Linear Terms in the Expanding Universe
Recall that we are solving for the (linear) perturbing parameters for an expanding universe using 3 equations:
(recall that by definition of ). The trick to solving these three equations is to go into Fourier space and change from physical coordinates to comoving coordinates (to get rid of the Hubble expansion component). We have 3 fields () that we need to move into Fourier space:
where is the comoving wavenumber, and are the physical coordinates. Thus:
and so on. Using the properties listed above, we get for the first equation:
The second equation becomes:
And finally, Poisson’s equation is:
To solve these equations, we first combine (1) and (2). Taking (1), we get:
then taking the derivative with respect to time:
We can rewrite this in terms of the comoving Jeans wavenumber :
The unstable regime for this equation is where :
Sometimes you’ll find this equation written with fewer dots (time derivatives). This is because people will occasionally redefine their time to be “conformal time”, which is normal time divided by the expansion parameter.
- Suppose we have a flat, matter dominated universe where . Then rewriting so we don’t get confused about perturbation parameters vs. present-day values, we get:
This equation has a polynomial solution. If , then:
Thus, there are 2 solutions–a growing mode:
and a decaying mode:
These are very important results for understanding how structure forms in an expanding universe.
Now we can solve for : . The perturbed gravitational potential is constant.
We also want to solve for . To do this, we’ll decompose , where is called the divergence free “rotational” or “transverse” mode. Then using:
we find that . For (the curl-free “longitudinal” mode), , so . Thus in k-space:
For the component, equation 2 gives us , so . Likewise, for the component, we get:
Using (1), this becomes:
So we worked out linear perturbations for a matter dominated universe.