4.5 Gravitational wave detection
A gravitational wave detector can be modelled as a body of mass
at a distance
from a fiducial
laboratory point, connected to the point by a spring of resonant frequency
and quality
factor
. From the equation of geodesic deviation, the infinitesimal displacement
of the
mass along the line of separation from its equilibrium position satisfies the equation of motion
where
and
are “beam-pattern” factors that depend on the direction of the source
and on a polarization angle
, and
and
are gravitational waveforms
corresponding to the two polarizations of the gravitational wave (for a review, see [255]). In a source
coordinate system in which the
plane is the plane of the sky and the
-direction points toward the
detector, these two modes are given by
where
represent transverse-traceless (TT) projections of the calculated waveform of Equation (68),
given by
where
is a unit vector pointing toward the detector. The beam pattern factors depend on the
orientation and nature of the detector. For a wave approaching along the laboratory
-direction, and for a
mass whose location on the
plane makes an angle
with the
axis, the beam pattern
factors are given by
and
. For a resonant cylinder oriented along the
laboratory
axis, and for source direction
, they are given by
,
(the angle
measures the relative orientation of the laboratory and source
-axes). For a laser interferometer with one arm along the laboratory
-axis, the other
along the
-axis, and with
defined as the differential displacement along the two arms,
the beam pattern functions are
and
.
The waveforms
and
depend on the nature and evolution of the source. For example, for
a binary system in a circular orbit, with an inclination
relative to the plane of the sky, and the
-axis
oriented along the major axis of the projected orbit, the quadrupole approximation of Equation (70) gives
where
is the orbital phase.