Space based Gravitational Wave Observations I – M Hewitson from Lake Como School of Advanced Studies

M Hewitson
Institut für Gravitationsphysik
der Leibniz Universität
Hannover
SIGRAV
May 2018
Albert Einstein Institute
Max Planck Institute for Gravitational Physics and
Leibniz Universität Hannover
Space based
Gravitational Wave
Observations
LISA and its pathﬁnder

M Hewitson, LISA Detector, SIGRAV, May 2018
Martin Hewitson: Who he?
• Undergrad and PhD in
Glasgow
• At AEI in Hannover since 2004
• Worked on:
• GEO600 calibration and
commissioning
• LISA Pathfinder Data Analysis
and operations
• LISA mission formulation and
instrumentation
2

Overview of lectures
• Lecture 1
• why low(er) frequency?
• low frequency, compare to ground based
• measurement principle for LISA
• discussion of noise sources
• Lecture 2
• TDI: the basics
• Building the LISA noise curve
• Lecture 3
• LPF: what did we test, and why?
3

Why go to low frequency?
• why go to low frequency?
• Larger mass systems merge at lower frequency
• Schwarzschild radius:
• Orbital frequency for binary system: innermost
stable circular orbit
4
r =
2Gm
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fISCO /
1
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GW Spectrum
5
http://rhcole.com/apps/GWplotter/
small
bighuge
= 3000 km
= 300 million km
= 30 ⇥ 1012
km
10 M☉
106 M☉>109 M☉

Compare space and ground strain
6
LISA
aLIGO

Challenges of observing at low frequency
• what limits ground based detectors at low frequency?
• Isolation:
• suspensions on ground, free-fall in space
• what limits at low frequencies?
• on ground: external displacements, external forces
• in space: small forces on test masses
• smaller, lighter
• Interferometry challenges
• temperature, thermo-mechanical stability
• readout noise
7

Remember the goal
• want to measure the relative proper motion of
free-falling test masses
• infer from that the effect of gravitational waves
8

Isolation on ground
• Complex pendulum style
isolation for test masses
• Get 1/f2 isolation from
each stage
9
Brett Shapiro, G1400964-v3
~free-fall along beam axis
at frequencies above the
pendulum resonance

Typical ground motion
10

Isolation on ground
11

aLIGO limiting noise sources (design)
12

ET low frequency Interferometer
13

Low frequency disturbances
• Going to lower frequency (mHz) is not feasible
on ground
• Go to space!
• What is the measurement principle in space?
• What are the challenges of low-frequency
observations in space?
14

Measurement principle: fundamentals
• Aim is to measure the relative motion of free-
falling test masses
• On ground test masses are suspended, but are
‘free-falling’ along the measurement axis and
above the suspension resonance
15
test mass
in free-fall

Measurement principle: fundamentals
• Compare different pairs of test masses
arranged orthogonally to combat technical
(frequency) noise
16

Ground based interferometers
17
Compare length changes of each arm
frequency noise is common to both
Laser wavelength (frequency) is our ‘ruler’
frequency noise is indistinguishable from GW signal
Actually use ~equal arm-
length interferometer,
insensitive to frequency noise

Length stability
• Advanced suspensions produce very quiet mirrors 
• Advanced optics and readout result in
displacement sensitivity < 10-19 m/√Hz at 100Hz!
• Achievable because mirrors are held at operating point
• Actuation noise is all at lower frequency
18

aLIGO displacement sensitivity
19

Homodyne versus heterodyne
• why can’t we lock on a fringe in space?
• orbital motion of free-falling bodies produces large
relative motions
• would need very strong (and noisy!) feedback forces to
maintain ‘fixed' arm-lengths
• this would need to be a mix of force on S/C (thrusters) and
force on test masses
• Use another concept instead: heterodyne
interferometry with phase tracking
• can measure over many fringes
20

The LISA Concept
21

The LISA Concept
• Probe the change in proper time between pairs
of free-falling test masses caused by GWs
• Free-falling test masses are housed in drag-
free spacecraft separated by a few million km
• Proper time is inferred by the time of flight of
photons exchanged between the satellites
• We have multiple transponder links from
which we can form Michelson-like signals
21

LISA Science
22

LISA is a mature concept
• 1993 M3 proposal for 4 S/C ESA/NASA
collaborative mission
• 1995 LISA selected as ESA Cornerstone
• 1997 3 S/C ESA/NASA LISA 
appears
• 2005-2011 Joint ESA-NASA Mission
Formulation study
• 2012-13 Mission reformulated as ESA-led eLISA
(evolving LISA)
• 2013 Gravitational Universe selected as L3
science theme
• 2017 LISA proposal selected as ESA’s L3
mission
23

LISA Proposal for ESA’s L3 mission slot
24
https://arxiv.org/abs/1702.00786
www.lisamission.org
Submitted
January 2017

20th June 2017
25
LISA selected as ESA's L3 mission!
“The LISA trio of satellites to detect gravitational waves from
space has been selected as the third large-class mission in
ESA’s Science programme, while the Plato exoplanet hunter
moves into development.
These important milestones were decided upon during a
meeting of ESA’s Science Programme Committee today, and
ensure the continuation of ESA’s Cosmic Vision plan through
the next two decades.”
http://www.esa.int/Our_Activities/Space_Science/

LISA Measurement concept: Challenges
• protect test masses from external forces which
move test masses away from perfect geodesics
• e.g. solar wind 
• can’t do interferometry as on ground (locked
cavities)
• high dynamics
• beam divergence means most of the light is lost
• need a transponder scheme
26

Test mass free-fall in space
• Shield free-falling test mass (TM) in a spacecraft
(S/C)
• Need to ensure local forces between S/C and
TM are ‘low enough’
• less relevant at higher frequencies on ground as they
fall off with 1/f2
• S/C has no mechanical connection to the TM
• drag-free S/C control
• issues of charge accumulation on the TM (see later)
27

Measurement principle
• make 3 measurements:
• TM to S/C
• S/C to S/C (contains the GW signal!)
• S/C to TM
• Combine to get TM to TM
28
~million km
m1 m2s1

What can possibly go wrong?
29
~million km
m1 m2s1
S/C are
‘shaking’
thrusters for drag-free control
external forces (e.g. solar wind)
local forces
disturb TM
magnetic force
thermal
residual gas
force gradients

Lasers over millions of km
30
w(x) defined where amplitude falls to 1/e ~ 0.37

• laser beam with gaussian profile diverges as  
 
30
w(x) = w0
s
1 +
✓
z
zR
◆2
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beam width
Rayleigh range
beam waist diameter

 
• where the Rayleigh range is given by  
 
30
w(x) = w0
s
1 +
✓
z
zR
◆2
beam width
Rayleigh range
beam waist diameter
zR =
⇡w2
0
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wavelength

 
 
• Far from the beam waist (large z), we have 
30
w(x) = w0
s
1 +
✓
z
zR
◆2
beam width
Rayleigh range
beam waist diameter
zR =
⇡w2
0
wavelength
w(x) ⇡
z
⇡w0<latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit>

 
 
• Far from the beam waist (large z), we have 
• For 2.5 million km, and laser wavelength of 1064 nm 
 
30
w(x) = w0
s
1 +
✓
z
zR
◆2
beam width
Rayleigh range
beam waist diameter
zR =
⇡w2
0
wavelength
w(x) ⇡
z
⇡w0<latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit>
w(x) =
2.5 ⇥ 109
[m] ⇥ 1064 ⇥ 10 9
[m]
⇡ 13.4[cm]
⇡ 6.4 km
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31
transmit aperture receive aperture

• if we send out a 2W beam from S/C 1, how much power do we intercept
at S/C 2?
• depends on:
• distance, receiving aperture (telescope), etc
31

at S/C 2?
• depends on:
• calculations on previous page assume we can get all the light out of the
transmitting telescope
• in practice, not practical, requires too large a telescope
31

at S/C 2?
• depends on:
• calculations on previous page assume we can get all the light out of the
transmitting telescope
• in practice, not practical, requires too large a telescope
• Rather we should optimise transmit beam properties to maximise
intensity over receiving telescope
31

Optimising beam waist and received power
32
[1] Holmes D, Avizonis P and Wrolstad K 1970 Appl. Opt. 9 2179–80
[2] S Barke et al 2015 Class. Quantum Grav. 32 095004

• According to [1], the intensity of light received on-axis from a
focused, apertured gaussian beam can be expressed as 
 
32
I0 =
2P
⇡!2
"
exp
⇥
(a/!)2
⇤
exp
⇥
(b/!)2
⇤ 2
+ 4 exp
⇥
(a2
+ b2
)/!2
⇤
sin2
⇥
⇡(b2
a2
)(z R)/2 zR
⇤
( z/⇡!2)2 + [(z R)/R]
2
#
,
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• According to [1], the intensity of light received on-axis from a
focused, apertured gaussian beam can be expressed as 
 
• Ouch… apply some assumptions:
• The arm-length is very long compared to any beam properties, so L→∞
• We set  
 
where dtel is the diameter of the transmitting telescope, and α is the waist
radius in units of the telescope radius
• The outer radius of the annulus, b, is set to dtel/2
• The intensity is considered at the far SC, i.e., where z=L
32
I0 =
2P
⇡!2
"
exp
⇥
(a/!)2
⇤
exp
⇥
(b/!)2
⇤ 2
+ 4 exp
⇥
(a2
+ b2
)/!2
⇤
sin2
⇥
⇡(b2
a2
)(z R)/2 zR
⇤
( z/⇡!2)2 + [(z R)/R]
2
#
,
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! = ↵
dtel
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sha1_base64="/KjNQ+C6Cjl5OWTL+p4gNaZ6F9Q=">AAACDXicdVA9SwNBEN3z2/gVtbRZjIJVuItKkkIQbSwjGBVyIcxt5pLF3btjd08Ix/0BG/+KjYUitvZ2/hs3H4KKPhh4vDfDzLwgEVwb1/1wpqZnZufmFxYLS8srq2vF9Y1LHaeKYZPFIlbXAWgUPMKm4UbgdaIQZCDwKrg5HfpXt6g0j6MLM0iwLaEX8ZAzMFbqFHf8WGIP6BH1QSR98EMFLOt2Ml9JalDkeVbJO8WSW3br+9WDfWrJCJbU67WKe0i9iVIiEzQ6xXe/G7NUYmSYAK1bnpuYdgbKcCYwL/ipxgTYDfSwZWkEEnU7G32T012rdGkYK1uRoSP1+0QGUuuBDGynBNPXv72h+JfXSk1Ya2c8SlKDERsvClNBTUyH0dAuV8iMGFgCTHF7K2V9sHkYG2DBhvD1Kf2fXFbKnlv2zg9KxyeTOBbIFtkme8QjVXJMzkiDNAkjd+SBPJFn5955dF6c13HrlDOZ2SQ/4Lx9AsUqnAc=</latexit>

Far-field intensity
• Applying the assumptions gives an expression
for the far-field intensity:
33
Irec =
⇡Pteld2
tel
2L2 2
↵2
exp
⇥
2/↵2
⇤
exp
⇥
1/↵2
⇤
1
2
,
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maximise this by choosing beam waist

Received power
• Received power depends on
• far-field intensity
• receiving aperture (telescope)
34
Prec = ⇡
✓
Dtel
2
◆2
Irec,
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Some typical values for LISA:
L = 2.5 Gm
Pout = 1.25 W
Dtel = 30 cm
Irec = 10.25 [nW/m^2]
Prec = 507.31 [pW]

So where are we?
35

So where are we?
• We use lasers to measure relative motion of
35

So where are we?
• But received power is too low to do a round-
trip measurement
35

So where are we?
trip measurement
• Homodyne detection is challenging due to
large dynamic range
35

So where are we?
trip measurement
• Homodyne detection is challenging due to
large dynamic range
• We need another scheme:
• transponder-like scheme
35

Transponder-like scheme in LISA
• send out one beam to far S/C
• in far S/C compare received beam to local beam
• control difference of the two laser frequencies
• Laser 2 is now locked to Laser 1
• f2-f1 is kept constant
• Far S/C returns this Laser 2 to S/C 1
• measure phase difference between two lasers
• encodes GW signal + S/C motion + frequency noise of Laser 1
36
f2f1
Δs
frequency
lock
polarisation optics omitted

Full constellation
• One master laser
locked to, e.g., a
cavity to stabilise its
frequency
• All other lasers locked
to this (in a chain)
37
Barke, Inter-Spacecraft Frequency Distribution, PhD Thesis 2015

Features of this approach
• Transmitted lasers are subject to doppler shifts
due to orbital motions
• Line-of-sight relative velocities of S/C impose
frequency shifts on received light
38
f = f0
v
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relative velocity
along line-of-sight
laser frequency
(emitted)
speed of light
frequency
shift

Orbital motions
39

Relative velocities
40

Doppler Shifts
41

Side bar: Heterodyne interferometry
• how does it work?
42
f1
f2
f2-f1
Phase of beat signal
depends on path
lengths

Interference
43
f1
f2
f2-f1
50/50 beam splitter
50% reflective
coating
2 light fields
combine at a
beam splitter

Detection
44
f1
f2
f2-f1
Note: carrier frequency is
neglected throughout. It is THz,
and can’t be detected. We only
detect average light-fields.
photodetector
Sum amplitudes
of light field!
photodetector is
a ‘square law’
detector
⌫ =
c
=
299792458
⇥
m s 1
⇤
1064 [nm]
=0.28 THz<latexit sha1_base64="MZ8LFLu90Nsn0FYWyrkumiGQd34=">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</latexit><latexit sha1_base64="MZ8LFLu90Nsn0FYWyrkumiGQd34=">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</latexit><latexit sha1_base64="MZ8LFLu90Nsn0FYWyrkumiGQd34=">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</latexit><latexit sha1_base64="MZ8LFLu90Nsn0FYWyrkumiGQd34=">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</latexit>

Do the math…
45
s1 = a1e(i!1t+ 1)
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s2 = a2e(i!2t+ 2)
<latexit sha1_base64="EsfXMQd+1iVZYBQZcTTxjkU+sTk=">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</latexit><latexit sha1_base64="k/5dNYhxTx5NlbcYMGDe2I8yfjk=">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</latexit><latexit sha1_base64="k/5dNYhxTx5NlbcYMGDe2I8yfjk=">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</latexit><latexit sha1_base64="xIw4UzqXq2Ak9bUNHlOCdlYGrSA=">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</latexit>
Ipd / |s1 + s2|2
<latexit sha1_base64="mU0tU3vIIZWgAxH+2NRTo/0kjv4=">AAACCXicdVA9SwNBEJ2LXzF+RS1tFqMgCOEuKTRd0Ea7COYDknjsbTbJkr27ZXdPCJe0Nv4VGwuDWAn+Azt/iL2bRDGKPhh4vDfDzDxPcKa0bb9Zibn5hcWl5HJqZXVtfSO9uVVRYSQJLZOQh7LmYUU5C2hZM81pTUiKfY/Tqtc7HfvVayoVC4NL3Re06eNOwNqMYG0kN43O3bghfSRaQ9QQMhQ6RAPlOugQKTc3uMq56YyTtSdA9jcp5Ar5Qh59WZni3vvoGQBKbvq10QpJ5NNAE46Vqju20M0YS80Ip8NUI1JUYNLDHVo3NMA+Vc148skQ7RulhdqhNBVoNFFnJ2LsK9X3PdPpY91Vv72x+JdXj3T7uBmzQESaBmS6qB1xZN4dx4JaTFKied8QTCQztyLSxRITbcJLzYbwP6nkso6ddS6cTPEEpkjCDuzCAThwBEU4gxKUgcAN3MEDjKxb6956tJ6mrQnrc2YbfsB6+QCQ7Jvk</latexit><latexit sha1_base64="WVpchXEjEv9JslRxBGlU9txHqNI=">AAACCXicdVDLSsNAFJ34rK2PqEs3g1UQhJKkC+2u6EZ3FewDmhom02k7dJIMM5NCSbt146+4cWERt/0Dd36Irp22ilX0wIXDOfdy7z0+Z1Qqy3o1FhaXlldWU2vpzPrG5pa5vVORUSwwKeOIRaLmI0kYDUlZUcVIjQuCAp+Rqt89n/jVHhGSRuG16nPSCFA7pC2KkdKSZ8JLL3FFAHlzCF0uIq4iOJCeDY+h9JzBjeOZWTtnTQGtb1JwCvlCHn5Z2eLB22jcy7yXPPPFbUY4DkioMENS1m2Lq0aChKKYkWHajSXhCHdRm9Q1DVFAZCOZfjKEh1ppwlYkdIUKTtX5iQQFUvYDX3cGSHXkb28i/uXVY9U6bSQ05LEiIZ4tasUM6ncnscAmFQQr1tcEYUH1rRB3kEBY6fDS8yH8TypOzrZy9pWdLZ6BGVJgD+yDI2CDE1AEF6AEygCDW3APHsHIuDMejCfjeda6YHzO7IIfMMYfiYydXg==</latexit><latexit sha1_base64="WVpchXEjEv9JslRxBGlU9txHqNI=">AAACCXicdVDLSsNAFJ34rK2PqEs3g1UQhJKkC+2u6EZ3FewDmhom02k7dJIMM5NCSbt146+4cWERt/0Dd36Irp22ilX0wIXDOfdy7z0+Z1Qqy3o1FhaXlldWU2vpzPrG5pa5vVORUSwwKeOIRaLmI0kYDUlZUcVIjQuCAp+Rqt89n/jVHhGSRuG16nPSCFA7pC2KkdKSZ8JLL3FFAHlzCF0uIq4iOJCeDY+h9JzBjeOZWTtnTQGtb1JwCvlCHn5Z2eLB22jcy7yXPPPFbUY4DkioMENS1m2Lq0aChKKYkWHajSXhCHdRm9Q1DVFAZCOZfjKEh1ppwlYkdIUKTtX5iQQFUvYDX3cGSHXkb28i/uXVY9U6bSQ05LEiIZ4tasUM6ncnscAmFQQr1tcEYUH1rRB3kEBY6fDS8yH8TypOzrZy9pWdLZ6BGVJgD+yDI2CDE1AEF6AEygCDW3APHsHIuDMejCfjeda6YHzO7IIfMMYfiYydXg==</latexit><latexit sha1_base64="peDpQyYi9p89+s+4G5zmJJIP9BU=">AAACCXicdVDLSsNAFJ34rPUVdelmsAiCUJJ0od0V3eiugn1AE8NkMmmHTpJhZiKUtFs3/oobF4q49Q/c+TdOH2IVPXDhcM693HtPwBmVyrI+jIXFpeWV1cJacX1jc2vb3NltyjQTmDRwylLRDpAkjCakoahipM0FQXHASCvon4/91i0RkqbJtRpw4sWom9CIYqS05Jvw0s9dEUMejqDLRcpVCofSt+ExlL4zvHF8s2SXrQmg9U2qTrVSrcAvqwRmqPvmuxumOItJojBDUnZsiysvR0JRzMio6GaScIT7qEs6miYoJtLLJ5+M4KFWQhilQlei4ESdn8hRLOUgDnRnjFRP/vbG4l9eJ1PRqZfThGeKJHi6KMoY1O+OY4EhFQQrNtAEYUH1rRD3kEBY6fCK8yH8T5pO2bbK9pVdqp3N4iiAfXAAjoANTkANXIA6aAAM7sADeALPxr3xaLwYr9PWBWM2swd+wHj7BMZ8mRU=</latexit>

Do the math…
45
s1 = a1e(i!1t+ 1)
s2 = a2e(i!2t+ 2)
Ipd / |s1 + s2|2
Ipd / |a1|2
+ |a2|2
+ 2|a1||a2| cos ((!2 !1)t + )<latexit sha1_base64="S2Hhk4/+IIjDEooVIkJU4lTn0dY=">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</latexit><latexit sha1_base64="Gu1EdMiggggElKTntSga7Byy/vo=">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</latexit><latexit sha1_base64="Gu1EdMiggggElKTntSga7Byy/vo=">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</latexit><latexit sha1_base64="9kY5xf0tTjSlns4PUe0j9scsxwI=">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</latexit>

Do the math…
45
s1 = a1e(i!1t+ 1)
s2 = a2e(i!2t+ 2)
Ipd / |s1 + s2|2
Ipd / |a1|2
+ |a2|2
+ 2|a1||a2| cos ((!2 !1)t + )<latexit sha1_base64="S2Hhk4/+IIjDEooVIkJU4lTn0dY=">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</latexit><latexit sha1_base64="Gu1EdMiggggElKTntSga7Byy/vo=">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</latexit><latexit sha1_base64="Gu1EdMiggggElKTntSga7Byy/vo=">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</latexit><latexit sha1_base64="9kY5xf0tTjSlns4PUe0j9scsxwI=">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</latexit>
d.c. power

Space based Gravitational Wave Observations I – M Hewitson from Lake Como School of Advanced Studies

Space based Gravitational Wave Observations I – M Hewitson from Lake Como School of Advanced Studies

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Space based Gravitational Wave Observations I – M Hewitson from Lake Como School of Advanced Studies