By definition, a stochastic background of gravitational waves is a superposition of waves arriving at random times and from random directions, overlapping so much that individual waves are not identifiable. We assume that there are so many sources (either astrophysical sources or the quantum fluctuations that create the CGWB) that individual ones are not distinguishable. Such a gravitational wave field will appear in detectors as a time-series noise, which by the central limit theorem should have a Gaussian-normal distribution function if there are enough overlapping sources. This kind of background will compete with instrumental noise. It will be detectable by a single detector, if it is stronger than instrumental noise, but a weaker background could still be detected by using a pair of detectors and looking for a correlated component of their “noise” output, on the assumption that their instrumental noise is not correlated.

As a random phenomenon, the gravitational wave fields at two different locations are uncorrelated, because gravitational waves arrive from all directions and at all frequencies. It might, therefore, be thought that two detectors’ responses would be correlated only if they were located at the same position. But if one considers one component of the wave field with a single frequency, then it is clear that there will be strong correlations between points if they are separated along the wave’s propagation direction by much less than a wavelength. We shall see that these frequency-dependent correlations allow one to detect a background by cross-correlating the output of two separated detectors, albeit with less sensitivity than if they were co-located. We shall consider cross-correlation as a detection method in Section 8.1.2.

Random gravitational waves are conventionally described in terms of their energy density spectrum , rather than their mean amplitude. It is convenient to normalize this energy density to the critical density required to close the universe, which is given in terms of the Hubble constant as

We then define

This can be interpreted as the fraction of the closure energy density that is in random gravitational waves between the frequency and . If the source of radiation is scale-free (which means that there is no preferred length or time scale in the process), then it will produce a power-law spectrum, i.e., one in which depends on a power of . Inflation, as we describe below, predicts a flat energy spectrum, one in which is essentially independent of frequency [30].The energy in the cosmological background is, of course, related to the spectral density of the noise that the background would produce in a gravitational wave detector. Since we describe the gravitational wave noise in terms of amplitude rather than energy, there are scaling factors involving the frequency between the two. An isotropic gravitational wave background incident on an interferometric detector will induce a strain spectral noise density equal to [362, 30]

Note that the explicit dependence on frequency is : two factors come from the relation of energy and squared-strain, and one factor from the fact that is an energy distribution per unit logarithmic frequency. Note also that there are no explicit factors of or needed in this formula if one wants to work in nongeometrized units.If we scale by , and we note that , then this equation implies that the strain noise is

To be observed by a single gravitational wave detector, the gravitational wave noise must be larger than the instrumental noise. This is a bolometric method of detection of the background, and it requires great confidence in the understanding of the detector, in order to believe that the observed noise is external. This is how the cosmic microwave background was originally discovered in a radio telescope by Penzias and Wilson.

If there are two detectors, then one may be able to get better sensitivity by cross-correlating their output, as mentioned in Section 4.7.3 above. This works best when the two detectors are close enough together to respond to the same random wave field. Even when they are separated, however, they are correlated well at lower frequencies.

From Equation 135 and the discussion in Section 4.7.3 it is straightforward to deduce that two co-located detectors, each with spectral noise density and fully uncorrelated instrumental noise, observing over a bandwidth at frequency for a time , can detect a stochastic background with energy density

The two LIGO detectors (separated by about 10 ms in light-travel time) are reasonably well placed for performing such correlations, particularly when upgrades push their lower frequency limit to 20 Hz or less. Two co-located first-generation LIGO instruments operating at 100 Hz could, in a one-year correlation, reach a sensitivity of . But the separation of the actual detectors takes its toll at this frequency, so that they can actually only reach . Advanced LIGO may improved this by two or three orders of magnitude, going well below the nucleosynthesis bound. The third-generation instrument ET, with instrumental noise as shown in Figure 5, can go even deeper. Two co-located ETs, observing at 10 Hz for three years, could reach . At this frequency the detectors could be as far apart as 5000 km without a substantial loss in correlation sensitivity. The numbers given here are reflected in the curves in Figure 5.

Correlation searches are also possible between resonant detectors or between one resonant and one interferometric detector [55]. This has been implemented with bar detectors [54] and between LIGO and the ALLEGRO bar detector [391].

LISA does not gain by a simple correlation between any two of its independent interferometers, since
they share a common arm, which contributes common noise that competes with that of the background. A
gravitational wave background of would compete with LISA’s expected instrumental noise.
However, using all three interferometers together can improve things for LISA at low frequencies, assuming
that the LISA instrumental noise is well behaved [196]. This might enable LISA to go below
10^{–11}.

Other less-direct methods are also being used to search for primordial gravitational waves. As we saw in Section 4.4.2, pulsar timing can, in principle, detect gravitational-wave–induced fluctuations in the arrival times of pulses. Millisecond pulsars are such stable clocks when averaged over years of observations that they are being used to search for gravitational waves with periods longer than one year. A single pulsar can set limits on a stochastic background by removing the slow spindown and looking for random timing residuals. Although one would never have enough confidence in the stability of a single pulsar to claim a detection, this sets upper limits in the important frequency range below that accessible to man-made instruments. The best such limits are on pulsar PSR B1855+09, with an upper limit (at 90% confidence) of at [213].

Arrays of pulsars offer the possibility of cross-correlating their fluctuations, which makes it
possible to distinguish between intrinsic variability and gravitational-wave–induced variability.
Pulsars are physically separated by much more than a wavelength of the gravitational waves
even with periods of 10 yrs, so that the correlated fluctuations come from the wave amplitudes
at Earth. It will soon be possible to monitor many pulsars simultaneously with multibeam
instruments, as mentioned in Section 4.4.2. This method could push the limits on
[cf. Equation (134)] down to 10^{–16} at 10 nHz [224], which translates into a limit on of around
10^{–12}.

Observations of the cosmic microwave background (CMB) may in fact make the first detections
of stochastic (or any other!) gravitational waves. The temperature fluctuations first detected
by the Cosmic Background Explorer (COBE) [345] and more recently measured with great
precision by the Wilkinson Microwave Anisotropy Probe (WMAP) [74] are produced by both
density perturbations and long-wavelength gravitational waves in the early universe (see the next
Section 8.2.1). Inflation suggests that the gravitational wave component may be almost as large as the
density component, but it can only be separated from the density perturbations by looking at the
polarization of the cosmic microwave background [216]. WMAP made the first measurements of
polarization [281], but it did not have the sensitivity to see the weak imprint of gravitational waves,
which appears in the B-component of the polarization, the part that is divergence-free on the
whole sky. The best limits on the B-component so far (early 2008) have been made by the
QUaD^{13}
detector [302], a cryogenic detector that operated for three seasons in Antarctica. These have not yet shown
any evidence for gravitational waves. Results are expected soon from the Background Imaging of Cosmic
Extragalactic Polarization (BICEP) detector, also in Antarctica [215]. The next satellite to study
the microwave background will be Planck, due for launch by the European Space Agency in
2009 [292].

The gravitational waves detectable in the CMB have wavelengths a good fraction of the horizon size at the time of decoupling, and today they have been redshifted to much longer wavelengths. They are, therefore, much lower frequency than the radiation that would be observed directly by LISA or ground-based detectors, or even by pulsar timing.

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