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5.8. Vacuum Fluctuations and Perturbations

Recall that the structures - clusters and superclusters of galaxies - we see on the largest scales in the universe today, and hence the observed fluctuations in the CMB, form from the gravitational instability of initial perturbations in the matter density. The origin of these initial fluctuations is an important question of modern cosmology.

Inflation provides us with a fascinating solution to this problem - in a nutshell, quantum fluctuations in the inflaton field during the inflationary epoch are stretched by inflation and ultimately become classical fluctuations. Let's sketch how this works.

Since inflation dilutes away all matter fields, soon after its onset the universe is in a pure vacuum state. If we simplify to the case of exponential inflation, this vacuum state is described by the Gibbons-Hawking temperature

Equation 216 (216)

where we have used the Friedmann equation. Because of this temperature, the inflaton experiences fluctuations that are the same for each wavelength delta phik = TGH. Now, these fluctuations can be related to those in the density by

Equation 217 (217)

Inflation therefore produces density perturbations on every scale. The amplitude of the perturbations is nearly equal at each wavenumber, but there will be slight deviations due to the gradual change in V as the inflaton rolls. We can characterize the fluctuations in terms of their spectrum AS(k), related to the potential via

Equation 218 (218)

where k = aH indicates that the quantity V3 / (V')2 is to be evaluated at the moment when the physical scale of the perturbation lambda = a / k is equal to the Hubble radius H-1. Note that the actual normalization of (218) is convention-dependent, and should drop out of any physical answer.

The spectrum is given the subscript "S" because it describes scalar fluctuations in the metric. These are tied to the energy-momentum distribution, and the density fluctuations produced by inflation are adiabatic - fluctuations in the density of all species are correlated. The fluctuations are also Gaussian, in the sense that the phases of the Fourier modes describing fluctuations at different scales are uncorrelated. These aspects of inflationary perturbations - a nearly scale-free spectrum of adiabatic density fluctuations with a Gaussian distribution - are all consistent with current observations of the CMB and large-scale structure, and have been confirmed to new precision by WMAP and other CMB measurements.

It is not only the nearly-massless inflaton that is excited during inflation, but any nearly-massless particle. The other important example is the graviton, which corresponds to tensor perturbations in the metric (propagating excitations of the gravitational field). Tensor fluctuations have a spectrum

Equation 219 (219)

The existence of tensor perturbations is a crucial prediction of inflation which may in principle be verifiable through observations of the polarization of the CMB. Although CMB polarization has already been detected [166], this is only the E-mode polarization induced by density perturbations; the B-mode polarization induced by gravitational waves is expected to be at a much lower level, and represents a significant observational challenge for the years to come.

For purposes of understanding observations, it is useful to parameterize the perturbation spectra in terms of observable quantities. We therefore write

Equation 220 (220)


Equation 221 (221)

where nS and nT are the "spectral indices". They are related to the slow-roll parameters of the potential by

Equation 222 (222)


Equation 223 (223)

Since the spectral indices are in principle observable, we can hope through relations such as these to glean some information about the inflaton potential itself.

Our current knowledge of the amplitude of the perturbations already gives us important information about the energy scale of inflation. Note that the tensor perturbations depend on V alone (not its derivatives), so observations of tensor modes yields direct knowledge of the energy scale. If large-scale CMB anisotropies have an appreciable tensor component (possible, although unlikely), we can instantly derive Vinflation ~ (1016 GeV)4. (Here, the value of V being constrained is that which was responsible for creating the observed fluctuations; namely, 60 e-folds before the end of inflation.) This is remarkably reminiscent of the grand unification scale, which is very encouraging. Even in the more likely case that the perturbations observed in the CMB are scalar in nature, we can still write

Equation 224 (224)

where epsilon is the slow-roll parameter defined in (196). Although we expect epsilon to be small, the 1/4 in the exponent means that the dependence on epsilon is quite weak; unless this parameter is extraordinarily tiny, it is very likely that Vinflation1/4 ~ 1015 - 1016 GeV.

We should note that this discussion has been phrased in terms of the simplest models of inflation, featuring a single canonical, slowly-rolling scalar field. A number of more complex models have been suggested, allowing for departures from the relations between the slow-roll parameters and observable quantities; some of these include hybrid inflation [167, 168, 169], inflation with novel kinetic terms [170], the curvaton model [171, 172, 173], low-scale models [174, 175], brane inflation [176, 177, 178, 179, 180, 181, 182] and models where perturbations arise from modulated coupling constants [183, 184, 185, 186, 187]. This list is necessarily incomplete, and continued exploration of the varieties of inflationary cosmology will be a major theme of theoretical cosmology into the foreseeable future.

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