Detecting gravitational waves with light Open Access

Results published in the summer of 2023 provide strong evidence for a low-frequency gravitational wave background from measurements using pulsar timing arrays (PTA).^1–4 In January 2024, the European Space Agency gave the go-ahead for the space-based gravitational wave detector LISA, slated for launch in 2037.⁵ Both developments provide a challenge for teaching about gravitational waves at an undergraduate level: Neither PTA nor LISA can be understood using the so-called short-arm approximation for interferometric detectors that is commonly employed when teaching about gravitational wave detectors.^6–9 

The aim of this article is to provide teachers with a comprehensive account of gravitational wave detection with electromagnetic radiation, that is., light, which encompasses not only both pulsar timing and a setup like that of LISA but also detectors like LIGO, at the level of introductory physics or astronomy courses. After a review of the basics of gravitational waves in Sec. II, we deduce their influence on light propagation in Sec. III. We then apply the results to different kinds of detection scenarios, demonstrating that all of them can be understood along the same basic principles: spacecraft transponders in Sec. IV, pulsar timing in Sec. V, and interferometric detectors in Sec. VI.

Visualizations of gravitational waves typically feature the basic quadrupole pattern shown in Fig. 1, where stretching of distances in one direction on the plane always coincides with shrinking in the orthogonal direction, and vice versa.

The pattern illustrates the effect of a passing gravitational wave on free-floating test particles, here arranged in a circle with one particle in the center. The wave in question is a plane wave: We can imagine three-dimensional space as “sliced up” into parallel planes, which are orthogonal to the gravitational wave's direction of propagation (here chosen to be the z direction), with the phase of the gravitational wave the same within each slice. Like their electromagnetic counterparts, gravitational waves are transverse: test particle accelerations are orthogonal to the direction of propagation. That the pattern of stretching and shrinking is strictly separated in the x and y directions (instead of changing direction over time) makes this particular example a linearly polarized wave.

For the following, let us concentrate on one of the parallel planes, say, the plane $z=0$⁠. Characteristically, test particle distances within that plane change by a (direction-specific) factor: If two particles are initially separated in a given direction by the distance L, a passing gravitational wave will change that distance in proportion to some factor $a(t)$⁠, as $a(t)·L$⁠. All separations between free-floating particles in the same direction will vary by the same factor; an initial distance 2L will vary as $a(t)·2L$⁠, and so forth. For the gravitational wave shown in Fig. 1, distances in the x direction are changed by a factor $ax(t)$⁠; distances in the y direction by another factor $ay(t)$⁠. As one would expect, changes in length that have both an x and a y component can be calculated using Pythagoras's theorem. The fact that this is a so-called quadrupole pattern can be expressed by $ax(t)=1/ay(t)$⁠.

If we want to describe the action of this particular gravitational wave in another of the parallel planes, at different value of the z coordinate, the generalization is simple: All we need to do is replace the time variable t by the delayed time $t−z/c$⁠, and consider $ax(t−z/c)$ instead of $ax(t)$⁠, and analogously for $ay$ and for h. This construction shows clearly that our wave is propagating at the speed of light c in the positive z direction.

Before we can examine the influence of our gravitational wave on light propagation, we need to give some thought to suitable coordinates. Imagine that our plane is densely filled with freely floating test particles, all at rest relative to each other prior to the arrival of the gravitational wave. Each particle's world line has an associated proper time: duration as measured by a co-moving ideal clock. Before the gravitational wave makes itself felt, we assume spacetime geometry to be governed by special relativity, and we synchronize the co-moving clocks accordingly. Once the gravitational wave has arrived, we continue to assign to each event $E$ in our plane the time t shown by the co-moving clock of the free-floating particle that is present at $E$⁠.

Similarly, before the gravitational wave arrives, we assign to each of our family of free-floating particles Euclidean x, y coordinates corresponding to its position, with the central particle in our circle as the spatial origin. We keep those coordinates for each particle fixed (“comoving coordinates”) even during the passage of the gravitational wave, and to each event $E$ in our plane, we assign the coordinates x, y of the unique free-floating particle from our family that is present at $E$⁠. Inter-particle distances calculated via Pythagoras's theorem in these co-moving coordinates only correspond to physical distances in the absence of gravitational waves. In the presence of gravitational waves, we need to multiply x and y coordinate differences with $ax(t)$ and $ay(t)$⁠, respectively, to obtain physical distances.

In the general-relativistic formalism for describing small-amplitude gravitational waves, this coordinate choice, with the time coordinate defined via co-moving clocks and spatial coordinates via distances at a given reference time, is known as the “TT gauge.” It is one of several possible gauge choices for describing linearized gravitational waves, and the physical effects on any of the detector configurations presented in the following are of course independent of the chosen gauge. For the purpose of this article, the TT gauge has the considerable advantage that it allows for a derivation of gravitational wave effects on light that requires little more than Einstein's equivalence principle: the fact that even within a gravitational field, physics in an infinitesimal spacetime region around an object that is in free fall is governed by the laws of special relativity. This allows for a derivation of the influence of gravitational waves on light that is suitable for students who are not familiar with the basic formalism of general relativity.

Before we examine the influence of gravitational waves on light, let us make explicit what we mean by light in this context. Following standard usage in astrophysics,¹⁰ we use “light” to refer to all varieties of electromagnetic radiation, not just the more specific “visible light.” Electromagnetic radiation, in turn, is a quantum phenomenon, reaching our detectors and interacting with optical elements such as mirrors as a stream of photons. For the practicalities of gravitational wave detection, those quantum properties play an important role. In detectors like LIGO, the fact that light is reflected at mirrors not as a smooth and continuous energy flow, but as the stochastic rat-a-tat of photons, is responsible for part of the noise that makes gravitational wave signals hard to detect. This “quantum noise,” together with the thermal noise associated with thermal fluctuations, defines the “noise floor” that fundamentally limits the sensitivity of a detector design.¹¹ The latest generation of ground-based detectors goes so far as to use so-called “squeezed light,”¹² manipulating the quantum properties of radiation in a way that suppresses the associated noise in a way that cannot be described by classical physics. Such noise estimates, however, and even more so nonclassical light, are beyond the scope of the present article.

Furthermore, even for the classical electromagnetic field, we do not require the full description in terms of electric and magnetic field vectors. Instead, we model interference effects in a simplified way that is commonly used when teaching about the basics of interference and interferometry:¹³ electromagnetic waves are modeled as scalar waves, usually taken to be sinusoidal, characterized at each location by a value for the displacement and a phase. For such a simplified wave, the displacement can be interpreted as representing the component of the electric field vector in the direction of polarization for a plane, linearly polarized electromagnetic wave. In that last respect, at least, the model is rather close to reality: Plane waves of this kind are particularly suitable for interferometry, and in detectors like LIGO, considerable technical effort is invested in creating electromagnetic waves with just the right properties: linearly polarized, high-intensity, singular-frequency, fundamental-mode (which roughly translates to: plane wave like) laser light.

In situations where our analysis does not require the wave properties of light, we will resort to an even simpler picture: We will model light pulses as point particles traveling at the speed of light, using the basic picture common in special and general relativity where light propagation is described in terms of “light-like” world lines. Note that, on the quantum level, there is no fundamental difference between a light pulse (essentially, a bunch of photons) and the traveling maximum of an electromagnetic wave (again, essentially, a bunch of photons). In our model, light pulses and the maxima (or any other fixed-phase points) of a sinusoidal wave travel at the same speed, in special relativity: at the usual speed of light c.

Consider a light pulse propagating in the x direction in one of our transverse planes. If we were in special relativity (or classical physics), we would automatically assume that a pulse that starts out in the x direction will keep propagating in the x direction. In general relativity, where light gets deflected under the influence of gravity, this statement should not be taken for granted, but in the special situation we have here, symmetries guarantee that our light pulse indeed keeps propagating in the x direction: The gravitational wave's effects are transverse, so we know our pulse will not deviate out of the plane. By construction, the gravitational wave's effects are symmetric about the x axis, so none of the directions in which the trajectory of our pulse could deviate within the x–y plane is preferred relative to the others. So even while we are not in the gravity-free realm of special relativity, the setup for our simple gravitational wave ensures that a light pulse that starts out in the x direction will continue to propagate in the x direction (and an analogous statement holds for a light pulse propagating in the y direction).

Wherever the light pulse passes, there will be one of the family of free-floating particles that we introduced in Sec. II. At this point, we make use of Einstein's equivalence principle, which encapsulates a realization Einstein had at the very beginning of his path to his theory of general relativity, and that he later called the happiest thought of his life: For an observer in free fall in a gravitational field, the immediate effects of gravity—such as the pull felt by an observer standing on the Earth's surface—are absent. Since locally, all objects fall at the same rate, such an observer would see other objects floating alongside themselves. Enclose the observer in a small cabin, and they would not be able to distinguish whether they were in free fall in a gravitational field, or else in deep space, far from all sources of gravity. Einstein generalized this statement to encompass all of physics, and the result is known as Einstein's equivalence principle: for an observer in free fall in a gravitational field, the local physical laws are those of special relativity—at least in an infinitesimally small neighborhood of space-time.

For our space-filling family of free-floating particles, this means we can proceed as follows. We had introduced comoving coordinates x and y, defined via physical distances in the absence of any gravitational wave. For a free-floating particle on our x axis, its (permanently assigned) x coordinate is its distance from the origin in the absence of gravitational waves. Once the gravitational wave arrives, the comoving coordinates can still serve as coordinates, that is, as a means of assigning a tuple x, y as a unique identifier to each point in our transverse plane. However, coordinate differences will no longer correspond to physical distances. Instead, as we saw in Sec. II, the gravitational wave stretches distances in the x direction by the factor $ax(t)$⁠, so a comoving coordinate interval $dx$ will correspond to a physical distance $ds=ax(t)·dx$⁠.

We had already associated a comoving clock with each of the free-floating particles, and used those clocks to define our time coordinate. Let us go one step further and for each of the particles, imagine a comoving observer, who can perform basic (local) measurements. Since the particle, and hence the comoving observer, are in free fall, Einstein's equivalence principle applies: For a light signal passing by at time t, such an observer will find that, as measured by their own clock (which is the same as the local time given by our time coordinate t) and their local meter stick, the signal travels at the usual constant speed c.

In addition to the various kinds of Doppler effect described in Sec. III C, we can use the light propagation Eq. (5) to deduce phase information for sinusoidal light waves. Later on, we will want to describe interferometers such as LIGO, we will model elements such as light sources, beam splitters, mirrors, and detectors, as free-floating particles, whose distances from each other change under the influence of a passing gravitational wave. We will restrict our analysis to the simplest case, when all these elements are in the same plane, transverse to the direction of propagation of the gravitational wave, and where light propagates as a sinus wave either in the x or in the y direction.

For simple sine waves like our light waves, time t and phase $φ$ are related as $φ=2πft+φ0,$ with f the frequency of the wave. Thus, to relate phases at different locations—say, the phase at the light source and at a distant detector—we will need to be able to tell how much time our light requires to travel from a starting point to an end point.

From a pedagogical point of view, the parallels between light propagation under the influence of a gravitational wave and in an expanding universe are worthy of note. Recall that, in the standard description of an expanding universe, distances between distant galaxies are proportional to a time-dependent cosmic scale factor $a(t)$⁠. The pattern of motion this imposes on galaxies is called the Hubble flow. We can introduce a space-filling family of point-like, idealized galaxies, “fundamental particles of cosmology.” whose motion follows the Hubble flow exactly. The usual cosmological time coordinate can then be defined with reference to that family: At each point in space, cosmic time is time as measured on the comoving clock of the local Hubble-flow galaxy. Global synchronization for those comoving clocks is provided by making use of the homogeneity of the universe in these cosmological models: the comoving clocks are synchronized so that at any given moment in cosmic time, all observers in Hubble-flow galaxies will measure the same value for the mean density of the universe around them.¹⁴ 

Comoving coordinates in an expanding universe can be introduced using that same family of idealized Hubble-flow galaxies. Consider a snapshot of the universe at some fixed cosmic time $t0$⁠. Spatial relations within that snapshot can be described using suitable coordinates; in the simplest case, that of a spatially flat universe, we could assign to each galaxy standard Cartesian coordinates. The only change in spatial relations relative to that $t=t0$ snapshot will be one of overall scale. That allows us to assign each galaxy its snapshot coordinate at $t=t0$ as a permanent, “comoving” coordinate, valid at arbitrary times t. The only drawback is that for $t≠t0$⁠, the distances associated with those spatial coordinates do not correspond to physical distances. That is easily remedied, though: We can go from comoving distances (which, after all, correspond to physical distances at $t=t0$⁠) to physical distances at any time t, simply by rescaling with $a(t)/a(t0)$⁠.¹⁵ 

The parallels with the description of gravitational waves in Sec. II should be evident, and the simplified calculation for light propagation based on the equivalence principle works in either case: Our family of free-floating test particles correspond to Hubble-flow galaxies; the time coordinate we defined corresponds to cosmic time, and both situations can be described using comoving coordinates. The direction-dependent scale factors $ax(t)$ and $ay(t)$ correspond to a single, universal scale factor $a(t)$ in cosmology, which governs cosmic expansion. The Doppler formula (11) yields the standard cosmological redshift, and the analog of Eq. (5) describes light propagation in an expanding cosmos. In the end, both seemingly different situations are governed by the same physics, rooted in general relativity.

The basic setup for our first example is shown in Fig. 2: an antenna on Earth (left) sends a radio signal with frequency f to a distant space probe (right); the probe's transponder immediately sends the radio signal back. We call the Earth-to-probe distance $D=cT/2,$ with T the total two-way travel time.

In the Doppler shift formula (23), there are two “copies” of that signal: first $h(t)$ and later on, delayed by a time interval T, the term $h(t−T)$⁠. If the delay T is large enough, $tb−ta<T$⁠, then the $h(t)$ copy of the signal will have run its course before the term $h(t−T)$ has even begun to diverge from zero. In that case, we see two copies of the transient gravitational wave signal cleanly separated from each other: first the transient signal in $h(t)$⁠, then a pause where nothing happens, and then a repeat of the signal via the term $h(t−T)$⁠.

The basic principles for this kind of detection were worked out half a century ago.^17,18 Since then, transponder measurements in search of gravitational waves have been made using a number of interplanetary spacecraft, notably the Cassini probe. So far, those measurements have not been sensitive enough to detect gravitational waves. However, whenever gravitational waves are searched for at a given sensitivity, but not detected, the results provide an upper limit for the strength of gravitational wave signals in the frequency range that was covered by the measurements.¹⁹ A positive result, that is, a direct detection of gravitational waves using this method, is likely to take another decade. Planned space missions to the ice giants, with a travel time of about 10 years, would provide an opportunity for these kinds of measurement.²⁰ Detections should be feasible for gravitational wave frequencies between about $10−5$ and $1 Hz$⁠. Between the start of such a mission around 2030 and its arrival at Uranus or Neptune about 10 years later, transponder measurements might detect merging supermassive black holes, a stellar-size black hole merging with a supermassive black hole (extreme mass ratio inspiral, EMRI), or merging stellar-mass black holes.

Figure 3 shows a logarithmic plot of the dependence of $(hc)2$ on $fgw$⁠. The T was chosen so as to roughly correspond to a probe near Neptune, at about $30 au$⁠, with a round-trip time of roughly 8 hours. This is our first encounter with a general feature of gravitational wave detectors: linear growth due to $sin(x)≈x$ in Eq. (24) as long as the detector time scale is small compared to the gravitational wave period, followed by periodic sensitivity drops where the gravitational wave period is an integer multiple of the detector time scale. Sensitivity curves in the literature, e.g., Fig. 2 in Ref. 20, will look similar, but usually not identical—they average over different orientations of the detector relative to the gravitational wave, whereas we only consider the orthogonal case.

Next, consider the case where our radio signal is not artificial, but the highly regular series of pulses reaching us from a distant pulsar. The situation is sketched in Fig. 4.²¹ The pulses we receive on Earth are created through a light-house effect: The pulsar emits intense radio waves in two opposite directions. Those directions are not aligned with the pulsar's rotational axis, and thus trace out two cones as the pulsar rotates. If the pulsar is aligned just right, we receive a radio pulse every time the pulsar's beam brushes across Earth.

For T the (one-way) travel time of the signal from the pulsar to Earth, the distance is $D=cT$⁠. Pulse times-of-arrival (TOA) are recorded by radio telescopes utilizing precise atomic clocks. Those TOA measurements require summing up a considerable number of consecutive pulses in a coherent way, and fitting a suitable template for the shape of the pulse to the result. Uncertainties in this fitting procedure are the main source of uncertainty for TOA measurements. The TOA sequence is then decomposed into a regular part, corresponding to the period P, the correction due to (constant) period drift $Ṗ$⁠, and a time-dependent correction $ΔP$ that changes the time interval between the arrival of each pulse and its successor. In the following simplified toy model, we will instead pretend that our radio telescope is determining times-of-arrival for single, separate pulses, that we have already determined P in the absence of gravitational waves, and that $Ṗ=0$⁠, so that we can now set out directly to measure the influence of the gravitational wave by considering the $ΔP$⁠.

For the moment, let us ignore $h(t−T)$⁠, focus on $h(t)$⁠, and consider the accuracy needed for detecting $ΔP$⁠. The current PTA measurements detect a “stochastic background,” the combination of many unresolved gravitational wave events in various locations in the cosmos. In our toy model, we replace this background by a continuous sine wave with maximum amplitude $h/2∼10−14$⁠. For a millisecond pulsar, say $P=5 ms$⁠, this would mean $ΔP=5×10−8 ns$⁠. Even in a fictitious world where we could determine pulse times-of-arrival with an accuracy of $1 ns$⁠, this order of magnitude would be impossible to detect.

However, there is hope. Let us, just for the moment, consider $ΔP>0$ as constant over the observation time—corresponding to an extremely low-frequency gravitational wave. Using our clock to document pulse arrival time, we find the second pulse is $ΔP$ later than expected. The third pulse will be late by $2·ΔP$⁠, and the $(n+1)$th pulse by $n·ΔP$⁠. Even where $ΔP$ itself is undetectable, a shift $n·ΔP$ with large n need not be. In our example, after 500 days (about $4×107$ seconds, or $n=8×109$ periods), the pulse is late by $400 ns$⁠, a shift that current TOA measurement techniques, whose accuracy can be better than $100 ns$⁠, would be able to detect.

For a realistic three-dimensional pulsar timing array, the analysis is more complicated, but an important part of what is going on is covered by the toy model: There, too, documenting timing residuals over a sufficiently long time will boost the signal. The more complicated geometry, with the pulsars distributed all over the sky, means that the toy model's simple adding up of residuals (29) is not sufficient. We require an additional step: First, pairs of pulsar signals are correlated. Then, those correlations are averaged, which has the same effect as in Eq. (29) of the pulsar terms averaging out. The usual averaging-out procedure requires the fact that the real gravitational wave background is not a sine wave from a well-defined direction, as in our toy model, but a stochastic mix of signals reaching us from all possible directions in the sky.^23,24 This kind of analysis is the basis for the June 2023 announcements of various PTA collaborations.^1–4 If instead of a stochastic background signal, the array were to look at a monochromatic signal from a localized source, then at least for nearby sources, a more complicated PTA analysis could make use of both the Earth term and the pulsar term to reconstruct $h(t)$⁠.²⁵ For a transient instead of a monochromatic signal, the same reasoning would apply as for the transponder method in Sec. IV: given a sufficiently brief transient signal, we would first detect the Earth term, and then after a pause (which, depending on the pulsar distance, would likely defy the time scale of any human research project), the pulsar term.

The basic setup of an interferometric detector like LIGO or Virgo is that of a Michelson interferometer as in Fig. 6. Monochromatic light from a light source S propagates to a beam splitter BS; half of the light takes a round trip via the mirror M1, the other via the mirror M2. On returning to BS, the light portions are combined coherently, and some portion of combined light reaches the photodetector D. In a detector like LIGO, the optical elements are suspended so that we can treat their motion, at least when it comes to the distances between BS and M1/M2, as free-fall motion. That is fortunate: For our simple linearly polarized gravitational wave propagating orthogonal to the detector plane, we can use the formalism we developed in Secs. II and III.

Let BS and M2 be separated in the x direction, BS and M1 in the y direction. When the two portions of light leave the beam splitter BS, they have the same phase. For concreteness, let us describe the amplitude of the light wave (e.g., the electric field in z direction) emanating from BS into each arm by a sine function $A· sin(2πf·t)$⁠, with f the frequency of the laser light.

Now, we extend our analysis to length changes caused by gravitational waves. We again consider the layout shown in Fig. 6, but this time with gravitational-wave-induced length changes given by Eqs. (1) and (3), respectively.

So what is happening here? There are two overall effects. One is that light travel time in each arm is changed as the overall arm length changes due to the influence of the gravitational wave. The other is that the wavelength of the traveling light is affected directly while the light spends propagating in the detector. The short-arm approximation is valid when the time light spends inside the detector is short relative to the gravitational wave period. In this case, the wavelength change given by the Doppler formula (14) is small, and can be neglected; the only remaining effect is that of the arm length difference at that particular time. In that scenario, the detector constantly uses new, “fresh,” unaffected light to map its differential arm length change—and as one would expect, the result (39) is the same as if we were dealing with a classical Michelson interferometer, its light completely unaffected, but its arm lengths changing in the characteristic quadrupole pattern.

Beyond the short-arm approximation, both travel time and the Doppler effects need to be accounted for, as in Eq. (40). The periodic points of complete insensitivity in that formula, and Fig. 7, can be made plausible, as well: When $τ·fgw=1$⁠, light propagating once throughout the detector experiences each phase of the gravitational wave exactly once, in a completely symmetric fashion: for every stretching of the distance, the corresponding shrinking; for every Doppler stretching of the waves themselves, a corresponding blueshift. A simplified animation of what happens outside the short-arm regime can be found in Ref. 28.

For pedagogical reasons, throughout our analysis in Secs. II–VI, we have concentrated on certain special cases. A comprehensive description would need to include all possible gravitational wave polarizations, more general wave forms, non-planar gravitational waves, and arbitrary orientations of the detectors and of the propagating light relative to the gravitational wave. However, even the simplified accounts of the various detection methods show the connections between seemingly different measurements such as pulsar timing residuals and interferometric phase comparisons. At the most fundamental level, all of the measurements that rely on gravitational waves' interaction with light and (approximately) free-fall particles can be understood in terms of the Doppler formula in its various guises (14)–(16) and the phase formula (22), the former the differential forms of the latter.

The unified view presented here is almost at the opposite end of the spectrum from a question like “If light waves are stretched by gravitational waves, how can we use light as a ruler to detect gravitational waves?”^29,30 Light waves being doppler-shifted is what all the detection methods have in common. It is the Michelson interferometer that is the odd one out, in that changes in overall path length play an important role. The short-arm approximation, neglecting the Doppler shifts altogether, is an extreme limiting case of a more general setup.

Almost all of the technical efforts in constructing detectors like LIGO, Virgo, KAGRA or LISA are directed toward the suppression of various sources of noise that would otherwise drown out the exceedingly weak gravitational wave signals. It is worth noting that some of the resulting design modifications do affect the applicability of the simple interferometer model from Sec. VI B: Consider the key dimensions of the German-British detector GEO600,³¹ the LIGO detectors,³² and LISA³³ summarized in Table I. For the ground-based detectors, GEO600 with its folded light path bringing the travel length to and from the mirrors to twice the 600 meter overall length that is part of the detector's name, ensures that the short-arm approximation is safely applicable.

The LIGO detectors, however, are not the simple Michelson interferometers of Fig. 6. LIGO boosts sensitivity by building each arm as a Fabry–Perot interferometer: with a probability of nearly 99%, light heading back in the direction of the beam splitter will be reflected at an extra mirror at the inner end of the arm, taking another turn up and down that arm. Most light will spend considerably longer than $27 μs$ in an arm. This increases laser power by a factor $Garm=270$⁠, corresponding to an average light storage time $τ=3.6 ms$⁠, the same as for a detector with an arm length of $Leff=270.4=1080 km$⁠.

This would seem to be well within LIGO's $τgw$ range, suggesting that the short-arm approximation is not applicable. However, it's not that simple, either: Different portions of laser light will spend a different amount of time in a LIGO arm, largely averaging out the L-specific effects in Eq. (38). Due to this averaging out, the short-arm approximation works well for LIGO (and Virgo, and KAGRA) after all, and is commonly used as the basis for analyzing the performance of such detectors.³⁴ Discussions of the limits of the short-arm approximation and the need for corrections have a long history within the community.^35–41 

For LISA, there is a different complication. With its three satellites forming a gigantic, approximately equilateral triangle 2.5 × 10⁶ km a side, LISA is clearly beyond the short-arm approximation, and the “arm length penalty” imposed by Eq. (38), with a periodic structure overlaid with overall-worsening sensitivity at higher frequencies, can be clearly seen in sensitivity plots.³³ 

However, in practice, LISA's performance is not directly based on the phase formula (22). Instead, LISA's laser signals are compared to a reference laser whenever they arrive at one of the spacecraft (by superimposing both signals and measuring the beat frequency), yielding Doppler shifts (16). In the analysis, the Doppler shifts are integrated up, in effect making the transition from the Doppler formula (16) back to the phase formula (22), but with a twist: For the integration, artificial time delays are introduced, yielding the phase formula not for LISA (whose arm lengths vary by about 1% over the course of a year) but for a virtual Michelson interferometer with equal-length arms. This process is called time-delay interferometry (TDI), and it is again related to noise suppression: In an equal-arm-length interferometer, noise due to the unavoidable jitter in the laser frequency cancels out. Suppressing laser frequency noise by about eight magnitudes in this way is part of what will make LISA's gravitational wave detections possible in the first place.⁴² 

The author would like to thank Thomas Müller and the anonymous referees for helpful comments on earlier versions of this text, and Norbert Wex for his help with the section on pulsar timing arrays.

The author has no conflicts to disclose.

The data that support the findings of this study are available within the article.