MODERN COSMOLOGY

Dynamics of structure formation 53

growth increases smoothly to the Einstein–de Sitterδ∝abehaviour (M ́esz ́aros
1974). The overall behaviour is therefore similar to the effects of pressure on
a coupled fluid: for scales greater than the horizon, perturbations in matter and
radiation can grow together, but this growth ceases once the perturbations enter
the horizon. However, the explanations of these two phenomena are completely
different.

2.6.3 The peculiar velocity field

The foregoing analysis shows that gravitational collapse inevitably generates
deviations from the Hubble expansion, which are interesting to study in detail.
Consider first a galaxy that moves with some peculiar velocity in an
otherwise uniform universe. Even though there is no peculiar gravitational
acceleration acting, its velocity will decrease with time as the galaxy attempts
to catch up with successively more distant (and therefore more rapidly receding)
neighbours. If the proper peculiar velocity isv, then after time dtthe galaxy
will have moved a proper distancex =vdtfrom its original location. Its near
neighbours will now be galaxies with recessional velocitiesHx=Hvdt, relative
to which the peculiar velocity will have fallen tov−Hx. The equation of motion
is therefore just

v ̇=−Hv=−

a ̇

a

v,

with the solutionv∝a−^1 : peculiar velocities of non-relativistic objects suffer
redshifting by exactly the same factor as photon momenta. It is often convenient
to express the peculiar velocity in terms of its comoving equivalent,v≡au,
for which the equation of motion becomesu ̇=− 2 Hu. Thus, in the absence of
peculiar accelerations and pressure forces, comoving peculiar velocities redshift
away through the Hubble drag term 2Hu.
If we now include the effects of peculiar acceleration, this simply adds the
accelerationgon the right-hand side. This gives the equation of motion

u ̇+

2 a ̇

a

u=−

g

a

,

whereg=∇δ/ais the peculiar gravitational acceleration. Pressure terms have
been neglected, soλλJ. Remember that throughout we are using comoving
length units, so that∇proper=∇/a. This equation is the exact equation of motion
for a single galaxy, so that the time derivative is d/dt=∂/∂t+u·∇. In linear
theory, the second part of the time derivative can be neglected, and the equation
then turns into one that describes the evolution of the linear peculiar velocity field
at a fixed point in comoving coordinates.
The solutions for the peculiar velocity field can be decomposed into modes
either parallel to g or independent of g (these are the homogeneous and
inhomogeneous solutions to the equation of motion). The interpretation of
these solutions is aided by knowing that the velocity field satisfies thecontinuity