Cosmology Lecture 19

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Finishing Linear Perturbation Theory

The last missing piece in the equations we’ve been discussing is the calculation of the speed of sound in these fluids. For this, we’ll focus on the baryon-photon fluid in two epochs:

  • Before Decoupling ():

The energy density of the baryon-photon fluid is given by:

where is a constant, is the mass of hydrogen, and because , which corresponds to , which was only valid very early on. The pressure of the baryon-photon fluid is dominated by , so:

Now we can calculate the adiabatic sound speed:

where is entropy. Now since , and , we can say , and . Therefore:

Thus, our equation for the adiabatic sound speed becomes:

If the energy density is dominated by photons (), then:

Using , , we can show that for .

  • After Decoupling:

After decoupling, the speed of sound in baryons are the speed of sound in neutral hydrogen (and Helium), which is an Ideal Gas to a good approximation. Thus, . We also know , where is small (again). Recall that for an ideal gas (), that , where . is the specific heat for constant pressure, volume. The proof for this is as follows:

Replacing , using , we get:

Thus, , where . Since we also know that , so using that :

We also know . We can use a previous equation to show , so: