2
Response of a Detector to a Gravitational-Wave Signal
There are two main methods to detect gravitational waves which
have been implemented in the currently working instruments. One
method is to measure changes induced by gravitational waves on
the distances between freely moving test masses using coherent
trains of electromagnetic waves. The other method is to measure
the deformation of large masses at their resonance frequencies
induced by gravitational waves. The first idea is realized in
laser interferometric detectors and Doppler tracking
experiments
[70, 55]
whereas the second idea is implemented in resonant mass
detectors
[10
]
.
Let us consider the response to a plane
gravitational wave of a freely falling configuration of masses.
It is enough to consider a configuration of three masses shown in
Figure
1
to obtain the response for all currently working and planned
detectors. Two masses model a Doppler tracking experiment where
one mass is the Earth and the other one is a distant spacecraft.
Three masses model a ground-based laser interferometer where the
masses are suspended from seismically isolated supports or a
space-borne interferometer where the three masses are shielded in
satellites driven by drag-free control systems.
Let
be the frequency of the coherent beam used in the detector
(laser light in the case of an interferometer and radio waves in
the case of Doppler tracking). Let us assume for simplicity that
the distance between the masses is the same and equal to
. Let
,
, be the unit vectors along the lines joining the test masses,
let
be the point lying in the plane of the three masses and
equidistant from the masses, and let
,
, be the vectors of length
joining
and the masses. Let
be the relative change
of frequency induced by a transverse, traceless, plane
gravitational wave on the coherent beam travelling from the mass
to the mass
, and let
be a relative change of frequency induced on the beam travelling
from the mass 3 to the mass 1. The equations for
and
are given by (see
[31, 8]
and also
[71]
for a coordinate free derivation)
where
In Equation (3)
is the three-dimensional matrix of the spatial metric
perturbation produced by the wave in the proper reference frame
of the detector,
is the unit vector in this reference frame directed from the
center
to the source of the gravitational wave and T denotes matrix
transposition.
In the source frame the three-dimensional
matrix
of the spatial metric perturbation produced by the gravitational
wave is given by
where
and
are the two polarizations of the wave. In general the detector
is moving with respect to the source of a gravitational wave and
the signal registered by the detector will be modulated by this
motion. To obtain the matrix
it is convenient to introduce a coordinate system with respect
to which the source is fixed. The origin of this coordinate
system is usually chosen to be the
solar system barycenter
(SSB). Then we obtain
by the transformation
Here
is the transformation matrix from the source frame to SSB and
is the transformation matrix from the detector frame to SSB
(see
[35, 19, 41
, 50
]
for details).
The difference of the phase fluctuations
measured, say, by a photo detector, is related to the
corresponding relative frequency fluctuations
by
For a standard Michelson, equal-arm interferometric configuration
is given in terms of one-way frequency changes
and
[see Equations (1) and (2)] by the expression
[80]
In the long-wavelength approximation
Equation (7) reduces to
Consequently the phase change is given by
where the function
is the response of the interferometer to a gravitational-wave
signal in the long-wavelength approximation. In this
approximation the response of a laser interferometer is usually
derived from the equation of geodesic deviation (where the
response is defined as the relative change of the length of the
two arms, i.e.,
). There are important cases where the long-wavelength
approximation is not valid. These include the space-borne LISA
detector for gravitational-wave frequencies larger than a few mHz
and satellite Doppler tracking measurements.
In the case of a bar detector the
long-wavelength approximation is very accurate and the detector’s
response
, where
is the change of the length
of the bar, is given by
where
is the unit vector along the symmetry axis of the bar.
In most cases of interest the response of the
detector to a gravitational-wave signal can be written as a
linear combination of four constant amplitudes
,
where the four functions
depend on a set of parameters
but are independent of the parameters
. The parameters
are called
extrinsic
parameters whereas the parameters
are called
intrinsic
. In the long-wavelength approximation the functions
are given by
where
is the phase modulation of the signal and
,
are slowly varying amplitude modulations.
Equation (12) is a model of the response of the space-based detector LISA to
gravitational waves from a binary system
[50
], whereas Equation (13) is a model of the response of a ground-based detector to a
continuous source of gravitational waves like a rotating neutron
star
[41
]
. The gravitational-wave signal from spinning neutron stars may
consist of several components of the form (12). For short observation times over which the amplitude
modulation functions are nearly constant, the response can be
approximated by
where
and
are constant amplitude and initial phase, respectively, and
is a slowly varying function of time. Equation (14) is a good model for a response of a detector to the
gravitational-wave signal from a coalescing binary system
[79, 18]
. We would like to stress that not all deterministic
gravitational-wave signals may be cast into the general
form (12).