Annu. Rev. Astron. Astrophys. 1994. 32: 319-70 Copyright © 1994 by Annual Reviews. All rights reserved

2. THEORY

2.1. Inflation

Perhaps the most well studied paradigm for producing density fluctuations in the early universe is the inflationary mechanism. A review of the inflationary predictions for primordial power spectra has recently been completed by Liddle & Lyth (1993). We will include here a brief overview for completeness (see also Narlikar & Padmanabhan 1991).

An inflationary phase drives the Universe towards flat spatial hypersurfaces, i.e. + / 3H² = 1, or ₀ = 1 if there is no cosmological constant today (which is the standard assumption, also generally adopted in this review). In the inflationary paradigm, the fluctuations that cause temperature anisotropies in the CMB are generated by fluctuations in quantum fields during the inflationary phase. The wavelength of the fluctuations is stretched by the general expansion until they represent modes outside the horizon. For such modes the equation of motion is simple: The amplitude is a constant or, in the common jargon, they are "frozen in ". When the period of inflation ends, the horizon grows faster than the scale factor, and eventually (today) these fluctuations "reenter" the horizon. For a massless field, since the amplitude of the fluctuation is frozen in while the energy of each quantum is redshifting with the general expansion, the number of quanta describing each state has to increase dramatically. Thus when the fluctuation reenters the horizon it can be considered to evolve classically.

During inflation, when the perturbations were generated, the scale is set by the Hubble constant, so we expect that the power spectrum of fluctuations is proportional to (H_k / m_Pl)² where H_k is the Hubble constant at the time mode k left the horizon during inflation and m_Pl is the Planck mass. This is the entire story for isocurvature fluctuations. For fluctuations in the inflaton field, which are adiabatic and are the primary source of density perturbations from inflation, a solution of the perturbed Einstein equations shows that the energy density associated with the fluctuation grows while it is outside the horizon. The growth depends on the details of the inflaton potential V. The final result is that adiabatic fluctuations are enhanced over their isothermal counterparts by a factor (V / V') (see e.g. Kolb & Turner 1990, Mukhanov et al 1992).

If, during inflation, the Hubble constant changes only slowly, then we expect that the spectrum should be nearly scale invariant (in horizon crossing coordinates) or Harrison-Zel'dovich. If H_k is nearly constant then so is V (using the Friedmann equation) and we expect the enhancement of the adiabatic fluctuations to be large. As the potential becomes steeper the spectrum becomes more tilted, and the isocurvature fluctuations can become comparable to their adiabatic counterparts. Since in general V decreases with time, the slope of the power spectrum from inflation generically has more power on larger scales (see, however, Mollerach et al 1993).

The inflationary paradigm generally assumes that the inflaton field, or other fluctuating fields (e.g. axions, gravity), are weakly coupled during the epochs under consideration. In this case the fluctuations are predicted to be Gaussian. Models in which the inflaton field couples strongly to other fields or has non-negligible self interactions can give rise to non-Gaussian fluctuations (Allen et al 1987). Also, if the evolution equations for the inflaton field are nonlinear, it is possible that fluctuations can be non-Gaussian (Kofman et al 1991, Moscardini et al 1991, Scherrer 1992).