2.6 Luminosity in gravitational waves
The general formula for the local stress-energy of a gravitational wave field propagating
through flat spacetime, using the TT-gauge, is given by the Isaacson expression [262
, 335]
where the angle brackets denote averages over regions of the size of a wavelength and times of the
length of a period of the wave. The energy flux of a wave in the
direction is the
component.
The gravitational wave luminosity in the quadrupole approximation is obtained by integrating the
energy flux from Equation (14) over a distant sphere. When one correctly takes into account the projection
factors mentioned after Equation (2), one obtains [262
]
where
is the trace of the matrix
. This equation can be used to estimate the backreaction effect
on a system that emits gravitational radiation.
Notice that the expression for
is dimensionless when
. It can be converted to normal
luminosity units by multiplying by the scale factor
This is an enormous luminosity. By comparison, the luminosity of the sun is only 3.8 × 1026 W, and that
of a typical galaxy would be 1037 W. All the galaxies in the visible universe emit, in visible light, on the
order of 1049 W. We will see that gravitational wave systems always emit at a fraction of
, but that
the gravitational wave luminosity can come close to
and can greatly exceed typical electromagnetic
luminosities. Close binary systems normally radiate much more energy in gravitational waves than in light.
Black hole mergers can, during their peak few cycles, compete in luminosity with the steady luminosity of
the entire universe!
Combining Equations (2) and (15) one can derive a simple expression for the apparent luminosity of
radiation
, at great distances from the source, in terms of the gravitational wave amplitude [335]:
The above relation can be used to make an order-of-magnitude estimate of the gravitational wave amplitude
from a knowledge of the rate at which energy is emitted by a source in the form of gravitational waves. If a
source at a distance
radiates away energy
in a time
, predominantly at a frequency
, then
writing
and noting that
, the amplitude of gravitational waves is
When the time development of a signal is known, one can filter the detector output through a copy
of the expected signal (see Section 5 on matched filtering). This leads to an enhancement in
the SNR, as compared to its narrow-band value, by roughly the square root of the number
of cycles the signal spends in the detector band. It is useful, therefore, to define an effective
amplitude of a signal, which is a better measure of its detectability than its raw amplitude:
Now, a signal lasting for a time
around a frequency
would produce
cycles. Using this we
can eliminate
from Equation (18) and get the effective amplitude of the signal in terms of the energy,
the emitted frequency and the distance to the source:
Notice that this depends on the energy only through the total fluence, or time-integrated flux
of the wave. As in many other branches of astronomy, the detectability of a source is
ultimately a function of its apparent luminosity and the observing time. However, one should not
ignore the dependence on frequency in this formula. Two sources with the same fluence are not
equally easy to detect if they are at different frequencies: higher frequency signals have smaller
amplitudes.