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

**2.4. Fluctuations**

Both inflation and defect models predict that the fluctuation spectrum
should be stochastic in nature. Thus we live in one sample of an
ensemble of "possible universes" which was drawn from a distribution
specified by the underlying theory. Due to the weak coupling nature of
most inflationary theories, the distribution of fluctuations is
predicted to be Gaussian. (The assumption of Gaussian *fluctuations* is
not to be confused with the further restrictive assumption of a
Gaussian *shape* for the power spectrum.) In contrast, most defect
models predict a non-Gaussian character for the fluctuations.

As discussed in Section 2.2, on large
(e.g. *COBE*) scales
fluctuations in almost all theories will look Gaussian. On smaller
scales there is the possibility of detecting non-Gaussian phase
correlations. There have been several tests for non-Gaussian
fluctuations proposed, e.g. the total curvature or genus of
isotemperature contours
(Coles 1988b,
Gott et al 1990,
Brandenberger et al 1993,
Smoot et at 1994),
the distribution of peaks
(Sazhin 1985,
Zabotin & Nasel'skii
1985,
Bond & Efstathiou 1987,
Vittorio & Juszkiewicz
1987,
Coles & Barrow 1987,
Coles 1988a,
Gutierrez de la Cruz et al
1993,
Cayon et al 1993c,
Kashlinsky 1993b),
skewness and
kurtosis of the observed temperature distribution
(Scaramella & Vittorio
1991,
1993a;
Luo & Schramm 1993a;
Perivolaropoulos 1993b;
Moessner et al 1993),
or the 3-point function
(Falk et al 1993,
Luo & Schramm 1993b,
Srednicki 1993,
Graham et al 1993).
Of these, the
3-point function currently seems the most promising, although more
work is needed to see if it can detect departures from Gaussianness at
the level predicted by the "defect" theories.

In general, the predictions of a theory are expressed in terms of
predictions for the
*a*_{m}. If the
fluctuations are Gaussian, the
predictions are fully specified by giving the 2-point function for the
*a*_{m} [just
as the matter spectrum
*P*_{mat}(*k*) is predicted by giving
(*r*)].
Using rotational symmetry, it is conventional to write

(7) |

where the angle brackets here represent an average over the ensemble
of possible universes. The prediction for CMB anisotropy measurements
of a theory can thus be expressed as a series of
*C*_{}s.
Alternatively
one can work in *k*-space and define a (3-dimensional) power
spectrum of fluctuations per wavelength interval. We describe this, and its
relation to the
*C*_{}s, in
Section 3.2. For large
, the
*C*_{}s are
approximately the same as the 2-dimensional power spectrum of
fluctuations
(Bond & Efstathiou
1987).

2.4.1 ADIABATIC Any initial density perturbation may be decomposed into a sum of an adiabatic and an isocurvature perturbation. Since inflation naturally predicts adiabatic density perturbations, we will consider these first.

Adiabatic modes are fluctuations in the energy density, or the
number of particles, such that the *specific entropy* is constant for
any species *i* (assumed nonrelativistic here):

(8) |

In terms of the perfect fluid stress-energy tensor of general relativity the assumption of an adiabatic perturbation is equivalent to assuming that the pressure fluctuation is proportional to the energy density perturbation. For a discussion of the classification and behavior of adiabatic perturbations in relativity see Kolb & Turner (1990), Efstathiou (1990), Mukhanov et al (1992), and Liddle & Lyth (1993).

It has been the standard assumption that inflation predicts a "flat"
or Harrison-Zel'dovich (*n* = 1) spectrum of adiabatic density
fluctuations. Recently, with the advent of the *COBE* measurement, it
has been emphasized that inflation generically predicts departures
from the simple *n* = 1 form. In "new" inflation, the departure is
logarithmic, e.g. *P*_{mat}(*k*)
*k* log^{3}(*k* / *k*_{0}), with
*k*_{0} ~ *e*^{60} Mpc^{-1} - this
can lead to an effectively tilted spectrum between say, *COBE* scales
and more traditional normalization scales (typically
*n* 0.95) even
though the spectrum is "flat" on large scales. In "power-law",
"chaotic", "natural", or "extended" inflation, one can have power law
spectra with *n* < 1. Exotic models of inflation even allow
*n* > 1
(Mollerach et al 1993).

On scales larger than the horizon size at last scattering (i.e.
~ 2°), the generation of temperature fluctuations from density
inhomogeneities is straightforward to analyze. In addition to
fluctuations in the temperature on the surface of last scattering (due
to fluctuations in the radiation energy density), the matter
perturbations give rise to potentials on the scattering surface and
possibly time-dependent perturbations in the metric. Any time
dependence (such as gravity waves, to be discussed later) leads to
energy nonconservation along the photon line of sight. The potentials
give rise to a "red"-shifting of photons as they leave the last
scattering surface. To first order in the perturbing quantity, the
total energy change of a photon (above and beyond the cosmological
redshifting) from the time it leaves the last scattering surface
(emission) is the integral along the unperturbed path of the
(conformal) time derivative of the metric perturbation
(*h*_{µ}),
plus the change in the potential between last scattering and observation
(see Appendix B):

(9) |

where *n*^{} is
the direction vector of the photon and
is a
parameter along the line of sight. If
*h*_{µ} is
due solely to density perturbations, the integrand is basically
4, where the overdot
represents a (conformal) time derivative. Either or both of these
terms are known as the Sachs-Wolfe effect. The simplest part is the
potential difference between the last scatterers and the
observer. [The other (integral) term is usually associated with a
background of gravitational waves, nonlinear effects, or
_{0} < 1
universes - see later.] To this energy shift must be added the
temperature fluctuation on the last scattering surface itself. For
fluctuations in the radiation field, we have

(10) |

where the second equality follows from the adiabatic
condition. Because an overdensity gives a larger gravitational potential
/
=
2
+
[(*k* /
*H*)^{2}] that a photon must climb out of,
for adiabatic fluctuations, the two terms partially cancel. One finds
that *T* / *T*
= - 1/3
. The minus sign means that
CMB hot spots are matter under-densities.

2.4.2 ISOCURVATURE Isocurvature modes are fluctuations in the number density of particles which do not affect the total energy density. They perturb the specific entropy or the equation of state,

(11) |

While such perturbations are outside the horizon, causality precludes them from becoming an energy density perturbation. Inside the horizon, however, pressure gradients can convert an isocurvature perturbation into an energy density fluctuation.

The possibility of scalar isocurvature fluctuations is not well
motivated by usual inflation models, although if more than one field
contributes significantly to the energy density during inflation one
can get isocurvature fluctuations (the energy density fluctuation is
no longer proportional to the pressure fluctuation). For isocurvature
fluctuations, a positive fluctuation in the matter density (and
therefore the gravitational potential) is compensated by a negative
fluctuation in the photon temperature. The Sachs-Wolfe effect and the
initial temperature fluctuation therefore add (rather than cancel as
in the adiabatic case), giving rise to six times more large-scale
*T* / *T*
for a given "matter" perturbation. For this reason, isocurvature cold
dark matter models that are normalized to give the observed peculiar
velocities predict too large a temperature anisotropy in the CMB.

Specifically, CDM isocurvature models with roughly scale-invariant
(i.e. *m* = - 3) power spectra (e.g. in the axion model of
Axenides et al 1993)
are probably ruled out
(Efstathiou & Bond
1986).
The situation is similar for HDM
(Sugiyama et al 1989).
Scale-invariant
baryon-dominated models are also in serious conflict with the
microwave background anisotropies
(Efstathiou 1988),
and cannot be
saved even by invoking a cosmological constant
(Gouda & Sugiyama
1992).
However, models with larger in are not as yet ruled out (see also
Efstathiou & Bond
1987).
Isocurvature fluctuations are these
days only discussed in terms of the Baryonic Dark Matter model. This
is an observationally-motivated model, with low
_{0} in baryons
only. The large fluctuations generated at small scales have to be
erased by the reheating due to some early collapsed objects. The
effects of such a reionization will be discussed later. Constraints
from anisotropies on scales
1°
(Peebles 1987b,
Sugiyama & Gouda
1992),
from the Vishniac effect at small scales
(Efstathiou 1988,
Hu et al 1994),
from spectral distortions
(Daly 1991,
Barrow & Coles 1991),
and from the clustering properties of galaxies
(Cen et al 1993)
imply that only models with
-1
*m* 0 are
viable. High values of
_{0}
and high values of *h* which enhance the "bump" also tend to be ruled
out. It has recently been shown
(Sugiyama & Silk 1994)
that the BDM picture generally leads to an effective slope
*n*_{eff}
2 for the
radiation power spectrum on large scales. Fluctuations on smaller
angular scales depend on a number of tunable parameters, making BDM
complicated to constrain in practice
(Hu & Sugiyama
1994b,
c).

2.4.3 GRAVITATIONAL WAVES Until now, we have focused on the anisotropies in the cosmic microwave background arising from density perturbations in the early universe. In many models, there is also the possibility that a stochastic background of long-wavelength gravitational waves (GW) can be produced (Starobinskii 1979); for a discussion of inflationary models in this context see Rubakov et al (1982), Adams et al (1992), and Liddle & Lyth (1993). If such a background were to exist, it would leave an imprint on the CMB at large scales through the Sachs-Wolfe effect (Fabbri & Pollock 1983, Abbott & Wise 1984c, Starobinskii 1985, Abbott & Schaefer 1986, Fabbri et al 1987, Linder 1988b, White 1992). With the advent of the COBE measurement of the power at large scales, many authors addressed the question of the interpretation in terms of scalar and tensor contributions (Krauss & White 1992, Liddle & Lyth 1992, Adams et al 1992, Salopek 1992, Lucchin et al 1992, Dolgov & Silk 1993).

If there is a sizable contribution from GW in the COBE-detected
anisotropies, this would lower the predicted value of
(*T*/*T*)_{rms} on
smaller scales. This should be kept in mind when comparing
degree-scale experiments or large-scale structure studies to power
spectra normalized to *COBE* on large scales.

Unlike the anisotropies generated by scalar fluctuations
(Section 3),
those generated by (isocurvature) tensor perturbations, or GW,
damp at scales comparable to the horizon (see e.g.
Starobinskii 1985,
Turner et al 1993,
Atrio-Barandela & Silk
1994),
which means ~
sqrt[1 + *z*_{rec}]
30
(see Appendix A). This can be understood
as due to the redshifting
of GW that entered the horizon before recombination. The maximal
contribution to the anisotropy on some scale comes from gravitational
waves with wavelengths comparable to that scale. GW begin to redshift
after they enter the horizon; thus scales that are smaller than the
horizon at last scattering are dominated by GW that have redshifted
before the photon begins to travel to us. The different behavior at
small scales leads one to hope that the two contributions could be
disentangled. A detailed numerical analysis of the anisotropy
generated by GW on both large and small scales has been carried out by
Crittenden et al (1993a).

In general, GW provide a small contribution to
*T* / *T* on
top of the
scalar anisotropy. One requires a comparison of both large- and
small-scale temperature anisotropies to isolate them. On large scales,
one must deal with cosmic variance; on small scales one has sample
variance and uncertainties due to cosmological parameters and history,
which are far from orthogonal. The situation with regard to
disentangling a gravitational wave signal is somewhat confused.
White et al (1993)
claim that cosmic variance and cosmological model
uncertainty makes such a detection extremely difficult, while
Crittenden et al (1993a)
predict that a definitive detection is
possible. [The analysis assumed a specific form for the relation
between the spectral index and the ratio of scalar and tensor
contributions to the quadrupole: *T* / *S* = 7(1 - *n*)
(R. Davis et al 1992).
This form requires correction for most theories
(Liddle & Lyth 1992,
Kolb & Vadas 1993)
and also biases the fit towards "detection"
of a tensor component. In addition, recent work
(Bond et al 1994)
suggests that including uncertainties in cosmological history may
alter Crittenden et al's conclusions regarding gravity waves.] This
question is of some importance, since any possible GW signal will
affect the power spectrum normalization inferred from *COBE*. Since the
GW production predicted in most theories is very small [for
*n* 0.9 as
required by *COBE* and Tenerife
(Hancock et al 1994)
*T* / *S*
1],
perhaps their only observable effect for some time will be in generating
large angular scale CMB anisotropies
(Sahni 1990,
Krauss & White 1992,
Souradeep & Sahni
1992,
Liddle 1994,
Turner et al 1993).
The possibility that GW lead to an observable polarization in the CMB
(Polnarev 1985)
has been shown to be very small
(Crittenden et al 1993b;
however, see
Frewin et al 1994).

In theories of inflation, the normalization of the spectrum of
scalar fluctuations depends on both the inflation potential and its
derivative at the epoch of fluctuation generation. In contrast, the
tensor spectrum depends only on the value of the inflation potential
at the same epoch. This fact coupled with the *COBE* measurement can be
used to limit the scale of inflation
(Rubakov et al 1982,
Lyth 1985,
Krauss & White 1992,
Liddle 1994).
In principle, one can also derive
information about the inflation potential from both the tensor and
scalar components of the CMB anisotropy
(Liddle & Lyth 1992,
R. Davis et al 1992,
Salopek 1992).
Recently, several authors have considered
the possibility of reconstructing the "inflation" potential from CMB
observations
(Hodges & Blumenthal
1990;
Copeland et al 1993a,
b,
1994;
Lidsey & Tavakol 1993;
Turner 1993;
see also
Carr & Lidsey 1993)
or of using relations between observable parameters as "tests" of inflation
(R. Davis et al 1992,
Bond et al 1994,
but see Liddle & Lyth
1992,
Kolb & Vadas 1993).