Dynamics of star clusters containing stellar mass black ...Dynamics of star clusters containing...
Transcript of Dynamics of star clusters containing stellar mass black ...Dynamics of star clusters containing...
Dynamics of star clusters containing stellar mass black holes:
1. Introduction to Gravitational Waves
July 25, 2017 Bonn
Hyung Mok Lee Seoul National University
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Outline• What are the gravitational waves?
• Generation of gravitational waves
• Detection methods
• Astrophysical Sources
• Detected GW Sources2
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Principle of Equivalence
• Principle of Equivalence: inertial mass = gravitational mass
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• The trajectory of a particle in gravity does not depend on the mass: geometrical nature
• In a freely falling frame, one cannot feel the gravity.
• However, the presence of the gravity will cause tidal force. Again, this is similar to the difference between flat and curved space.
F = mia = mgg
g = �GM
r3r
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Curved Spacetime
• General geometry of space-time can be characterized by metric
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• In the absence of gravity, flat spacetime
• The presence of gravity causes deviation from flat spacetime: curved spacetime
ds
2 = gµ⌫dxµdx
⌫
gµ⌫ = ⌘↵� =
0
BB@
�1 0 0 00 1 0 00 0 1 00 0 0 1
1
CCA
ds
2 = �dt
2 + dx
2 + dy
2 + dz
2
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Tidal gravitational forces• Let the positions of the two nearby particles be x and x+𝝌, then the equations of motions are
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• If 𝝌 is small, one can expand 𝚽
d
2(xi + �
i)
dt
2= ��
ij @�(x+ �)
@x
j
• Then the relative acceletation between two particles becomes
@�(x+ �)
@x
j⇡ @�(x)
@x
j+
@
@x
k
✓@�(x)
@x
j
◆�
k + ...
Tidal acceleration tensor
d
2x
i
dt
2= ��
ij @�(x)
@x
j,
d
2�
i
dt
2= ��
ij
✓@
2�
@x
j@x
k
◆
x
�
k
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
In GR, similar expression can be derived
• "Geodesic deviation" equation
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• In the weak field limit, v≪c, |𝝫|≪c2. The only remaining components are 𝜇=ν =0. Therefore,
i.e.,
Riemann curvature tensorD
2�
�
D⌧
2= R
�⌫µ⇢
dx
⌫
d⌧
dx
⇢
d⌧
�
µ
d2�i
dt2= Ri
0j0�j
R
i0j0 = � @
2�
@x
i@x
j
• Ricci tensor can be defined by contraction of Riemann tensor:Rµ⌫ = R�
µ�⌫
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Einstein's Field Equation• In Newtonian dynamics, the motion of a particle is governed by the
gravitational potential which can be computed by the Poisson's equation:
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i.e., summation over the quantities that describe the geodesic deviation (=tidal acceleration tensor).
• The Ricci tensor is similarly defined with Φ, i.e., summation of all tidal components.
• In vacuum,
r2� = �
ij
✓@
2�
@x
i@x
j
◆= 4⇡G⇢
Rµ⌫ = 0, similar to r2� = 0
• In the presence of energy (mass, etc.), Einstein's field equation becomes
Rµ⌫ � 1
2Rgµ⌫ = �8⇡G
c4Tµ⌫
Curvature scalar
Energy-momentum tensor
R = Rµµ
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Field equation in weak field limit
• In the weak field limit,
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• Einstein's field equation becomes
• Perform coordinate transformation,
gµ⌫ = ⌘µ⌫ + hµ⌫ , |hµ⌫ | ⌧ 1
where
flat part small perturbation
⇤hµ⌫ =@
2
@x
�@x
µh
�⌫ � @
2
@x
�@x
⌫h
�µ +
@
2
@x
µ@x
⌫h
�� = �16⇡GSµ⌫
Sµ⌫ ⌘ Tµ⌫ � 1
2⌘µ⌫T
�� and ⇤ = � @2
@t2+r2
x
µ ! x
0µ = x
µ + ⇠
µ
• Then, h
0µ⌫ = hµ⌫ � @⇠µ
@x
⌫� @⇠⌫
@x
µ
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Choice of guage (i.e., coordinates)
• Define a trace-reversed perturbation
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and impose the Lorentz guage condition
then equation for becomes
hµ⌫ = hµ⌫ � 1
2h⌘µ⌫
@hµ⌫
@x⌫= 0
hµ⌫
⇤hµ⌫ = �16⇡GTµ⌫
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Further Properties• In vacuum (i.e., T𝜇𝜈=0),
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⇤hµ⌫ = 0
• The solution can be written in the form
¯
hµ⌫ = Re{Aµ⌫ exp(ik�x�)}
• One can choose a coordinate system by rotating so that
Aµµ = 0 (traceless), A0i = 0 purely spatial
• The wave is transverse in Lorentz gauge. Therefore the metric perturbation becomes very simple in transverse-traceless (TT) gauge. Also in TT coordinates.hµ⌫ = hµ⌫
i.e, equation for plane waves
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Effects of GWs in TT gauge• In TT guage, GWs traveling along z-direction can be written with only
two components, h+ and hx: they are called ‘plus’ and ‘cross’ polarizations
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hTTµ⌫ = h+(t� z)
0
BB@
0 0 0 00 1 0 00 0 �1 00 0 0 0
1
CCA+ h⇥(t� z)
0
BB@
0 0 0 00 0 1 00 1 0 00 0 0 0
1
CCA
• In these coordinates, the line element becomes
ds
2 = �dt
2 + (1 + h+)dx2 + (1� h+)dy
2 + dz
2 + 2h⇥dxdy
• The lengths in x- and y- directions for h+ then oscillate in the following manner
L
x
=
Zx2
x1
p1 + h+dx ⇡ (1 +
1
2h+)Lx0;
L
y
=
Zy2
y1
p1� h+dy ⇡ (1� 1
2h+)Ly0
h+ hx
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Generation of gravitational waves• The wave equation
gives formal solution of
• One can show that
hjk(t,x) =2rd2Ijk(t�r)
dt2
• Using the identities@T
tt
@t
+@T
kt
@x
k= 0, and
@
2T
tt
@t
2=
@
2T
kl
@x
k@x
l
⇤hµ⌫ = �16⇡GTµ⌫
hµ⌫(t,x) = 4
ZTµ⌫(t� |x� x
0|,x0)
|x� x
0| d
3x ⇡ 4
r
ZTµ⌫(t� |x� x
0|,x0)d3x
whereIjk =
Z⇢x
jx
kd
3x
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Order of magnitude estimates of GW amplitude
• Note: the moment of inertia tensor is not exactly the same as the quadrupole moment tensor, but in TT guage, it does not matter.
• Quadrupole kinetic energy
¨Ijk ⇠ (mass)⇥(size)2
(transit time)2 ⇠ quadrupole Kinetic E. = ✏Mc2
• In most cases, 𝝐 is small, but it could become ~0.1
hjk ⇠ 10�22�
✏0.1
� ⇣MM�
⌘⇣100Mpc
r
⌘
Qjk =
Z⇢
✓x
jx
k � 1
3r
2�jk
◆d
3x
Ijk =
Z⇢x
jx
kd
3x
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
How small is h~10-21
which was the detected amplitude of the GW150914?
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Simulation created by T. Pyle, Caltech/MIT/LIGO Lab
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Measurement of GWs: 1. Tidal forces• Equation of geodesic deviation becomes GW tidal acceleration:
d
2�x
j
dt
2= �Rj0k0x
k =1
2hjkx
k
• Riemann tensor Rj0k0 is a gravity gradient tensor in Newtonian limit
• Gravity gradiometer can be used as a gravitational wave detector
Rj0k0 = � @
2�
@x
j@x
k= �!
2GWhjk
• Tidal force can cause resonant motion of a metallic bar = bar detector
Force field
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Measurement of GWs: 2. Laser Interferometer
• Consider a simple Michelson interferometer with lx ≈ ly ≈l.
• The phase difference of returning lights reflected by x and y ends
• Toward the laser:
• Toward the photo-detector:
• For small Δ𝜙,
�� = �x
� �y
⇡ 2!0 [lx � ly
+ lh(t)]
Elaser / ei�x + ei�y
EPD / ei�x � ei�y = ei�y (ei�� � 1)
IPD
/ |ei�� � 1|2 ⇡ |��|2 ⇡ 4!20(lx � l
y
)2 + 8!20(lx � l
y
)lh(t)
Time varying part
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Astrophysical sources• Compact Binary Coalescence (neutron stars
and black holes) • Strong signal, but rare • Computable waveforms
• Continuous (~ single neutron stars) • Weak, but could be abundant • Deviation from axis-symmetry is not known
• Burst (supernovae or gamma-ray bursts) • GW amplitudes are now well known, not
frequent • Stochastic
• Superposition of many random sources • Could be useful to understand distant
populations of GW sources or early universe
Black hole binaries
Ijk =
Z⇢x
jx
kd
3x
Confirmation of GW with Binary Pulsar
• Orbit of binary neutron stars shrinks slowly
• Hulse & Taylor received Nobel Prize in 1993 for the discovery of a binary pulsar Weisberg & Taylor 2005
P=7.75 hrs a=1.6 x 106 km M1=1.4 Msun M2=1.35 Msun Time to merge=3x108 yrs
PSR 1913+16
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
About 300 million years after, the following event is expected (Movie credit: Gwanho Park [SNU])
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
The final moment of merger of black holes (Movie credit: Han-Gil Choe [SNU])
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Comparison of waveforms between NS and BH mergers: Note differences in frequencies and shape .
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Waveform tells many thingsNeutron star binary
Black hole binary
high f
low f
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
GW from a merging black hole
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LIGO-G1600341
Principles of GW DetectorImage credit: LIGO/T. Pyle
LIGO-G1600341
Gravitational Wave Detector 1990
K. Thorne, R. Weiss, R. Drever
• 2002-2010 Initial LIGO • 2015.9 ~ Advanced LIGO (10times better sensitivity)
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Simple Estimates of Sensitivity of Interferometers
• If the length resolution is 𝛌laser, detectable strain is
• However, due to quantum nature of the photons, the length resolution coud be as small as N -1/2
photons𝛌laser. Thus sentivity could reach
h ⌘ �l
l=
�laser
l=
10�6m
103m= 10�9
• Optical path length can be significantly increased by adopting optical cavity, but should be smaller than GW wavelength (~1000 km for 300 Hz)
h ⇠ �l
leff⇠ �laser
�GW⇠ 10�6m
106m= 10�12
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h ⇠ N�1/2photons
�laser
�GW
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Shot Noise• Collect photons for a time of the order of the period of GW
wave
Nphotons
=Plaser
hc/�laser
⌧ ⇠ Plaser
hc/�laser
1
fGW
⌧ ⇠ 1/fGW
• For 1W laser with λlaser=1 µm, fGW=300Hz, Nphotons=1016
• By adopting high power laser (20W for O1) and power recycling, we can reach ‘astrophysical sensitivity’ of ~10-22.
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h ⇠ �l
leff
⇠N�1/2
photons
�laser
�GW
⇠ 10�8 ⇥ 10�6m
106m= 10�20
LIGO-G1601201-v3
Other Noises• Radiation pressure noise
• Make mirrors heavier • Suspension thermal noise/ mirror coating
brownian noise • Increase beam size, monolithic
suspension structure • Seismic noise
• Multi-stage suspension, underground • Newtonian Noise
• So far difficult to avoid. • Seismic and wind measurement and
careful modeling
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
GW Events from O1/O2 • GW150914
(FAR<6x10-7 yr-1)
• LVT151012 (Candidate, FAR~0.37 yr-1)
• GW151226 (FAR<6x10-7 yr-1)
• GW170104 (<5x10-5 yr-1
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
What made these detections interesting?
• LIGO detected gravitational waves from merger of black hole binaries, instead of neutron star binaries
• Black hole mass range was quite large: GW150914 is composed of 36 and 29 times of the mass of the Sun
• Most of the known black holes are much less massive (~10 times of the sun)
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0
1
2
3
4
5
6
7
5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
Num
ber
Mass(Msun)
BHMassDistribu3on
LVT151012
GWsources
X-rayBinaries
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
What these results tell us? (Abbott et al., 2016, ApJL, 828, L22; PHYSICAL REVIEW X 6, 041015 (2016)
PRL 118, 221101 (2017))
• Proof of the existence of the black holes
• Existence of stellar mass black holes in binaries
• Individual masses in wide range (7-35 Msun)
• Formation of ~60 Msun BH
• BH appears to be much more frequent than previously thought
• Current estimation of the rate 12-210 yr-1 Gpc-1
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Summary of the GW sources• GW150914:
• First Unambiguous detection of stellar mass black holes and a BH binary
• Accurate measurement of black hole masses (within ~10%) • Higher mass of stellar mass BH than previously thought: low
metallicity environment? • GW151226:
• Lower masses than GW150914, similar to the X-ray binary BH mass
• Lower mass progenitor or high metallicity environment? • GW170104
• High mass (~50 Msun) • Spin may not be aligned
• Origin • Isolated or dynamical?
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Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
References• S. Weinberg, "Gravitation and Cosmology: Principles and applications of the general theory of
relativity", 1972, John Wiley & Sons • J. B. Hartle, "Gravity, an introduction to Einstein's General Relativity". 2003, Addison Wesley • Lecture notes by K. Thorne:
• https://www.lorentz.leidenuniv.nl/lorentzchair/thorne/Thorne1.pdf • http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/ (together with R. Blandford,
Chapters 25 & 27) • Recent LIGO papers
• Abbott et al. 2016, "Abbott et al. 2016, "Observation of Gravitational Waves from a Binary Black Hole Merger", 2016, PRL, 116, 061102
• Abbott et al., 2016, "ASTROPHYSICAL IMPLICATIONS OF THE BINARY BLACK HOLE MERGER GW150914" , ApJL, 828, L22
• Abbott et al. 2016, "GW150914: Implications for the Stochastic Gravitational-Wave Background from Binary Black Holes", PRL, 115, 131102
• Abbott et al. 2016, "GW150914: The Advanced LIGO Detectors in the Era of First Discoveries", 2016, PRL, 116, 131103
• Abbott et al., 2017, “GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence”, PRL, 118, 221101
at Re
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