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Gravitational Wave Detection #2:
How detectors work
Peter Saulson
Syracuse University
8 June 2004
Summer School on Gravitational
Wave Astronomy
1
Outline
1. How does an interferometer respond to
gravitational waves?
2. How does a resonant bar respond to
gravitational waves?
3. A puzzle: If light waves are stretched by
gravitational waves, how can we use light as
a ruler to detect gravitational waves?
8 June 2004
Summer School on Gravitational
Wave Astronomy
2
Three test masses
8 June 2004
Summer School on Gravitational
Wave Astronomy
3
Solving for variation in light
travel time: start with x arm
ds 2  c 2 dt 2  1  h11 dx 2  0
First, assume h(t) is constant during light’s
travel through ifo.
Rearrange, and replace square root with 1st two
terms of binomial expansion
1 
1 
 dt  c  1  2 h11  dx
and integrate from x = 0 to x = L:
t  h11L / 2c
8 June 2004
Summer School on Gravitational
Wave Astronomy
4
Solving for variation in light
travel time (II)
In doing this calculation, we choose coordinates
that are marked by free masses.
“Transverse-traceless (TT) gauge”
Thus, end mirror is always at x = L.
Round trip back to beam-splitter:
t  h11L / c
y-arm (h22 = - h11 = -h):
t y   hL / c
Difference between x and y round-trip times:
  2hL / c
8 June 2004
Summer School on Gravitational
Wave Astronomy
5
Multipass, phase diff
To make the signal larger, we can arrange for N
round trips through the arm instead of 1.
More on this in a later lecture.
2 NL
  h
 h stor
c
It is useful to express this as a phase difference,
dividing time difference by radian period of
light in the ifo:
  h stor
8 June 2004
2c

Summer School on Gravitational
Wave Astronomy
6
Sensing relative motions of
distant free masses
Michelson
interferometer
8 June 2004
Summer School on Gravitational
Wave Astronomy
7
How do we make the
travel-time difference visible?
In an ifo, we get a change in output power as a
function of phase difference.
At beamsplitter, light beams from the two arms
are superposed. Thus, at the port away from
laser
Eout  E0 cos 
and at the port through which light enters
Erefl  E0 sin 
8 June 2004
Summer School on Gravitational
Wave Astronomy
8
Output power
We actually measure the optical power at the
output port
Pout  Pin cos 2 
Pin
1  cos 2 

2
Note that energy is conserved:


Pout  Prefl  Pin cos 2   sin 2   Pin
8 June 2004
Summer School on Gravitational
Wave Astronomy
9
Interferometer output vs.
arm length difference
1
0.9
0.8
Output power
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
8 June 2004
-1
-0.5
0
0.5
1
1.5
Optical Path Difference, modulo n wavelengths (cm) -5
x 10
Summer School on Gravitational
Wave Astronomy
10
Ifo response to h(t)
Free masses are free to track time-varying h.
As long as stor is short compared to time scale of h(t),
then output tracks h(t) faithfully.
If not, then put time-dependent h into integral of
slide 4 before carrying out the integral.
Response “rolls off” for fast signals.
This is what is meant by interferometers being broadband detectors.
But, noise is stronger at some frequencies than others.
(More on this later.) This means some frequency
bands have good sensitivity, others not.
8 June 2004
Summer School on Gravitational
Wave Astronomy
11
Interpretation
Recall grav wave’s effect on one-way travel
time:
hL
t 
2c
Just as if the arm length is changed by a fraction
L h

L 2
In the TT gauge, we say that the masses didn’t
move (they mark coordinates), but that the
separation between them changed.
Metric of the space between them changed.
8 June 2004
Summer School on Gravitational
Wave Astronomy
12
Comparison with rigid ruler,
force picture
We can also interpret the same physics in a
different picture, using different coordinates.
Here, define coordinates with rigid rods, not
free masses.
With respect to a rigid rod, masses do move
apart.
In this picture, it is as if the grav wave exerts
equal and opposite forces on the two masses.
8 June 2004
Summer School on Gravitational
Wave Astronomy
13
Two test masses and a spring
The effect of a grav wave on this system is as if
it applied a force on each mass of
1
Fgw  mLh
2
Character of force:
Tidal (prop to L,) and gravitational (prop to m)
August 10, 1998
SLAC Summer Institute
14
Resonant-mass detector response
Impulse excites high-Q resonance
Impulse Response (N-kg/m)
7
1
x 10
0.5
0
-0.5
-1
0
0.001
0.002
0.003
200
300
Impulse Response (N-kg/m)
7
1
0.004 0.005 0.006
Time (sec)
0.007
0.008
0.009
0.01
700
800
900
1000
x 10
0.5
0
-0.5
-1
August 10, 1998
0
100
400
500
600
Time (sec)
SLAC Summer Institute
15
Strategy behind
resonant detectors
Ideally, an interferometer responds to an
impulse with an impulsive output.
Not literally, but good approximation unless signal
is very brief.
Bar transforms a brief signal into a long-lasting
signal.
As long as the decay time of the resonant mode.
Why? It is a strategy to make the signal stand
out against broad-band noise.
More on this in a later lecture.
8 June 2004
Summer School on Gravitational
Wave Astronomy
16
First generation resonant-mass
system
PZT
Low noise
amp
Rectifier or lock-in
To chart recorder
Room temperature, hrms ~ 10-16
August 10, 1998
SLAC Summer Institute
17
Second generation resonant-mass
system
Tuned
resonator
Superconducting
transducer
SQUID
amplifier
Lock-in
amplifier
To computer
T = 4 K, hrms ~ 10-18
August 10, 1998
SLAC Summer Institute
18
How are tests masses realized?
Interferometer: test masses are 10 kg (or so)
cylinders of fused silica, suspended from fine
wires as pendulums
Effectively free for horizontal motions at frequencies
above the resonant frequency of the pendulum
(~1 Hz)
This justifies modeling them as free masses.
Bar: “test masses” are effectively the two ends
of the bar.
Resonant response, due to “spring” made up of the
middle of the bar, is key.
8 June 2004
Summer School on Gravitational
Wave Astronomy
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How to find weak signals in
strong noise
Look for something that looks more like the
signal than noise, at suspiciously large level.
s1(t)
5
0
-5
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time t
60
70
80
90
100
s2(t+tau)
5
0
xcorr(s1,s2)
-5
100
0
-100
August 10, 1998
SLAC Summer Institute
20
The “rubber ruler puzzle”
If a gravitational wave stretches space, doesn’t
it also stretch the light traveling in that space?
If so, the “ruler” is being stretched by the same
amount as the system being measured.
And if so, how can a gravitational wave be
observed using light?
How can interferometers possibly work?
8 June 2004
Summer School on Gravitational
Wave Astronomy
21
Handy coordinate systems
In GR, freely-falling masses play a special role, whose
motion is especially simple.
It makes sense to use freely-falling masses to mark out
coordinate systems to describe simple problems in
gravitation.
In cosmology, we describe the expansion of the
Universe using comoving coordinates, marked by
freely-falling masses that only participate in the
Hubble expansion.
For gravitational waves, the transverse-traceless gauge
used on previous slide is marked by freely-falling
masses as well.
8 June 2004
Summer School on Gravitational
Wave Astronomy
22
Cosmological redshift
ds2  c2dt 2  R2  t dx 2  0


8 June 2004
1
 dt  c  R t  dx
1 Rt1 

0 Rt 0 
Summer School on Gravitational
Wave Astronomy
23
Heuristic interpretation
Cosmological redshift:
Wavefronts of light are adjacent to different
comoving masses. As Universe expands, each
wavefront “thinks” it is near a non-moving mass. As
separations between masses grow, separations
between wavefronts must also grow. Hence,
wavelength grows with cosmic scale factor R(t).
Especially vivid if we were to imagine the Universe
expanded from R(t0) to R(t1) in a single step function.
All light suddenly grows in wavelength by factor R(t1)/
R(t0).
8 June 2004
Summer School on Gravitational
Wave Astronomy
24
A gravitational wave
stretches light
Imagine many freely-falling masses along arms of
interferometer.
Next, imagine that a step function gravitational wave,
with polarization h+, encounters interferometer.
Along x arm, test masses suddenly farther apart by
(1+h).
Wavefronts near each test mass stay near the mass. (No
preferred frames in GR!)
The wavelength of the light in an interferometer is
stretched by a gravitational wave.
8 June 2004
Summer School on Gravitational
Wave Astronomy
25
Are length changes real?
Yes.
The Universe does actually expand.
Interferometer arms really do change length.
We could (in principle) compare arm lengths to
rigid rods.
It is important not to confuse our coordinate
choices with facts about dynamics.
“Eppur si muove”
8 June 2004
Summer School on Gravitational
Wave Astronomy
26
OK, so how do interferometers
actually work?
Argument above only proves that there is no
instantaneous response to a gravitational
wave.
Wait.
Arm was lengthened by grav wave. Light
travels at c. So, light will start to arrive late, as
it has to traverse longer distance than it did
before the wave arrived.
Delay builds up until all light present at wave’s
arrival is flushed out, then delay stays
constant at   h(2NL/c).
8 June 2004
Summer School on Gravitational
Wave Astronomy
27
New light isn’t stretched, so it
serves as the good ruler
In the end, there is no puzzle.
The time it takes for light to travel through
stretched interferometer arms is still the key
physical concept.
Interferometers can work.
8 June 2004
Summer School on Gravitational
Wave Astronomy
28