362 Cosmic microwave background anisotropies
and we obtain the result that
(
δT
T
+
)
=const, (9.21)along null geodesics. The influence of the gravitational potential on the microwave
background fluctuations is known as the Sachs–Wolfe effect.
In reality, radiation is a small but not completely negligible fraction of the en-
ergy density immediately after recombination, and so is actually slowly time-
varying. Consequently, the combination(δT/T+ ), according to (9.20), varies
by an amount proportional to the integral of∂ /∂ηalong the geodesic of the pho-
ton. The contribution to the temperature fluctuations induced by the change of the
gravitational potential due to the residual radiation after equality is called the early
integrated Sachs–Wolfe effect. When dark energy, either quintessence or vacuum
density, overtakes the matter density in more recent epochs, the gravitational po-
tential begins to vary once again, causing a further contribution to the temperature
fluctuations. This phenomenon is referred to as the late integrated Sachs–Wolfe ef-
fect. To simplify the final formulae, we neglect both integrated Sachs–Wolfe effects
which in any case never contribute more than 10–20% to the resulting amplitudes
of the fluctuations. The reader can easily generalize the formulae derived below to
include these effects.
Free-streamingLet us consider an initial distribution of free photons with temper-
atureT+δTwhere
δT
T(ηi,x,l)=Aksin(kx)g(l). (9.22)In this case, the radiation energy density is inhomogeneous and its spatial variation
is proportional to
〈δT/T〉l=Aksin(kx)〈g〉l,where〈〉ldenotes the average over directionsl.
Problem 9.2Neglecting the gravitational potential in (9.20), show that at later
times
〈
δT
T
(η)〉
l=
sin(k(η−ηi))
k(η−ηi)〈
δT
T(ηin)〉
l. (9.23)
Thus, atηηi, the initial spatial inhomogeneities of the energy density of free
photons will be suppressed by the ratio of the inhomogeneity scale to the Hubble
radius. This damping effect is known as free-streaming. The suppression occurs
as a result of the mixing of photons arriving at a given point from regions with
different temperatures.