3 Detectors

3.1 Gravitational-wave interferometers

Kilometer-scale gravitational-wave interferometers have been in operation for over a decade. These types of detectors use laser interferometry to monitor the locations of test masses at the ends of the arms with exquisite precision. Gravitational waves change the relative length of the optical cavities in the interferometer (or equivalently, the proper travel time of photons) resulting in a strain

h = ΔL-, L
where ΔL is the path length difference between the two arms of the interferometer.

Fractional changes in the difference in path lengths along the two arms can be monitored to better than 1 part in 1020. It is not hard to understand how such precision can be achieved. For a simple Michelson interferometer, a difference in path length of order the size of a fringe can easily be detected. For the typically-used, infrared lasers of wavelength λ ∼ 1μm, and interferometer arms of length L = 4 km, the minimum detectable strain is

λ h ∼ --∼ 3 × 10 −10. L

This is still far off the − 20 10 mark. In principle, however, changes in the length of the cavities corresponding to fractions of a single fringe can also be measured provided we have a sensitive photodiode at the dark port of the interferometer, and enough photons to perform the measurement. This way we can track changes in the amount of light incident on the photodiode as the lengths of the arms change and we move over a fringe. The rate at which photons arrive at the photodiode is a Poisson process and the fluctuations in the number of photons is 1∕2 ∼ N, where N is the number of photons. Therefore we can track changes in the path length difference of order

ΔL ∼ --λ--. N 1∕2
The number of photons depends on the laser power P, and the amount of time available to perform the measurement. For a gravitational wave of frequency f, we can collect photons for a time t ∼ 1∕f, so the number of photons is
P N ∼ f-h-ν , p
where hp is Planck’s constant and ν = c∕λ is the laser frequency. For a typical laser power P ∼ 1 W, a gravitational-wave frequency f = 100 Hz, and λ ∼ 1μm the number of photons is
N ∼ 1016,
so that the strain we are sensitive to becomes
h ∼ 10−18.

The sensitivity can be further improved by increasing the effective length of the arms. In the LIGO instruments, for example, each of the two arms forms a resonant Fabry–Pérot cavity. For gravitational-wave frequencies smaller than the inverse of the light storage time, the light in the cavities makes many back and forth trips in the arms, while the wave is traversing the instrument. For gravitational waves of frequencies around 100 Hz and below, the light makes about a thousand back and forth trips while the gravitational wave is traversing the interferometer, which results in a three-orders-of-magnitude improvement in sensitivity,

h ∼ 10−21.
For frequencies larger than 100 Hz the number of round trips the light makes in the Fabry–Pérot cavities while the gravitational wave is traversing the instrument is reduced and the sensitivity is degraded.

The proper light travel time of photons in interferometers is controlled by the metric perturbation, which can be expressed as a sum over polarization modes

∑ A hij(t,⃗x) = hij(t,⃗x), (49 ) A
where A labels the six possible polarization modes in metric theories of gravity. The metric perturbation for each mode can be written in terms of a plane wave expansion,
∫ ∫ A ∞ i2πf(t− ˆΩ⋅⃗x) A A hij(t,⃗x) = df dˆΩe &tidle;h (f, ˆΩ)𝜖ij(ˆΩ ). (50 ) −∞ S2
Here f is the frequency of the gravitational waves, ⃗k = 2πf ˆΩ is the wave vector, ˆΩ is a unit vector that points in the direction of propagation of the gravitational waves, eA ij is the Ath polarization tensor, with i,j = x,y,z spatial indices. The metric perturbation due to mode A from the direction ˆ Ω can be written by integrating over all frequencies,
∫ ∞ hA (t − Ωˆ ⋅⃗x) = df ei2πf(t− ˆΩ⋅⃗x)&tidle;hA (f, ˆΩ )𝜖A (ˆΩ). (51 ) ij −∞ ij
By integrating Eq. (50*) over all frequencies we have an expression for the metric perturbation from a particular direction ˆΩ, i.e., only a function of t − ˆΩ ⋅⃗x. The full metric perturbation due to a gravitational wave from a direction ˆΩ can be written as a sum over all polarization modes
∑ hij(t − ˆΩ ⋅⃗x ) = hA (t − Ωˆ⋅ ⃗x)𝜖Aij(ˆΩ ). (52 ) A

The response of an interferometer to gravitational waves is generally referred to as the antenna pattern response, and depends on the geometry of the detector and the direction and polarization of the gravitational wave. To derive the antenna pattern response of an interferometer for all six polarization modes we follow the discussion in [329*] closely. For a gravitational wave propagating in the z direction, the polarization tensors are as follows

( ) ( ) + 1 0 0 × 0 1 0 𝜖ij = ( 0 − 1 0) ,𝜖ij = ( 1 0 0) , 0 0 0 0 0 0 ( ) ( ) x 0 0 1 y 0 0 0 𝜖ij = ( 0 0 0) ,𝜖ij = ( 0 0 1) , 1 0 0 0 1 0 ( 1 0 0) ( 0 0 0) b ( ) ℓ ( ) 𝜖ij = 0 1 0 ,𝜖ij = 0 0 0 , (53 ) 0 0 0 0 0 1
where the superscripts +, ×, x, y, b, and ℓ correspond to the plus, cross, vector-x, vector-y, breathing, and longitudinal modes.
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Figure 1: Detector coordinate system and gravitational-wave coordinate system.

Suppose that the coordinate system for the detector is xˆ= (1,0,0), yˆ= (0, 1,0), ˆz = (0,0, 1), as in Figure 1*. Relative to the detector, the gravitational-wave coordinate system is rotated by angles (𝜃,ϕ), ˆx ′ = (cos𝜃 cosϕ, cos𝜃sinϕ, − sin 𝜃), ˆy′ = (− sin ϕ,cosϕ, 0), and zˆ′ = (sin𝜃 cosϕ,sin𝜃 sin ϕ,cos 𝜃). We still have the freedom to perform a rotation about the gravitational-wave propagation direction, which introduces the polarization angle ψ,


The coordinate systems (xˆ, ˆy, ˆz) and (ˆm, ˆn,Ωˆ) are also shown in Figure 1*. To generalize the polarization tensors in Eq. (53*) to a wave coming from a direction ˆ Ω, we use the unit vectors mˆ, ˆn, and ˆΩ as follows

𝜖+ = ˆm ⊗ ˆm − ˆn ⊗ ˆn, × 𝜖 = ˆm ⊗ ˆn + ˆn ⊗ ˆm, 𝜖x = ˆm ⊗ ˆΩ + ˆΩ ⊗ ˆm, y ˆ ˆ 𝜖 = ˆn ⊗ Ω + Ω ⊗ ˆn, 𝜖b = ˆm ⊗ ˆm + nˆ⊗ ˆn, ℓ ˆ ˆ 𝜖 = Ω ⊗ Ω. (55 )
For LIGO and VIRGO the arms are perpendicular so that the antenna pattern response can be written as the difference of projection of the polarization tensor onto each of the interferometer arms,
F A(ˆΩ, ψ) = 1-(ˆxiˆxj − ˆyiˆyj)𝜖A(Ωˆ, ψ). (56 ) 2 ij
This means that the strain measured by an interferometer due to a gravitational wave from direction ˆΩ and polarization angle ψ takes the form
h(t) = ∑ h (t − Ωˆ⋅ x)FA (ˆΩ,ψ ). (57 ) A A
Explicitly, the antenna pattern functions are,
+ 1- 2 F (𝜃,ϕ,ψ ) = 2(1 + cos 𝜃 )cos2ϕ cos2 ψ − cos𝜃 sin 2ϕ sin 2ψ, 1 F ×(𝜃,ϕ,ψ ) = − -(1 + cos2𝜃) cos2ϕ sin 2ψ − cos 𝜃sin2 ϕcos 2ψ, x 2 F (𝜃,ϕ,ψ ) = sin 𝜃(cos𝜃 cos2ϕ cosψ − sin 2ϕ sin ψ), F y(𝜃,ϕ,ψ ) = − sin𝜃(cos 𝜃cos2ϕ sinψ + sin 2ϕ cosψ), 1 F b(𝜃,ϕ) = − --sin2 𝜃cos 2ϕ, 2 F ℓ(𝜃,ϕ) = 1-sin2 𝜃cos 2ϕ. (58 ) 2
The dependence on the polarization angles ψ reveals that the + and × polarizations are spin-2 tensor modes, the x and y polarizations are spin-1 vector modes, and the b and ℓ polarizations are spin-0 scalar modes. Note that for interferometers the antenna pattern responses of the scalar modes are degenerate. Figure 2* shows the antenna patterns for the various polarizations given in Eq. (58*) with ψ = 0. The color indicates the strength of the response with red being the strongest and blue being the weakest.
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Figure 2: Antenna pattern response functions of an interferometer (see Eqs. (58*)) for ψ = 0. Panels (a) and (b) show the plus (|F+ |) and cross (|F ×|) modes, panels (c) and (d) the vector x and vector y modes (|F | x and |F | y), and panel (e) shows the scalar modes (up to a sign, it is the same for both breathing and longitudinal). Color indicates the strength of the response with red being the strongest and blue being the weakest. The black lines near the center give the orientation of the interferometer arms.

3.2 Pulsar timing arrays

Neutron stars can emit powerful beams of radio waves from their magnetic poles. If the rotational and magnetic axes are not aligned, the beams sweep through space like the beacon on a lighthouse. If the line of sight is aligned with the magnetic axis at any point during the neutron star’s rotation the star is observed as a source of periodic radio-wave bursts. Such a neutron star is referred to as a pulsar. Due to their large moment of inertia pulsars are very stable rotators, and their radio pulses arrive on Earth with extraordinary regularity. Pulsar timing experiments exploit this regularity: gravitational waves are expected to cause fluctuations in the time of arrival of radio pulses from pulsars.

The effect of a gravitational wave on the pulses propagating from a pulsar to Earth was first computed in the late 1970s by Sazhin and Detweiler [378, 145]. Gravitational waves induce a redshift in the pulse train

i j z (t,Ωˆ) = 1---ˆp-ˆp---Δh , (59 ) 2 1 + Ωˆ ⋅ ˆp ij
where ˆp is a unit vector that points in the direction of the pulsar, ˆΩ is the unit vector of gravitational wave propagation, and
Δhij ≡ hij(te,Ωˆ) − hij(tp, ˆΩ ), (60 )
is the difference in the metric perturbation at the pulsar when the pulse was emitted and the metric perturbation on Earth when the pulse was received. The inner product in Eq. (59*) is computed with the Euclidean metric.

In pulsar timing experiments it is not the redshift, but rather the timing residual that is measured. The times of arrival of pulses are measured and the timing residual is produced by subtracting off a model that includes the rotational frequency of the pulsar, the spin-down (frequency derivative), binary parameters if the pulsar is in a binary, sky location and proper motion, etc. The timing residual induced by a gravitational wave, R(t), is just the integral of the redshift

∫ t R (t) ≡ dt′z(t′). (61 ) 0
Times-of-arrival (TOAs) are measured a few times a year over the course of several years allowing for gravitational waves in the nano-Hertz band to be probed. Currently, the best timed pulsars have residual root-mean-squares (RMS) of a few 10 s of ns over a few years.

The equations above ((59*)ff) can be used to estimate the strain sensitivity of pulsar timing experiments. For gravitational waves of frequency f the expected induced residual is

h R ∼ --, f
so that for pulsars with RMS residuals R ∼ 100 ns, and gravitational waves of frequency f ∼ 10 −8 Hz, gravitational waves with strains
h ∼ Rf ∼ 10 −15
would produce a measurable effect.

To find the antenna pattern response of the pulsar-Earth system, we are free to place the pulsar on the z-axis. The response to gravitational waves of different polarizations can then be written as

1 ˆziˆzj F A(ˆΩ, ψ) = -----------𝜖Aij(ˆΩ,ψ ), (62 ) 2 1 + cos𝜃
which allows us to express the Fourier transform of (59*) as
( ) ˆ −2πifL(1+ˆΩ⋅ˆp) ∑ &tidle; ˆ A ˆ &tidle;z(f,Ω ) = 1 − e hA(f,Ω )F (Ω), (63 ) A
where the sum is over all possible gravitational-wave polarizations: A = +, ×,x, y,b,l, and L is the distance to the pulsar.


+ 2 𝜃 F (𝜃,ψ) = sin 2 cos 2ψ, (64 ) 𝜃 F× (𝜃,ψ) = − sin2 -sin2 ψ, (65 ) 2 F x(𝜃,ψ) = − 1--sin2𝜃---cosψ, (66 ) 2 1 + cos𝜃 y 1 sin 2𝜃 F (𝜃,ψ) = -----------sin ψ, (67 ) 2 1 + cos𝜃 F b(𝜃) = sin2 𝜃-, (68 ) 2 ℓ 1 cos2 𝜃 F (𝜃) = -----------. (69 ) 2 1 + cos𝜃
Just like for the interferometer case, the dependence on the polarization angle ψ, reveals that the + and × polarizations are spin-2 tensor modes, the x and y polarizations are spin-1 vector modes, and the b and ℓ polarizations are spin-0 scalar modes. Unlike interferometers, the antenna pattern responses of the pulsar-Earth system do not depend on the azimuthal angle of the gravitational wave, and the scalar modes are not degenerate.

In the literature, it is common to write the antenna pattern response by fixing the gravitational-wave direction and changing the location of the pulsar. In this case the antenna pattern responses are [284*, 22*, 99*]

&tidle;+ 2 𝜃p F (𝜃p,ϕp) = sin 2 cos2ϕp, (70 ) 𝜃 F&tidle;×(𝜃p,ϕp) = sin2-p sin 2ϕp, (71 ) 2 &tidle;x 1--sin2-𝜃p-- F (𝜃p,ϕp) = 2 1 + cos𝜃p cosϕp, (72 ) F&tidle;y(𝜃p,ϕp) = 1--sin2-𝜃p--sin ϕp, (73 ) 2 1 + cos𝜃p b 2 𝜃p &tidle;F (𝜃p) = sin --, (74 ) 22 &tidle;F ℓ(𝜃p) = 1--cos--𝜃p-, (75 ) 2 1 + cos𝜃p
where 𝜃p and ϕp are the polar and azimuthal angles, respectively, of the vector pointing to the pulsar. Up to signs, these expressions are the same as Eq. (69*) taking 𝜃 → 𝜃p and ψ → ϕp. This is because fixing the gravitational-wave propagation direction while allowing the pulsar location to change is analogous to fixing the pulsar position while allowing the direction of gravitational-wave propagation to change – there is degeneracy in the gravitational-wave polarization angle and the pulsar’s azimuthal angle ϕp. For example, changing the polarization angle of a gravitational wave traveling in the z-direction is the same as performing a rotation about the z-axis that changes the pulsar’s azimuthal angle. Antenna patterns for the pulsar-Earth system using Eqs. (75*) are shown in Figure 3*. The color indicates the strength of the response, red being the largest and blue the smallest.
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Figure 3: Antenna patterns for the pulsar-Earth system. The plus mode is shown in (a), breathing modes in (b), the vector-x mode in (c), and longitudinal modes in (d), as computed from Eq. (75*). The cross mode and the vector-y mode are rotated versions of the plus mode and the vector-x mode, respectively, so we did not include them here. The gravitational wave propagates in the positive z-direction with the Earth at the origin, and the antenna pattern depends on the pulsar’s location. The color indicates the strength of the response, red being the largest and blue the smallest.

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