Gravitational wave spectrum
Whereas astrophysical electromagnetic waves are typically much
smaller than their sources, ranging from a few kilometres down to
sub-nuclear wavelengths, gravitational waves are larger than their
sources, with wavelengths starting at a few kilometres and ranging up
to the size of the Universe. A gravitational perturbation larger than
the Universe would not be called a wave because it would not have any
detectable oscillation; in fact, it would not be detectable at all.
The following diagram illustrates some typical amplitudes and
wavelengths of gravitational waves across this entire spectrum, and
the sensitivities of several detection methods. Some of these sources
are quite speculative, or have highly uncertain amplitudes. There are
also many more speculative sources that have not been included here.
The h axis is not the raw, instantaneous strain of the
source. (In particular some sources are stochastic in nature and have
no well-defined "instantaneous" strain.) Instead, it is the
"characteristic" strain that one obtains by accumulating the signal
over some timescale. For rotating sources the strain is also averaged
over possible inclination angles. Specifically:
- For sources with a well-defined h(t), one can
accumulate the power in the signal over many cycles N, giving
hc² ≈ ∫ h(t)² fdt ≈ Nh0²/8, where h0 is
the peak amplitude and 1/8 approximates the time and angle averaging.
For nearly constant-frequency sources, we use
N = f ×1 year. For sources that
sweep through a large frequency range in one year, the number of
cycles it spends near a given frequency is
N ≈ f ²/(df/dt).
In the latter case, we plot one year's worth of signal, sweeping
through amplitude and frequency, from the strongest such source that
one might expect in any given year.
- For sources of a stochastic nature, we do not have a well-defined
h(t). Instead, such signals have a power spectral
density Sh(f) such that the
mean squared fluctuations in h are given by
h² = ∫ Sh(f) df. By correlating the
signal between two detectors, one can achieve a further improvement in
the signal strength, by a factor of N1/2, where N is the number of cycles
correlated. We therefore define a characteristic strain hc² ≈ N1/2f Sh(f), where
N = f ×1 year, or
N = 1 for f < 1/year.
- For the various detectors, their sensitivities are defined by the
power spectral density of their noise Sn(f). As above we can define a
characteristic noise strain hn = f Sn(f). A signal is detectable in a given
detector if hc exceeds hn by a factor of a few (the signal-to-noise
- Relic background: A stochastic
signal from the Big Bang itself, this consists of quantum fluctuations
in the initial explosion that have been amplified by the early
expansion of the Universe. While the spectral shape of this source
can be predicted, its overall strength is highly uncertain, but is
constrained by the fact that gravitational wave perturbations are one
of several components contrbuting to the observed temperature
fluctuations in the cosmic microwave background. This limits the
maximum strength of gravitational waves at cosmological lengthscales.
Two curves are shown: one at the upper limit of the observational
constraints, and another an order of magnitude weaker.
- Binary background: Another
stochastic signal, this one arising from thousands of binary systems
emitting gravitational waves continuously in overlapping frequency
bands. The individual signals are unresolveable. At long wavelengths
(larger than 1014m), the binaries in question
are pairs of supermassive black holes (millions of times the mass of
the Sun) orbiting in the centres of galaxies. The hump at shorter
wavelengths (1013 to 1011m) is contributed by binary white dwarf stars
within our own Galaxy.
- SMBHB (Super-Massive Black Hole
Binaries): Occasionally, one of the supermassive black hole
systems mentioned above will merge, producing a huge burst of
gravitational waves at millihertz frequencies. Such bursts would be
detectable throughout most of the known Universe, though the rate is
highly uncertain: one per year is an optimistic estimate.
- WDB (White Dwarf Binaries):
Above the white dwarf stochastic background are a few thousand
individually-resolveable white dwarf binary systems in our Galaxy.
Some of these systems have already been charted with conventional
astronomy, and thus would be known callibrators for future
- EMRI (Extreme-Mass-Ratio
Inspirals): These are compact stellar remnants (white
dwarfs, neutron stars, or stellar-mass black holes only a few times
more massive than our Sun) in the process of being captured and
swallowed by a supermassive black hole (millions of times more massive
than the Sun).
- BHB (Black Hole Binaries): These
are binary systems consisting of two stellar-mass black holes (a few
times the mass of the Sun).
- NSB (Neutron Star Binaries):
These are binary systems consisting of two neutron stars.
- NS (Neutron Stars): This refers
to the gravitational waves generated by individual neutron stars as
they spin. In order to generate gravitational waves, the neutron star
must deviate from pure axisymmetry. Several mechanisms have been
proposed for generating or sustaining such asymmetries, but their
magnitudes are highly uncertain; the plot indicates some optimistic
- Cosmic microwave background:
Several thousand years after the Big Bang, when the hot plasma of
protons and electrons cooled and combined to form the first atoms,
electromagnetic radiation was released into the newly-transparent
Universe. Today, this cooled and redshifted radiation is seen as a
pervasive microwave background. Density fluctuations in the plasma
resulted in small fluctuations in observed temperature across the sky,
but long-wavelength gravitational waves will also contribute their own
perturbations to the spectrum. At present these contributions are
difficult to separate out, so the total observed fluctuations place an
upper limit on the size of gravitational-wave fluctuations.
- Pulsar timing: Pulsars are
spinning neutron stars that emit beams of electromagnetic radiation,
seen as "pulses" when they sweep over the Earth. Since the spin of a
neutron star is very stable, these pulses can be predicted and fit
with high precision. A passing gravitational wave alters the path
length between the pulsar and the Earth, changing the pulse arrival
times in a fluctuating manner. The lack of such fluctuations can be
interpreted as an upper limit on gravitational waves that have wave
periods shorter than the total duration of the pulse observations
(years or decades). A gravitational wave could be detected if two or
more pulsars show a correlated pattern of fluctuations in pulse
- LIGO (Laser Interferometer
Gravitational-wave Observatory): This consists of two
facilities in separated locations in North America. Each facility has
an L-shaped vacuum tube 4 kilometres long, with masses hanging at the
corner and ends of each arm, carefully shielded against vibrations or
other outside disturbances. A passing gravitational wave changes the
relative distances between the masses in the two arms, which can be
detected by interfering laser beams traveling along each arm.
Present-day sensitivity is at a level where detection of gravitational
waves is plausible, if not likely.
- Pulsar timing array: In the next
decade, it is expected that discoveries of new pulsars, improvements
in the precision of pulse timing measurements, and longer observations
of pulsars, will result in a dramatic improvement in the sensitivity
of pulsar timing to gravitational waves. The "pulsar timing array"
refers to this coordinated detection effort.
- LISA (Laser Interferometer Space
Antenna): This is a space mission, planned to launch within
the next ten years or so, that would place three spacecraft in Solar
orbit 5 million kilometres apart. The spacecraft would use laser
ranging to monitor their relative separations, and thus would be
sensitive to changes caused by passing gravitational waves. This is
essentially similar to LIGO, only a million times larger in size, and
sensitive to wavelengths several million times longer. Unlike other
detectors, LISA will have several known (through optical astronomy)
gravitational-wave sources that it will be able to use as
- Advanced LIGO: Continual
improvements to the LIGO detectors will result in an
order-of-magnitude improvement in sensitivity over the next few years.
At that level of sensitivity, gravitational-wave detections are likely
to occur on a daily or weekly basis.
Sections marked with provide optional
additional mathematical detail.
Start: Gravitational waves demystified
Analogy: Electromagnetic fields
Electromagnetic field of an accelerated charge
Derivation of the radiative electromagnetic field
Gravitational tidal field
Equivalence between dipole and tidal field
Formulae and details
Differences between gravitational and electromagnetic radiation
Gravitational wave spectrum