theoretical sciences - rhodes university · 2019. 2. 19. · p. ajith international center for...

Approximation methods in general relativity for gravitational-wave astrophysics

P. Ajith International Center for Theoretical Sciences, Bangalore, India

Lecture 3

24th Chris Engelbrecht Summer School Rhodes University, Grahamstown, South Africa. 15 to 24 January 2013

Lecture 3: Overview

• Computation of the phase evolution of inspiralling compact binaries in the adiabatic approximation.

• Inclusion of post-Newtonian corrections in the "phasing formula".

• Inclusion of spin effects.

• Validity of the adiabatic approximation. Need to go beyond the adiabatic approximation.

2

Computing the GW phase evolution: Adiabatic approximation

• In the “adiabatic approximation” (energy loss is negligible over one orbit) we can compute the orbit-averaged quantities:

• Thus, the conservation of energy gives:

3

m

orbit without RR(conserved E and L)

with RR

r

dE

}

hF i = 32G5c5 µ

2Dr4!6E

*dE

orb

dt

+= � hF i

hEorb

i =*

1

2

µr2!2 � V(r)

+= �1

2

µDr2!2

E


• Validity of adiabatic approximation

From Kepler’s third law: r3 = Gm/ω2,

The approximation is valid when

4

m

r

r!

r = �23

(r!)✓ !!2

◆

radial velocity tangential velocity

r ⌧ r!, or,✓ !!2

◆⌧ 1


• Validity of adiabatic approximation

• If , the inspiral orbit can be well-approximated by an adiabatic sequence of quasi-circular orbits.

5

!

!2 =965µ

m

✓Gm!c3

◆5/3

a dimensionless velocity parameter

!/!2 ⌧ 1

v

c⌘✓Gm!

c3

◆1/3

inaccurate approximation

accurate approximation

!/!

2v

c⌘✓Gm!

c3

◆1/3


• Time evolution of the orbital phase φorb can be computed by solving the set of coupled ODEs, derived from the energy balance:

6

“phasing formula”

G = c = 1dvdt= � F

dEorb

/dv

d'orb

dt= ! ⌘ v

3

m


• GW signals can be computed by plugging φ(t) in the quadrupolar polarizations:

where

7

• Non-zero components of are

where

hTTi j

A ⌘ 4d

Gc4 µ (r!)2

h+(t) ⌘ hTT

11

= �hTT

22

= �A cos 2! (t � d/c)

h⇥(t) ⌘ hTT

12

= hTT

21

= �A sin 2! (t � d/c)

GW freq = 2 orb freq

[from Lecture 2]h+(t) = �A cos'(t)h⇥(t) = �A sin'(t)

A =4µdv2

' = 2'orb

GW phase:


Signals with increasing frequency and amplitude -- “chirps”

time (sec)

pola

riza

tions

h+

hx

GW signals from the inspiral of two neutron stars at 1 Mpc

Gravitational waveforms including post-Newtonian corrections

• So far, we have computed the conservative dynamics using Newtonian equations of motion, and included the leading-order effects of radiation reaction (using the quadrupole formula).

• There are relativistic corrections to the conservative dynamics of the binary (e.g. precession of the periastron, as in the case of Mercury’s orbit).

• There are higher order corrections to the radiation reaction (contribution from the higher multipoles of the source).

• The higher order corrections can be included in terms of a “post-Newtonian” expansion around the Newtonian/quadrupolar limit.

• Post-Newtonian expansion Expansion in terms of a small parameter

9

v

c⌘✓Gm!

c3

◆1/3


•With PN corrections, the phasing formula becomes:

where

10E

PN

= Eorb

h1 + +(. . . ) v + (. . . ) v2 + (. . . ) v3 + . . .

i

FPN = Fh1 + +(. . . ) v + (. . . ) v2 + (. . . ) v3 + . . .

i

leading order terms higher order corrections

dvdt= � FPN

dEPN/dvdvdt= � F

dEorb

/dv ➾


• Phasing formula is derived only from the conservation of energy. We did not use the conservation of angular momentum.

• This is due to the fact that, for (quasi) circular orbits, orbital angular momentum can be expressed in terms of the orbital energy.

11

E = �G2µ2m3/2L2

Circular orbits are entirely described by the energy. Angular momentum can be written in terms of energy.

We will use this fact later while computing the GW signals from (non-spinning) compact binaries

[from Lecture 2]


• Astrophysical measurements & theory predict that compact objects in nature can be rapidly spinning. Need to consider the non-negligible spin angular momenta, in addition to the orbital angular momentum.

12

Cygnus X-1 hosts a Kerr BH of spin S/m2 ~0.91-0.97 [Gou et al 2011]

Chandra

Michael K

ramer

PSR J0737-3039: Spin periods 23 ms (S/m2 ~ 0.02) and 2.8 s.

J

LN

S1

S2

m1

m2

orbital plane

LN

S1

S2

m1

m2

orbital plane

Inclusion of spin effects in GW signals

• If spins are misaligned, spin-orbit, spin-spin interactions (frame dragging by the spins) will cause precession of the spins & the orbital plane.

13

torbit

⌧ tprecession

⌧ tinspiral

Time scales[Remember Levin’s first lecture]

J

LN

S1

S2

m1

m2

orbital plane

LN

S1

S2

m1

m2

orbital plane



14

⌦i

torbit

⌧ tprecession

⌧ tinspiral

Time scales

5

+ v5⇤�

�a · LN

��35⇤⌅

2� 732985⇤

2016

⇥+ �s · LN

�85⌅2

2+

6065⌅18

� 7329852016

⇥� 65⇧⌅

8+

38645⇧672

⇥ln(v)⌅

+ v6⇤�127825⌅3

5184+

76055⌅2

6912+

2255⇧2⌅

48� 15737765635⌅

12192768� 1712⇥E

21� 160⇧2

3+

1234861192645118776862720

� 1712 ln(4v)21

⌅

+v7⇤�74045⇧⌅2

6048+

378515⇧⌅12096

+77096675⇧

2032128

⌅⌃, (3.4)

where ⌥0 is a certain reference phase.Thus the phasing formula can be reduced to one di⇤erential

equation describing the evolution of the orbital frequency, andone explicit expression of the orbital phase:

dvdt=

⇤�mE⌃(v)F (v)

⌅�1

, ⌥(v) = ⌥0 �1

32v5⌅

⇧1 + . . .

⌃. (3.5)

We call this particular way of solving the phasing formula the“TaylorT5” approximant [93]. The choice of this particularapproximant as the target signal family is motivated by thefollowing reason: Since the spin-dependent terms in the PNexpansion of the energy and flux functions are available onlyup to a rather low 2.5PN order, the di⇤erent approximants givesomewhat di⇤erent results. We want to isolate this issue fromthe issue of the e⇤ect of spin-precession in the target signals.Thus, we construct the target waveforms in such a way thatthey are as close to the non-precessing frequency-domain tem-plate family as possible in the limit of non-precessing spins.Since the frequency-domain template family used in this pa-per (“TaylorF2” approximant [22]) is constructed based on are-expansion of E⌃(v)/F (v), we choose to construct the time-domain target waveforms also based on this re-expansion.Note that the e⇤ectualness of the template family (althoughweakly) depends on the particular approximation used in theconstruction of the target and template waveforms. This is anindication of the level of truncation error in the PN expansion,and points to the need of computing the higher PN order spinterms. This is being explored in an ongoing work [64]. Alsonote that the “TaylorT5” approximant has an additional advan-tage (over TaylorT1 and TaylorT4) that only one di⇤erentialequation needs to be solved numerically; the orbital phase iscomputed as an explicit expansion in v.

If the spin vectors are misaligned with the orbital angularmomentum, the spin-orbit and spin-spin coupling cause thespins and orbital angular momentum to precess around thenearly constant direction of the total angular momentum J,constantly changing the angle between the spins and angularmomentum [9]. The evolution equations for the orbital angu-lar momentum and spins, including the next-to-leading-orderspin-orbit terms, are given by [27, 53]:

||L|| dLN

dt= � d

dt(S1 + S2), (3.6)

dSi

dt= �i ⇤ Si , i = 1, 2, (3.7)

where

�1 =v5

m

⇧�34+⌅

2� 3⇤

4

⇥LN

+v

2m2

⌥�3 (S2 + q S1).LN LN + S2

�

+ v2�

916+

5⌅4� ⌅

2

24� 9⇤

16+

5⇤⌅8

⇥LN

⌃,

�2 =v5

m

⇧�34+⌅

2+

3⇤4

⇥LN

+v

2m2

⌥�3 (S1 + q�1 S2).LN LN + S1

�

+ v2�

916+

5⌅4� ⌅

2

24+

9⇤16� 5⇤⌅

8

⇥LN

⌃. (3.8)

Above, q ⌅ m2/m1 is the mass ratio. The instantaneous pre-cession frequency of the individual spins is ||⇥i||.

The orbital frequency, spins and orbital angular momentumcan be evolved by solving the di⇤erential equations Eqs. (3.5),(3.6) and (3.7). In order to perform the evolution, we adoptthe coordinate system proposed by Finn and Cherno⇤ [65],and subsequently used by several authors. The z-axis of this(fixed) source coordinate system is determined by the initialtotal angular momentum vector J ⇧ LN + S1 + S2 and the x-axis is chosen in such a way that the detector lies in the x � zplane.

In the absence of precession, the phase evolution of the(dominant harmonic) GWs is twice the orbital phase ⌥(t). Butthe precession of the orbital plane introduces additional mod-ulation in the orbital phase. If we define �(t) as the orbitalphase with respect to the line of ascending nodes (the point atwhich the orbit crosses the x � y plane from below) then thethe phase evolution of the (dominant harmonic) GWs is givenby 2�(t), where �(t) is given by

d�dt= ⌃ � d�

dtcos i, (3.9)

where � and i are the angles describing the evolution of theorbital angular momentum vector LN in the Finn-Cherno⇤ co-ordinate system:

� ⌅ arctan(LNy/LNx), i ⌅ arccos(LNz). (3.10)

Computation of gravitational waveforms in the detector framefollows the description of BCV [27] (Section IIC). In the re-stricted PN approximation, the resulting gravitational wave-form observed at the detector can be written as:

h(t) = CQ(t) cos 2 [�(t) + �0] + S Q(t) sin 2 [�(t) + �0] ,(3.11)

Precession of spins

precession frequency

spin vector



15

J

LN

S1

S2

m1

m2

orbital plane

LN

S1

S2

m1

m2

orbital plane

if the spins are (anti) aligned with L, no precession

torbit

⌧ tprecession

⌧ tinspiral

Time scales

5

+ v5⇤�

�a · LN

��35⇤⌅

2� 732985⇤

2016

⇥+ �s · LN

�85⌅2

2+

6065⌅18

� 7329852016

⇥� 65⇧⌅

8+

38645⇧672

⇥ln(v)⌅

+ v6⇤�127825⌅3

5184+

76055⌅2

6912+

2255⇧2⌅

48� 15737765635⌅

12192768� 1712⇥E

21� 160⇧2

3+

1234861192645118776862720

� 1712 ln(4v)21

⌅

+v7⇤�74045⇧⌅2

6048+

378515⇧⌅12096

+77096675⇧

2032128

⌅⌃, (3.4)



dvdt=

⇤�mE⌃(v)F (v)

⌅�1

, ⌥(v) = ⌥0 �1

32v5⌅

⇧1 + . . .

⌃. (3.5)



||L|| dLN

dt= � d

dt(S1 + S2), (3.6)

dSi

dt= �i ⇤ Si , i = 1, 2, (3.7)

where

�1 =v5

m

⇧�34+⌅

2� 3⇤

4

⇥LN

+v

2m2

⌥�3 (S2 + q S1).LN LN + S2

�

+ v2�

916+

5⌅4� ⌅

2

24� 9⇤

16+

5⇤⌅8

⇥LN

⌃,

�2 =v5

m

⇧�34+⌅

2+

3⇤4

⇥LN

+v

2m2

⌥�3 (S1 + q�1 S2).LN LN + S1

�

+ v2�

916+

5⌅4� ⌅

2

24+

9⇤16� 5⇤⌅

8

⇥LN

⌃. (3.8)




d�dt= ⌃ � d�

dtcos i, (3.9)





Precession of spins

⌦i



16

torbit

⌧ tprecession

⌧ tinspiral

Time scales

5

+ v5⇤�

�a · LN

��35⇤⌅

2� 732985⇤

2016

⇥+ �s · LN

�85⌅2

2+

6065⌅18

� 7329852016

⇥� 65⇧⌅

8+

38645⇧672

⇥ln(v)⌅

+ v6⇤�127825⌅3

5184+

76055⌅2

6912+

2255⇧2⌅

48� 15737765635⌅

12192768� 1712⇥E

21� 160⇧2

3+

1234861192645118776862720

� 1712 ln(4v)21

⌅

+v7⇤�74045⇧⌅2

6048+

378515⇧⌅12096

+77096675⇧

2032128

⌅⌃, (3.4)



dvdt=

⇤�mE⌃(v)F (v)

⌅�1

, ⌥(v) = ⌥0 �1

32v5⌅

⇧1 + . . .

⌃. (3.5)



||L|| dLN

dt= � d

dt(S1 + S2), (3.6)

dSi

dt= �i ⇤ Si , i = 1, 2, (3.7)

where

�1 =v5

m

⇧�34+⌅

2� 3⇤

4

⇥LN

+v

2m2

⌥�3 (S2 + q S1).LN LN + S2

�

+ v2�

916+

5⌅4� ⌅

2

24� 9⇤

16+

5⇤⌅8

⇥LN

⌃,

�2 =v5

m

⇧�34+⌅

2+

3⇤4

⇥LN

+v

2m2

⌥�3 (S1 + q�1 S2).LN LN + S1

�

+ v2�

916+

5⌅4� ⌅

2

24+

9⇤16� 5⇤⌅

8

⇥LN

⌃. (3.8)




d�dt= ⌃ � d�

dtcos i, (3.9)





Precession of spins

dLdt= �

dS1

dt+

dS2

dt

!Precession of the orbital plane

orbital ang. momentum conservation of ang. momentum over precession timescales

J

LN

S1

S2

m1

m2

orbital plane

LN

S1

S2

m1

m2

orbital plane


• Time evolution of the GW phase can be computed by solving the following set of coupled ODES.

17

dvdt= � F

dEorb

/dv

5

+ v5⇤�

�a · LN

��35⇤⌅

2� 732985⇤

2016

⇥+ �s · LN

�85⌅2

2+

6065⌅18

� 7329852016

⇥� 65⇧⌅

8+

38645⇧672

⇥ln(v)⌅

+ v6⇤�127825⌅3

5184+

76055⌅2

6912+

2255⇧2⌅

48� 15737765635⌅

12192768� 1712⇥E

21� 160⇧2

3+

1234861192645118776862720

� 1712 ln(4v)21

⌅

+v7⇤�74045⇧⌅2

6048+

378515⇧⌅12096

+77096675⇧

2032128

⌅⌃, (3.4)



dvdt=

⇤�mE⌃(v)F (v)

⌅�1

, ⌥(v) = ⌥0 �1

32v5⌅

⇧1 + . . .

⌃. (3.5)



||L|| dLN

dt= � d

dt(S1 + S2), (3.6)

dSi

dt= �i ⇤ Si , i = 1, 2, (3.7)

where

�1 =v5

m

⇧�34+⌅

2� 3⇤

4

⇥LN

+v

2m2

⌥�3 (S2 + q S1).LN LN + S2

�

+ v2�

916+

5⌅4� ⌅

2

24� 9⇤

16+

5⇤⌅8

⇥LN

⌃,

�2 =v5

m

⇧�34+⌅

2+

3⇤4

⇥LN

+v

2m2

⌥�3 (S1 + q�1 S2).LN LN + S1

�

+ v2�

916+

5⌅4� ⌅

2

24+

9⇤16� 5⇤⌅

8

⇥LN

⌃. (3.8)




d�dt= ⌃ � d�

dtcos i, (3.9)





dLdt= �

dS1

dt+

dS2

dt

!

d'orb

dt= ! ⌘ v

3

m

Evolution of the spins

Evolution of the orb. angular momentum

Evolution of the orbital phase

Evolution of the orbital frequency


• GWs will be beamed along the direction of the orbital angular momentum.

18

time ➝

S1

S2

LJ

S1

S2L

S1S2

LJ

L

S1

S2

J

J = S1 + S2 + L

m1m2


• GWs will be beamed along the direction of the orbital angular momentum.

19

time ➝

LL

L

L


• GWs will be beamed along the direction of the orbital angular momentum... waveform observed by a fixed detector will contain amplitude & phase modulations.

20

time ➝

! !"# !"$ !"% !"! !"& !"'!("#

!("(&

(

("(&

("#

h(t)

LL

L

L


• GWs propagating along the instantaneous orbital angular momentum vector are:

21

h+(t) = �4µ

dv2 cos'(t)

h⇥(t) = �4µ

dv2 sin'(t)

e1S

e2S

e3S ≡LNezS

exS

eyS

LN

instantaneous

orbital plane

detector

⇥ 'orb

e1S

e2S

e3S ≡LNezS

exS

eyS

LN

instantaneous

orbital plane

detector

⇥ 'orb


• GWs propagating along the instantaneous orbital angular momentum vector are:

• GWs propagating along a fixed direction in space are:

where the polarization tensors T+ and Tx are

22

h+(t) = �4µ

dv2 cos'(t)

h⇥(t) = �4µ

dv2 sin'(t)

with ni and ! i the unit vectors along the separation vector ofthe binary r and along the corresponding relative velocity v.These unit vectors are related to the adiabatic evolution ofthe dynamical variables by

n!e1S cos"S"e2

S sin"S , !!#e1S sin"S"e2

S cos"S ;#16$

the vectors e1,2S form an orthonormal basis for the instanta-

neous orbital plane, and in the FC convention they are givenby

e1S!

ezS$LN

sin i , e2S!

ezS#LN cos isin i . #17$

The vector e1S points in the direction of the ascending node of

the orbit on the (x ,y) plane. The quantity "S is the orbitalphase with respect to the ascending node; its evolution isgiven by

"S!%#& cos i , #18$

where i and & are the spherical coordinates of LN in thesource frame, as shown in Fig. 1. Using Eqs. #14$ and #16$,we can write Eq. #15$ as

Qci j!#2#'e"

S ( i jcos 2"S"'e$S ( i j sin 2"S$, #19$

where the polarization tensors e"S and e$

S are given by

e"S )e1

S! e1

S#e2S

! e2S , e$

S )e1S

! e2S"e2

S! e1

S . #20$

For a detector lying in the direction N!ezScos*"ex

Ssin*, itis expedient to express GW propagation in the radiation co-ordinate system with unit vectors +ex

R ,eyR ,ez

R, 'see our Fig. 1together with, for instance, Eq. #4.22$ of Ref. '16(( given by

exR!ex

S cos*#ezS sin* , #21$

eyR!ey

S , #22$

ezR!ex

S sin*"ezS cos*!N. #23$

In writing Eqs. #21$–#23$ we used the fact that for a genericbinary-detector configuration, the entire system consisting ofthe binary and the detector can be always rotated along the zaxis in such a way that the detector will lie in the (x ,z)plane. Later in this paper #in Sec. IV$ we shall find it conve-nient to conserve the explicit dependence of our formulas onthe azimuthal angle - that specifies the direction of the de-tector.In the transverse-traceless #TT$ gauge, the metric pertur-

bations are

hTT!h"T""h$T$ , #24$

where

T")exR

! exR#ey

R! ey

R , T$)exR

! eyR"ey

R! ex

R #25$

and

h"!12 h

i j'T"( i j , h$!12 h

i j'T$( i j . #26$

The response of a ground-based, interferometric detector#such as LIGO or VIRGO$ to the GWs is '15(

h resp!F"h""F$h$

!#2.

DMr 'e"

Si jcos 2"S"e$Si jsin 2"S(

$#'T"( i jF""'T$( i jF$$, #27$

where F" and F$ are the antenna patterns, given by

F" ,$!12 'e x! e x#e y! e y( i j'T" ,$( i j #28$

with e x ,y the unit vectors along the orthogonal interferometerarms. For the geometric configuration shown in Fig. 2, withdetector orientation parametrized by the angles / , 0 , and 1 ,we have

FIG. 1. Source and radiation frames in the FC convention '15(.

FIG. 2. Detector and radiation frames in the FC convention '15(.

DETECTING GRAVITATIONAL WAVES FROM . . . PHYSICAL REVIEW D 67, 104025 #2003$

104025-5

with ni and ! i the unit vectors along the separation vector ofthe binary r and along the corresponding relative velocity v.These unit vectors are related to the adiabatic evolution ofthe dynamical variables by

n!e1S cos"S"e2

S sin"S , !!#e1S sin"S"e2

S cos"S ;#16$

the vectors e1,2S form an orthonormal basis for the instanta-

neous orbital plane, and in the FC convention they are givenby

e1S!

ezS$LN

sin i , e2S!

ezS#LN cos isin i . #17$

The vector e1S points in the direction of the ascending node of

the orbit on the (x ,y) plane. The quantity "S is the orbitalphase with respect to the ascending node; its evolution isgiven by

"S!%#& cos i , #18$

where i and & are the spherical coordinates of LN in thesource frame, as shown in Fig. 1. Using Eqs. #14$ and #16$,we can write Eq. #15$ as

Qci j!#2#'e"

S ( i jcos 2"S"'e$S ( i j sin 2"S$, #19$

where the polarization tensors e"S and e$

S are given by

e"S )e1

S! e1

S#e2S

! e2S , e$

S )e1S

! e2S"e2

S! e1

S . #20$

For a detector lying in the direction N!ezScos*"ex

Ssin*, itis expedient to express GW propagation in the radiation co-ordinate system with unit vectors +ex

R ,eyR ,ez

R, 'see our Fig. 1together with, for instance, Eq. #4.22$ of Ref. '16(( given by

exR!ex

S cos*#ezS sin* , #21$

eyR!ey

S , #22$

ezR!ex

S sin*"ezS cos*!N. #23$

In writing Eqs. #21$–#23$ we used the fact that for a genericbinary-detector configuration, the entire system consisting ofthe binary and the detector can be always rotated along the zaxis in such a way that the detector will lie in the (x ,z)plane. Later in this paper #in Sec. IV$ we shall find it conve-nient to conserve the explicit dependence of our formulas onthe azimuthal angle - that specifies the direction of the de-tector.In the transverse-traceless #TT$ gauge, the metric pertur-

bations are

hTT!h"T""h$T$ , #24$

where

T")exR

! exR#ey

R! ey

R , T$)exR

! eyR"ey

R! ex

R #25$

and

h"!12 h

i j'T"( i j , h$!12 h

i j'T$( i j . #26$

The response of a ground-based, interferometric detector#such as LIGO or VIRGO$ to the GWs is '15(

h resp!F"h""F$h$

!#2.

DMr 'e"

Si jcos 2"S"e$Si jsin 2"S(

$#'T"( i jF""'T$( i jF$$, #27$

where F" and F$ are the antenna patterns, given by

F" ,$!12 'e x! e x#e y! e y( i j'T" ,$( i j #28$

with e x ,y the unit vectors along the orthogonal interferometerarms. For the geometric configuration shown in Fig. 2, withdetector orientation parametrized by the angles / , 0 , and 1 ,we have

FIG. 1. Source and radiation frames in the FC convention '15(.

FIG. 2. Detector and radiation frames in the FC convention '15(.

DETECTING GRAVITATIONAL WAVES FROM . . . PHYSICAL REVIEW D 67, 104025 #2003$

104025-5

inaccurate approximation

accurate approximation

!/!

2

v

c⌘✓Gm!

c3

◆1/3post-Newtonian parameter

adia

batic

par

amet

er

Validity of adiabatic approximation

• Adiabatic approximations becomes inaccurate for v/c ≫ 0. Need to be replaced by semi-analytical computations without relying on the adiabatic approximation, or numerical relativity.

23

Summary of the lecture

•When the energy loss is small compared to the energy of the orbit ("adiabatic parameter" is small), phase evolution can be computed from the energy balance (conservation of energy). Higher order corrections to the conserved energy and radiation-loss can be included in terms of post-Newtonian expansions.

• If the compact objects contain significant spin angular momenta, use, in addition, the conservation of the total angular momentum.

• At late inspiral (adiabatic parameter ~ 1), adiabatic approximation becomes inaccurate. Need to be replaced by non-adiabatic approximations or numerical relativity.

• Further reading M. Maggiore, “Gravitational Waves, Vol. 1. Theory and Experiments”, Oxford University Press (2007).

A. Buonanno, “Gravitational Waves”, Lecture Notes from Les Houches Summer School (2006). arXiv: 0709.4682v1.

A. Buonanno, Y. Chen, M. Vallisneri, Physical Review D 67, 104025 (2003). gr-qc/0211087v4

24

theoretical sciences - rhodes university · 2019. 2. 19. · p. ajith international center for...

Documents