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3.4. Tensor and vector perturbations

Gravitational wave perturbations, also known as tensor perturbations, are inevitably produced at some level by inflation, but the level depends on the model under consideration and it is perfectly possible, and perhaps even likely [20], that the level is too small to be detected by currently envisaged experiments. This prevents them acting as a test.

In standard inflation models, the gravitational waves are directly observable only by the microwave background anisotropies they induce. Assuming Einstein gravity, the Hubble parameter always decreases during inflation which leads to a spectrum which decreases with decreasing scale; the upper limit set by these anisotropies places the amplitude on short scales orders of magnitude below planned detectors (and probably well below the stochastic background from astrophysical sources). The exception is the pre big bang class of models [11] (implemented in extensions of Einstein gravity), where the gravitational wave spectrum rises sharply to short scales and is potentially visible in laser interferometer experiments.

As well as the direct effect on the microwave background, gravitational waves evidence themselves as a deficit of short-scale power in the density perturbation spectrum of COBE-normalized models. Presently the combination of large-scale structure data with COBE gives the strongest upper limit on the fractional contribution r on COBE scales, at r ltapprox 0.5 [21]. There is no evidence to favour the tensors, but this constraint is fairly weak. Eventually the PLANCK satellite is expected to be able to detect (at 95% confidence) a contribution above r ~ 0.1 [22], and may perhaps do better if there is early reionization and/or the foreground contamination turns out to be readily modellable. Conceivably, high-precision observations of the polarization of the microwave background might improve this further.

The verdict, therefore, is that if a tensor component is seen, corresponding to gravitational waves on scales bigger than the Hubble radius at decoupling, that is extremely powerful support for the inflationary paradigm. This would be stronger yet if the observed spectrum could be shown to satisfy an equation known as the consistency equation to some reasonable accuracy; this relates the tensor spectral index to the relative amplitude of tensors and scalars, and signifies the common origin of the two spectra from a single inflationary potential V(phi) [23, 24]. However, the tensor perturbations do not provide a test for inflation in the formal sense, since no damage is inflicted upon the inflationary paradigm if they are not detected.

While known inflationary models generate both scalar and tensor modes, it appears extremely hard to generate large-scale vector modes. There are two obstacles. The first is that massless vector fields are conformally invariant, which means that perturbations are not excited by expansion; this has to be evaded either by introducing a mass (which suppresses the effect of perturbations) or an explicit coupling breaking conformal invariance [25]. The second obstacle is that vector perturbations die off rapidly as the Universe expands, and to survive until horizon entry their initial value would have to be considerably in excess of the linear regime. In consequence, a significant prediction of inflation is the absence of large-scale vector perturbations. If they are seen, it seems likely to be impossible to make them with inflation alone, though I am not aware of a cast-iron proof. By contrast, topological defect models generically excite vector perturbations.

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