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 ratio).

Sources:

• 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 gravitational-wave detectors.
• 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 upper limits.

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. 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 arrival times.
• 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.

Future detectors:

• 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 callibrators.
• 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
Electromagnetic waves
Gravitational tidal field
  Equivalence between dipole and tidal field
Gravitaional waves
  Formulae and details
Differences between gravitational and electromagnetic radiation
Gravitational wave spectrum