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 diagram below shows the amplitudes of some known and
expected sources across the full gravitational-wave spectrum,
along with the sensitivities of some current and planned
detectors. The horizontal
axis 𝑓𝑔𝑤
(or λ𝑔𝑤)
is straightforward, but the vertical axis requires some
explanation: we want to define a characteristic
strain ℎ𝑐
that can be compared to a detector
noise ℎ𝑛
to indicate a source's detectability. This necessarily depends
on the nature of the source's
waveform ℎ(𝑡) and how it
can be identified in the detector output.
- For sources with a
well-defined ℎ(𝑡), one
can accumulate the power in the signal over many
cycles 𝑁,
giving ℎ2𝑐 =
∫ℎ(𝑡)2𝑑𝑡
= ½𝑁ℎ20
where ℎ0
is the peak amplitude. For nearly constant-frequency sources,
𝑁 = 𝑓𝑇
where 𝑇 is the observation
time. For sources that sweep through a large frequency
range in one year, the number of cycles it spends near a given
frequency is 𝑁 ∼
𝑓2/𝑓̇.
Conventionally, ℎ(𝑡)
refers to strain measured in an optimally-oriented frame,
so typical
(angle-averaged) ℎ𝑐
may be reduced by a factor ∼ 0.4.
- Stochastic sources have no
well-defined ℎ(𝑡) but
instead have a power spectral
density 𝑆ℎ(𝑓),
such that the mean squared fluctuations
in ℎ are
⟨ℎ2⟩
=
∫𝑆ℎ(𝑓) 𝑑𝑓.
Note
that 𝑆ℎ(𝑓)
has units of strain2/Hz; we define
a corresponding
quantity √𝑓 𝑆ℎ(𝑓)
with units of strain. By correlating the signal between
multiple detectors over many cycles, one can achieve a
further improvement, but only by a factor of
𝑁1/2
in power
(𝑁1/4
in strain amplitude), giving a characteristic strain for
stochastic
sources ℎ𝑐
≈
𝑁1/4√𝑓 𝑆ℎ(𝑓)
where 𝑁 =
𝑓𝑇.
- A detectors's sensitivity to gravitational waves is
determined by its instrumental noise divided by
its response to an (optimally oriented) incident wave.
The result is an effective noise power spectral
density 𝑆𝑛(𝑓)
and characteristic noise
ℎ𝑛 =
√𝑓 𝑆𝑛(𝑓).
The signal-to-noise ratio of a given source in a given
detector is (approximately) given
by ℎ𝑐/ℎ𝑛.
Typically this depends on the observation
time 𝑇. In the following
diagram we assume a canonical value
of 𝑇 = 1 year: for
continuous sources, this sets the number of cycles of
integration 𝑁; for transient
sources, we show the final 1 year of the signal from the
loudest transient that we expect to occur in a given 1 year
period. (For gravitational-wave frequencies below 1/year we
treat 𝑁≈1: signals are
only detectable if their instantaneous strain exceeds
the estimated noise.)
Sources:
- Primordial 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 slope of this source can be predicted, its overall
strength is highly uncertain, but is constrained at
cosmological lengthscales by observations of the Cosmic
Microwave Background (below).
- Foreground: A stochastic
signal from thousands of binary systems emitting overlapping
gravitational waves. The individual signals are
unresolveable, and effectively obscure any weaker signals at
these frequencies. At long wavelengths
(≳ 1014 m) the
sources are supermassive black hole binaries (SMBHB, below);
at shorter wavelengths they are white dwarf binaries (WDB) in
our Galaxy. As of 2025, preliminary indications of a
stochastic foreground (presumed to be SMBHBs) have been
observed via pulsar timing (below).
- SMBHB (Super-Massive Black Hole
Binaries): Sweeping signals from the inspiral and merger
of pairs of black holes, each one millions or billions of
times the mass of our Sun. Such systems may be found at the
centres of distant galaxies. As their orbit speeds up, the
gravitational-wave signal will emerge from the confusion
foreground (above) and be detectable throughout the known
Universe by space-based detectors. The rate, however, is
highly uncertain; one per year might be optimistic.
- WDB (White Dwarf Binaries):
Continuous signals from pairs of compact stellar remnants
between 0.5 and 1.5 times the mass of our Sun, left over when
ordinary Sunlike stars exhaust their nuclear fuel. Most such
binaries in our Galaxy will be part of the stochastic
foreground (above), but a few hundred to a few thousand should
be individually resolveable by space-based detectors, with
higher amplitude or frequency than the foreground. WDB
systems may eventually inspiral and merge, but such events are
rarely detectable (perhaps once every few centuries in our
Galaxy).
- BHB (Black Hole Binaries):
Sweeping signals from the inspiral and merger of black holes
with masses of order tens of Solar masses
(𝑀☉).
These are the predominant source of gravitational wave signals
observed to date. The first detected gravitational wave
signal in 2015 came from the merger of a
29 𝑀☉
and a
36 𝑀☉
Solar mass black hole, to form a new
62 𝑀☉
black hole (radiating the remaining
3 𝑀☉𝑐2
of energy in less than a second in the form of gravitational
waves). As of 2025, similar inspirals are being detected by
LIGO at a rate of over 100/year out to distances of over
109 parsecs.
- NSB (Neutron Star Binaries):
Sweeping signals from the inspiral and merger of pairs of
neutron stars (NS, below), with masses around 1.5 to 2 Solar
masses. Originally expected to be the primary sources for
LIGO, they have proven to be somewhat elusive: as of 2025,
only two events have been observed, plus one intermediate case
(where the more massive member might be a NS or a BH).
- NS (Neutron Stars):
Continuous signals from individual neutron stars. These are
compact remnants formed from supernova explosions at
the end of the active lives of certain massive stars. They
are (essentially) giant atomic nuclei,
≈ 20 km across and
≈ 1.5× as massive as the Sun, spinning at
up to hundreds of rotations per second. Individual neutron
stars may emit spin-generated gravitational waves if they are
non-axisymmetric, due to "mountains", magnetic distortions, or
other irregularities. As of 2025 these signals have not been
detected, and their amplitudes are highly uncertain; the plot
indicates some optimistic upper limits from known
objects.
Detectors:
- 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.
Cosmic-scale density fluctuations and gravitational waves
resulted in small variations in background temperature. These
contributions are difficult to separate out, so the observed
temperature variations place an upper limit on
long-wavelength gravitational waves.
- Pulsar timing: Pulsars are
spinning neutron stars (above) 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 with high precision. A
passing gravitational wave alters the path length between the
pulsar and the Earth, changing the pulse arrival times. While
many effects could alter an individual pulsar's
observed timing, correlated perturbations of multiple
pulsars in a given direction can be used to infer the presence
of gravitational radiation. The first observations of such an
effect were announced in 2020.
- LIGO (Laser Interferometer
Gravitational-wave Observatory):
This observatory comprises two detectors at separate locations
in North America. Each has an L-shaped vacuum tube
4 km long, with masses hanging at the corner and ends
of each arm, isolated from outside disturbances. Similar
detectors exist at other locations worldwide. 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. In 2015, LIGO made the
first direct detection of gravitational waves, which came from
a binary black hole inspiral. In the subsequent decade,
nearly 300 more inspirals have been detected.
Future detectors:
- LISA (Laser Interferometer Space
Antenna):
This planned mission will place three spacecraft in Solar
orbit 2.5 million kilometres apart. The spacecraft will
use laser ranging to monitor their relative separations, and
so will be sensitive to changes caused by passing
gravitational waves. This is similar to LIGO, but a million
times larger in size, and sensitive to wavelengths millions of
times longer. Unlike other detectors, LISA has multiple known
continuous gravitational-wave sources (confirmed white dwarf
binaries) that it can use as callibrators.
- ET (Einstein Telescope): This
is one of several proposed "3rd Generation"
laser-interferometric gravitational-wave detectors, with
comparable sensitivities. Their design is similar to LIGO,
but with somewhat longer arms, and with improved isolation and
readout systems.