Density Fluctuations 227Linear Approximation. Much of the interesting physics of density fluctuations can
be captured by a Newtonian linear perturbation analysis of a viscous fluid. Small per-
turbations grow slowly over time and follow the background expansion until they
become heavy enough to separate from it and to collapse into gravitationally bound
systems. As long as these perturbations are small they can be decomposed into Fourier
components that develop independently and can be treated separately. For fluctua-
tions in the linear regime,|훿푘|<1, where
휌m=휌푚+훥휌m,푝=푝+훥푝, 푣푖=푣푖+훥푣푖,휙=휙+훥휙, (10.16)the size of the fluctuations and the wavelengths grows linearly with the scale푎,
whereas in the nonlinear regime,|훿푘|>1, the density fluctuations grow faster, with
the power푎^3 , at least (but not exponentially). The density contrast can also be
expressed in terms of the linear size푑of the region of overdensity normalized to the
curvature radius,
훿≈(
푑
푟U
) 2
. (10.17)
In the linear regime푟Uis large, so the Universe is flat. At the epoch when푑is of the
order of the Hubble radius, the density contrast is
훿H≈
(
푟H
푟U
) 2
, (10.18)
free streaming can leave the region and produce the CMB anisotropies. Structures
formed when푑≪푟H, thus when훿≪1. Although훿may be very small, the fluctuations
may have grown by a very large factor because they started early on (see Problem 3
in Chapter 7).
When the wavelength is below the horizon, causal physical processes can act and
the (Newtonian) viscous fluid approximation is appropriate. When the wavelength is
of the order of or larger than the horizon, however, the Newtonian analysis is not
sufficient. We must then use general relativity and the choice of gauge is important.
Gauge Problem. The mass density contrast introduced in Equation (9.4) and the
power spectrum of mass fluctuations in Equation (10.9) represented perturbations to
an idealized world, homogeneous, isotropic, adiabatic, and described by the FLRW
model. For subhorizon modes this is adequate. For superhorizon modes one must
apply a full general-relativistic analysis. Let us call the space-time of the world just
described. In the real world, matter is distributed as a smooth background with
mass perturbations imposed on it. The space-time of that world is not identical to,
so let us call it′.
To go from′, where measurements are made, to, where the theories are defined,
requires agauge transformation. This is something more than a mere coordinate
transformation—it also changes the event inthat is associated to an event in′.Aper-
turbation in a particular observable is, by definition, the difference between its value at
some space-time event inand its value at the corresponding event in the background
(also in). An example is the mass autocorrelation function휉(푟)in Equation (10.8).