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 u = HR. Thus its equation of motion is
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 any time. The universe expands at an ever-slower rate without ever reaching a maximum expansion (or contracts at an ever-faster rate from infinite radius, but we don't live in that solution!). Using the identity H = (dR/dt)/R, Eq. (4) implies that
a(t) 2/3 (5) R(t) = R(t_0) ------ = R(t_0) (t/t_0) , a(t_0)where the dimensionless scale factor a(t) completely describes how the radius of the universe changes with time.
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 r is the position vector, u is the velocity field, Phi is the gravitational potential, and the ds represent partial derivatives. Explicit index notation is used here to make subsequent transformations painfully obvious. These equations are valid even if the density fluctuations are large, but because pressure forces have been neglected they do not describe the dynamics in collapsed regions where orbits have intersected.
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 Lambda merely to show that such a constant has no overt influence on the local dynamics.
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 comoving position x, rather than a fixed spatial position r (Peebles 1993, Eq. 5.101). However, most of the additional terms generated by this shift cancel, leaving Eqs. (13) & (14) very similar in structure to Eqs. (6) & (7). The main difference is the second term on the LHS of Eq. (14), which has no obvious analog in Eq. (7). This term represents the decay of peculiar velocities v ~ 1/a which comes about because in an expanding universe a particle with nonzero peculiar velocity tends to catch up with unperturbed particles with the same initial space velocity (e.g. Peebles 1993, Eq. 5.44).
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 general solution to this equation has the form
2/3 -1 (20) delta(x,t) = A(x) t + B(x) t ,where A(x) and B(x) are functions of position only. The first term, proportional to t^2/3, is the growing mode, while the second term, proportional to 1/t, is the decaying mode. Even if the early universe contained a mixture of growing and decaying perturbations, only the growing ones would be left today.
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 c_s = (dp/drho)^1/2 is the sound velocity. This is no longer a completely local equation: the rate of change of delta now depends on its spatial gradient. We can understand the implications of the pressure term by expanding delta in a Fourier series:
--- \ 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 rho_b and sound speed c_s. Perturbations with wavelengths much greater than lambda_J grow unimpeded by pressure forces, while those with wavelengths much smaller behave like sound waves. Detailed computations (e.g. Peebles 1993, Fig. 6.9) show that before the decoupling epoch density perturbations are `frozen' in place by radiation drag, and that those with wavelengths greater than the Jeans wavelengths at decoupling can subsequently grow. Perturbations with wavelengths less than lambda_J are damped by friction between matter and radiation as the universe becomes transparent, and consequently have amplitudes much reduced after decoupling.
Last modified: April 5, 1995