Chapter 15

Gravitational waves from binarysystems

In this chapter we will apply the formulae derived in the previous chapter to study thegravitational wave emission of a binary system composed by two point masses.

In this section we shall show that the orbital period P and the orbital distance l0 of abinary system decrease in time, due to gravitational wave emission. P ⌘ dP/dt was indeedmeasured for PSR 1913+16 and this measure – in very good agreement with the predictions ofGeneral Relativity – provided the first indirect proof of the existence of gravitational waves.The orbital evolution driven by gravitational wave emission quietly proceeds bringing thestars closer. As their distance decreases, the process becomes faster and the two stars spiraltoward their common center of mass until they coalesce.

We shall now describe how the binary system evolves, explicitely computing P (t), l0(t)and the emitted gravitational wave signal, up to the point when the quadrupole approxima-tion is violated and the theory developed in this chapter can no longer be applied.

We shall start by computing the gravitational luminosity defined in eq. (14.54); usingthe reduced quadrupole moment of a binary system given by eqs. (14.55) and (14.56) wefind

3X

k,n=1

...Qkn

...Qkn = 32 µ2 l40 !6

K = 32 µ2 G3 M3

l50.

and by direct substitution in eq. (14.102)

LGW ⌘ dEGW

dt=

32

5

G4

c5µ2M3

l50. (15.1)

This expression has to be considered as an average over several wavelenghts (or equivalently,over a su�ciently large number of periods), as stated in eq. (14.102); therefore, in orderLGW to be defined, we must be in a regime where the orbital parameters do not changesignificantly over the time interval taken to perform the average. This assumption is calledadiabatic approximation, and certainly applies to systems like PSR 1913+16 or PSR J0737-3039 which are very far from coalescence. In the adiabatic regime, the system has the timeto adjust the orbit to compensate the energy lost in gravitational waves with a change in

204

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 205

the orbital energy, in such a way that

dEorb

dt+ LGW = 0. (15.2)

This equation allows to compute the quantities we need.The orbital energy is

Eorb = EK + U

where the kinetic and the gravitational energy, EK and U , respectively are

EK =1

2m1!

2K r21 +

1

2m2!

2K r22 =

1

2!2K

"

m1m22l

20

M2+

m2m21l

20

M2

#

=1

2!2Kµl

20 =

1

2

GµM

l0(15.3)

and

U = �Gm1m2

l0= �GµM

l0.

Therefore

Eorb = �1

2

GµM

l0(15.4)

and its time derivative is

dEorb

dt=

1

2

GµM

l0

1

l0

dl0dt

!

= �Eorb

1

l0

dl0dt

!

. (15.5)

Substituting eq. (15.5) in eq. (15.2), using eqs. (15.4) and (15.1), we find

1

l0

dl0dt

=LGW

Eorb= �

"

64

5

G3

c5µ M2

#

· 1l40

. (15.6)

Assuming that at some initial time t = 0 the orbital separation is l0(t = 0) = lin0 , byintegrating eq. (15.6) we get

l40(t) = (lin0 )4 � 256

5

G3

c5µ M2 t , (15.7)

and defining

tc =5

256

c5

G3

(lin0 )4

µM2, (15.8)

eq. (15.7) finally gives how the orbital separation l0 changes in time

l0(t) = lin0

1� t

tc

�1/4

. (15.9)

From this equation we see that the orbital separation decreases in time and, when t = tcbecomes zero; this is due to the fact that we assumed that the bodies composing the binarysystem are pointlike. Of course, stars and black holes have finite sizes1, therefore they start

1for black holes we can consider twice the horizon radius as a characteristic size.

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 206

merging before t = tc is reached. In addition, when the two bodies are close enough boththe slow motion approximation and the weak field assumption, on which the quadrupoleformalism relies, fail and strong field e↵ects have to be considered. However, the value of tcgives a reliable estimate of the time the system needs to merge starting from a given initialdistance lin0 . Indeed, the last phases of coalescence are very very fast, and do not contributesignificantly to the total time needed to merge.

Using eq. (15.9) we can now compute how the angular velocity and the orbital periodchange in time. Indeed,

!K =(GM)1/2

l3/20

! !K(t) =(GM)1/2

(lin0 )3/2

1� t

tc

��3/8

i.e.

!K(t) = !inK

1� t

tc

��3/8

, !inK =

(GM)1/2

(lin0 )3/2. (15.10)

Moreover, since !K = 2⇡P�1, we get

P (t) = P in

1� t

tc

�3/8

. (15.11)

To compare the predictions of general relativity with binary pulsars observations, it isuseful to evaluate the rate at which the orbital period changes due to gravitational waveemission. Since

!2K = GMl�3

0 ! P 2 =4⇡2

GMl30 ! 2 lnP = ln

4⇡2

GM+ 3 ln l0 ! 1

P

dP

dt=

3

2

1

l0

dl0dt

,

and by eq. (15.6) we find

1

P

dP

dt=

3

2

LGW

Eorb= �64

5

G3

c5µ M2

l40. (15.12)

By using the relation between l0 and P eq.(15.12) becomes

1

P

dP

dt= �64

5

G2

c5µ M

l0

GM

l30

!

= �64

5

G2

c5µ M

l0

4⇡2

P 2

!

(15.13)

= �96

5

G5/3

c5µ M2/3

✓

2⇡

P

◆8/3

(15.14)

For example if we consider PSR 1913+16, assuming the orbit is circular and using thedata given in (14.67), we find

dP

dt⇠ �2.0 · 10�13.

As mentioned in section 14.4, the orbit of the real system has a quite strong eccentricity✏ ' 0.617. Repeating the calculation using the equations of motion appropriate for eccentricorbits we would find

1

P

dP

dt= �96

5

G5/3

c5µ M2/3

✓

2⇡

P

◆8/3

⇥ 1

(1� e2)7/2

✓

1 +73

24e2 +

37

96e4◆

, (15.15)

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 207

i.e.dP

dt= �2.4 · 10�12.

PSR 1913+16 has now been monitored for decades and the rate of variation of the period,measured with very high accuracy, is

dP

dt= � (2.4184 ± 0.0009) · 10�12.

(J. M. Weisberg, J.H. Taylor Relativistic Binary Pulsar B1913+16: Thirty Years of Observa-tions and Analysis, in Binary Radio Pulsars, ASP Conference series, 2005, eds. F.AA.Rasio,I.H.Stairs).

Thus, the prediction of General Relativity are confirmed by observations. This resultprovided the first indirect evidence of the existence of gravitational waves and for this dis-covery Hulse and Taylor have been awarded of the Nobel prize in 1993.For the recently discovered double pulsar PSR J0737-3039

P = 8640 s, Eorb ⇠ �2.55 · 1048 erg, LGW ⇠ 2.24 · 1032 erg/s (15.16)

anddP

dt⇠ �1.2 · 10�12,

which is also in agreement with observations.

15.1 The gravitational wave signal emitted by inspi-ralling compact bodies

Equation (15.10) shows how the orbital frequency changes as the two compact objects inspiralaround the common center of mass in the adiabatic regime, i.e. assuming that the orbitevolves through a sequence of stationary circular orbits.

In section 14.4 we showed that the gravitational wave signal emitted by a binary systemin circular orbit, located at at distance r, is

hTT

ij (t, r) = � h0

rATT

ij (t� r

c), (15.17)

where h0 is the wave amplitude

h0 =4 µ M G2

l0 c4, ATT

ij (t� r

c) =

PijklAkl(t�r

c)�

, (15.18)

and

Aij(t) =

0

B

@

cos 2!Kt sin 2!Kt 0sin 2!Kt � cos 2!Kt 0

0 0 0

1

C

A

. (15.19)

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 208

The wave is monocromatic, and is emitted at twice the orbital frequency, i.e.

⌫GW = 2⌫orb =!K

⇡. (15.20)

Consequently, as the two bodies approach the wave frequency increases according to eq.(15.10), i.e.

⌫GW (t) = ⌫inGW

1� t

tc

��3/8

, ⌫inGW =

1

⇡

s

GM

(lin0 )3. (15.21)

In a similar way, since the orbital distance decreases according to eq. (15.9), the amplitudeof the emitted signal also changes with time

h0(t) =4µMG2

l0(t)c4=

4µMG2

c4· !2/3

K (t)

G1/3M1/3

(15.22)

=4⇡2/3 G5/3 M5/3

c4⌫2/3GW (t),

where we have defined

M5/3 = µ M2/3 ! M = µ3/5 M2/5 =(m1m2)3/5

M1/5. (15.23)

Eqs. (15.21) and (15.22) show that both the amplitude and the frequency of the gravitationalsignal emitted by a coalescing binary system during the inspiralling increase with time. Thisbehaviour is typical of the chirp of a singing bird, and for this reason the waveform emittedby the coalescing binary is named chirp, and M is said the chirp mass.

Since !K depends on time, the phase appearing the polarization tensor (15.19) must bereplaced by the integrated phase

�(t) =Z t

2!K(t)dt =Z t

2⇡⌫GW (t) dt+ �in, where �in = �(t = 0)

To compute the integral it is convenient to use the following relations. First of all we writeeq. (15.21) in the form

⌫GW (t) =⌫inGW t3/8c

[tc � t]3/8; (15.24)

Remembering that tc =5

256c5

G3 (lin0 )4/(µM2), and being 2563/8 = 8, it is easy to show that

⌫inGW t3/8c =

53/8

8⇡

c3

GM

!5/8

, (15.25)

so that ⌫GW (t) can be written as

⌫GW (t) =53/8

8⇡

c3

GM

!5/8

1

tc � t

�3/8

. (15.26)

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 209

The integrated phase therefore is

�(t) = �2

"

c3 (tc � t)

5GM

#5/8

+ �in (15.27)

which shows that the phase of the signal dependes on the chirp mass.In conclusion, the signal emitted during the inspiralling will be

hTT

ij = �4⇡2/3 G5/3

c4⇥ M

r⇥ (M⌫GW (t))2/3

PijklAkl(t�r

c)�

(15.28)

where

Aij(t) =

0

B

@

cos �(t) sin �(t) 0sin �(t) � cos �(t) 0

0 0 0

1

C

A

15.2 September 15th, 2015: the detection of gravita-tional waves

On September 15th, 2015 the interferometric antennas of the American experiment LIGOdetected, for the first time, the signal emitted in the coalescence of two black holes. Thetwo antennas are located in Livingston (Luisiana) and in Hanford (Washington). The signal,which is shown in figure 15.1, was very loud and was detected with a high signal to noiseratio of 23.7. In the upper panel of the figure the output of the two detectors is shown asa function of time; the second panel shows the signal extracted from the raw data usingsuitable filtering techniques; the third panel shows the residual noise after data filtering, andthe lower panel shows how the frequency of the signal changes in time.

The analysis of the data showed that this signal, named “GW150914”, has been emittedby two black holes with masses

m1 = 29.1+3.8�4.4 M�, m2 = 35.7+5.4

�3.8 M�, (15.29)

which coalesced and formed a single black hole with mass and angular momentum

M = 61.8+4.2�3.5 M�, a = J/M = 0.68+0.05

�0.06. (15.30)

A comparison of the total mass of the two black holes with the mass of the final black holeshows that ⇠ 3 M� have been radiated in gravitational waves. This is equivalent to a hugeamount of energy:

EGW ⇠ 3 M�c2 ! 3⇥ 1.989 · 1033g ⇥ (2.998 · 1010cm/s)2 = 5.4 · 1054 erg.

The estimated luminosity distance and redshift of the source are

DL = 420+150�180 Mpc, z = 0.088+0.031

�0.038. (15.31)

Let us see how these conclusions have been reached.

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 210

Figure 15.1: From: B.P. Abbott et al., Observation of gravitational waves from a binaryblack hole merger, Physical review Letters 116, 061102, 2016.

15.3 How to extract the chirp mass and the luminositydistance of the source from the observed signal

As we know, the first part of the signal –the chirp– is emitted during the inspiralling, and ithas the form described in the previous section. As the two bodies approch each other andstart merging, the interaction becomes strongly non linear and the emitted waveforms haveto be computed solving Einstein’s equations in the fully non linear regime. However, alreadyfrom the chirp we can extract interesting information about the source. If we set

A =53/8

8⇡

c3

GM

!5/8

, (15.32)

eq. (15.26) becomes

⌫GW =A

(tc � t)3/8and ⌫GW =

3

8

A

(tc � t)11/8! ⌫GW ⌫�11/3

GW =3

8A�8/3. (15.33)

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 211

Using the definition of A the last equation gives

"

8⇡

53/8

✓

GMc3

◆5/8#8/3

=8

3⌫GW ⌫�11/3

GW (15.34)

from which we get (note that 88/3 = 256)

M =c3

G

5

96

1

⇡8/3⌫GW ⌫�11/3

GW

�3/5

. (15.35)

From this equation we see that the chirp mass can be measured by measuring the wavefrequency and its time derivative at some time.

Equation (15.35) is evaluated in the source frame. However, frequencies are measured inthe detector frame, i.e. the quantity which we really measure is

M0 =c3

G

"

5

96

1

⇡8/3

d

dtobs⌫obsGW

!

(⌫obsGW )�11/3

#3/5

. (15.36)

The observed frequency is related to the emission frequency by

⌫obsGW =

⌫GW

1 + z, (15.37)

and dtobs = dt(1 + z), where z is the cosmological redshift, i.e. the redshift due to theexpansion of the Universe. It follows that

"

d

dtobs⌫obsGW

!

(⌫obsGW )�11/3

#3/5

= [⌫�11/3GW ⌫GW ]3/5 ⇥ (1 + z)

therefore, the quantity M0 which we measure is related to the true chirp mass by the relation

M0 =c3

G

5

96

1

⇡8/3⌫GW ⌫�11/3

GW

�3/5

⇥ (1 + z) i.e. M0 = M(1 + z). (15.38)

M0 is said “redshifted chirp mass”.

For a source at a cosmological distance the wave amplitude appearing in eq. (15.28)should be written as

h0 =4⇡2/3 G5/3

c4⇥ M

D(z)⇥ (M⌫GW (t))2/3 (15.39)

where D(z) is the luminosity distance given by

D(z) =2

H0⌦20

⌦0z � (2� ⌦0)(q

1 + ⌦0z � 1)�

. (15.40)

Here H0 is the Hubble constant, and ⌦0 =8G⇡3

⇢m0

H20and ⇢m0 is the actual matter density.

Since M0⌫obsGW = M⌫GW , and M0 = M(1+ z), from the measured wave amplitude h0 we

can only extract an e↵ective distance

deff = D(z)⇥ (1 + z). (15.41)

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 212

The problem now is that we do not know the source redshift. This information wouldbe available if we had detected an electromagnetic counterpart of this event, which wouldallow us to identify the redshift of the hosting galaxy. But unfortunately these sources arenot expected to have such e.m. counterparts, because according to what is known abouttheir formation, they should not be surrounded by an accretion disk su�ciently massive toproduce a significant electromagnetic emission at coalescence. In addition, with the twoLIGO detectors it is practically impossible to localize the source position. Indeed the area ofthe sky were the source could be located is of the order of 230 deg2, which is too large to bespanned by any existing electromagnetic detector. In the future, when more detectors willbe operational2 it will be possible to restrict the source area to about 5 deg2; but to reachthis goal at least four detectors will be needed.

In order to estimate the true chirp mass and the luminosity distance of the source, sincewe do not know the redshift, we need to adopt a cosmological model. The LIGO-Virgocollaboration assumes a ⇤CDM cosmology, with the values of the cosmological parametersmeasured by the Planck mission (Planck Collaboration, Ade, P. A. R., Aghanim, N., et al.,Planck 2015 results. XIII. Cosmological parameters, A&A 594, A13, 2016):

H0 = 67.9 km s�1Mpc�1, ⌦0 = 0.306. (15.42)

By substituting these values in the expression of the luminosity distance in eq. (15.41), andusing the measured value of deff deduced from the wave amplitude, the value of the sourceredshift has been found to be

z = 0.088+0.031�0.038, (15.43)

and from this, the chirp mass and the luminosity distance of the coalescing black holes hasfinally been evaluated

M = 27.9+2.1�3.9 M�, D(z) = 410+160

�180 Mpc. (15.44)

(The LIGO Scientific Collaboration and The Virgo Collaboration, Properties of the binaryblack hole merger GW150914, Phys. Rev. Lett, 116, 241102, 2016).

15.4 A lower bound for the total mass of the system

In figure (15.2) we plot the total mass M = m1+m2 of the system as a function of the massof one of the two bodies, say m1, assuming Mchirp = 27.9 M� and using the equation

(m1m2)3/5 M1/5 = 27.9 M�. (15.45)

From the figure we see that to be compatible with the observed chirp mass, the total massof the sistem must be larger than M = 63.7 M�. From this value we understand that thecoalescing bodies cannot be two neutron stars, since the maximum mass observed for neutronstars is ⇠ 2 M�.

2The Virgo detector in Cascina (Italy) will be fully operational in 2017, and within a decade a detectorbased in India, Indico, and one in Japan, KAGRA, will join the network.

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 213

Figure 15.2: The total mass of the coalescing system, M = m1 +m2, is plotted versus themass m1, assuming M = 27.9 M�.

A further information can be obtained by evaluating the distance between the two objectsjust before merging. We know that the instantaneous wave frequency is related to the orbitalfrequency by

⌫GW = 2⌫orb =1

⇡

v

u

u

t

G(m1 +m2)

l30.

Over 0.2 s the observed wave frequency increases from 35 to 150 Hz, from which we inferthat just before merging, the distance between the two objects is

l0 =

G(m1 +m2)

(⌫GW⇡)2

!1/3

! l0(150 Hz) = 338.7 km,

where we have assumed (m1 + m2) = 63.7 M�. Considering how large is the mass of thetwo bodies, this distance is extremely small, and this indicates that they must be extremelycompact.

At this point we have exploited all the information we can obtain from the chirp: thevalue of the chirp mass and of the luminosity distance of the source, a lower bound on thetotal mass, which excludes the possibility that the two bodies are neutron stars, and thefact that just before merging they are extremely close, which means that they are extremelycompact.

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 214

15.5 How do we understand that the two coalescingcompact objects are black holes?

In order to identify the nature of the sources, we now need to analyze the part of the signalemitted at later times, when the two bodies merge and form a single compact object.

In figure (15.3) we show the plot of a typical signal emitted in the coalescence of blackholes. The first part of the signal is the chirp, emitted during the inspiralling up to the pointwhen the orbital velocities are no longer much smaller than the speed of light; in addition,tidal interactions between the two bodies become dominant, and the quadrupole approachcan no longer be applied to describe the waveform.

The approximate value of the orbital distance at which our description of the waveformusing the quadrupole formalism becomes no longer appropriate is given by the value of l0 atthe Innermost Stable Circular Orbit (ISCO), which is about

l0 ISCO ⇠ 6G(m1 +m2)

c2. (15.46)

At distances comparable to l0 ISCO the quadrupole waveform must be corrected with moreterms in the Post-Newtonian expansion in v/c of the equation of motion of the two bodies.Thus, l0 ISCO must be considered as an upper limit for the validity of the chirp waveform.The corresponding frequency is

⌫ISCOGW =

!K

⇡=

1

⇡

v

u

u

t

G(m1 +m2)

l30 ISCO

=c3

⇡Gp63

1

(m1 +m2). (15.47)

The part of signal between red arrows in figure (15.3) is emitted when the two bodies getvery close to each other and finally merge, and the last part, indicated with green arrows, isemitted by the final black hole.

As anticipated above, the merging part of the signal has to be found by solving numer-ically Einsteins equations in the fully non linear regime These studies started in the late1990s with the Grand Challenge project, which aimed at simulating the head-on collision oftwo black holes.

After decades of studies a bank of templates (still far to be complete) has been obtainedwhich can be compared with the observed signal. From these numerical waveforms, whichinclude the spins of the bodies and initially eccentric orbits, some very useful fitting formulaehave been derived which allows to estimate the masses and the spins of the two coalescingbodies, and the mass and spin of the final black hole.

For the detected signal GW150914 the quality of the data did not allow to obtain areliable estimate of the spins. In principle, their values could also be estimated by usingmore accurate waveforms for the chirp. Indeed, the formulae we derived are only the lowestorder terms in a Post-Newtonian expansions of the equations of motion in the parameter v/c,where v are the velocities of the two bodies. The spin contribution appears at highest orderand its evaluation goes beyond the scope of this course. However, it may be noted that sinceit appears at a power of (v/c) larger than one, it starts to be significant when the velocitiesbecome large, the two bodies are very close to merging, and a large signal-to-noise ratio is

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 215

100 1000Frequency (Hz)

1e-26

1e-25

1e-24

1e-23

1e-22

1e-21

Gra

vita

tiona

l Stra

in

0 0.05 0.1 0.15Time (s)

-5e-21

0

5e-21

Gra

vita

tiona

l Stra

in

ç inspiral (chirp) èinspiral

çèç Mergingè

çè

ç è

ringdown

the ringdown isasuperposition ofdampedsinusoids: theQuasi-NormalModes(QNM)oftheblackholewhicheventuallyforms

perturbativeapproaches

thispartisfoundbynumericallyintegrating Einstein’sequations

THELASTPARTOFTHESIGNAL:theringdownofthefinalblackhole

Chandrasekhar&Detweiler1975:hundreds ofpaperssincethen.Stillaveryactivefield

M =nM¤ νo ~(12/n)kHzτ ~n 5.510-5 s

willfuturemeasurementsofthesefrequenciesconfirmthepredictionsofgeneralrelativity?

frequencyincreasesupto30%iftheBHrotates

theQNMsfrequenciesdependonlyontheBHmassandangularmomentum

es.M=10M¤ νo ~1,2kHzτ ~ 0.55ms (VIRGO/LIGO)M=106 M¤ νo ~12mHz τ ~ 55s(LISA)

Figure 15.3: The signal emitted by coalescing black holes. The part of the signal betweenred arrows is emitted during the merging, the last part, the ringing tail indicated in green,is emitted by the final black holes oscillating in its quasi-normal modes.

needed to extract its contribution from the noisy data. Indeed, even though GW150914 wasdetected with a high signal-to-noise ratio, it has not been possible to measure the individualspins of the coalescing bodies with su�cient accuracy.

A further contribution of the spins to the emitted waveform would appear if they are notaligned with the orbital angular momentum. In this case the orbit would show a precessionmotion around the direction of the orbital angular momentum which, however, has not beenobserved in the detected signal.

The last part of the signal, the ringing tail, is emitted by the black hole which forms as aresult of the merging. It is a superposition of damped sinusoids, that are the eigenfrequenciesof its Quasi-Normal Modes (QNM). These are the proper modes of oscillation of a black hole,and the corresponding frequencies are complex. The real part is the frequency at which theblack hole oscillates and emits gravitational waves. The imaginary part is the inverse ofthe damping time; indeed the oscillations are damped because the black hole looses energyemitting gravitational waves.

The QNM frequencies can be found by perturbing Einstein’s equations around the Schwarzshildsolution (or the Kerr solution if the black hole is rotating) and solving these equations im-posing that the solution is a pure ingoing wave at the horizon, because nothing can escapefrom it, and that at radial infinity the solution behaves as a pure outoing wave. Theseboundary conditions are satisfied only for a discrete set of complex frequencies which are the

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 216

frequencies of the quasi-normal modes3

It has been shown that the QNM frequencies depend only on the mass and on the angularmomentum of the black hole. This result is known as the no hair theorem.

For example, if the black hole is non rotating, and the black hole mass is n times themass of the Sun, the frequency and damping time of the lowest quasi-normal mode is

M = n M�, ⌫0 ⇠12

nkHz, ⌧0 ⇠ n 5.5 · 10�5 s; (15.48)

i.e. the larger is the mass, the smaller is the frequency and the longer is the damping time.As an example, if the black hole has a mass of 10 M� or if it is a supermassive black holewith M = 106 M�, the frequencies and damping times of the lowest QNMs are, respectively,

M = 10M�, ⌫0 = 1, 2 kHz, ⌧ = 5.55 · 10�4s, (15.49)

M = 106M�, ⌫0 = 1, 21 · 10�2Hz, ⌧ = 55.5s.

If the black hole rotates, the frequencies split in two families, said co-rotating and counter-rotating. The co-rotating (counter-rotating) frequencies increase (decrease) up to 30% if theBH rotates.

The QNM frequencies are important because the gravitational signal emitted by a per-turbed black hole will, during its last stages, decay as a superposition of the first few quasinormal modes. Therefore, we expect to see these damped sinusoids in the last part of thesignal emitted by coalescing black hole binaries, and from them we would like to infer themass and angular momentum of the final black hole. From the data of GW150914 it hasbeen possible to extract only the frequency of the lowest QNM; the mass and the angularmomentum have then be estimated, and their values agree with those estimated from themerging part of the signal, within the experimental errors.

Thus, we can say that we have a complete description of the waveforms emitted in blackhole binary coalescence in framework of General Relativity which, compared to the data,allow to interpret the detected signal as due to the coalescence of two black holes with themasses given in eq. (15.29) which, after merging, gave birth to a rotating black hole, withmass and angular momentum given in eq. (15.30).

3For a complete account on black hole quasi-normal modes see The mathematical theory of black holes

(Oxford: Claredon Press 1984).

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 217

15.6 More signals from binary black hole coalescence

Figure 15.4: The three signals detected during the first observational run of Advanced LIGOin 2015 are plotted versus the time they spend in the detector bandwidth.

During the first observational run of LIGO, other signals have been detected. One ofthem, GW151226, has been identified as a binary black hole coalescence; the two black holeshave masses

m1 = 7.5+2.3�2.3 M�, m2 = 14.2+8.3

�3.7 M�, (15.50)

and formed a black hole with mass and angular momentum

M = 20.8+6.1�1.7 M�, a = J/M = 0.74+0.06

�0.06. (15.51)

The source is at the distance

DL = 440+180�190 Mpc, z = 0.009+0.03

�0.04. (15.52)

This event has been detected with a signal to noise ratio of 13, and the energy radiated ingravitational waves is about EGW ⇠ 1 M�.

CHAPTER 15. GRAVITATIONAL WAVES FROM BINARY SYSTEMS 218

A further detected signal, LVT151012, is weaker than the previous two. Its statisticalsignificance is 2�, and the signal-to noise ratio is 9.7, therefore it is considered only a“candidate” gravitational wave event. Assuming that it may be due to the coalescence oftwo black holes, the parameters have been estimated and are

m1 = 13+4�5 M�, m2 = 23+18

�6 M�, (15.53)

and the mass and angular momentum of the final black hole are

M = 35+14�4 M�, a = J/M = 0.66+0.09

�0.10. (15.54)

The source is at a larger distance

DL = 1+0.5�0.5 Gpc, z = 0.20+0.09

�0.09. (15.55)

In figure (15.4) we show the three signals plotted versus the time they spend in the detectorbandwidth. The reason why GW150914 spans a time interval shorter than the other twosignals is due to the following reason. The chirp, emitted during the inspiralling of the twobodies, ends approximately when the they are separated by the orbital distance l0 ISCO givenin eq. (15.46) and the wave frequency is given in eq. (15.47). From the latter we see that alarger total mass corresponds to a smaller wave frequency at the ISCO.

For the three signals the frequency of the ISCO is:

GW150914 l0 ISCO = 574, 1 km ⌫ISCOGW = 67.9 Hz (15.56)

GW151226 l0 ISCO = 192, 3 km ⌫ISCOGW = 202.6 Hz

LVT151012 l0 ISCO = 319 km ⌫ISCOGW = 122.1 Hz. (15.57)

Considering that the frequency region where LIGO is sensitive is in the range⇡ (10Hz, 2 kHz),we see that the chirp of GW151226 will span a frequency interval larger than GW150914,and therefore will stay in the detector bandwidth for a longer time, as shown in figure (15.3).The duration of the merging part of the signal and of the ringing tail is much shorter thanthat of the chirp.