Structure Formation

Assume that some process produces fluctuations in the local density of the Universe. Random fluctuations will not production perturbations large enough to lead to the formation of galaxies--something must generate larger fluctuations than predicted by random processes. The idea is that these overdense regions of the Universe collapse (due to gravity) and form the galaxies, stars, clusters of galaxies, Great Walls, ... , and so on, which are found in the Universe. Comments:

• Most of the structures we see in the Universe are composed of baryons. As a consequence, most of the structures in the Universe will not start to form before the Universe makes the transition from opaque to transparent with respect to the photons. This is due to the fact that when the Universe is opaque, baryons and radiation are strongly coupled (interact strongly), and the radiation is able to prevent the collapse of the overdense regions.
• As implied by the tiny fluctuations in the CMBR, dT/T ~ 0.00001, the material which coupled strongly to the photons could not be strongly clumped at the Epoch of Recombination.

Baryonic structures that we see in the Universe must have started forming after the Epoch of Recombination and so the timescales for structure formation have to be fairly short.

TIMESCALES

A Newtonian analysis leads to the result that density fluctuations, in the absence of pressure support, grow as

d/dt(dF/dt) + 2/H(t) dF/dt - 4 pi G rho(t) F = 0

(in the co-moving frame). If we cross-out the term with H(t), that is, we consider a static universe we get the familiar star formation result that the density fluctuation grows exponentially as

F = Fluctuation ~ exp[time x sqrt{G rho}]

where 1 / sqrt[G rho] is the free-fall timescale. This is very fast. For a static Universe relative density fluctuations of size ~ 10**(-35), galaxy sized structures would form over the lifetime of the Universe. The bugaboo is that the Universe is not static--it is expanding. What happens is that the Universe tries to pull the fluctuation apart at the same time that gravity tries to amplify the fluctuation. The competition leads to a smaller rate for the amplification of density fluctuations. For an Einstein-de Sitter universe,

Fluctuation ~ constant(1) x time**(2/3) + constant(2) / time

The growing fluctuation depends only weakly on the time. In order for galaxies to form by z ~ 2, the sizes of the fluctuations at recombination must have been on the order of 0.01 ---> 0.0003. This is large, recall that COBE only detected temperature fluctuations on the order of ~ 0.00003 Kelvin out of 2.73 Kelvin. What's the scoop?

TEMPERATURE FLUCTUATIONS

We have been assuming that there is a direct relationship between density fluctuations and temperature fluctuations. Is this reasonable?

• For baryonic material which is strongly coupled to photons, this makes a certain amount of sense. For example, suppose that we look at the hot early Universe around recombination;

  the gas pressure is P = n k T ===> if the density goes up ===> the temperature must go down to maintain constant pressure. We find that dn/n ~ dT/T. For what is known as an adiabatic perturbation, we have dn/n = 1/3 (dT/T) -- dT/T and dn/n are roughly the same.

  For a mix of radiation and matter, the same sort of relationship holds but the coupling ("1/3") is not as strong. Is this a way out? No, because the Universe is matter dominated at the time of recombination. [The Universe becomes matter dominated at around z ~ 20,000 - 30,000. Recombination occurs around z ~ 1,000.]

• For non-baryonic material which de-couples before the baryons de-couple, the temperature fluctuations do not have to be as strongly coupled to the temperature fluctuations. For example, say there are weakly interacting massive particles (WIMPs) which start to grow around the time the Universe becomes matter dominated. Such fluctuations can start growing before Recombination and not affect the small scale smoothness of the temperature of the CMBR.