PSR 1913+16
Assuming the orbit is circular
P = 27907 s,
Eorb ∼ −1.4·1048 erg,
LGW ∼ 0.7·1031 erg/s
and
dP
3 P
=
LGW
dt
2 Eorb
→
dP
∼ −2.2 · 10−13.
dt
The orbit of the real system has a quite strong eccentricity
0.617. Using the quadrupole formalism
with the appropriate orbit:
dP
= −2.4 · 10−12.
dt
PSR 1913+16
Assuming the orbit is circular
P = 27907 s,
Eorb ∼ −1.4·1048 erg,
LGW ∼ 0.7·1031 erg/s
and
dP
3 P
=
LGW
dt
2 Eorb
→
dP
∼ −2.2 · 10−13.
dt
The orbit of the real system has a quite strong eccentricity
0.617. Using the quadrupole formalism
with the appropriate orbit:
dP
= −2.4 · 10−12.
dt
Measured value
dP
= − (2.4184 ± 0.0009) · 10−12.
dt
(J. M. Wisberg, J.H. Taylor Relativistic Binary Pulsar
B1913+16: Thirty Years of Observations and Analysis,
in Binary Radio Pulsars, ASP Conference series, 2004, eds.
F.AA.Rasio, I.H.Stairs).
PSR 1913+16
Assuming the orbit is circular
P = 27907 s,
Eorb ∼ −1.4·1048 erg,
LGW ∼ 0.7·1031 erg/s
and
dP
3 P
=
LGW
dt
2 Eorb
→
dP
∼ −2.2 · 10−13.
dt
The orbit of the real system has a quite strong eccentricity
0.617. Using the quadrupole formalism
with the appropriate orbit:
dP
= −2.4 · 10−12.
dt
Measured value
dP
= − (2.4184 ± 0.0009) · 10−12.
dt
(J. M. Wisberg, J.H. Taylor Relativistic Binary Pulsar
B1913+16: Thirty Years of Observations and Analysis,
in Binary Radio Pulsars, ASP Conference series, 2004, eds.
F.AA.Rasio, I.H.Stairs).
Residual differences due to Doppler corrections, due to the
relative velocity between us and the pulsar induced by the
differential rotation of the Galaxy.
Ṗcorrected
= 1.0013(21)
ṖGR
For the recently discovered double pulsar PSR J07373039
P = 8640 s,
and
Eorb ∼ −2.55·1048 erg,
dP
∼ −1.2 · 10−12,
dt
which also agrees with observations.
LGW ∼ 2.24·1032 erg/s
Thus, this prediction of General Relativity is
confirmed by observations. This result provided
the first indirect evidence of the existence of
gravitational waves and for this discovery R.A.
Hulse and J.H. Taylor have been awarded of the
Nobel prize in 1993.
WAVEFORM: AMPLITUDE AND PHASE
Orbital separation
1/4
t
in
,
l0(t) = l0 1 −
tcoal
5 l in 4
5 c 0
.
where tcoal =
256 G3 µM 2
Wave frequency:
ωK
=
νGW (t) =
π
in
νGW
1−
t
3/8 ,
in
νGW
=
1
π
GM
(l0 in)3 ,
tcoal
The instantaneous amplitude of the emitted signal is
4µM G2 4G5/3µM 2/3 2/3
h0(t) =
=
· ωK (t),
4
4
rl0(t)c
rc
2
ωK
= GM/l03
if we define the quantity M, called chirp mass,
M5/3 = µ M 2/3
and remember that the wave frequency νGW is twice the
orbital frequency νK = ωK /2π we find
4π 2/3 G5/3 M5/3 2/3
νGW (t) .
h0(t) =
4
cr
The amplitude and the frequency of the gravitational signal emitted by a coalescing system
increase with time. For this reason this peculiar waveform is called chirp, like the chirp of a
singing bird
20
15
10
chirp
5
0
-5
-10
-15
-20
retarded time
4π 2/3 G5/3 M5/3 2/3
νGW (t) .
h0(t) =
c4 r
if we know the signal phase we can measure the
chirp mass.
hTijT
4π 2/3 G5/3 M5/3 2/3 r νGW (t) Pijkl Akl (t − )
=−
c4 r
c
the signal phase must be integrated
t
t
Φ(t) =
2ωK (t)dt =
2πνGW (t) dt + Φin,
where Φin = Φ(t = 0)

r
Aij (t − ) =
c

r
r
cos Φ(t − ) sin Φ(t − ) 0
c
c






r
r


 sin Φ(t − ) − cos Φ(t − ) 0 
c
c




0
0
0
LIGO[40 Hz − 1 − 2 kHz] LISA[10−4 − 10−1] Hz
VIRGO[10 Hz − 1 − 2 kHz]
Let us consider 3 binary system
a) m1 = m2 = 1.4 M
b) m1 = m2 = 10 M
c) m1 = m2 = 106 M
LIGO[40 Hz − 1 − 2 kHz] LISA[10−4 − 10−1] Hz
VIRGO[10 Hz − 1 − 2 kHz]
Let us consider 3 binary system
a) m1 = m2 = 1.4 M
b) m1 = m2 = 10 M
c) m1 = m2 = 106 M
Let us first calculate what is the orbital distance
between the two bodies on the Innermost Stable
Circular Orbit (ISCO) and the corresponding
emission frequency
6GM
GM
ISCO
∼
, ωK =
= π νGW
l0
c2
l03
1
GM
ISCO
→ νGW =
π (l0ISCO )3
LIGO[40 Hz − 1 − 2 kHz] LISA[10−4 − 10−1] Hz
VIRGO[10 Hz − 1 − 2 kHz]
Let us consider 3 binary system
a) m1 = m2 = 1.4 M
b) m1 = m2 = 10 M
c) m1 = m2 = 106 M
Let us first calculate what is the orbital distance
between the two bodies on the Innermost Stable
Circular Orbit (ISCO) and the corresponding
emission frequency
6GM
GM
ISCO
∼
, ωK =
= π νGW
l0
c2
l03
1
GM
ISCO
→ νGW =
π (l0ISCO )3
a) l0ISCO = 24, 8 km
νGW = 1570.4 Hz
b) l0ISCO = 177, 2 km
νGW = 219.8 Hz
c) l0ISCO = 17.720.415, 3 km
νGW = 2.2·10−3 Hz
LIGO[40 Hz − 1 − 2 kHz] LISA[10−4 − 10−1] Hz
VIRGO[10 Hz − 1 − 2 kHz]
Let us consider 3 binary system
a) m1 = m2 = 1.4 M
b) m1 = m2 = 10 M
c) m1 = m2 = 106 M
Let us first calculate what is the orbital distance
between the two bodies on the Innermost Stable
Circular Orbit (ISCO) and the corresponding
emission frequency
6GM
GM
ISCO
∼
, ωK =
= π νGW
l0
c2
l03
1
GM
ISCO
→ νGW =
π (l0ISCO )3
a) l0ISCO = 24, 8 km
νGW = 1570.4 Hz
b) l0ISCO = 177, 2 km
νGW = 219.8 Hz
νGW = 2.2·10−3 Hz
c) l0ISCO = 17.720.415, 3 km
a) and b) are interesting for LIGO and VIRGO,
c) will be detected by LISA
Let us consider LIGO and VIRGO; we want to compute
the time a given signal spends in the detector bandwidth
before coalescence.
From
3/8
νGW (t) =
we get
t = tcoal 1 −
in
νGW
tcoal
[tcoal − t]3/8
in
νGW
νGW (t)
8/3
.
Let us consider LIGO and VIRGO; we want to compute
the time a given signal spends in the detector bandwidth
before coalescence.
From
3/8
νGW (t) =
we get
in
νGW
tcoal
[tcoal − t]3/8
t = tcoal 1 −
in
νGW
8/3
νGW (t)
.
Putting :
in
νGW
= lowest frequency detectable by the antenna, and
max
ISCO
= νGW
we find
νGW
LIGO
VIRGO
a) (m1 = m2 = 1.4 M) [40 − 1570.4 Hz] [10 − 1570.4 kHz
∆t = 24.86 s
b) (m1 = m2 = 10 M)
∆t = 16.7 m
[40 − 219.8 Hz] [10 − 219.8 kHz]
∆t = 0.93 s
∆t = 37.82 s
Let us consider LIGO and VIRGO; we want to compute
the time a given signal spends in the detector bandwidth
before coalescence.
From
3/8
νGW (t) =
we get
in
νGW
tcoal
[tcoal − t]3/8
t = tcoal 1 −
in
νGW
8/3
νGW (t)
.
Putting :
in
νGW
= lowest frequency detectable by the antenna, and
max
ISCO
= νGW
we find
νGW
LIGO
VIRGO
a) (m1 = m2 = 1.4 M) [40 − 1570.4 Hz] [10 − 1570.4 kHz
∆t = 24.86 s
b) (m1 = m2 = 10 M)
∆t = 16.7 m
[40 − 219.8 Hz] [10 − 219.8 kHz]
∆t = 0.93 s
∆t = 37.82 s
VIRGO catches the signal for a longer time.
Chirp at 100 Mpc
1e-20
ν1/2 h(ν) [ Hz-1/2 ]
100-100 MSun BHs
1e-21
10-10 MSun BHs
1e-22
Adv Virgo
1e-23
Sn
1/2
,
Virgo+
1e-24
νISCO
νISCO
100
1000
ν [ Hz ]
Planned sensitivity curve for VIRGO+ and ADVANCED
VIRGO
The plotted signal is the
strain amplitude = ν 1/2h(ν) ,
evaluated for the chirp.
Source located at a distance of 100 Mpc.
Chirp at 100 Mpc
1e-20
ν1/2 h(ν) [ Hz-1/2 ]
100-100 MSun BHs
1e-21
10-10 MSun BHs
1e-22
Sn1/2,
Virgo+
Adv Virgo
1e-23
1e-24
νISCO
νISCO
100
1000
ν [ Hz ]
Planned sensitivity curve for VIRGO+ and ADVANCED
VIRGO
The plotted signal is the
strain amplitude = ν 1/2h(ν) ,
evaluated for the chirp.
Source located at a distance of 100 Mpc.
The chirp is only a part of the signal emitted
during the binary coalescence.
Black hole-black hole coalescence
Hybrid waveform at 100 Mpc
1e-20
ν1/2 h(ν) [ Hz-1/2 ]
100-100 MSun BHs
1e-21
10-10 MSun BHs
1e-22
Sn1/2,
Virgo+
Adv Virgo
1e-23
1e-24
νISCO
νISCO
100
1000
ν [ Hz ]
This hybrid waveform is obtained by matching three signals:
1) chirp waveform for ν < νISCO
2) waveform emitted during the merger phase of two black
holes, obtained by numerical integration of Einstein’s equations.
3) waveform emitted after the final black hole is formed,
due to ringdown oscillations.
P. Ajith et al., Phys. Rev. D 77, 104017 2008
WHAT ABOUT LISA?
LISA [10−4 − 10−1] Hz
1e-20
Detection threshold
LISA 1-yr observation
1e-21
1e-22
1e-23
1e-24
0.0001
0.001
0.01
ν [ Hz ]
0.1
1
Let us consider 2 BH-BH binary systems
a) m1 = m2 = 102 M
b) m1 = m2 = 106 M
Orbital distance between the two bodies on the innermost
stable circular orbit (ISCO) and the corresponding emission frequency
6GM
1
GM
GM
ISCO
ISCO
∼
, ωK =
= π νGW → νGW =
l0
3
2
c
l0
π (l0ISC
a) l0ISCO = 1772 km
b) l0ISCO = 17.720.415, 3 km
νGW = 21.98 Hz
νGW = 2.2 10−3 Hz
Time a given signal spends in the detector bandwidth
before coalescence.
a) m1 = m2 = 102 M
b) m1 = m2 = 106 M
t = tcoal 1 −
νin
νGW (t)
8/3
LISA
a)
[10−4 − 10−1 Hz]
∆t = 556.885 years
b)
[10−4 − 2.2 · 10−3 Hz]
∆t = 0, 12 years = 43 d 18 h 43 m 24 s
1e-14
LISA
10Msun BH-BH
6
10 Msun BH-BH
1e-16
1e-17
1e-18
1e-19
Sn
1/2
, ν
1/2
h(ν) [ Hz
-1/2
]
1e-15
1e-20
1e-21
1e-05
1e-04
0.001
ν [ Hz ]
0.01
0.1