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.