Introduction to Cosmology

Structure Formation 229

a 1 a 2

Inflation

(No inflation)

H

-^1 ~t~a

1/n

H–^1

Standard cosmology

log (length)

Ngal =

In(aH/a 2 )

Nhor = In(aH/a 1 )

λgal

λhor

aH

log(a)

aeq a 0

Figure 10.1The comoving length scale of fluctuations versus. the cosmic scale of the Universe
푎. The wavelength휆horcorresponds to fluctuations which disappeared over the horizon at푎 1
and re-entered only recently, at≈푎 0. The growth in the scale factor푎while the fluctuations are
outside the horizon, from푎 1 to the end of inflation at푎H, is for our present horizon,푁hor=
ln(푎H∕푎 1 )≈53. Fluctuations exiting at푎 2 will correspond to galaxies entering at푎H, when the
scale factor has grown by푁gal=ln(푎H∕푎 2 )≈45.

locally and to attract surrounding matter which can be seen as inflow. In other regions
where the pressure forces dominate, the fluctuations move as sound waves in the fluid,
transporting energy from one region of space to another.
The dividing line between these two possibilities can be found by a classical argu-
ment. Let the time of free fall to the center of an inhomogeneity in a gravitational field
of strength퐺be

푡G= 1 ∕

√

퐺휌. (10.19)

Sound waves in a medium of density휌and pressure푝propagate with velocity

푐s=

√

휕푝

휕휌

,

so they move one wavelength휆in the time

푡s=휆∕푐s. (10.20)