The growth of fluctuations in an expanding universe is properly treated in General Relativity, but with a few hints from that field the basics can be understood using Newtonian dynamics. Here it is shown that fluctuations in pressureless `dust' grow as a power of the time. Gas pressure causes small-scale fluctuations to behave like sound waves, while fluctuations with wavelengths much greater than the Jeans length grow like their pressureless counterparts.

Before contemplating the evolution of fluctuations, consider the
dynamical evolution of an expanding but homogeneous universe.
Newtonian theory cannot handle an infinite universe because the
gravitational potential is not defined. But a theorem due to Birkhoff
(*e.g.* Peebles 1993) implies that a *finite* spherical
volume carved from such a universe evolves just as it would if left in
place; in effect, the rest of the universe can be considered a nested
set of spherical shells, and Newton already knew that the force
*inside* a spherical shell is zero.

Thus at some initial time `t_0` let the sphere have radius
`R(t_0)` and uniform density `rho_b(t_0)`, and at radius
`r` let the outward velocity be `u(r) = H(t_0) r`, where
`H(t_0)` is Hubble's `constant'. With these initial conditions
the sphere's density remains uniform at later times -- the center of
the sphere is *not* a special place in the universe it was carved
from, so the density there can be no different than the density
elsewhere. Thus the evolution of the sphere, and the universe it was
carved from, is given by the function `R(t)`.

Focus on a test particle at the edge of the sphere. It moves in the potential of a mass

4pi 3 (1) M = --- R rho_b , 3with initially outward-directed velocity

2 d R GM (2) --- = - -- . 2 2 dt RThe energy per unit mass of the test particle is a conserved quantity, given by

1 2 M (3) E = T + U = - v - G - 2 R 1 2 4pi 2 = - (HR) - --- G R rho_b . 2 3

The sign of `E` determines the character of the motion of the
particle -- and by extension that of the sphere and indeed even the
universe from which that sphere is carved. If `E > 0` the
particle escapes with energy to spare -- thus at late times the universe
goes over into free expansion. If `E > 0` the particle reaches
a maximum radius before falling back -- and the universe does likewise,
ending in a `big crunch'.

We are mostly interested in the so-called `critical' case where `E
= 0` and the particle is neither bound nor unbound, first because
our own universe is not *very* far from this case, and second
because at early times when structures started forming, the dynamics of
*any* universe are closely approximated by those of the critical
case. Putting `E = 0` in Eq. (3), we find that

2 8pi (4) H = --- G rho_b . 3This is the law relating the expansion rate to the density in a critical or `Einstein-de Sitter' universe, and it is true at

a(t) 2/3 (5) R(t) = R(t_0) ------ = R(t_0) (t/t_0) , a(t_0)where the dimensionless

Now suppose that the sphere contains some density fluctuations; that
is, the density is not everywhere equal to the average density
`rho_b`. We will first consider the case where pressure forces
are negligible, implying that the temperature is zero everywhere. This
is also the case if the sphere is composed of collisionless particles
with zero velocity dispersion -- sometimes referred to as `cold dust'
although it has nothing to do with interstellar material.

The Newtonian equations for the evolution of the density, velocity, and potential are

d d (6) -- rho + ---- (rho u_i) = 0 , dt dr_i du_i du_i d (7) ---- + u_j ---- = - ---- Phi , dt dr_j dr_i d d (8) ---- ---- Phi = 4 pi G rho , dr_i dr_iwhere

Because we want to discuss inhomogeneous *departures* from the
unperturbed solution given in Eq. (5), it is convenient to work in a
coordinate system where the unperturbed solution does not change with
time. In effect, we will adopt a frame of reference which contracts at
just the right rate to cancel out the uniform expansion, the better to
see non-uniform structures develop. In terms of the scale factor
`a(t)`, let

(9) x = r/a , . (10) v = u - a x , (11) delta = rho/rho_b - 1be the `comoving' position vector, peculiar velocity, and density contrast at each place within the sphere. It is convenient to also use the peculiar potential

1 2 2 (12) phi = Phi - - (4 pi G rho_b - Lambda) a x , 6written here with a cosmological constant

With the coordinate transformation defined by Eq. (9), the equations of motion become

d 1 d (13) -- delta + - ---- [(1 + delta) v_i] = 0 , dt a dx_i . dv_i a 1 dv_i 1 d (14) ---- + - v_i + - v_j ---- = - - ---- phi , dt a a dx_j a dx_iwhile Poisson's equation becomes

d d 2 (15) ---- ---- phi = 4 pi G rho_b a delta . dx_i dx_iA subtle point to note here is that the time derivatives are now to be evaluated at a fixed

The next step is to linearize the equations of motion by assuming
that `|delta|` is much less than unity. It follows from Eq.
(13) that `|v_i|` is also small, so we can just drop all
terms proportional to `delta v_i` or v_i^2 above,
obtaining

d 1 dv_i (16) -- delta + - ---- = 0 , dt a dx_i . dv_i a 1 d (17) ---- + - v_i - - ---- phi = 0 . dt a a dx_iCombining the time derivative of Eq. (16) with the gradient of Eq. (17) and inserting Eq. (5), we get

2 . d a d (18) --- delta + 2 - -- delta = 4 pi G rho_b delta . 2 a dt dtThis equation is valid for a closed, open, or critical cosmology. For the critical case derived above (Eqs. 4 & 5), it reduces to

2 d 4 d 2 delta (19) --- delta + --- -- delta = ------- . 2 3 t dt 2 dt 3 tThe

2/3 -1 (20) delta(x,t) = A(x) t + B(x) t ,where

Note that fluctuations grow as a power of the time `t`.
Instabilities encountered in everyday life, such as the development of
turbulence in flowing water, typically grow exponentially with time.
Exponential growth is so rapid that there is no real difficulty in
accounting for the `seed' fluctuations. But the growth seen in Eq.
(20) is relatively slow; for example, a fluctuation which is just
entering the nonlinear regime (`delta ~ 1`) can only have grown
by a factor of `1+z = 10^3` since the epoch of decoupling
probed by the COBE observations. Thus if galaxies formed by
gravitational instability, significant density fluctuations must have
been present in the early universe.

The effects of gas pressure can be included by adding a pressure term to Eq. (7). Carrying this term through the subsequent analysis, we get

2 2 d 4 d 2 delta c_s d d (21) --- delta + --- -- delta = ------- + --- ---- ---- delta , 2 3 t dt 2 2 dx_i dx_i dt 3 t awhere

--- \ i k.x (22) delta(x,t) = | delta_k(t) e . / --- kInserting this expansion in Eq. (21), we find

2 2 2 d 4 d k c_s (23) --- delta_k + --- -- delta_k = (4 pi G rho_b - -----) delta_k . 2 3 t dt 2 dt aThe RHS of this equation vanishes for Fourier waves with the proper wavelength

2 pi a pi c_s^2 1/2 (24) lambda_J = ------ = (--------) , k_J G rho_bwhich is just the Jeans length for a gas with density

- Peebles, P.J.E. 1993,
*Principles of Physical Cosmology*.

Joshua E. Barnes (barnes@zeno.ifa.hawaii.edu)

Last modified: April 5, 1995