Cosmology Lecture 14

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Jeans Instability Continued

So far we’ve written down 3 equations: the Continuity Equation, Euler’s Equation, and Poisson’s Equation. From these we derived a dispersion relation for a static medium. We evaluated two wavelength regimes for this relation and found that for wavelengths longer than the Jeans wavelength, we had exponential collapse due to self-gravity, and for wavelengths shorter than the Jeans wavelength, we have normal, stable oscillation.

Basically, we are seeing a balance between pressure outward and gravity inward. Collapse occurs when gravity wins out over pressure. Consider a sphere of density , and a piece of volume at distance . Then the force of gravity on that piece is:

since . The pressure force on that volume is:

Comparing these forces, we find that gravity wins if:

Solving for :

So this is a quick-and-dirty way of deriving the Jeans length scale. The timescale for collapse is also a balance between gravity and pressure. The time for gravitational free fall is given by:

Thus, the free fall time is independent of distance. The pressure timescale is the sound crossing time, given by:

We will have gravitational collapse if , which means:

So again we get out the Jeans wavelength.

Gravitational Instability in an Expanding Fluid

Density is obviously affected by expansion. To zeroth order:

where is our familiar scale factor. To examine first order perturbations, we define the density field , where , so that:

Our velocities are also changed by expansion:

where is in physical coordinates. This velocity is purely from Hubble expansion. Note that:

We’ll use to describe the peculiar velocity (motion with respect to the expansion). Now we need to solve our fluid equations using these parameters. The continuity equation, to zeroth order, says:

To first order:

Using that , and :

Euler’s equation gives us, to zeroth order:

So there is no Jeans swindle in this case. Expanding this to first order, we get:

To figure out what is, let’s expand it:

Thus our equation above becomes:

Finally, we have Poisson’s Equation: