of 17
Detecting continuous gravitational waves with superfluid
4
He
S. Singh,
1, 2,
L.A. De Lorenzo,
3
I. Pikovski,
2, 4
and K.C. Schwab
3
1
Department of Physics, College of Optical Sciences and B2 Institute,
University of Arizona, Tucson, Arizona 85721, USA
2
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
3
Applied Physics, California Institute of Technology, Pasadena, CA 91125 USA
4
Department of Physics, Harvard University, Cambridge, MA 02138, USA
(Dated: June 17, 2016)
Direct detection of gravitational waves is opening a new window onto our universe. Here, we
study the sensitivity to continuous-wave strain fields of a kg-scale optomechanical system formed by
the acoustic motion of superfluid helium-4 parametrically coupled to a superconducting microwave
cavity. This narrowband detection scheme can operate at very high
Q
-factors, while the resonant
frequency is tunable through pressurization of the helium in the 0.1-1.5 kHz range. The detector can
therefore be tuned to a variety of astrophysical sources and can remain sensitive to a particular source
over a long period of time. For reasonable experimental parameters, we find that strain fields on the
order of
h
10
23
/
Hz are detectable. We show that the proposed system can significantly improve
the limits on gravitational wave strain from nearby pulsars within a few months of integration time.
I. INTRODUCTION
The recent detection of gravitational waves (GW)
marks the beginning of gravitational wave astronomy
1
.
The first direct detection confirmed the existence of grav-
itational waves emitted from a relativistic inspiral and
merger of two large black holes, at a distance of 400
M
parsecs (pc). Indirect evidence for gravitational radia-
tion was previously attained by the careful observation
since 1974 of the decay of the orbit of the neutron star
binary system PSR B1913+16 at a distance of 6.4 kpc,
which agrees with the predictions from general relativ-
ity to better than 1%
2
. In this paper, we discuss the
potential to use a novel superfluid-based optomechani-
cal system as a tunable detector of narrow-band grav-
itational wave sources, which is well suited for probing
nearby pulsars at a distance of less than 10kpc. As we
discuss below, in the frequency range exceeding
500 Hz,
this novel scheme has the potential to reach sensitivities
comparable to Advanced LIGO.
FIG. 1.
Left:
Schematic of the proposed gravity wave sensor
based on acoustic modes of superfluid helium. Two cylindrical
geometries considered here are Gen1 (radius
a
= 11cm, length
L
= 50cm, mass
M
= 2
.
7kg) and Gen2 (
a
= 11cm,
L
= 3m,
M
= 16kg).
Right:
Prototype of the detector with
a
=
1
.
8cm,
L
= 4cm,
M
= 6g and resonant frequency 10 kHz.
The GW detector under consideration is formed by
high-
Q
acoustic modes of superfluid helium parametri-
cally coupled to a microwave cavity mode in order to de-
tect small elastic strains. This setup was initially studied
in Ref.
3
, and is shown in fig. 1. The helium detector ef-
fectively acts as a Weber bar antenna
4
for gravitational
waves, but with two important differences. Firstly, the
Q/T
-factor of the helium is expected to be much larger
than that of metals, where
Q
is the acoustic quality fac-
tor, and
T
is the mode temperature. Secondly, the acous-
tic resonance frequency can be changed by up to 50% by
pressurization of helium without affecting the damping
rate, making the detector both narrowband and tunable.
The power spectrum of gravitational waves is expected
to be extremely broad and is estimated to range from
10
16
to 10
3
Hz
5–7
for known sources. Ground-based
optical interferometers (such as LIGO, Virgo, GEO,
TAMA) allow for a broad-band search for gravitational
waves in the frequency range 10 Hz - 1 kHz. These de-
tectors are expected to be predominantly sensitive to the
chirped, transient, GW impulse resulting from the last
moments of coalescing binaries involving compact objects
(black holes(BH) and/or neutron stars (NS))
8
. Space-
based interferometric detectors can in principle be sensi-
tive to lower frequency gravitational waves, as they are
not limited by seismic noise
9
.
Unlike broadband impulse sources, rapidly rotating
compact objects such as pulsars are expected to generate
highly coherent, continuous wave gravitational wave sig-
nals due to the off-axis rotating mass, with frequencies
spanning from
1 kHz for millisecond pulsars (MSPs) in
binaries, to 1 Hz for very old pulsars
5,10–13
. Given the
unknown mass distribution of the pulsar, one can only
estimate the strain field here at earth. However, several
mechanisms give upper bounds to the strength of gravi-
tational waves on earth. One such limit is the “spin down
limit”, which is given by the observed spin-down rate of
the pulsar, and the assumption that all of the rotational
kinetic energy which is lost is in the form of gravitational
waves
14
. Another limit is given by the yield strength of
the material which makes up the neutron star, and how
arXiv:1606.04980v1 [gr-qc] 15 Jun 2016
2
much strain the crust can sustain before breaking apart
due to centripetal forces
15
. The presence of strong mag-
netic fields indicate a potential mechanism for producing
and sustaining such strains due to deformation of the
neutron star
16
. However, without knowing the strength
and direction of the internal magnetic fields in a pulsar,
it is difficult to estimate a lower limit on the size of grav-
itational wave signal. The measurement of gravitational
waves from pulsars would therefore give us crucial infor-
mation about the interior of neutron stars.
Since pulsars should emit continuous and coherent
gravity waves at specific and known frequencies, we can
use a narrowband detector and integrate the signal for
long times, averaging away the incoherent detector noise.
We show that for reasonable parameters, the super-
fluid helium detector can approach strain sensitivities
of 1
5
×
10
23
/
Hz at around 1 kHz, depending on
the size and
Q
factor of the detector. Pulsar frequen-
cies are observed to vary slightly due to random glitches
f/f
10
6
10
11
(older, millisecond pulsars be-
ing more stable)
17
, and due to the motion of the earth
around the sun and resulting doppler frequency shifts.
The tunability of the acoustic resonance will be essential
to track these shifts during long detection integration
times. Simultaneous monitoring of the targeted pulsar
electromagnetically can facilitate the required precision
frequency tracking. The frequency agility can also allow
for using the same acoustic resonator to look for signals
from multiple pulsars in a similar frequency range.
Recent measurements with LIGO and Virgo have un-
successfully searched for the signals from 179 pulsars and
have limited the strain field
h
.
10
25
for most pul-
sars after nearly a year of integration time
18
. In a par-
allel development, hundreds of new pulsars have been
discovered in the last few years by analyzing Gamma-
Ray sources observed by the Fermi Large Area Telescope
(Fermi-LAT), some less than 0.5 kpc from earth
19,20
.
Together, these developments are signaling a promising
path towards gravitational wave astronomy of pulsars.
This paper is organized as follows. We start with an
overview of continuous gravitational waves from pulsars
to get an estimate for the strains produced on earth in
Section II. We then describe the superfluid helium de-
tector and show how it functions as a detector for grav-
itational waves in Section III. In Section IV, we provide
the detection system requirements. We then compare
this detector with other functional gravity wave detec-
tors, and show the key fundamental differences between
these detectors and our proposed detector in Section V.
Finally, we conclude with a brief summary of the key fea-
tures of this detector and outlook in Section VI. A review
of the relevant concepts and derivations are relegated to
the appendices for the interested reader.
II. SOURCES OF CONTINUOUS
GRAVITATIONAL WAVES
The generation of gravitational waves can be studied
by considering the linearized Einstein equations in the
presence of matter
21
. The computations are similar to
the analogous case in electromagnetism
22
, see Appendix
A for details. However, in the absence of gravitational
dipoles, a quadrupole moment
Q
ij
is necessary to source
gravitational waves. The emitted power of gravitational
waves is found to be
23
P
=
G
5
c
5
...
Q
ij
...
Q
ij
,
(1)
i.e.
it depends on the third time derivative of the
quadrupole moment of the system, where
Q
ij
:=
ρ
body
x
i
x
j
dV
for a body of density
ρ
.
In the far-field limit where size of the source (
GM/c
2
)

wavelength of gravity wave (
c/ω
)

distance to detec-
tor (
d
), the gravitational metric perturbation becomes
h
ij
=
2
G
c
4
d
̈
Q
ij
,
(2)
where
h
is the gravitational perturbation tensor in
transverse-traceless gauge. Since
G/c
4
10
44
N s
4
/
kg
2
,
one needs events with relativistic changes in mass
quadrupole moment to have a measurable source of grav-
itational radiation on earth. As an estimate, if all the ob-
served slowdown of the Crab pulsar was converted into
gravitational radiation, the power would correspond to
P
4
.
5
×
10
31
W (10
5
times the electromagnetic radia-
tion power from the sun)
24
. However, at a distance 2 kpc
away from the pulsar (distance to earth), the power flux
is 10
9
W
/
m
2
and the metric perturbation is
h
10
24
.
Even though the power flux is macroscopic and easily de-
tectable in other forms (acoustic, electromagnetic, etc.),
the resulting strain is very small due to the remarkably
high impedance of space-time. This is at the heart of the
difficulty with laboratory detection of gravity waves.
The several astrophysical candidates for gravitational
waves considered so far can be broadly classified into
three categories: stochastic background, broadband im-
pulses, and continuous sources
25,26
. It is estimated that
there will be a broadband background of gravitational
waves from the expansion of the early universe. Further-
more, there is a low frequency stochastic background due
to gravitational waves emitted by masses moving in the
galaxies. Impulse sources could stem from supernovae or
mergers of compact objects. The latter is the primary
source being searched for by most ground based detec-
tors, and was recently observed by the LIGO detectors
1
.
Lastly, continuous gravitational waves can be expected
from stellar binaries (albeit at very low frequencies), or
from pulsars. We now discuss the generation of gravita-
tional waves from asymmetric pulsars and the limits on
the signal set on earth.
Estimates of gravitational radiation from pulsars is an
active area of theoretical research that goes back to early
3
observations of pulsars
13
. The mechanism for gravita-
tional wave generation is assumed to be an asymmet-
ric mass distribution. Several mechanisms are proposed
for the deviation from axial symmetry in mass distribu-
tion, for example magnetic deformations, star quakes or
instabilities due to gravitational or viscous effects
17,27
.
However, due to the unknown equation of state, there
is significant variability in estimates of mass asymmetry
and thus gravitational wave strain from pulsars. Null re-
sults from measurements of GW strain from pulsars have
already put limits on the equation of state
15
.
Assuming that the emission of gravitational waves is a
contributing mechanism towards the observed slowdown
of pulsars enables us to set upper bounds on the GW
metric strain here on earth. We now estimate the grav-
itational wave perturbation strain from measured spin-
down rates and briefly discuss the validity of this limit.
We then present relevant numbers for a few millisecond
pulsars (MSPs) of interest for our detector. Details about
these derivations and typical parameters for other pulsars
of interest are presented in Appendix B.
For an ellipsoidal pulsar rotating about the
z
-axis with
frequency
ω
p
, the two polarizations of
h
are given by
h
+
=
4
G
c
4
d
I
zz
ω
2
p
cos 2
ω
p
t,
(3)
h
×
=
4
G
c
4
d
I
zz
ω
2
p
sin 2
ω
p
t,
(4)
where
I
zz
is the moment of inertia along the
z
-axis and

characterizes the mass quadrupole ellipticity (

= (
Q
xx
Q
yy
)
/I
zz
). Thus, the strain changes at twice the rotation
frequency of the pulsar. The energy flux for a continuous
GW of polarization
A
from a pulsar source is given by
s
A
=
c
3
16
πG
̇
h
A
(
t
)
2
,
(5)
where
A
∈{
+
,
×}
and the bar indicates time-averaging.
Typical neutron stars have mass 1-1.5
M
(where
M
= 2
×
10
30
kg is the solar mass) and have a radius of
around 10 km. Using these values, the moment of iner-
tia amounts to 10
38
kg-m
2
, the estimate used in previous
GW searches, c.f. Ref.
18
. The ellipticity parameter is
estimated by assuming that the observed slow-down rate
( ̇
ω
p
) of a pulsar is entirely due to emission of gravita-
tional waves. This estimate is then used to compute the
upper-limit estimate for gravitation perturbation strain
known as the spin down strain,
h
sd
=
4
G
c
4
d
I
zz
ω
2
p
=
5
GI
zz
̇
ω
p
2
c
3
d
2
ω
p
.
(6)
From Eq. 6, it is clear that given the values of distance
(
d
), rotational frequency (
ω
p
) and spin-down frequency
( ̇
ω
p
) from astronomical observations, we can put limits
on gravitational wave strains due to pulsars.
While
h
sd
is a useful first-principles upper limit, it
over-estimates the strength of gravitational waves, par-
ticularly from young pulsars that are highly active elec-
tromagnetically. This has already been confirmed by
braking index measurements
28,29
and the negative results
from recent GW detector data
18
. However, long-lived
and stable millisecond pulsars (
̇
ω
p
/
2
π <
10
14
Hz s
1
)
such as the ones considered here have been proposed as
likely sources of continuous gravity waves
30
. The fre-
quency stability and relatively low magnetic field indi-
cate that unlike young pulsars like Crab and Vega, the
dominant spin down mechanism in MSPs is more likely
to be quadrupolar gravitational radiation.
We can also set limits on GW generation mechanism
by considering specific models of the interior of neutron
stars, as discussed in ref.
27
, and reviewed briefly in Ap-
pendix B. Assuming standard nuclear matter and break-
ing strain for elastic forces, the ellipticity sustained can
be limited to less than 6
×
10
7
, irrespective of the physics
leading to deformations
15,31
. This can also be used to
evaluate the GW strain amplitude limit
h
n
. Since the
strain limits
h
sd
and
h
n
come from different physics
(conservation of angular momentum and balancing forces
in stellar interior), we use the lower of the two as the up-
per limit for metric strain.
Table IV details parameters for pulsars of interest with
rotational frequency higher than 500 Hz, along with cur-
rent limitations on GW strain from LIGO+VIRGO col-
laboration. Theoretical estimates of metric strain assum-
ing spin-down limit, and elastic crust breakdown limit on
ellipticity (

= 6
×
10
7
) from Ref.
15
are also given. Table
IV also gives the strain estimate set by the helium detec-
tor outlined in figure 1 (Gen1) that we will discuss in
detail in the following sections. Several of these pulsars
were discovered recently by analyzing gamma-ray sources
from Fermi-LAT. The number of known fast spinning
pulsars is expected to grow significantly as more sources
are discovered and analyzed. Furthermore, there is grow-
ing evidence that the GeV excess emission in our galactic
center is in fact due to hundreds of unresolved MSPs and
not from dark-matter annihilation
33,34
.
Since the strain due to gravity waves from pulsars is
expected to be very small but coherent, one needs to
integrate the signal from these detectors for a long time
(typically several days, the last result being a compilation
4
Pulsar
ω
p
/
2
π
f
GW
(Hz)
̇
ω
p
/
2
π
(Hz s
1
)
d
(kpc)
h
sd
h
n
h
95%
0
h
95%
He
,
1
[l,m,n]
J0034-0534
532.71
1065.43
1
.
5
×
10
16
0
.
5
±
.
1
2
.
7
×
10
27
1
.
4
×
10
24
1
.
8
×
10
25
1
.
1
×
10
26
[020]
J1301+0833
542.38
1084.76
3
.
1
×
10
15
0
.
7
±
.
1
2
.
8
×
10
27
1
.
1
×
10
24
1
.
1
×
10
25
1
.
0
×
10
26
[020]
J1747-4036
609.76
1219.51
4
.
9
×
10
15
3
.
4
±
0
.
8
6
.
7
×
10
28
2
.
8
×
10
25
no data
8
.
8
×
10
27
[020]
J1748-2446O
596.44
1192.87
9
.
4
×
10
15
5
.
9
±
.
5
5
.
4
×
10
28
1
.
5
×
10
25
2
.
6
×
10
25
9
.
1
×
10
27
[020]
J1748-2446P
578.50
1157
8
.
7
×
10
14
5
.
9
±
.
5
1
.
7
×
10
27
1
.
4
×
10
25
1
.
6
×
10
25
9
.
5
×
10
27
[020]
J1748-2446ad
716.36
1432.7
1
.
7
×
10
14
5
.
9
±
.
5
6
.
7
×
10
28
2
.
2
×
10
25
1
.
8
×
10
25
1
.
3
×
10
26
[201]
J1810+1744
601.41
1202.82
1
.
6
×
10
15
2
.
0
±
.
3
6
.
6
×
10
28
4
.
6
×
10
25
1
.
8
×
10
25
9
.
0
×
10
27
[020]
J1843-1113
541.81
1083.62
2
.
8
×
10
15
1
.
7
±
.
2
1
.
1
×
10
27
4
.
4
×
10
25
1
.
1
×
10
25
1
.
0
×
10
26
[020]
J1902-5105
574.71
1149.43
3
.
0
×
10
15
1
.
2
±
.
2
1
.
5
×
10
27
7
.
0
×
10
25
no data
9
.
6
×
10
27
[020]
J1939+2134
641.93
1283.86
4
.
3
×
10
14
3
.
6
±
.
3
5
.
8
×
10
28
2
.
9
×
10
25
1
.
3
×
10
25
8
.
1
×
10
27
[020]
J1959+2048
622.12
1244.24
4
.
4
×
10
15
2
.
5
±
.
5
8
.
6
×
10
28
3
.
9
×
10
25
1
.
5
×
10
25
8
.
5
×
10
27
[020]
TABLE I. Table of millisecond Pulsars with rotation frequency greater than 500 Hz:
ω
p
is the rotational frequency,
f
GW
=
ω
p
is the frequency of gravitational waves, ̇
ω
p
is the measured spin down rate, and
d
is the distance to the pulsar in kilo-parsecs,
h
sd
and
h
en
are the spin-down and elastic strain limits. These values are compared to the strain limit set by recent continuous
GW surveys by interferometric detectors
h
95%
0
18
, and the strain limit for two identical Gen1 helium resonant detectors (with an
integration time of 250 days) as shown in Eq. 20. Pulsars indicated by superscript
were discovered by the Fermi gamma ray
telescope. Pulsars J1747-4036 and J1902+5105 were only discovered in 2012
32
, and were not included in the LIGO+VIRGO
analysis presented in Ref.
18
of
250 days of integration over multiple detectors
18
).
Also, in order to rule out noise we need to detect a gravi-
tational wave signal from at least two different detectors.
Strain sensitivity also improves as
N
d
, where
N
d
is the
number of detectors
35
. There has been computationally
intensive analysis of many days of data from broadband
detectors like LIGO to search for such gravitational wave
signals. However, all such searches have so far been un-
successful, although they have improved the upper bound
on the emitted wave amplitudes. These upper bounds in
turn constrain the equation of state of exotic neutron
stars. In the following, we outline the proposal for a sim-
ple, low-cost, narrowband detector for these gravitational
waves based on a superfluid helium optomechanical sys-
tem. Being relatively simple and economical, superfluid
helium detectors can also be set up in multiple locations
to improve overall detection sensitivity.
As a precursor to subsequent discussions, we present
the central result of our work in figure 2, showing the lim-
its set by different detectors for ten MSPs of interest from
Table IV. Along with the spin-down limit and the strain
limit set up the previous LIGO measurement
18
, we also
show the limits set up two different geometries of helium
detectors that we discuss in detail in Sections III and IV.
Since we are detecting a continuous GW signal, the strain
sensitivity improves with integration time. Here, we have
assumed that the resonance frequency of the same acous-
tic mode can be tuned by up to 200 Hz without changing
the
Q
-factor (of 10
11
), thereby resonantly targeting each
of these pulsars with the same detector.
FIG. 2. Strain sensitivity of various detectors for 10 MSPs of
interest for the helium detector versus measurement time for
two helium detectors with same sensitivity operating simul-
taneously, assuming a bath temperature of 5mK, (
l,m,n
) =
(0
,
2
,
0), and
Q
factor of 10
11
for both Gen1(mass=2.66 kg,
in blue) and Gen2(mass= 15.9 kg, in red) detectors. We also
show the limits set by three interferometric sensors operating
at LIGO-S6 sensitivity for 250 days
18
and at Advanced LIGO
design sensitivity for one year
36
. As seen in the figure, the
current limit on pulsar GW strains can be surpassed within a
few days of integration time for Gen1, and in under a day for
Gen2. Also shown is the spin-down limit for these pulsars.
III. SUPERFLUID HELIUM GRAVITY
DETECTOR
The gravitational wave strain detector we propose is
a resonant mass detector formed by acoustic modes of
superfluid helium in a cavity parametrically coupled to a
microwaves in a superconducting resonator. For the pur-
5
pose of our calculations, we will treat the superfluid as
an elastic medium with zero dissipation. At the tempera-
tures we expect to operate this detector,
T <
10mK, the
normal fluid fraction
ρ
n
is expected to be
ρ
n
0
<
10
8
,
where
ρ
0
is the total density of the fluid
37
. For tem-
peratures below
T <
100mK, the dissipation of audio
frequency acoustic waves is expected and found to be
dominated by a three-phonon process, falling off as
T
4
.
An elastic body (with dimensions

λ
GW
) in a gravi-
tational field will undergo deformation due to changes in
space-time as a gravity wave passes by. For distances far
away from the source of radiation, the space-time pertur-
bation acts like an external tidal force
38
, as also discussed
in Appendix A. The equation of motion for the displace-
ment field
u
(
r,t
) of an elastic body is given by
39
ρ
2
u
∂t
2
μ
L
2
u
(
λ
L
+
μ
L
)
(
.
u
) =
1
2
ρ
̈
hx
,
(7)
where
ρ
is the density,
λ
L
L
are the Lam ́e coefficients
for the elastic body and
̈
hx
is the effective amplitude of
the wave for a particular orientation of the detector that
exerts an effective tidal force on the detector.
This acoustic deformation can be broken into its eigen-
modes
u
(
r
,t
) =
n
ξ
n
(
t
)
w
n
(
r
). For this analysis, we
assume our acoustic antenna is in a single eigenmode of
frequency
ω
m
, thus dropping index
n
. In this analysis,
we have used the notation where
w
n
(
r
) is a dimension-
less spatial mode function with unit amplitude, and the
actual amplitude of the displacement field is in
ξ
(
t
).
Rigid boundary walls and absence of viscosity enables
us to describe the acoustic modes accurately via a sim-
ple wave equation as opposed to Navier-Stokes equa-
tions typically used to describe fluid flow. The spatial
modes are obtained by solving the acoustic equations of
motion
40
. For elastic deformations in enclosed spaces,
the change of pressure
p
(
r
) is described by
2
p
1
c
2
s
2
p
∂t
2
= 0
(8)
with the speed of sound in the material (here helium)
being
c
s
. The particle velocity
v
=
̇u
is related to pres-
sure via
v
/∂
t
=
−∇
p
. Thus each vector component
of the velocity
v
also satisfies the same wave equation
as the pressure, but the components are not indepen-
dent of each other. The full solution can be equivalently
expressed in terms of the Helmholtz potential for the ve-
locity,
v
=
Φ(
r
). In terms of the potential, the acoustic
pressure becomes
p
=
ρ∂
Φ
/∂t
, and the potential satis-
fies the same wave equation
2
Φ
1
c
2
s
2
Φ
∂t
2
= 0
(9)
As before, the time dependence can be explicitly sepa-
rated via Φ
Φ(
r
)
ξ
(
t
). For cylindrical symmetry the
solution for the spatial part of the potential is
Φ(
r,θ,z
) =
J
m
(
k
m
(
n
)
r
) cos(
) cos
(
k
z
(
l
)(
z
+
L
2
)
)
,
(10)
where the wavevectors are found from the rigid bound-
ary conditions
Φ
/∂z
= 0 at
z
=
±
L/
2 and
Φ
/∂r
= 0
at r=a, such that
k
z
(
l
) =
lπ/L
with
l
= 0
,
1
,
2
...
and
k
m
(
n
) follows from the n roots of
J
m
(
k
m
(
n
)
a
) = 0.
Having the solution for the potential, one can obtain
the velocity vector field, and thus the spatial modes,
via
w
(
r,θ,z
) =
Φ(
r,θ,z
)
/
|
w
max
|
, where
|
w
max
|
is the
maximum value of
Φ(
r,θ,z
). These acoustic modes of
helium in a superconducting cavity were experimentally
studied by some of the authors in Ref.
3
. We found these
modes to be well-modeled by this theory and to have
extremely high
Q
-factors (
Q >
10
8
) at 45 mK.
FIG. 3.
The first few pressure modes with non-zero
quadrupolar tensors for the cylindrical cavity. While the form
of quadrupolar tensor (shown on the right) is similar for many
modes, the constant could be different for each acoustic mode.
For the purposes of this paper, we will simply add the
finite linear dissipation to the acoustic resonance, param-
eterized as a finite
Q
. For a damped acoustic resonator,
eq. 7 can be simplified to show that the displacement
field
ξ
(
t
) satisfies the equation of motion
μ
(
̈
ξ
+
ω
m
Q
He
̇
ξ
+
ω
2
m
ξ
)
=
1
4
ij
̈
h
ij
q
ij
,
(11)
where
Q
He
is the
Q
-factor associated with the acoustic
mode,
μ
is the reduced mass for the particular eigenmode,
μ
=
ρ
w
2
dV,
(12)
and
q
ij
is the dynamic part of the quadrupole moment,
q
ij
=
ρ
(
w
i
x
j
+
x
i
w
j
2
3
δ
ij
w
·
r
)
dV .
(13)
Figure 3 shows the first few pressure modes of the cylin-
drical cavity that have a non-zero quadrupolar tensor,
6
FIG. 4.
The directivity patterns for acoustic mode (
l,m,n
) = (0
,
2
,
0) of the cylindrical cavity. The +
,
×
polarization and
total directivity functions are given for two different polarizations of the detector (Euler angle
ψ
= 0
,π/
2). The orientation of
the detector can be adjusted to optimize the directivity function for the astrophysical source in consideration.
along with the form of the tensor. As can be deduced
from eq. 10, several modes have a zero quadrupole mo-
ment due to symmetry.
In their analysis of various antenna geometries for grav-
itational radiation detection, Hirakawa and co workers in-
troduced two quantities to compare GW antennas span-
ning different size and symmetry groups
41
. These are
the effective area of the antenna (
A
G
) characterizing the
GW-active part of the vibrational mode, and the direc-
tivity function (
d
A
), which characterizes the directional
and polarization dependence of the GW sensor. They are
defined as
A
G
=
2
μM
q
2
ij
(14)
and
d
A
(
θ,φ
) =
5
4
(
q
ij
e
A
ij
(
k
)
)
2
q
2
ij
(15)
where
M
is the total mass of the antenna and
e
A
ij
is
the unit vector for incoming GW signal polarization
A
(
A
∈{
+
,
×}
) in arbitrary direction
k
(
θ,φ,ψ
). The Euler
angles (
θ,φ,ψ
) transform from the pulsar coordinate sys-
tem to the detector co-ordinate system and are discussed
in Appendix A, along with the explicit form of
e
A
ij
. In
sum, the angles
θ
and
φ
describe the direction of the in-
coming gravitational wave, and
ψ
defines the polarization
of the detector (rotation of the
x
y
plane of the source).
An important distinction between the proposed detector
and other gravitational wave sensors, particularly the in-
terferometric ones is that the orientation of the detector
can be adjusted to optimize the directivity function for
the astrophysical source in consideration due to its small
size. This acts as another tunable parameter that can
give significant enhancement in sensitivity for a particu-
lar source, as shown in fig. 4 for a specific acoustic mode
(
l,m,n
) = (0
,
2
,
0).
In terms of previously defined expressions, the mean
squared signal force from a continuous gravity wave
source of polarization
A
is given by
f
2
G
=
2
πG
5
c
3
Mμω
2
G
A
G
d
A
(
θ,φ
)
s
A
,
(16)
=
1
40
Mμω
4
G
A
G
d
A
h
A
(
t
)
2
(17)
where
ω
G
= 2
ω
p
is the frequency of gravity wave. Here,
we have assumed a delta-function gravity wave spectrum.
7
As an example, we choose a cylindrical cavity of radius
a
= 10
.
8 cm, length
L
= 50 cm (from now onwards re-
ferred to as Gen1 or with subscript (He
,
1)). We focus
on acoustic mode
f
(0
,
2
,
0)
= 1071 Hz, which has an ef-
fective mass
μ
= 0
.
625
M
, and a large GR-active area of
A
G
= 0
.
629
πa
2
due to its quadrupolar shape, shown in
figure 3. Another geometry considered in this work is a
cylindrical cavity of the same radius, but length
L
= 3 m
(from now onwards referred to as Gen2 or with subscript
(He
,
2)). Since the resonance frequency of the [020] mode
is independent of length, it has the same frequency. How-
ever, increasing the mass gives us a larger effective mass
for the same area. Figure 4 shows the various directivity
functions for this acoustic mode that capture the angular
dependence of the sensitivity of the detector.
IV. NOISE MECHANISMS AND MINIMUM
DETECTABLE STRAIN
The system we are proposing and have been exploring
in the laboratory
3
is a parametric transducer
42
and es-
sentially similar to other optomechanical systems
43
: the
acoustic motion of the superfluid and resulting perturba-
tion of the dielectric constant modulates the frequency
of a high-
Q
superconducting microwave resonator. The
details of the coupled acoustic and microwave system,
sources of dissipation (phonon scattering, effect of iso-
topic impurities, radiation loss,) microwave and signal
detection limits, effects of electrical dissipation, require-
ments on thermal stability, etc. will be the subject of
another manuscript
44
. Here we take a few central results
of this analysis.
The noise sources relevant to this system are the Brow-
nian motion of the fluid driven by thermal/dissipative
forces, the additive noise of the amplifier which is used
to detect the microwave field, the added noise of the
stimulating microwave field (phase noise), and possible
back-action forces due to fluctuations of the field inside
the microwave cavity (due to phase noise and quantum
noise). We will assume for the purpose of this discussion
that the challenging job of seismically isolating the super-
fluid cell from external vibrations has been accomplished
as has been done for other gravitational wave detectors.
Due to the high frequency and narrow bandwidth of the
astrophysical source of interest, the strain noise due to
Newtonian gravity fluctuations are expected not to be
relevant for this detector
26
. The effect of vortices in su-
perfluid helium due to earth’s rotation on the
Q
-factor is
unclear. However, using an annular cylinder or an equa-
torial mount allows for long integration times without
the possibly detrimental effects due to vortices.
For a sufficiently intense microwave pump, with suffi-
ciently low phase noise, the thermal Brownian motion of
the helium will dominate the noise. Assuming the de-
vice is pumped on the red-sideband,
ω
pp
=
ω
c
ω
m
, and
that the system is the side-band resolved limit,
ω
m
> κ
c
,
the upconversion rate of microwave photons is given by:
Γ
opt
= 4(∆
p
SQL
·
g
0
)
2
n
p
c
, where
ω
pp
,
ω
c
, and
ω
m
are
the pump, cavity, and acoustic mode frequency respec-
tively,
κ
c
=
ω
c
/Q
Nb
is the cavity damping rate, ∆
p
SQL
is the amplitude of the zero-point fluctuation of the pres-
sure of the acoustic field,
n
p
is the amplitude of the
pump inside the cavity measured in quanta, and
g
0
is
the coupling between the acoustic and microwave field.
For the geometry we consider here, Gen1:
l
= 0
.
5 m,
d
= 0
.
108 m,
ω
m
= 1071
·
2
π
Hz,
ω
c
= 1
.
6
·
2
π
GHz, and
g
0
=
7
.
5
·
10
11
·
2
π
Hz.
To achieve a readout with noise temperature of 1mK,
which means that the added noise of the amplifier is equal
to the thermal noise amplitude when the helium is ther-
malized at 1mK, requires
n
p
= 6
·
10
9
microwave pump
photons and a phase noise of
145
db
c
/Hz. To begin
to dampen and cool the acoustic resonance with cavity
backaction force, would require
n
p
= 10
12
, and a phase
noise of
145
db
c
/Hz. Microwave sources have been re-
alized using whispering gallery modes of sapphire with
phase noise of
180
db
c
/Hz. Together with a tunable
superconducting cavity, it is possible to realize a source
with sufficient low noise to broaden and cool this mode
with backaction. Furthermore, as we will detail in our
future work
44
,
3
He impurities diluted into the
4
He are ex-
pected to add acoustic loss, additional to the 3-phonon
process. To achieve
Q
He
= 10
11
, we estimate that an
isotopic purity of
n
3
/n
4
= 10
11
is required.
Due to the very low dielectric constant of helium
(

He
= 1
.
05), the bare optomechanical coupling constant
is small compared to typical micro-scale optomechanical
systems:
g
0
= ∆
p
SQL
·
∂ω
c
/∂
p
=
7
.
5
·
10
11
·
2
π
Hz:
this is the frequency shift of the Nb cavity,
ω
c
, due to the
zero-point fluctuations of the acoustic field of the helium,
p
SQL
. However, the relevant quantity is cooperativity,
C
= Γ
opt
He
, which compares the rate of signal photon
up-conversion, Γ
opt
, to the loss rate of acoustic quanta
to the thermal bath,
γ
He
=
ω
He
/Q
He
. With quantum
limited microwave detection (now possible with a num-
ber of new amplifiers), detection at the SQL is achieved
when
C
= 1, and is the onset of significant backaction ef-
fects such as optomechanical damping and cooling. The
key point is that for this system we expect to be able
to realize very large
n
p
. This is due to the very high
Q
possible in Nb, (
Q
Nb
10
11
is now routine for acceler-
ator cavities
45,46
, even when driven to very high inter-
nal fields of 10
7
V/m
corresponding to
n
p
= 10
23
,) and
dielectric losses and resulting heating at microwave fre-
quency in liquid helium are expected to be negligible up
to very high pump powers. Assuming the dielectric loss
angle in helium is less than 10
10
, our estimates suggest
that
n
p
= 10
16
should be achievable before dissipative
effects lead to significant heating of the helium sample at
5mK, far beyond the internal pump intensity used with
micro-optomechanical systems and far above the onset
of backaction effects,
C
= 1 for
n
p
= 8
·
10
11
. As a re-
sult, we are optimistic that SQL limited detection and
significant backaction cooling and linewidth broadening
are possible.
8
Since the frequency and phase of the pulsar’s grav-
ity wave signal should be known through observations
of the electromagnetic signal, single quadrature back-
action evading, quantum non-demolition measurement
techniques could be implemented
47
. This has the advan-
tage of avoiding the back-action forces from the cavity
field fluctuations and can lower the phase noise require-
ments of the microwave pump.
For a damped harmonic oscillator with
γ
He
=
ω
m
/Q
He
in equilibrium with a thermal bath at overall effective
temperature
T
, the position noise spectral density is
given by
S
th
ξξ
[
ω
] =
k
B
T
μω
2
m
{
γ
He
/
2
(
ω
+
ω
m
)
2
+
γ
2
He
/
4
+
γ
He
/
2
(
ω
ω
m
)
2
+
γ
2
He
/
4
}
.
(18)
Assuming that noise at the detection frequency is dom-
inated by the thermal noise of the acoustic mode, the
force noise spectral density
S
FF
is given by the re-
lation
S
ξξ
[
ω
] =
|
χ
(
ω
)
|
2
S
FF
[
ω
], with the susceptibility
χ
(
ω
) = [
μ
((
ω
2
m
ω
2
) +
He
ω
)]
1
.
For gravitational strain, using eq. 11, we find
S
hh
[
ω
] =
40
S
FF
[
ω
]
/
(
μMω
4
G
d
A
A
G
) for a continuous gravity wave
source at frequency
ω
G
. Combining these, we find that
for a resonant mass detector at
ω
G
=
ω
m
,
S
hh
[
ω
] =
80
k
B
T
Md
A
A
G
Q
He
ω
3
m
.
(19)
The strain sensitivity of our detector is simply
S
hh
[
ω
], and the minimum noise is
S
hh
[
ω
]
int
after
an integration time
τ
int
. The minimum detectable strain
field with 2
σ
certainty is therefore given by
30
h
min
2
S
hh
[
ω
]
τ
int
=
320
k
B
T
3
G
A
G
d
A
Q
m
1
τ
int
.
(20)
The 2
σ
uncertainty limit is used to be consistent with
previously reported limits on
h
min
set by LIGO
18
.
As an example, both cylindrical cavities considered
in section III have acoustic mode
f
(0
,
2
,
0)
1071 Hz.
This mode of the detector can easily be tuned (by un-
der
±
15 Hz) to be in resonance with pulsars J0034-0534,
J1301+0833, and J1843-1113. Similarly, another acous-
tic mode (
f
[2
,
0
,
1]
= 1425 Hz) is found to have resonant
frequencies in the vicinity (
<
8 Hz) of the frequency of
gravitational waves from pulsar J1748-2446ad. Taking
into account the different quadrupole tensors, effective
mass and directivity functions for the different acous-
tic modes, Table 2 lists the minimum detectible strain
for several pulsars for cylindrical detector Gen1 after
250 days of integration time (same time as the current
LIGO+VIRGO estimates in Ref.
18
). Here we have as-
sumed an acoustic
Q
-factor of 10
11
and thermal
T
th
= 5
mK for both geometries. Since the detector is small
enough to be rotated or moved geographically to opti-
mize signal from a particular pulsar, we have assumed
ψ
= 0 and (
θ,φ
) that maximizes the directivity.
Pulsar
ω
p
/
2
π
f
GW
(Hz)
h
sd
h
95%
0
- LIGO
h
95%
He
,
1
[l,m,n]
h
95%
He
,
2
[l,m,n]
J0034-0534
532.71
1065.43
2
.
7
×
10
27
1
.
8
×
10
25
1
.
1
×
10
26
[020]
3
.
6
×
10
27
[020]
J1301+0833
542.38
1084.76
2
.
8
×
10
27
1
.
1
×
10
25
1
.
0
×
10
26
[020]
3
.
5
×
10
27
[020]
J1843-1113
541.81
1083.62
1
.
1
×
10
27
1
.
1
×
10
25
1
.
0
×
10
26
[020]
3
.
5
×
10
27
[020]
J1748-2446ad
716.36
1432.7
6
.
7
×
10
28
1
.
8
×
10
25
1
.
3
×
10
26
[201]
no coupling
TABLE II. Table of millisecond Pulsars of interest for helium detector Geo1 and Geo2. Here,
ω
p
is the oulsar rotational
frequency,
f
GW
=
ω
p
is the frequency of gravitational waves, as given in Table 1. These values are compared to the strain
limit set by recent continuous GW survey by interferometric detectors
18
, along with the strain limit for the helium resonant
detectors Gen1 with an integration time of 250 days and Gen 2 with one year integration time, as shown in eq. 20. Here,
ψ
= 0,
Q
-factor is 10
11
, and the acoustic mode is given in square brackets.
In order to compare the sensitivity of our proposed
detector with other GW sensors, we pick a specific ex-
pected astrophysical source: gravity waves from pulsar
J1301+0833, with
ω
G
= 2
π
×
1084
.
76 Hz. Gen1 (mode
[020]) gives us sensitivity of
h
min
= 3
.
4
×
10
23
/
Hz,
which is significantly below the sensitivity of LIGO, and
comparable (within a factor of 2) to current sensitivity of
advanced LIGO. Such a detector can surpass the LIGO
+VIRGO estimate on minimum strain
h
95%
0
= 1
.
1
×
10
25
in under a week of integration time (under a month if
Q
-
factor is 10
10
instead). Increasing the mass by a factor
of 6 (by choosing Gen2), while assuming the same
Q
-
9
factor and noise characteristics, we can get sensitivity of
1
.
4
×
10
23
/
Hz, which is below the strain sensitivity of
advanced LIGO for this frequency range. Figure 5 shows
the minimum detectable strain as a function of integra-
tion time for various
Q
factors for two resonant detectors
operating at the same sensitivity. Figure 5 also shows the
sensitivity estimates for three interferometeric detectors
operating at LIGO-S6 sensitivity, and at advanced LIGO
design sensitivity, as used in ref.
18
.
As figure 5 and Table 2 demonstrate, Gen2 can come
within a factor of 2 of the spin-down limit for pulsar
J1301+0833 (and several other pulsars) in a year of in-
tegration time. Considering the conjecture that the pri-
mary spin-down mechanism for MSPs is the emission of
gravitational radiation, our detector seems a promising
candidate for searches of continuous gravitational waves
from this and similar other pulsars.
FIG. 5. Strain sensitivity versus measurement time for two
helium detectors with same sensitivity operating simulta-
neously, assuming a bath temperature of 5mK, (
l,m,n
) =
(0
,
2
,
0), for Geometry 1(blue) and 2(red). We also show
the limits set by three interferometric detectors operating at
LIGO-S6 sensitivity (solid black), and the design sensitivity
of advanced LIGO (dashed black). The stars shows the cur-
rent limit on minimum strain set by LIGO, and the projected
limit by Advanced LIGO. As seen in the figure, the current
limit can be surpassed within a few days of integration time
for Gen1, and under a day for Gen2. Also shown is the spin-
down limit for pulsar J1301+0833.
We would like to note that several noise suppression
mechanisms (such as squeezed light injection) currently
used in LIGO can also be employed here. More impor-
tantly, there are ways to squeeze the mechanical motion
of the detector
48–50
. This can significantly relax the size,
Q
-factor and microwave noise requirements, increasing
the sensitivity of our proposed detector significantly. For
example, exploring methods to squeeze mechanical mo-
tion by changing the speed of sound periodically, and
exploring other effects arising from parametric coupling
between the helium acoustic modes and the microwave
resonator container is a straight forward extension of the
current setup, since the helium is already being pressur-
ized and parametrically coupled to microwaves for reso-
nant force detection. A detailed analysis of implementing
these protocols for improved gravity wave sensing will be
the subject of future research.
V. COMPARISON WITH OTHER DETECTORS
The basic principle of the superfluid helium detector
is analogous to that of other resonant mass sensors, such
as Weber bars. The use of resonant mass GW detectors
has a 50 year history, dating back to early experiments
by Weber
4
. There have been several proposals of using
resonant mass detectors to search for GW from pulsars,
for example Ref.
30
, and a few continuous GW searches
targeting specific pulsars, the most notable one being the
Tokyo group experiment looking for signal from Crab
Pulsar
51
. Here we highlight several key differences in the
implementation using superfluid helium.
Mass:
We discuss a kg-scale sample of helium which
is 10
3
times smaller than the typical resonant bar
detectors. The low mass limits the utility of the he-
lium detector to CW sources, where as the massive
detectors are useful for burst sources. Nonetheless,
there is high sensitivity for CW sources and the low
mass makes a helium detector economical and small
scale. One could deploy a few such detectors to seek
coincidence and further improve sensitivity.
T/Q
He
temperature and quality factor:
It is pos-
sible to cool an isolated sample of helium to tem-
peratures less than 10mK and we are anticipating
very low loss. For instance, helium at 25mK with
Q
He
= 10
9
has a ratio
T/Q
He
10
3
times smaller
than the best value found in the literature, and po-
tentially 10
6
times smaller at lower temperature
44
.
Optomechanical damping:
It appears possible to
substantially increase the acoustic resonance
linewidth without decreasing the force sensitivity
by parametrically coupling to microwaves
3
. While
parametric transducers are also used in other res-
onant mass detectors
52,53
, the particular geometry
and mechanism used in helium detector is expected
to have lower noise characteristics
44
.
Frequency tunability:
It is possible to change the
speed of sound in helium by 50% by pressuriza-
tion. This allows the apparatus to be frequency
agile; thus searching several pulsars with the same
detector. It also allows for long term tracking the
same pulsar in the presence of deleterious frequency
shifts. For example, the estimated Doppler shift of
the GW signal from Crab pulsar is
30 mHz/year
due to earth’s motion. Our detector can be tuned
to track this shift, allowing for months of integra-
tion time. By resonantly tracking the pulsar we also
reduce SNR, and thereby the detection threshold.
10
A standard figure of merit used in literature to compare
various bar detectors of different materials is
η
=
Qρc
3
s
25
.
Typical values of
η
range from 10
21
10
24
kg s
3
. Accord-
ing to this metric, helium may seem like a poor choice
for a bar detector, (
η
Q
He
×
10
9
kg s
3
). This figure
of merit is made of the material specific parameters in
the minimum detectable strain, as given in eq. 20. How-
ever, adding the temperature dependence, and the signif-
icantly large
Q
-factors make the helium sensor compara-
ble to the resonant bar detector. In addition, due to it’s
smaller size, temperature stability, seismic and acoustic
isolation are much easier to maintain.
Unlike interferometric detectors like LIGO conducting
a broadband search for gravitational waves, the helium
detector is narrowband, and works best for detection
of continuous waves such as pulsars. Nevertheless, as
highlighted in fig. 5, around 1kHz the setup described
above has strain sensitivity within a factor of 4 (Gen
1), or in principle even surpassing the sensitivity of ad-
vanced LIGO by considering a larger volume of super-
fluid helium (Gen 2). This allows us to surpass the lim-
its from previous CW searches of VIRGO+LIGO exper-
iments (
h
min
10
25
) within a week, or less depending
on the detector size and
Q
-factor.
There are several ongoing and proposed detectors for
gravitational waves, for example space-based interfero-
metric detector eLISA
9,54
, atom interferometry based de-
tector AGIS-LEO
55
, and Pulsar Timing Arrays
56
. These
detectors operate at different frequency ranges, typically
much lower than the ones considered here. The astro-
physical sources of interest are therefore different from
those of the helium detector.
Finally, an important advantage of considering super-
fluid helium as a resonant GW sensor is that by designing
different geometries and exploring different types of res-
onances, one could build detectors for a range of astro-
physical sources. For example, by considering smaller
containers or Helmholtz resonances in micro or nano-
fluidic channels
57
, it may be possible to build a resonant
detectors for high frequency sources of gravity waves as
explored in other devices
58,59
. Alternatively, larger con-
tainers or low-frequency Helmholtz resonances may be
used to detect continuous GWs from young pulsars or bi-
nary systems. Since the technology required for the pro-
posed superfluid helium gravity detector is space-friendly,
it may be possible to design low frequency detectors for
space missions if seismic noise becomes a deterrent.
VI. CONCLUSIONS AND OUTLOOK
As discussed in Section IV, there are several strin-
gent requirements for low-noise operation of our pro-
posed helium detector: isotopically pure sample, sub-
10mK cryogenic environment, very low phase-noise mi-
crowave source, and isolation from environmental vibra-
tions. Furthermore, due to the low density and speed of
sound, a reasonable size (
1m) bar detector made of he-
lium can only be used for detection of continuous gravity
waves.
Despite these extreme requirements, using superfluid
4
He does have several advantages. The low intrinsic dissi-
pation and dielectric loss and wide acoustic tunability are
direct manifestations of the inherent quantum nature of
the acoustic medium. Furthermore, due to the mismatch
between the speed of sound in helium and niobium, there
is an inherent acoustic isolation from the container. Since
the container itself is in a macroscopic quantum state
(superconductor), it further contributes to the extremely
low-noise, high sensitivity nature of the proposed device
by making an extremely high
Q
microwave resonator with
very high power-handling.
Several ideas for future work are outlined in the
manuscript at various places. They include investigating
more complex geometries for stronger coupling to grav-
itational strain, or investigating other high-
Q
acoustic
resonances (Helmholtz resonances) in helium to detect
other sources of continuous gravity waves. Also, many
ideas from quantum optics and quantum measurement
theory can be implemented in this system to increase
bandwidth or sensitivity. For example, by periodically
modulating the acoustic resonance frequency, it will be
possible to upconvert out of resonance signals into he-
lium resonance signals, thereby increasing the frequency
tunability of our detector. Several techniques from quan-
tum measurements can be applied to our proposed trans-
duction scheme to avoid measurement backaction or to
squeeze acoustic noise, thereby increasing the sensitivity
further.
Even without these techniques, the extreme displace-
ment sensitivity (
10
23
/
Hz
) of this meter-scale de-
vice corresponds to a measurement of the width of milky
way to cm-scale precision! This is again made possible
by combining two macroscopic quantum states in the
measurement scheme (a superfluid coupled to a super-
conductor). The resulting hybrid quantum sensor is an
extremely low noise detector at low temperatures due to
the robustness of the quantum state involved. As these
experiments develop, proposing a more broadly function-
ing gravity wave detector may be possible, as well as the
detection of other extremely small laboratory forces.
ACKNOWLEDGMENTS
We would like to acknowledge helpful conversations
with Rana Adhikari, Yanbei Chen, Dan Lathrop, Pierre
Meystre, David Blair and Nergis Mavalvala. We ac-
knowledge funding provided by the Institute for Quan-
tum Information and Matter, an NSF Physics Frontiers
Center (NSF IQIM-1125565) with support of the Gor-
don and Betty Moore Foundation (GBMF-1250) NSF
DMR-1052647, the NSF ITAMP grant, and DARPA-
QUANTUM HR0011-10-1-0066.
11
Appendix A: Brief introduction to Gravitational
waves
Gravitational waves are solutions to the linearized Ein-
stein equations, where the perturbed metric can be writ-
ten as
g
μν
=
η
μν
+
h
μν
. Here,
η
μν
= diag[
1
,
1
,
1
,
1] is
the Minkowski metric (which is a good approximation
for our solar system) and
|
h
μν
| 
1 is a small pertur-
bation of the metric. In free space, Einstein’s equations
of motion, which describe the dynamics of space-time,
reduce to
R
μν
= 0, where
R
μν
is the Ricci-tensor con-
structed from the metric. Since only the weak-field limit
is considered, terms that are of higher order in
h
μν
can
be neglected. In addition, general relativity has an inher-
ent gauge freedom related to the choice of coordinates.
In the Lorentz-gauge the equations of motion reduce to
a wave equation as in electromagnetism:
R
μν
=

h
μν
=
(
2
t
+
c
2
2
)
h
μν
= 0
.
(A1)
This is the wave equation for gravitational waves,
which are small perturbations of flat space-time that
propagate at the speed of light. A general plane-wave
solution has the form
h
μν
(
~x,t
) =
A
μν
cos(
ωt
~
k
·
~x
+
φ
),
with the dispersion relation
ω
=
c
|
~
k
|
. Choosing the spe-
cific transverse-traceless gauge, and a coordinate system
in which the wave propagates only in the z-direction, the
only non-vanishing components of the gravitational wave
tensor are the spatial components
h
ij
=
h
+
(
t
z
c
)
e
+
ij
z
) +
h
×
(
t
z
c
)
e
×
ij
z
)
(A2)
where
h
+
and
h
×
are the two polarization components
with the polarization tensors given by
e
+
(
ˆz
) =
1 0 0
0
1 0
0 0 0
(A3)
and
e
x
(
ˆz
) =
0 1 0
1 0 0
0 0 0
(A4)
Gravitational waves carry energy and have observable
effects on matter. For test particles at a distance much
shorter than the wavelength of the gravitational wave,
the wave induces an effective time-dependent tidal force.
To see this, it is convenient to use gauge-invariant quan-
tities, such as the Riemann tensor which is invariant
to linear order. Its only non-vanishing component is
R
μ
0
ν
0
=
1
2
̈
h
μν
, where the dot denotes differentiation
with respect to coordinate time
t
. The Riemann tensor
captures how neighboring geodesics (i.e. world lines of
free particles) change with respect to each other: the vec-
tor
x
μ
that connects two geodesics follows the geodesic
deviation equation ̈
x
μ
=
R
μ
0
ν
0
x
ν
=
1
2
̈
h
μν
x
ν
. This equa-
tion holds for geodesics that are close to each other as
compared to the wave length
λ
of the gravitational wave,
i.e.
x << λ
. From this equation follows the equation
of motion for the distance between two neighboring test
particles:
̈
x
=
1
2
(
̈
h
+
x
+
̈
h
×
y
)
,
̈
y
=
1
2
(
̈
h
×
x
̈
h
+
y
)
.
(A5)
The equations of motion are equivalent to the presence of
an effective tidal force
F
i
=
̈
h
ij
x
j
/
2 that acts on the par-
ticles. The corresponding effective force is conservative
and can therefore be represented by force lines, shown
in fig. 6 for a purely plus-polarized wave. For a general
polarization, the force line diagram is rotated counter-
clockwise by the angle Ψ where tan(2Ψ) =
̈
h
×
/
̈
h
+
.
For only a plus-polarized wave (
h
×
= 0), the solution
to lowest order in
h
is
x
(
t
) =
x
(0)
(
1 +
h
+
(
t
)
2
)
y
(
t
) =
y
(0)
(
1
h
+
(
t
)
2
)
.
(A6)
The distances between nearby points oscillate in the
x
and
y
directions, i.e. perpendicular to the gravita-
tional wave. A cross-polarized wave has the same effect
but with the x-y-plane rotated by
π/
4 (see also fig. 7).
x
y
FIG. 6. Force lines for the effective tidal force produced by a
plus-polarized gravitational wave (A). The force acts perpen-
dicular to the direction of propagation of the wave. The same
force lines, rotated counter clock-wise by Ψ =
π/
4, represent
the effect of a cross-polarized wave.
The detector co-ordinate axis (
x,y,z
) is not necessarily
aligned with the gravitational wavefront emitted from the
source (
x
,y
,z
). To account for the angular dependence,
the strain at the detector can be written as
h
(
t
) =
F
+
(
θ,φ,ψ
)
h
+
(
t
) +
F
×
(
θ,φ,ψ
)
h
×
(
t
)
(A7)
where (
θ,φ,ψ
) are the Euler angles that convert from
the pulsar co-ordinate system to the the detector plane,
as shown in fig. 8, and
F
+
/
×
(
θ,φ,ψ
) are known as the