# Cosmology Lecture 16

Recall that for :

We did different cases for this equation:

- For flat , we found two solutions:

We also found that was independent of , and that , and .

- Suppose we have an open universe:

Using the equation we derived previously:

and we can do some algebra to come up with the equation:

This equation again has two solutions:

- For a closed universe (we will play with this on the homework), we get the following equation:

This has solutions:

### Linear Perturbation Theory of Gravitational Instabilities

We’re just going to do a sketch of the full story, which grew out of Lifshitz (1946). Additionally, look at Weinberg section , Dodelson Ch. 4, and Peebles (1980). The full theory is described by linearized, coupled Einstein equations (for metric perturbations), and the Boltzmann equations (for density perturbations).

### Metric Perturbations (in “Synchronous” Gauge)

Recall our space-time interval in expanding space:

where . We are going to make perturbations to this equation. To do this, we are going to define a “gauge”. This is analogous to the vector in E&M (called the Coulomb gauge), where:

and . For metric perturbations, we will create a metric perturbation tensor , which is a symmetric tensor with 6 degrees of freedom. This is added to our equation for the space-time interval:

We then decompose this perturbation tensor into its trace and traceless components:

We will make the following definitions:

- (which has 1 degree of freedom)
- is transverse: (2 degrees of freedom)
- The divergences of the vectors (with 1 degree of freedom) and (with 2 degrees of freedom) are longitudinal (curl-free) and transverse (div free) respectively. We will write the longitudinal component as (recall ):

and the transverse component:

These definitions together give us:

where is some scalar field and is some vector field where .

Now we will discuss three different modes of perturbations:

- Scalar Mode, described by (recall , ), which accounts for galaxy formation.
- Vector Mode, described by (which is )
- Tensor Mode, described by , which accounts for gravitational waves/radiation.

How did we get 6 degrees of freedom? Einsteins equations tell us that , where is the metric (), and is the energy-momentum tensor. Since these tensors are symmetric, we have 10 degrees of freedom here. However, energy conservation (the Bianchi identity) states that , which imposes 4 more constraints, leaving us with 6 degrees of freedom.

It should also be said that we have other gauge choices besides the one we defined. For example, the *Conformal Newtonian gauge* (which is restricted for *scalar* modes) is:

and has 2 scalar fields () which are related by gauge transformation to ().