The Growth of Primordial Fluctuations

What are our earliest hints of structure formation in the Universe? Can we get there? If we let the very small density fluctuations we see in the CMB evolve with time, will they form the very large density fluctuations we see in high redshift galaxies and quasars?

Let's look at how fluctuations in density evolve with time. For simplicity, we will concentrate on a flat, matter-dominated universe (so OmegaM=1). Look at the Friedman equation:

Now let's look at a small piece of this universe which has a little more matter than the rest. It evolves slightly differently, because of the excess matter:
Subtract one equation from the other, and we get

Let's define a quantity called delta, which is the fractional overdensity of the region:

and look how delta depends on the evolving quantities, R and rho:
So as the universe gets bigger, the overdensity grows along with it. But since size and redshift are related R = (1+z)-1, we can rewrite this as

or we can relate the density fluctuations at two times as

The CMB is a picture of the universe at a redshift of z ~ 1000. So if we let the CMB fluctuations (delta ~ 10-5) grow from z=1000 to z=5, they will have evolved into density fluctuations on the order of

Egad! That's too low by orders of magnitude!! By themselves, these density fluctuations don't have enough mass to grow into galaxies.

But there's more than just baryons -- there's dark matter. How does that affect things?