244	Gravitational radiation

to Eq. (9.98), eliminate l0 in favor of the mass and frequency, and be left with just one unknown: the distance r to the source. Remarkably, this property holds even if the binary system does not have equal masses: chirping binary signals contain enough information to deduce the distance to the source (Schutz 1986). Gravitational wave astronomers call these systems ‘standard sirens’, by analogy with the usual standard candles of optical astronomy, which we will discuss in Ch. 12. We will see there that gravitational wave observations of black-hole binaries by LISA may assist astronomers measure the large-scale dynamics of the universe.

Spinning neutron stars

Neutron stars are very compact stars formed in gravitational collapse. We will study them as relativistic stellar objects in the next chapter. Here we simply note that many neutron stars are pulsars, whose spin sweeps a beam of electromagnetic radiation past the Earth each time they turn. Many spin rapidly, at frequencies above 20 Hz, and if these radiate gravitational waves, then they would be in the observing band of ground-based detectors. There could in principle be many stars not known as pulsars that also spin this rapidly, because their beams do not cross the Earth. Moreover, radio surveys for pulsars only cover the near neighborhood of the Sun in our Galaxy; there could be more distant pulsars that are not yet known.

Such stars could radiate gravitational waves if they are not symmetric about the rotation axis. Pulsars are clearly not symmetric, since they beam their radiation somehow. But it is not clear how much mass asymmetry is required to produce the beaming. Other asymmetries could come from frozen-in irregularities in the semi-solid outer layer of a neutron star (called its ‘crust’), or in a possible solid core. It is also known that spinning neutron stars are vulnerable to a gravitational-wave driven instability called the r-mode instability, which could produce significant radiation. We can compute the radiation due to mass asymmetry from Eqs. (9.84)–(9.86). If the star is nearly axisymmetric, then we can approximate the amplitude of either of the polarizations radiated along the spin axis by the formula

h 2ε 2INS/r,

where INS is the moment of inertia of the spherical neutron star and ε is the fractional asymmetry of the star about the spin axis. If we use typical values of INS = 1038 kg m2, r = 1 kpc (about 3 × 1022 m), = 2π f with f = 60 Hz, and ε = 10−5, then we get h 10−25. This is a very small amplitude, but not impossibly small. Scientists find such small signals by taking long stretches of data and filtering for them, essentially by performing a Fourier transform. The Fourier transform concentrates the power of the signal in one frequency band, while distributing the noise power of the data stream over the whole observing band. To go from the Advanced LIGO broad-band sensitivity of around 10−22 to a sensitivity of 10−25 for a narrow-band signal like the one we are considering here, the data analyst must have taken a number of cycles of the waveform equal to at least the square of the ratio of these two numbers, or 106. For this frequency, this would take less than a day.

245	9.5 Astrophysical sources of gravitational waves

Until a spinning neutron star has been observed, we won’t know what a reasonable value for ε is. However, for many known pulsars we can already set limits from radio observations. This is because, as for the binaries we considered above, a radiating pulsar loses energy. This causes it to spin down. Most pulsars are observed to spin down, and so their observed slowing rate sets an upper limit on the possible amplitude of gravitational waves. It is only an upper limit, because it is very likely that the spindown is dominated by other effects, such as losses to electromagnetic radiation and particle emission, so that gravitational waves play a minor role. For known pulsars the limits obtained on ε this way range from 10−3 to below 10−7.

It is therefore desirable to do searches for gravitational wave pulsars using months rather than days of data. When the pulsar’s position and frequency are known from radio observations, this is not a difficulty, but when gravitational wave astronomers try to search the entire sky for unknown neutron stars, the computational demands become enormous. This is because the apparent frequency of the pulsar signal is strongly Doppler modulated by the Earth’s spin and orbital motion during a period as long as a month or more, and the details of the modulation depend on the star’s location on the sky. Data analysts therefore have to search many different locations separately to perform their filtering. At present, this is a problem that would overwhelm the most powerful computers in existence. The gravitational wave projects are getting help in this analysis from the general public, using the screen-saver called Einstein@Home.

Gravitational col lapse

The objective that motivated Joseph Weber to develop the first bar detector was to register waves from a supernova. The spectacular optical display of a supernova explosion masks what really happens inside: the compact core of a giant star, having exhausted its supply of energy from nuclear reactions, collapses inward, and the subsequent dynamics can convert some of the energy released into the explosion that blows off the envelope of the star. But what happens to the collapsing core, and how that energy is converted into the explosion, is not well understood because it is impossible to observe the core directly. Gravitational waves, along with neutrinos, provide the only probes that come to us directly from the core.

The amplitude of gravitational waves to be expected is very uncertain. It is sensitive to the initial state of rotation of the core, to instabilities that develop during collapse, and to poorly understood details of the physics of dense matter. Modeling collapse on a computer is difficult, and the predictions so far are only approximate. However, there is wide agreement that the amplitudes are likely to be far smaller than the order-of-magnitude estimate we made in the opening paragraph of § 9.3.

What is more, it is not possible at present to predict a detailed waveform, so that the data analysts cannot dig so deeply into the noise of the detector as they can for binaries or spinning neutron stars. All of these circumstances make it seem less likely that the first detected signal will be that of a supernova explosion. That expectation would be reversed, however, if the Galaxy experienced another supernova explosion like SN1987a, which occurred in its satellite galaxy, the Large Magellanic Cloud.

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Gravitational waves from the Big B ang

The study of the large-scale structure of the universe, and its history, is called cosmology, and it will be the subject of Ch. 12 below. Cosmology has undergone a revolution since the 1980s, with a huge increase in data and in our insight into what went on in the early universe. Part of that revolution impacts on the study of gravitational waves: it seems very probable that the very early universe was the source of a random sea of gravitational radiation that even today forms a background to our observations of other sources.

The radiation originated in a host of individual events too numerous to count. The waves, superimposed now, have very similar character to the random noise that comes from instrumental effects. Although the radiation was intense when it was generated, the expansion of the universe has cooled it down, and one of the most uncertain aspects of our understanding is what intensity it should have today. It is possible that it will be strong enough that, as detectors improve their sensitivity, they will encounter a ‘noise’ that does not go away, and that can be shown to be isotropic on the sky. In exactly this way, Penzias (1979) and Wilson (1979) discovered the cosmic microwave background radiation in a radio receiver at Bell Labs, an event for which they were awarded the Nobel Prize for Physics.

However, it is more likely that the radiation is weaker and will remain below the noise in our detectors for some time to come. How, then, can we find it? The answer is that, while it is a random noise in any one detector, the randomness is correlated between detectors. Two detectors in the same place experience exactly the same noise. If we make a correlation of their output (simply multiplying them and integrating in time) we should obtain a nonzero result much larger than we expect from the variance of the correlation of two statistically independent noise fields. In practice, the most sensitive pairs of detectors are the two LIGO installations, and the VIRGO-GEO600 pair. Both have separations between them so that the correlations in their random wavefields would not be perfect. However, gravitational waves with wavelengths longer than the separation will still be well correlated, and this allows these detectors to search for a background.

At present, the only limits we have are from the two LIGO detectors, and they are not surprising. Cosmologists express the strength of backgrounds in terms of the energy density they carry, as a fraction of the total energy density of all the material in the universe, averaged over large volumes. We know from present observations that the energy density in random waves in the LIGO observing band is not larger than a fraction 10−5 of the total. It is hoped that Advanced LIGO may approach a limit around 10−10 of the total.

LISA can also make observations of the background. In its case, the background would have to be stronger than instrumental noise: correlation gains it nothing. But LISA’s sensitivity in its waveband is great, and it seems likely that it would be able to detect a background around 10−10 of the total. Observations of the cosmic microwave background could also detect this radiation, at very low frequencies.

Pulsar timing might be able to detect a random background of gravitational waves from astrophysical systems, but it seems likely that these backgrounds will be larger than the