318 Gravitational instability in General Relativity
is of order the photon diffusion scale at a given cosmological timet. To estimate
the diffusion scale, we note that the photons undergo aboutN∼t/τγscatterings
during timet. After every scattering the direction of propagation is completely
random, so the photon trajectory is similar to that of a “drunken sailor.” Therefore,
afterN steps of lengthτγ, the typical distance travelled (the diffusion scale) is
aboutτγ
√
N∼
√
τγt.Consequently, the ratio of the physical damping scale to the
horizon scale is
λphD
t∼(kDη)−^1 ∼√
τγ
t. (7.129)
This simple estimate is in agreement with the more rigorous result (7.128).
Before recombination, the mean free path of the photons is determined by
Thomson scattering on free electrons:
τγ=1
σTne, (7.130)
whereσT 6. 65 × 10 −^25 cm^2 is the Thomson cross-section andneis the number
density of free electrons. We are mainly interested in the dissipation scale at re-
combination time when the universe is dominated by cold dark matter. Taking into
account thattr∝
(
(^) mh^2
)− 1 / 2
z−r^3 /^2 andne∝(
(^) bh^2
)
z^3 r,we infer from (7.129) that(kDηr)−^1 ∼(σTnetr)−^1 /^2 ∝(
(^) mh^2
) 1 / 4 (
(^) bh^275
)− 1 / 2
zr−^3 /^4. (7.131)Problem 7.18Using the exact formula (7.128) withc^2 s= 1 /3 and assuming in-
stantaneous recombination, calculatekDand show that
(kDηr)−^1 0. 6(
(^) mh^2
) 1 / 4 (
(^) bh^2
)− 1 / 2
z−r^3 /^4. (7.132)
The dissipation scale can never exceed the curvature scaleH−^1 ≈t, because
there is insufficient time for radiation to rearrange itself on those scales. This im-
poses a limit on the range of validity of (7.132), namely,(kDηr)−^1 < 1 .Ifτγgrows
and begins to exceed the cosmological timet,then we have to use the kinetic de-
scription for photons. Our analysis of viscous damping is also not valid on scales
smaller than the mean free path of the photons since the hydrodynamical description
fails in this limit. On scales smaller than the mean free path, another effect,free
streaming, becomes important. Free streaming refers to the propagation of photons
without scattering. On scales smaller than the mean free path, photons coming
from different directions with different temperatures intermingle, smearing spatial
inhomogeneities in the radiation energy density distribution. However, in contrast
with viscous damping, free streaming does not remove the angular temperature
anisotropy of the radiation at a given point (see Problem 9.2). As with viscous
damping, free streaming has no effect on scales larger than the horizon scale.