(ed.). Gravitational waves (IOP, 2001)(422s).

22 Elements of gravitational waves2.4.2 Pulsar timingMany pulsars, in particular old millisecond pulsars, are extraordinarily regularclocks, whose random timing irregularities are too small for even the bestatomic clocks to measure. Other pulsars have weak but observable irregularities.Measurements of or even upper limits on any of these timing irregularities forsingle pulsars can be used to set upper limits on any background gravitationalwave field with periods comparable to or shorter than the observing time. Here thethree-term formula is replaced by a simpler two-term expression (see exercise (b)at the end of this chapter), because we only have a one-way transmission fromthe pulsar to Earth. Moreover, the transit time of a signal to Earth from the pulsarmay be thousands of years, so we cannot look for correlations between the twoterms in a given signal. Instead, the delay time is a combination of the effectsof uncorrelated waves at the pulsar when the signal was emitted and at the Earthwhen it is received.If one simultaneously observes two or more pulsars, the Earth-based part ofthe delay is correlated between them, and this offers a means of actually detectinglong-period gravitational waves. Observations require a timescale of several yearsin order to achieve the long-period stability of pulse arrival times, so this methodis suited to looking for strong gravitational waves with periods of several years.2.4.3 InterferometryAn interferometer essentially measures changes in the difference in the returntimes along two different arms. It does this by looking for changes in theinterference pattern formed when the returning light beams are superimposedon one another. The response of each arm will follow the three-term formulain equation (2.20), but with a different value of θ for each arm, depending in acomplicated way on the orientation of the arms relative to the direction of traveland the polarization of the wave. Ground-based interferometers are small enoughto use the small-L formulae we derived earlier. However, LISA, the space-basedinterferometer that is described by Bender in this book, is larger than a wavelengthof gravitational waves for frequencies above 10 mHz, so a detailed analysis of itssensitivity requires the full three-term formula.2.5 Exercises for chapter 2Suggested solutions for these exercises are at the end of chapter 7.(a) 1. Derive the full three-term return equation, reproduced here:dt returndt= 1 2{(1 − sin θ)hxx + (t + 2L) − (1 + sin θ)hxx + (t)+ 2 sin θh+ xx [t + L(1 − sin θ)]}. (2.21)