 

g µν + dzµ 

dτ 

dz ν 

dτ 

2∇ρ dz ρ 

h tail 


ρσ 

dτ 

dz σ 

dτ 

Alternatively, we can see this as geodesic motion in a 

perturbated background geometry. 








We neglect 

BH 

the spin of 

SCO 

Needs 

F µ 

the SCO! 


D 2 z µ 

Dτ 2 = −1 2 

g µν + dzµ 

dτ 

dz ν 

dτ 

2∇ρ dz ρ 

h tail 


ρσ 

dτ 

dz σ 

dτ 

Alternatively, we can see this as geodesic motion in a 

perturbated background geometry. 



Then, we have reduced the problem to the computation of the 

self-force, which constitutes a big computational challenge. 





Up to now, the self-force has only been computed for nonrotating 

BHs (Barack and collaborators), and there are some 

computations for circular equatorial orbits in Kerr (Shah et 

al). 








al). 

However, for a typical EMRI inspiral for eLISA, it is likely 

that the 1st-order self-force is not enough. We may need some 

pieces of the second-order self-force [Hinderer & Flanagan, 

PRD 78, 064028 (2008)]. 








al). 

However, for a typical EMRI inspiral for eLISA, it is likely 

that the 1st-order self-force is not enough. We may need some 

pieces of the second-order self-force [Hinderer & Flanagan, 

PRD 78, 064028 (2008)]. 

It may also be that some finite-size effects become nonnegligible 

near plunge. 



On top of all this, we have to produce “consistent” EMRI 

waveforms. This also requires to go to 2nd-order 

perturbations. 






In conclusion, there is a lot of theoretical work to be done. 






In conclusion, there is a lot of theoretical work to be done. 

OBVIOUS REMARK: The smaller the mass 

ratio the better this perturbative scheme works. 



What about IMRIs Given that the mass ratio is not as small 

as in EMRIs, can we also rely on perturbation theory 





Factors that we may become very important with respect to 

EMRI modeling: 







Non-linearities (higher-order perturbative terms). 








Spin of the small component of the binary. 









Physical Consequences of this additional effects: 









Physical Consequences of this additional effects: 

Additional time scales (spin-orbit and spin-spin 

interactions) -> More fundamental frequencies. 



Whereas BH + IMBH systems will be practically circular, 

eccentricity may play an important role in IMBH+MBH 

systems. 



Whereas BH + IMBH systems will be practically circular, 

eccentricity may play an important role in IMBH+MBH 

systems. 

Some very good news (K is the periastron advance): 

The PN and EOB results 

are valid at 3PN order. 

The shaded area marks 

the error margin of 

the NR data. 

[Le Tiec et al. PRL 107, 141101(2011)] 



Bad/Good News: Transient 

Resonances for generic 

orbits: 

Ω θ 

Ω r 

→ 3 2 

Flanagan & Hinderer arXiv:1009.4923 [gr-qc] 

∆φ ∼ q −1 



In practice, we will need fast algorithms to generate IMRI/ 

EMRI waveforms. These algorithms will probably be based on 

approximate methods fed by self-force/NR/... computations. 



Newtonian [Peters & Mathews (1963)]: Newtonian 

trajectories + Quadrupolar waveforms. 





Analytic Kludge [Barack & Cutler (2004)]: (Newtonian 

motion + pN corrections)+ Quadrupolar waveforms. 







Teukolsky [Drasco & Hughes (2004)]: Kerr geodesics + 

Teukolsky fluxes + Teukolsky waveforms. 









Numerical Kludge [Babak et al (2007); Gair & Glampedakis 

(2006)]: Kerr geodesics + pN fluxes + Multipolar waveforms. 









Numerical Kludge [Babak et al (2007); Gair & Glampedakis 

(2006)]: Kerr geodesics + pN fluxes + Multipolar waveforms. 

EOB [Yunes et al (2010)]: EOB fitted to the Teukolsky 

Method. 



In Greek mythology, the Chimera was a monstrous 

fire-breathing creature of Lycia (in Asia Minor), 

composed of the parts of multiple animals: upon the 

body of a lioness with a tail that terminated in a 

snake's head, the head of a goat arose on her back at the 

center of her spine. 

[CFS & N Yunes, PRD 

84 (2011) 124060 ] 





Kerr geodesics + 

Harmonic Coordinates 





Radiative Self-Force 

from post-Minkowskian 

approximation + Asymptotic 

matched expansions. 





Radiative Self-Force 

from post-Minkowskian 

approximation + Asymptotic 

matched expansions. 

Multipolar Expansion 

of the Gravitational 

Waveforms 



Radiative Self-From from post-Minkowskian 

approximation + Asymptotic matched expansions 

[Blanchet & Damour (1984); Iyer & Will (1995); 

Blanchet (1997)] 







Basic Idea: To determine the gravitational field 

both at the “near zone” and the “exterior zone” and to 

match the two solutions at the overlapping region. 

These solutions for the gravitational fields are 

expanded in G (post-Minkowskian expansion) and 

in harmonics (multipoles). 







Using the half retarded minus half advanced 

solution, the matching of the solutions provides an 

expression for a dissipative (radiative) self-force. 

h RR 

αβ ←− ∇ α V RR , ∇ α V i RR 

a α RR = − 1 2 

 

g αλ + u α u λ u µ u ν 2∇ µ h RR 

νλ −∇ λ h RR 

µν 



At 1PN order, the radiative self-force is 

determined from a scalar and a vector radiationreaction 

potentials: 

V RR (t, x) = − 1 5 xij M (5) 

ij (t) + 1 

189 xijk M (7) 

ijk (t) 

− 1 70 x2 x ij M (7) 

ij (t) , 

V i RR (t, x) = 1 21 ˆxijk M (6) 

jk (t) − 4 45 ijkx jl S (5) 

kl (t) , 



At 1PN order, the radiative self-force is 

determined from a scalar and a vector radiationreaction 

potentials: 

Burke-Thorne 

Potential 

V RR (t, x) = − 1 5 xij M (5) 

ij (t) + 1 

189 xijk M (7) 

ijk (t) 

− 1 70 x2 x ij M (7) 

ij (t) , 

V i RR (t, x) = 1 21 ˆxijk M (6) 

jk (t) − 4 45 ijkx jl S (5) 

kl (t) , 



Comment: All this assumes a harmonic gauge 

(harmonic coordinates for the multipolar expansion). 

V RR (t, x) = − 1 5 xij M (5) 

ij (t) + 1 

189 xijk M (7) 

ijk (t) 

− 1 70 x2 x ij M (7) 

ij (t) , 

V i RR (t, x) = 1 21 ˆxijk M (6) 

jk (t) − 4 45 ijkx jl S (5) 

kl (t) , 



The “local” radiative self-force has two pieces: 

(i) Gradients of the potentials [this implies we need 

up to eight-order time derivatives of the trajectory]. 

(ii) Interaction terms of the radiative potentials with 

the near-zone potentials (Kerr gravitational field). 



Kerr geodesics +Harmonic Coordinates[Ding 

(1983); Abe, Ichinose & Nakanishi(1987)] [See also 

Ruiz (1986)]: 

t H = t , 

x H = 

 

(r − M • ) 2 + a 2 sin θ cos[φ − Φ(r)] , 

y H = 

 

(r − M • ) 2 + a 2 sin θ sin[φ − Φ(r)] , 

z H = (r − M • ) cos θ . 





Ruiz (1986)]: 

Known 

t H = t , 

 

Function 

x H = (r − M • ) 2 + a 2 sin θ cos[φ − Φ(r)] , 

 

y H = (r − M • ) 2 + a 2 sin θ sin[φ − Φ(r)] , 

z H = (r − M • ) cos θ . 





Ruiz (1986)]: 

Known 

t H = t , 

 

Function 

x H = (r − M • ) 2 + a 2 sin θ cos[φ − Φ(r)] , 

 

y H = (r − M • ) 2 + a 2 sin θ sin[φ − Φ(r)] , 

z H = (r − M • ) cos θ . 

x α H =0 



Kerr geodesics +Harmonic Coordinates: An 

alternative to this is to use Asymptotically Cartesian 

and Mass Centered coordinates (ACMC) [Thorne 

(1980)] 

This may be a convenient path to deal with the case 

in which the massive object is not described by the 

Kerr metric. 



Multipolar Expansion of the Gravitational 

Waveforms [Thorne (1980)] 

h TT 

ij = 4 r 

∞ 

=2 

1 

! M() ijI −2 

(t − r)N I−2 

+ 

 

2 

TT 

( + 1)! ε kl(i J() (t − r)n 

j)kI −1 l N I−2 

Here, we also use harmonic coordinates. 



Multipolar Expansion of the Gravitational 

Waveforms [Thorne (1980)] 

h TT 

ij = 4 r 

∞ 

=2 

1 

! M() ijI −2 

(t − r)N I−2 

Radiative 

Multipole 

Moments 

+ 

 

2 

TT 

( + 1)! ε kl(i J() (t − r)n 

j)kI −1 l N I−2 

Here, we also use harmonic coordinates. 



Implementation: Method of Osculating Orbits: 

z α (τ) =(t(τ),r(τ),θ(τ),ϕ(τ)) −→ z α Geodesic (τ,Pα (τ), I α (τ)) 





Positional Orbital Elements: (x o ,y o ,z o ) 






Principal Orbital Elements: (E,L z ,Q)/(e, p, ι) 







Evolution of the Principal Orbital Elements (E,Lz,C): 

Ė = −ξ (t) 

µ a µ RR 

, 

˙L z = ξ (ϕ) 

µ a µ RR 

, 

˙Q = 2 ξ ρµ u ρ a µ RR 

. 







Evolution of the Principal Orbital Elements (E,Lz,C): 

Ė = −ξ (t) 

µ a µ RR 

, 

˙L z = ξ (ϕ) 

µ a µ RR 

, 

Symmetries of a 

Kerr Black Hole 

˙Q = 2 ξ ρµ u ρ a µ RR 

. 





The main difficulty is the evaluation of the 

numerical time derivatives that enter in the Self- 

Force: up to eight-order derivatives (six-order if we use 

analytical expressions for the second-order 

derivatives of the multipole moments). 



The main difficulty is the evaluation of the 

numerical time derivatives that enter in the Self- 

Force: up to eight-order derivatives (six-order if we use 

analytical expressions for the second-order 

derivatives of the multipole moments). 

We use the information of the local geodesic 

(fundamental frequencies) in combination with a 

least-squares fitting: 

f[zGeodesic](t) α 

= f k,m,n e −i (k Ω r +m Ω θ +n Ω φ ) t 

k,m,n 







(p, e, θ inc )=(6.0, 0.6, 0.2) 



(p, e, θ inc )=(6.0, 0.6, 0.2) 


Some Conclusions and Future Work 

IMRI/EMRI modelling presents many challenges 

and there is a lot of work to be done. 

This work can potentially provide substantial 

revenues in the analysis of the data of space-based 

and advanced ground detectors. IMRI/EMRI 

observations will impact Astrophysics, Cosmology, 

and Fundamental Physics. 

New schemes to provide template banks are 

underway and there is also a lot of work to be done. 

The Chimera scheme is an attempt to fill the gap. 

It is valid for generic orbits and can also be used to 

study the dynamics of EMRI resonances.

On the modeling of Intermediate- and Extreme-Mass-Ratio Inspirals

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