| Annu. Rev. Astron. Astrophys. 1994. 32:
319-70 Copyright © 1994 by . All rights reserved |
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
(Hk / mPl)2 where
Hk is the Hubble
constant at the time mode k left the horizon during inflation and
mPl
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 Hk 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).