Effect of ratio of energy density of matter and radiation on redshift

[cp05]

Homework Statement

If the ratio of the energy densities of matter and radiation was much higher than it is today (by several orders of magnitude), in a few sentences describe what effect this would have on the redshift (or scale factor) when the universe switched from being "radiation-dominated" to "matter-dominated", and hence on the damping of any "acoustic waves" within the "particle horizon"

Homework Equations

There's no equations, but the question says to look at a figure in the text. If you google "evolution of baryonic jeans mass and baryonic mass within particle horizon with scale factor", the first link that comes up takes you right to that page.

The Attempt at a Solution

First off, I don't understand what the figure we're supposed to look at has anything to do with the question.

I'm assuming the question is asking what will happen to the redshift when the universe switched from being radiation to matter dominated if there is a lot more matter in the universe than radiation, correct? I'm guessing the redshift would be higher (or larger...so the universe would make the switch earlier in its lifetime). And now I'm supposed to say how this affects the perturbations...if there are more perturbations therefore more large-scale structures formed or if the perturbations are damped within the particle horizon.

But I don't understand how the particle horizon and Jean's mass works...because if you're outside of the particle horizon then you can collapse, but if you are inside you can't. But in a matter-dominated universe the jean's mass is inside the particle horizon and we still have growing perturbations if their mass is greater than Jeans mass. Can someone explain how this works to me??

Right now I will guess that there is less damping of the perturbations because the era of recombination is happening earlier in time which makes jeans mass way smaller and more perturbations can collapse. But I really can't explain why this is correct (if it even is)

Thanks for all the help :)

 

[cepheid]

Welcome to PF cp05!

The Attempt at a Solution

First off, I don't understand what the figure we're supposed to look at has anything to do with the question.

Is there any way that you can show us the figure?

I'm assuming the question is asking what will happen to the redshift when the universe switched from being radiation to matter dominated if there is a lot more matter in the universe than radiation, correct? I'm guessing the redshift would be higher (or larger...so the universe would make the switch earlier in its lifetime).

Yeah. This is really easy to show quantitatively. How does the matter density vary with scale factor? Hint: for matter, in any given volume, the total particle number is conserved, and this volume increases with the cube of the scale factor. Now, how does the radiation density vary with scale factor? What happens when you equate the two of them?

And now I'm supposed to say how this affects the perturbations...if there are more perturbations therefore more large-scale structures formed or if the perturbations are damped within the particle horizon.

I'm a little less certain on this stuff, but here's what I do know. The trick is that during the radiation dominated era, the Jeans mass is on the same order as the mass enclosed in a horizon volume. Therefore, as soon as a perturbation enters the horizon during the radiation dominated era, it is stabilized against collapse (because it is less massive than the Jeans mass), and it begins to undergo acoustic oscillations. In contrast, perturbations on super-horizon scales continue to grow slowly during this era. Therefore, for perturbations with size scales less than or equal to the horizon scale at the epoch of equality, there is an attenuation. These perturbations didn't grow during the radiation dominated era, whereas the ones larger than the horizon scale at equality did grow. If you change when the epoch of equality occurred, you change which size scales are affected by this attenuation. That's all I know.