## 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

Fractional changes in the difference in path lengths along the two arms can be monitored to better than 1 part in . 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 , and interferometer arms of length , the minimum detectable strain is

This is still far off the 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 , where is the number of photons. Therefore we can track changes in the path length difference of order

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,

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

where 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, Here is the frequency of the gravitational waves, is the wave vector, is a unit vector that points in the direction of propagation of the gravitational waves, is the th polarization tensor, with spatial indices. The metric perturbation due to mode from the direction can be written by integrating over all frequencies, By integrating Eq. (50*) over all frequencies we have an expression for the metric perturbation from a particular direction , i.e., only a function of . The full metric perturbation due to a gravitational wave from a direction can be written as a sum over all polarization modesThe 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 direction, the polarization tensors are as follows

where the superscripts , , , , , and correspond to the plus, cross, vector-x, vector-y, breathing, and longitudinal modes.Suppose that the coordinate system for the detector is , , , as in Figure 1*. Relative to the detector, the gravitational-wave coordinate system is rotated by angles , , , and . We still have the freedom to perform a rotation about the gravitational-wave propagation direction, which introduces the polarization angle ,

The coordinate systems and are also shown in Figure 1*. To generalize the polarization tensors in Eq. (53*) to a wave coming from a direction , we use the unit vectors , , and as follows

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, This means that the strain measured by an interferometer due to a gravitational wave from direction and polarization angle takes the form Explicitly, the antenna pattern functions are, The dependence on the polarization angles reveals that the and polarizations are spin-2 tensor modes, the and polarizations are spin-1 vector modes, and the and 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 . The color indicates the strength of the response with red being the strongest and blue being the weakest.### 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

where is a unit vector that points in the direction of the pulsar, is the unit vector of gravitational wave propagation, and 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, , is just the integral of the redshift

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 the expected induced residual is

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

which allows us to express the Fourier transform of (59*) as where the sum is over all possible gravitational-wave polarizations: , and is the distance to the pulsar.Explicitly,

Just like for the interferometer case, the dependence on the polarization angle , reveals that the and polarizations are spin-2 tensor modes, the and polarizations are spin-1 vector modes, and the and 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*]

where and 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 and . 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 . For example, changing the polarization angle of a gravitational wave traveling in the -direction is the same as performing a rotation about the -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.