Gravitational waves from an extreme mass-ratio binary

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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
Index 1 Introduction 2 Self-force in a
Linear Perturbation 3 Regularization of the
fields 4 Radiation Reaction and
Adiabatic Evolution 5 Summary and Future
Yasushi Mino (? ??) WUGRAV, Washington University
at St. Louis E-mail mino_at_wugrav.wustl.edu
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
1 Introduction
We want to calculate the gravitational wave
form from an extreme mass-ratio binary system for
LISA project.
The central black hole is considered to be a Kerr
black hole. For its extreme mass-ratio, we
expect that a linear perturbation is an effective
method of investigation.
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System

• One can calculate the gravitational wave form by
  a linear perturbation, given an orbital evolution
  of the binary system.
• We need the orbital evolution of 108 rotations
  for one-year observation of gravitational waves.

Beyond 103 rotations, the orbit deviates from a
geodesic by the secular effect of radiation
reaction.
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
We want to know the evolution equation of the
orbit, namely, we want to solve The self-force
problem. We can use the linear perturbation
since the instantaneous deviation from a geodesic
is small.
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10 years ago
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
2 Self-force in A Linear Perturbation
We consider a linear perturbation induced by a
point mass.
Use of a point particle may be a good
approximation, but, it causes a difficulty.
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
The metric perturbation diverges around the
particle.
R spatial distance between the field point
and the particle location

• The geodesic equation in the perturbed metric
  diverges at R -gt 0.
• Because of the divergence, the linear
  perturbation becomes invalid.
• These problems were solved by the
  mass-regularization, the matched asymptotic
  expansion method, and an axiomatic approach.

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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
Self-Force (MiSaTaQuWa Force)
full Metric perturbation induced by a
point particle
S-Part of perturbation, singular
R-Part of perturbation, regular
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5 years ago
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
3 Regularization of the fields
We need a regularization calculation.

• A new type of regularization calculation
• A global technique to derive S-part is not known.
• We cannot use the spatial Fourier transformation.

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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
We have developed a formula By the local
coordinate expansion of S-part, we can decompose
the divergent and non-vanishing part of it.
If we have the full metric perturbation as a sum
of harmonics, we can derive the self-force.
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System

• A self-force calculation is successful in a
  Schwarzschild background in a simple manner.
• A calculation in a Kerr background comes to be
  possible in principle.
• The practical application is difficult,
  especially, in calculating the full metric
  perturbation.

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1 years ago
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
4 Radiation Reaction and Adiabatic Evolution
Poor-mans method really poor?

• A calculation by the energy balance equation
• We approximate the orbit at a given instant by a
  geodesic of constants (E,L,C).
• Instead of integrating the orbital equation, we
  consider the evolution of these constants.

E
L
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
Geodesics around a Kerr black hole are
characterized by 6 constants

1. We consider an evolution equation of (E,L,C)
   based on our understanding of the self-force.
2. By integrating the orbital equation
   perturbatively, we derive the evolution of the
   rest of constants.
3. We derive the adiabatic evolution of the orbit.

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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
We consider a family of geodesics bounded by the
BH gravitational potential.
r
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System

• r/q-motions are independent periodic motions

integral constants

• t-motion and f-motion

integral constant

• A family of geodesics is characterized by 7
  constants.

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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
Because of the periodicity of the orbit, the
self-force can be expanded as
l
Linear Perturbation becomes invalid
dE, dL, dC
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
In the short time scale (of the order of the
dynamical time scale), the orbit just exchanges
the energy with radiation. In the long time
scale, the orbital energy radiates away, and the
orbital energy tends to lose.
l
Linear Perturbation becomes invalid
dl, dl, dt, df
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
By taking the dominant part, we can define the
adiabatic evolution by the self-force.
These equations give a correct prediction of the
orbit up 109 (106) rotation.
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
1. One can calculate the time-averaged radiation
reaction to E,L,C, by using the radiative Green
function. The calculation method of this Green
function is known.
2. We prove that it is consistent in any gauge
condition.
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Now
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Gravitation A Decennial Perspective
(6/8-6/12/2003, PennState) Gravitational Waves
from Extreme Mass-Ratio Binary System
5 Summary and Future
We now have finished a theoretical foundation
to make gravitational wave templates.

• We need
• to make the data analysis strategy.
• to make a program to generate a template bank.

... 108? 1014 templates? ... A semi-analytic
method?
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We still have a long way