Gravitational Waves: Ground-Based Interferometric Detectors

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Marie Anne Bizouard and Nelson Christensen (2023), Scholarpedia, 18(12):53893. doi:10.4249/scholarpedia.53893 revision #201571 [link to/cite this article]
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Curator: Nelson Christensen




A century after their prediction by Albert Einstein, gravitational waves, produced from a binary black hole merger, were directly detected for the first time in 2015 by the two Michelson interferometers that form the Laser Interferometer Gravitational-wave Observatory (LIGO). LIGO, along with the European detector Virgo, have commenced a global network of gravitational-wave detectors that have launched the era of gravitational-wave astronomy. Together LIGO and Virgo have announced the detection of more than 90 gravitational-wave events from their first three observational runs in the advanced detector era (O1, O2 and O3; see Figure 16 for the dates). The Japanese detector, KAGRA, joined the network in 2020. The majority of the observations have come from binary black hole mergers, but binary neutron star mergers and neutron star - black hole mergers have also been observed, and are presented in catalog GWTC-3. LIGO, Virgo and KAGRA started their fourth observational run, O4, in mid 2023.

With arm lengths of 3 km for Virgo and KAGRA, and 4 km for LIGO, these detectors can measure a relative displacement of the interferometers’ arm lengths of \( 10^{-18} \) m or smaller. Presented here is a description of the properties of these Michelson interferometers that can measure disturbances of spacetime created by merging binary neutron star and black holes systems millions to billions of light-years away.

Albert Einstein postulated the existence of gravitational waves in 1916, and Joe Taylor and Joel Weisberg indirectly confirmed their existence through observations of the orbital decay of the binary pulsar 1913+16 system. The direct detection of gravitational waves has been difficult, and has taken decades of demanding experimental work to accomplish. The only possibility for producing detectable gravitational waves comes from extremely massive and non-symmetrical objects accelerating up to relativistic velocities. For example, the gravitational waves from the first detection GW150914 were produced by the merger of a \( 29 M_{\odot} \) black hole and a \( 36 M_{\odot} \) black hole some \( 1.3 \times 10^{9} \) light-years away. Note that \( M_{\odot} \) is the mass of our sun, or \( \sim 2 \times 10^{30} \) kg. The total energy radiated in gravitational waves was equivalent to \( 3 M_{\odot} c^2 \), with a peak luminosity of \( 3.6 \times 10^{56} \) ergs/s. The mergers of two neutron stars have also been observed, GW170817, in 2017. Gravitational waves from the final stages of coalescing binary neutron stars help to accurately determine the size of these objects and constrain the equation of state of nuclear matter. GW170817 also confirmed one of the mechanisms for the production of short gamma-ray bursts, and provided a new way to estimate the Hubble constant.

Yet, other sources of gravitational waves are possible: core collapse supernovae, pulsars, neutron star binary systems, newly formed black holes, or even early universe inflation. The observation of these types of events would be extremely significant for contributing to knowledge in astrophysics and cosmology. Gravitational waves from the Big Bang would provide unique information of the universe at its earliest moments. Observations of core-collapse supernovae will yield a gravitational snapshot of these extreme cataclysmic events. Pulsars are neutron stars that can spin on their axes at frequencies up to hundreds of Hz, and the signals (due to a slight non-spherical shape) from these objects will help to decipher their characteristics and aid in studying ultra-dense nuclear matter.

LIGO, Virgo and KAGRA search for gravitational waves from 10 Hz up to a few kHz. The current detector target sensitivities allow them to observe signals from the coalescence of binary neutron star systems (\( 1.4 M_{\odot} - 1.4 M_{\odot} \)) possibly out to distances of 325 Mpc for LIGO and 260 Mpc for Virgo, and 128 Mpc for KAGRA. The mergers of more massive binary black holes systems can be sensed to much greater distances, as described here.

Some General Relativity

Electromagnetic radiation has an electric field transverse to the direction of propagation, and a charged particle interacting with the radiation will experience a force. Similarly, gravitational waves will produce a transverse force on massive objects, a tidal force. Explained via general relativity it is more accurate to say that gravitational waves will deform the fabric of space-time. Let us imagine a linearly polarized gravitational wave propagating in the z-direction, \( h(z,t) = h_{0+} e^{i(kz - \omega t)} \). The fabric of space is stretched due to the strain created by the gravitational wave. Consider a length \( L_{0} \) of space along the x-axis. In the presence of the gravitational wave the length oscillates like\[ L(t) = L_{0} + \frac{L_{0} h_{0+}}{2} cos(\omega t) \] hence there is a change in its length of\[ \Delta L_{x} = \frac{L_{0} h_{0+}}{2} cos(\omega t) ~. \] A similar length \( L_{0} \) of the y-axis oscillates, like\[ \Delta L_{y} = - \frac{L_{0} h_{0+}}{2} cos(\omega t) ~. \] The amplitude of a gravitational wave is thus the amount of strain that it produces on spacetime. Just like electromagnetic radiation there are two polarizations for gravitational waves. The other gravitational wave polarization (\( h_{0\times} \)) produces a strain on axes \( 45^o \) from (x,y); contrast this with electromagnetic waves where the polarizations are at \( 90^o \) to each other. Imagine some astrophysical event produces a gravitational wave that has amplitude \( h_{0+} \) on Earth; in order to detect the small distance displacement \( \Delta L \) one should have a detector that spans a large length \( L_0 \). The first gravitational wave observed by LIGO, GW150914, had an amplitude of \( h \sim 10^{-21} \) with a frequency at peak gravitational-wave strain of 150 Hz. The amplitude of a gravitational wave falls off as \( 1/r \), so it will be impossible to observe events that are too far away. However, when the detectors' sensitivity is improved by a factor of \( n \), the rate of signals should grow as \( n^3 \) (the increase of the observable volume of the universe). This is because the gravitational-wave detectors to be discussed below measure signals from almost all directions; they cannot be pointed, but reside in a fixed position on the surface of the Earth.

A Michelson interferometer can measure small phase differences between the light in the two arms. Therefore, this type of interferometer can turn the length variations of the arms produced by a gravitational wave into changes in the interference pattern of the light exiting the system. This was the basis of the idea from which modern laser interferometric gravitational wave detectors have evolved. After many years of upgrades the LIGO, Virgo and KAGRA detectors currently measure distance displacements that are of order \( \Delta L \sim 10^{-18} \) m or smaller, much smaller than an atomic nucleus.

Interferometric Gravitational-Wave Detectors

Presently numerous collaborations are operating second generation interferometers in order to detect gravitational waves. LIGO in the United States consists of two 4 km observatories located in Livingston, Louisiana and Hanford, Washington. With the current 'Advanced' interferometers, LIGO started observations in 2015, and will be working over the coming years to achieve its design sensitivity, with the goal to reach it by the later 2020s. The European Advanced Virgo is a 3 km interferometer near Pisa, Italy, and started acquiring data in 2017, and will also be aiming for its target sensitivity in the coming years. GEO 600, a German-British collaboration, is a 600 m detector near Hanover, Germany, and is currently operational. KAGRA is the Japanese 3 km interferometer that commenced observations in 2020. There will be a third 4 km LIGO observatory, LIGO-India, located in India, with the goal to be operational in the early 2030s. All of the km-length terrestrial detectors will be attempting to detect gravitational waves with frequencies from 10 Hz up to a few kHz.

Next generation detectors are also being planned, with sensitivity increases of about 10 over the second generation detectors via greater length (10 to 40 km) and evolving interferometer technology. These include the Einstein Telescope in Europe, Cosmic Explorer in the United States, and NEMO in Australia. The hope is to have these detectors operational in the late 2030s.

There are a number of terrestrial noise sources that limit the performance of the interferometric detectors. In addition, the sensitivity of detection increases linearly with interferometer arm length, which implies that there could be advantages to constructing a gravitational-wave detector in space. This is the goal of the Laser Interferometer Space Antenna (LISA). The plan is to deploy three satellites in a heliocentric orbit with a separation of about \( 2.5 \times 10^{6} \) km. LISA is led by European Space Agency, with a target launch date in the mid 2030s. LISA will observe gravitational waves in a frequency band from roughly \( 10^{-4} \) Hz to \( 10^{-1} \) Hz. Due to the extremely long baseline, LISA is not strictly an interferometer, as most light will be lost as the laser beams expand while traveling such a great distance. Instead, the phase of the received light will be detected and used to lock the phase of the light that is re-emitted by another laser.

Interferometer Configurations

The Michelson interferometer is an optical configuration that is ideal to detect a gravitational wave. Figure 1 shows a basic optical setup. A single-frequency continuous wave laser illuminates a beamsplitter which transmits half the light and reflects the other half. The light is reflected back by mirrors on the surface of the end test masses (the freely suspended mirrors, using the terminology of Einstein's theory of general relativity), and recombines at the beamsplitter, effectively comparing the round-trip time in (and thus length of) the two arms. The beamsplitter and the end mirrors would be suspended by wires, and effectively free to move in the plane of the interferometer. The arms have lengths \( L_1 \) and \( L_2 \) that are roughly equal on a kilometer scale. With a laser of power \( P \) and wavelength \( \lambda \) incident on the beamsplitter, the light exiting the dark port of the interferometer (namely the light moving toward the photodetector) is\[ P_{out} = P ~ sin^2\bigg[\frac{2 \pi}{\lambda} (L_1 - L_2)\bigg] ~ . \]

Figure 1: A basic Michelson interferometer. The photodetector receives light exiting the dark port of the interferometer and hence the signal.

The interferometer operates with the condition that in the absence of excitation the power of the light exiting the dark port is very close to zero. This would be the case for a simple and basic interferometer. If \( E_0 \) is the amplitude of the electric field for the laser light, and assuming the use of a 50-50 beamsplitter, the electric field (neglecting unimportant common phase shifts) for the light incident on the photodetector would be\[ E_{out} = \frac{E_0}{2} (e^{i \delta \phi_1} - e^{i \delta \phi_2}) \approx i \frac{E_0}{2} (\delta \phi_1 - \delta \phi_2) = i \frac{E_0}{2} \frac{2 \pi}{\lambda} (L_1 - L_2) ~ . \]

A gravitational wave of optimal polarization normally incident upon the interferometer plane will cause one arm to decrease in length while the other increases. The Michelson interferometer acts as a gravitational wave transducer; the stretching and squeezing of the spacetime between the mirrors results in changes in the light intensity exiting the interferometer dark port. The suspended mirrors are free to move under the influence of the gravitational wave, acting like freely-falling masses as described in Einstein's relativity.

An interferometer's sensitivity to gravitational waves increases with arm length (until the arms are comparable to the gravitational wavelength), but geographical, physical, and financial constraints will limit the size of the arms. If there could be some way to bounce the light back and forth to increase the effective arm length it would increase the detector performance. Fabry-Perot cavities do just that. These Fabry-Perot cavities are formed from the end test mass optical surface and an additional optic or input test mass optical surface, and the input optic has some transmission to couple light in and out of the cavity. When the cavity length is an integral number of half-laser-wavelengths, a standing wave is set up, and it is said to be 'on resonance'. The resulting storage time for the light of\[ \tau_s = \frac{2 L (R_1 R_2)^{1/4}}{[c(1 - \sqrt{R_1 R_2})]} ~ , \] where (\( R_1 \) and \( R_2 \) are power reflection coefficients). Figure 2 shows the system of a Michelson interferometer with Fabry-Perot cavities. The far mirror \( R_2 \) has a very high reflectivity (\( R_2 \sim 1 \)) in order to ultimately direct the light back towards the beamsplitter. The front mirror reflectivity \( R_1 \) is such that the detetor's effective arm length increases from \( L = 4 \) km to \( L \sim 560 \) km. A LIGO test mass (and therefore a Fabry-Perot mirror) can be seen in Figure 3. An Advanced Virgo test mass can be seen on the right in Figure 4.

Figure 2: A Michelson interferometer with Fabry-Perot cavities in each arm. The front cavity mirrors have reflectivity \( R_1 \) while the end mirrors have \( R_2 \sim 1 \). By using Fabry-Perot cavities Advanced LIGO will increase the effective arm length by a factor of 140.
Figure 3: A picture of the input test mass (mirror \( R_1 \) ) for Advanced LIGO within its vibration isolation suspension system. The fused silica mirror is 40 kg, 34 cm in diameter and 20 cm thick. Photograph courtesy of LIGO/Caltech/MIT.
Figure 4: One of the Advanced Virgo Fabry-Perot cavity mirrors, which also serves as a test mass for the gravitational-wave detector. The mirror appears pink due to a protection film, removed before use. Two fused silica wires suspend the 42 kg, 35 cm diameter, 20 cm thick mirror. Photograph courtesy of the Virgo Collaboration.

In 1888 Michelson and Morley, with their interferometer, had a sensitivity that allowed the measurement of 0.02 of a fringe, or about 0.126 radian. The Advanced LIGO interferometers have already demonstrated a phase noise spectral amplitude of\[ \phi(f) = 5 \times 10^{-11} \textrm{radian}/\sqrt{\textrm{Hz}} ~ \] for frequencies around 150 Hz. Note that a spectral density is a function that describes how the power of a signal is distributed across different frequencies, and the square root of the spectral density is called the spectral amplitude. Assuming a 150 Hz signal with a 150 Hz bandwidth this implies a phase sensitivity of \( \Delta \phi = 6.1 \times 10^{-10} \) radian. There has been quite an evolution in interferometry since Michelson's time.

The noise sources that limit the interferometer performance are discussed below. However, let us consider one's ability to measure the relative phase between the light in the two arms. The Heisenberg uncertainty relation for light with phase \( \phi \) and photon number \( N \) is \( \Delta \phi ~ \Delta N \sim 1 \). For a measurement lasting time \( \tau \) using laser power \( P \), the photon number is \( N = P \lambda \tau / hc \) (here \( h \) is Planck's constant), and with Poisson statistics describing the light \( \Delta N = \sqrt{N} = \sqrt{P \lambda \tau / hc} \). Therefore\[ \Delta \phi ~ \Delta N = \frac{2 \pi}{\lambda} \Delta L \sqrt{P \lambda \tau /hc} = 1 \] implies that\[ \Delta L = \frac{1}{2 \pi} \sqrt{h c \lambda /P \tau} ~ . \]

With more light power the interferometer can measure smaller distance displacements and achieve better sensitivity. Advanced LIGO and Advanced Virgo will use about 200 W of laser light. However, there is a nice trick one can use to produce more light circulating in the interferometer, namely power recycling. Figure 5 displays the power recycling interferometer design. The interferometer operates such that virtually none of the light exits the interferometer dark port, and the bulk of the light returns towards the laser. An additional partially transmitting mirror, \( R_r \), recycles the light, effectively making a 'super Fabry-Perot' cavity with the interferometer as one 'mirror'. The Advanced LIGO goals are to have 125 W actually impinging on the recycling mirror \( R_r \), creating 5.2 kW upon the beamsplitter, and 750 kW within the Fabry-Perot cavities in each of the interferometer's arms. Advanced Virgo has a similar design. The higher circulating light power therefore improves the sensitivity of these interferometric detectors.

Figure 5: A power recycled Michelson interferometer with Fabry-Perot cavities in each arm. Normally light would exit the interferometer through the light port and head back to the laser. Installation of the recycling mirror with reflectivity \( R_r \) sends the light back into the system. A Fabry-Perot cavity is formed between the recycling mirror and the first mirror ( \( R_1 \) ) of the arms. For Advanced LIGO this strategy will increase the power circulating in the interferometer by a factor of 42.

There is one additional modification to the interferometer that can further improve sensitivity, but only at a particular range of frequencies. A further Fabry-Perot system can be made by installing what is called a signal recycling mirror; this would be mirror \( R_s \) in Figure 6. Imagine the light in arm 1 of the interferometer, and that it acquires additional phase as the arm expands due to a gravitational wave. The traveling gravitational wave's oscillation will subsequently cause arm 1 to contract while arm 2 expands. If the light that was in arm 1 could be sent to arm 2 while it is expanding, then the beam would acquire yet more phase. This process could be repeated over and over. Mirror \( R_s \) serves this purpose, with its reflectivity defining the storage time for light in each interferometer arm. The storage time defined by the cavity formed by the signal recycling mirror, \( R_s \), and the mirror at the front of the interferometer arm cavity, \( R_1 \), determines the resonance frequency. Signal recycling will give a substantial boost to interferometer sensitivity in a particular range of frequencies, and is implemented in all the main ground based interferometric detectors. It must be added that the Advanced LIGO, Advanced Virgo and KAGRA interferometers are significantly more complex than the relatively simple systems displayed in the figures of this article.

Figure 6: A signal recycled and power recycled Michelson interferometer with Fabry-Perot cavities in each arm. Normally light containing the gravitational-wave signal would exit the interferometer via the dark port and head to the photodetector. Installation of the signal recycling mirror with reflectivity \( R_s \) sends the light back into the system. The phase of the light acquired from the gravitational wave will build up at a particular frequency determined by the reflectivity \( R_s \).

Figure 7 presents an aerial view of the LIGO site at Hanford, Washington State in the USA. The magnitude of the 4 km system is apparent. Figure 8 displays the Virgo detector with its 3 km, located near Pisa, Italy.

Figure 7: Aerial view of the LIGO Hanford, Washington site. The vacuum enclosure at Hanford contains the 4 km interferometer. Photograph courtesy of LIGO/Caltech/MIT.
Figure 8: Aerial view of the Virgo detector, with 3 km arms, located near Pisa, Italy. Photograph courtesy of the Virgo Collaboration.

Noise Sources and Interferometer Sensitivity

If the interferometers are to detect distance displacements less than \( 10^{-18} \) m then they must be isolated from a host of deleterious noise sources. Seismic disturbances should not shake the interferometers. Thermal excitation of components can mask the gravitational-wave signal and should be minimized. The entire interferometer must be in an adequate vacuum in order to avoid fluctuations in gas density that would cause changes in the index of refraction and hence a modification of the optical path length. The laser intensity and frequency noise must be minimized. The counting statistics of photons is a sensing noise which must be reduced however possible. A figure showing the principal limitations to the sensitivity for the Advanced LIGO interferometers is displayed in Figure 9.

Figure 9: The target spectral density of the noise for the Advanced LIGO system. Advanced LIGO will be limited by seismic noise at low frequencies (\( \sim 10 \) Hz), thermal noise (from the suspension system and the coatings of the mirrors) in the intermediate regime (\( \sim 100 \) Hz}). Radiation pressure will also be an important noise source from \( \sim 10 \) Hz} to \( \sim 100 \) Hz}, while photon shot noise will be the limiting noise at frequencies higher than a few hundred Hz. Together the radiation pressure noise and the shot noise are referred to as quantum noise. Other sources of noise are also noted in the figure. Figure courtesy of LIGO/Caltech/MIT.

The expected noise sensitivity for Advanced Virgo is displayed in Figure 10, along with the sum of its expected noise sources.

Figure 10: The target Advanced Virgo sensitivity due to expected noise sources. This has been calculated for the configuration optimized for binary neutron star detection with 125 W of input into the interferometer laser power. Figure courtesy of the Virgo Collaboration.

Shot Noise

An ideal situation would be to have the interferometer sensitivity limited by the counting statistics of the photons. A proper functioning laser will have its photon number described by Poisson statistics, or shot noise; if the mean number of photons arriving per unit time is \( N \) then the uncertainty is \( \Delta N = \sqrt{N} \), which as noted above implies an interferometer displacement sensitivity of\[ \Delta L = \frac{1}{2 \pi} \sqrt{\frac{h c \lambda}{P \tau}} \] where \( P \) is the light power impinging on the beamsplitter, or a spectral density of\[ \Delta L(f) = \frac{1}{2 \pi} \sqrt{\frac{h c \lambda}{P}} \] in units of \( \textrm{m}/\sqrt{\textrm{Hz}} \). Note also that the sensitivity increases as the light power increases. The reason for this derives from the statistics of repeated measurements. The relative lengths of the interferometer arms could be measured, once, by a photon. However, the relative positions are measured repeatedly with every photon from the laser, and the variance of the mean decreases as \( \sqrt{N} \) where \( N \) is the number of measurements (or photons) involved. The uncertainty in the difference of the interferometer arm lengths is therefore inversely proportional to the square root of photon number, and hence the square root of the laser power. In terms of strain sensitivity this would imply\[ h(f) = \frac{1}{2 \pi L} \sqrt{\frac{h c \lambda}{P}} ~ , \] which has units of \( 1/\sqrt{\textrm{Hz}} \). This assumes the light just travels down the arm and back once. With Fabry-Perot cavities the light is stored, and the typical photon takes many trips back and forth before exiting the system. In order to maximize light power the end mirrors (\( R_2 \sim 1 \)) and the strain sensitivity is improved to\[ h(f) = \frac{1}{4 \pi \tau_s} \sqrt{\frac{\pi \hbar \lambda}{P c}} ~ , \] where the Fabry-Perot cavity storage time \( \tau_s \) was defined above.

As the frequency of gravitational waves increases the detection sensitivity decreases. If the gravitational wave causes the interferometer arm length to lengthen and shorten, while the photons are still in the arm cavity, then the phase acquired from the gravitational wave will be washed away. This is the reason why interferometer sensitivity decreases as frequency increases, and explains the high-frequency behavior seen in Figure 9 and Figure 10. Taking this into account, the strain sensitivity is\[ h(f) = \frac{1}{4 \pi \tau_s} \sqrt{\frac{\pi \hbar \lambda}{P c}} \left(1 + \left(4 \pi f \tau_s \right)^2\right)^{1/2} \] and \( f \) is the frequency of the gravitational wave.

Seismic Noise

If the gravitational wave is to change the interferometer arm length then the mirrors that define the arm must be free to move like freely-falling masses. Seismic noise will be troublesome for the detector at low frequencies. In systems like Advanced LIGO, Advanced Virgo and KAGRA, wires suspend the mirrors; each mirror is like a pendulum. The mirrors and the wires that suspend them are a monolithic fused silica assembly, with the wires annealed and welded to the sides of the mirrors. The pendulum itself is the first component of an elaborate vibration isolation system. The spectral density of the seismic noise is about \[ x(f) = \left(10^{-9} m/\sqrt{\textrm{Hz}}\right) \left(10 \textrm{Hz}/f\right)^2 \] for \( f > 10 \textrm{Hz} \) (the low frequency observational limit for Advanced LIGO, Advanced Virgo and KAGRA). A simple pendulum, by itself, acts as a motion filtering device. Above its resonance frequency a pendulum filters motion with a transfer function like \( T(f) \propto \left(f_0/f\right)^2 \), where \( f_0 \) is the resonant frequency for the pendulum. The mirrors for Advanced LIGO are actually suspended by a four-stage pendulum system. The various gravitational-wave detector collaborations have different vibration isolation designs. The mirrors in these interferometers are suspended in elaborate vibration isolation systems, which may include multiple pendulums, isolation stacks, and isolated optical tables. Active feedback is used on some parts of the isolation system to control seismic noise below \( \sim 10 \) Hz.

The mirror suspension chain used in Advanced Virgo is shown on Figure 11. This so called Superattenuator is composed of an inverted pendulum fixed to the ground, and a series of wires and mechanical filters attached to the top of the inverted pendulum. Their purpose is to filter the ground motion by repeatedly applying the \( (f_0/f)^2 \) factor to all degrees of freedom, horizontal for the inverted pendulum and wires, and vertical for the mechanical filters which are basically vertical springs. The resonant frequencies of these stage are in the 10 mHz - 2 Hz range, so well below the intended sensitive band 10 Hz - 10 kHz, and the quality factors are rather low (\( Q < 100 \)). The quality factor, \( Q \), is a measure of the damping of a resonant system, indicating the sharpness of the resonance peak. A damped oscillator decreases the root mean square motion by a factor of \( Q \), so low quality factors are chosen in the filtering chain to reduce the motion of the high frequency part of the signal, while keeping the remaining motion (dominated by the low frequency part) small. Seismic noise, and the induced gravity gradient noise (changes in the local gravitational field due to changes in the matter density near to the detector), will be the limiting factor for terrestrial interferometers seeking to detect gravitational waves in the vicinity of \( \sim 10 \) Hz, as can be seen in the sensitivity curves presented in Figure 9 and Figure 10.

Figure 11: Schematic of the Advanced Virgo Superattenuator. It is composed of an inverted pendulum fixed to the ground, and a series of wires and mechanical filters attached to the top of the inverted pendulum. The total height is approximately 8 m. The purpose of this chain is to isolate the mirror suspended at the bottom of the apparatus from the ground motion. Figure courtesy of the Virgo Collaboration.

Thermal Noise

Thermal noise is a type of random electrical or mechanical signal that arises due to the thermal motion of particles. It is an inherent property of any physical system and sets limitations on the sensitivity of measurements. Due to the extremely small distance displacements that these systems are trying to detect it should come as no surprise that thermal noise is a problem. This noise enters through a number of components in the system. The two most serious thermal noise sources are the wires suspending the mirrors in the pendulum, and the mirrors themselves, especially the optical coatings on the mirror surfaces. Consider the wires; there are a number of notes at which they can oscillate (i.e. violin modes). At temperature \( T \) each mode will have energy of \( k_{B}T \), but distributed over a band of frequencies determined by the quality factor (or \( Q \)) of the material. Low-loss (or high-\( Q \)) materials work best; for the violin modes of the wires there will be much noise at particular frequencies (in the hundreds of Hz). For the Advanced LIGO mirrors the first violin mode is at 510 Hz, while the vertical stretching mode of the wires is at \( \sim 9 \) Hz.

The mirror is a cylindrical object, which will have normal modes of oscillation that can be thermally excited. Advanced Virgo and Advanced LIGO have test masses composed of fused silica, which is typical for optical components. The \( Qs \) for the internal modes are greater than \( 2 \times 10^6 \). It is interesting to note that KAGRA, which will eventually be operating at a temperature of 20 K, will be using test masses composed of sapphire. Sapphire has a lower thermal noise at 20 K as compared to fused silica. The best sensitivity for Advanced Virgo and Advanced LIGO occurs around \( \sim 100 \) Hz. The limiting source of noise in this region (along with quantum noise) is due to Brownian noise in the optical coatings on the mirror surfaces. This is another reason why KAGRA will ultimately be cooling the test masses of their interferometer. Note from Figure 10 that in the important band from 80 Hz to 400 Hz Advanced Virgo will be limited by the coating thermal (Brownian) noise; this will also be true for Advanced LIGO, see Figure 9. There is a tremendous amount of on-going research to try and reduce the mechanical dissipation in the optical coatings.

The High Power Laser and Various Noise Issues

The frequency noise and intensity noise of the laser can couple into the system to produce length displacement noise in the interferometer. With arm lengths of 4 km for LIGO and 3 km for Virgo, it is impossible to hold the length of the two arms of the interferometer absolutely equal. The slightly differing arm spans means that the light sent back from each of the two Fabry-Perot cavities have slightly arrival times (hence differing phases and intensities, since the laser output is sampled at different times)). As a consequence, great effort is made to stabilize the frequency and intensity of the light entering the interferometer. The Advanced LIGO laser can be seen in Figure 12. The primary laser is a non-planar ring-oscillator (NPRO). This beam is then amplified to 35 W with a medium power oscillator, and then up to 220 W with a high power oscillator. For Advanced LIGO, the laser is locked and held to a specific frequency by use of signals from a reference cavity, a mode cleaner cavity, and the interferometer. For low-frequency stabilization the temperature of the NPRO is adjusted. At intermediate frequencies adjustment is made by signals to a piezo-electric transducer within the NPRO cavity. At high frequencies the noise is reduced with the use of an electro-optic crystal. The Advanced LIGO lasers currently have a frequency noise of \( 1 \times 10^{-6} Hz/\sqrt{Hz} \) at 100 Hz; this requirement is needed by Advanced Virgo too. Advanced Virgo commenced observations in O2 with a medium-power laser that is a Nd:YVO4 oscillator (\( \lambda = 1.06 \mu m \)) that amplifies a 20 W injection-locked laser to a power of 60 W; this was increased to 100 W of available power for O3. In the O4 run Advanced Virgo is using a new technology, namely a fiber laser amplifier, that can provide 130 W.

Figure 12: The Advanced LIGO laser system. This is a multi-stage Nd:YAG system that can deliver 200 W. Photograph courtesy of LIGO/Caltech/MIT.

It is also important to worry about the stability of the laser power for the interferometric detectors. The goal is to be quantum noise limited at frequencies within the observational frequency band. The Nd:YAG power amplifiers used are pumped with an array of laser diodes, so the light power is stabilized through controlling the power of the laser diodes. The Advanced LIGO requirements for the fluctuations on the power \( P \) are \( \Delta P/P < 2 \times 10^{-9} /\sqrt{Hz} \) at 10 Hz; Advanced Virgo's requirements are similar.

In these laser interferometric gravitational-wave detectors, the spatial quality of the light is ensured through the use of an input mode cleaning cavity; this is a Fabry-Perot cavity where both input and output mirrors have a few percent transmission, and the light the exits is constrained to be of the stable optical mode supported by the cavity. Advanced LIGO uses an isosceles triangular array of mirrors with the two base mirrors separated by 0.465 m and the third mirror displaced by 16.24 m. Advanced Virgo uses a triangular array of mirrors (all suspended); the triangular cavity has a length of 143 m and a finesse of 1200. The entire input optics arrangement for Advanced Virgo is quite complicated; a diagram showing all of the components is displayed in Figure 13.

Figure 13: A display of the input optics layout for Advanced Virgo. Photograph courtesy of the Virgo Collaboration.

The optical system for Advanced LIGO is displayed in Figure 14. Aside from the laser and the phase modulator, the entire optical system is in an ultra-high vacuum. Note that at the output of the interferometer there is a signal recycling mirror (SRM). Given the reflectivity of the mirror, and the phase of the light when arriving at there, it is possible to amplify the gravitational-wave signal in a particular range of frequencies by feeding it back into the interferometer.

Figure 14: The Advanced LIGO optical system. The laser (\( \sim 200 \) W) light propagates from the stabilized laser through a phase modulator (\( \phi_m \)) to the input mode cleaner, then through a Faraday Isolator (FI) to the power recycling mirror (PRM). The folding mirrors, PR2 and PR3, direct the light to the beamsplittler (BS) input of the interferometer. Note that approximately 125 W of light impinges upon the power recycling mirror, resulting in 5.2 kW at the input port to the beam splitter. The main arm Fabry-Perot cavities are formed with the input test mass (ITM), which is paired with a compensation plate (CP), and the end test mass (ETM) which is paired with a end reaction mass (ERM). The arm Fabry-Perot cavities will contain 750 kW of light power. Note too that the output signal from the interferometer can itself be recycled and amplified at specific frequencies, dependent on the reflectivity and exact position of the signal recycling mirror (SRM); SR2 and SR3 are folding mirrors. The output beam also has its spatial features cleaned with the output mode cleaner before the light falls upon a photodetector (PD). Photograph courtesy of LIGO/Caltech/MIT.

When Advanced LIGO attains its target sensitivity there will be 750 kW within the arm Fabry-Perot cavities. Advanced Virgo's light powers will be similar. With such a large amount of power the mirrors will actually get heated and lightly change their shape. Ring heaters encircle the input test masses and the end test masses, and are used to correct the shape of the masses. A CO\(_2\) laser beam is also sent onto the surface of the compensation plate to provide further corrections to the thermal lens of the input test mass. The compensation plate serves as a reaction mass for the input test mass in its isolation suspension system; the same is true for the end reaction mass with respect to the end test mass.

Squeezed States of Light

The immense number of photons, coupled with the fact that the photon arrival times are random (Poisson statistics) means that radiation reaction noise will be important; this is the random motion due to transfer of momentum from the photons to the test mass. This can be seen in the low frequency component of the quantum noise in Figure 9 and Figure 10. The low-frequency radiation reaction noise, plus the high frequency shot noise, combine to create the total quantum noise in the interferometer. At high frequencies the shot noise decreases with laser power, while at low frequencies radiation reaction noise increases with the power; the interferometer's quantum noise is a tradeoff between these two effects. While this quantum noise seems to be an unavoidable noise source, quantum states of light (namely squeezed states of light) can reduce this noise.

Squeezed light is a type of light in which the quantum fluctuations in one of the light's properties, such as its amplitude or phase, are reduced below the standard quantum limit. This reduction of noise was implemented by Advanced LIGO where squeezed light reduced the noise below the shot noise level for frequencies above 50 Hz by as much as 3 dB; Advanced Virgo has also achieved similar results. Advanced LIGO and Advanced Virgo are presently using a method called frequency dependent squeezing to reduce both the shot noise at high frequencies, and the radiation reaction noise at low frequencies. The use of quantum states of light is one of the ways that LIGO, Virgo and KAGRA plan to reduce their noise in the years to come.

Gravitational-Wave Detections by LIGO and Virgo

On September 14, 2015, at 09:50:45 UTC a gravitational wave was detected directly for the first time. The gravitational wave was first observed at the LIGO Livingston Observatory (Louisiana), and then 7 ms later at the LIGO Hanford Observations (Washington). An on-line transient search algorithm identified the signal in 3 minutes. An off-line examination of the data using a template based search for compact binary coalescence signals identified the gravitational wave with a signal-to-noise ratio of 24. Parameter estimation routines were used to determine that the gravitational-wave signal was emitted from the merger of two black holes with masses of 36 \( M_{\odot} \) and 29 \( M_{\odot} \). The newly created black hole had a mass of 62 \( M_{\odot} \), meaning that the total energy of gravitational waves emitted was equivalent to 3 \( M_{\odot} c^2 \). The system was 1.3 billions light years away from us when it merged.

The time-varying stretching and squeezing of space due to the gravitational-wave signal, GW150914, from the two LIGO detectors is displayed in Figure 15 and Figure 16. The peak amplitude of GW150914 is \( h \sim 10^{-21} \) which corresponds to a displacement of the interferometers' arms of \( \Delta L \sim 2 \times 10^{-18} \) m. The exquisite sensitivity of these interferometers can be seen from these numbers.

Figure 15: The measured gravitational-wave signal GW150914 as observed at the LIGO Hanford interferometric detector. The data has been bandpass filtered 35 Hz to 350 Hz; the gravitational-wave signal is clearly observable by eye. Figure courtesy of the Gravitational Wave Open Science Center.
Figure 16: The measured gravitational-wave signal GW150914 as observed at the LIGO Livingston interferometric detector. The data has been bandpass filtered 35 Hz to 350 Hz. The signal as observed from the LIGO Livingston detector (blue). In addition, the Hanford signal (red) is superimposed after it has been displaced by 7 ms and inverted (due to the relative orientation of the two detectors). The similarity of the two measured signals is clearly visible. Figure courtesy of the Gravitational Wave Open Science Center.

In their first three observing runs Advanced LIGO and Advanced Virgo reported the observations of more than 90 gravitational-wave events; the majority of these were from binary black hole mergers, but there were also binary neutron star mergers, and neutron star - black hole mergers. See the Scholarpedia review, "Gravitational Waves: Science with Compact Binary Coalescences", for a more complete description. The binary neutron star event, GW170817, was also observed by both LIGO and Virgo detectors, and as such, the position of the event could be estimated to a spot size of 28 deg\(^{2}\) on the sky and a distance of \( 40^{+8}_{-14} \) Mpc. There was also a gamma-ray signal from the Fermi Gamma-ray Space Telescope and the INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL), observed 1.7 s after the merger time of the binary neutron star system. From the gravitational-wave and gamma-ray position estimates it was possible to find the location of the source of these signals with optical telescopes. GW170817 signaled the birth of gravitational-wave multimessenger astronomy.


The sensitivity improvements of the Advanced LIGO and Advanced Virgo detectors continue to evolve with improvements in their sensitivity. An important point is the arrival of the KAGRA gravitational-wave detector to the world-wide network. KAGRA commenced observations with LIGO and Virgo at the end of the third observing run O3 in 2020. A third LIGO detector, located in India, should join the world-wide gravitational-wave network in the late 2020s. The LIGO-Virgo-KAGRA observing scenario calls for a sucession of data taking periods separated with detector upgrades and commissioning periods to reach the detectors' target sensitivities. Figure 17, Figure 18 and Figure 19 show the evolution of the strain amplitude spectral densities for the three detectors in the coming years. The Advanced LIGO detectors are expected to reach a 190 Mpc BNS range (the average distance over which a binary neutron star merger can be detected) circa 2024. The target Advanced Virgo sensitivity is slightly lower, with a BNS range of 115 Mpc due to the shorter arm length. KAGRA will eventually reach its design sensitivity of 125 Mpc after LIGO and Virgo; the addition of a fourth detector is anticipated to improve the sky localization of the sources before KAGRA reaches its design sensitivity, as GW170817 has demonstrated with Virgo. We note that plans for next-generation observatories in Europe and the US are advancing, with the prospect of ten times greater sensitivity, or 10,000 greater volume, of space accessible.

Figure 17: Advanced LIGO target strain sensitivity as a function of frequency. The binary neutron star inspiral range is given in Mpc. The (best guess) detector sensitivity evolution is presented for the future observing runs as the detectors approach their final design sensitivity targets and the binary neutron star optimized sensitivities. Observing run O5 will have the implementation of upgrades to the Advanced LIGO and Advanced Virgo target designs. Figure from the LIGO-Virgo-Kagra observing plan.
Figure 18: Same as Figure 17, but for Advanced Virgo. Figure from the LIGO-Virgo-Kagra observing plan.
Figure 19: Same as Figure 17, but for KAGRA. Figure from the LIGO-Virgo-Kagra observing plan.

The observing run planning for the future is presented in Figure 20. LIGO and Virgo release their data to the public after a proprietary period; all of the data from observing runs O1, O2, and O3 can be downloaded from the Gravitational Wave Open Science Center.

Figure 20: The planned sensitivity evolution and observing runs for the Advanced LIGO, Advanced Virgo and KAGRA detectors over the coming years. The colored bars show the observing runs, with the expected sensitivities given by the data in Figure 17, Figure 18 and Figure 19 for future runs, and the achieved sensitivities in O1, O2 and O3. Figure from the LIGO, Virgo, KAGRA observing run plans.

LIGO, Virgo, KAGRA and GEO are creating a new type of telescope to peer into the heavens. With every new means of looking at the sky there has come unexpected discoveries. This has started with the unexpected observation of gravitational waves produced by binary black hole systems with tens of solar masses. Short gamma-ray bursts come from the coalescence of binary neutron stars, as seen with GW170817. Neutron star - black hole binary systems have also been observed. Physicists do know that there will be other signals that they can predict. A core collapse supernova will produce a burst of gravitational waves that will hopefully rise above the noise. Pulsars, or neutron stars spinning about their axes at rates sometimes exceeding hundreds of revolutions per second, will produce continuous sinusoidal signals that can be seen by integrating for sufficient lengths of time. Gravitational waves produced by the Big Bang will produce a background stochastic noise that can possibly be extracted by correlating the outputs from two or more detectors. These are exciting physics results that will come through tremendous experimental effort. The exciting initial observations of gravitational waves have been made, but it is just the beginning of a new astronomy.


See also

Gravitational Waves: Science with Compact Binary Coalescences

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