**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

(216) |

where we have used the Friedmann equation. Because of this
temperature, the inflaton experiences fluctuations that are the same
for each wavelength
_{k} =
*T*_{GH}. Now, these fluctuations
can be related to those in the density by

(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
*A*_{S}(*k*), related to the potential via

(218) |

where *k* = *aH* indicates that the quantity
*V*^{3} / (*V'*)^{2} is to be
evaluated at the moment when the physical scale of the perturbation
= *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

(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

(220) |

and

(221) |

where *n*_{S} and *n*_{T} are the "spectral
indices". They are related to the slow-roll parameters of the potential by

(222) |

and

(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 *V*_{inflation} ~ (10^{16} 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

(224) |

where is the
slow-roll parameter defined in (196).
Although we expect
to be small, the 1/4 in the exponent means that the dependence on
is quite weak;
unless this parameter is extraordinarily tiny, it is very likely that
*V*_{inflation}^{1/4} ~ 10^{15} -
10^{16} 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.