The Sound Speed of Baryons
Last time we showed that after decoupling, , where for a monotonic ideal gas at constant entropy. As a result:
We also showed that using the third law of thermodynamics (, for ), our expression for becomes:
Where are the temperature and scaling factor at the time of decoupling. Thus, after decoupling. Moreover, the baryonic sound speed around decoupling is given by:
Using that , we have:
So at decoupling, the sound speed of baryons drops from to . This causes the Jeans mass to drop dramatically, and structure formation can begin. We’ll investigate this in PS#9.
Statistical Properties of Perturbation Fields
Recall our definition of for “ripples about the mean”:
In Fourier space, this looks like:
We should note that is real, so .
Now we’ll examine the power spectrum of these perturbations as a function of :
where is the Dirac delta function. Note that in , we dropped the vector information about . This is because the power spectrum should only depend on due to isotropy. The average we are taking in this equation is supposed to be an average over many universes. However, we only have one universe to work with, so we approximate that over a large amount of space, the average should be similar to that of several universes. Thus, we interpret this as a spatial average, even though that’s not exactly what was meant in the equation.
The Importance of :
- If is a Gaussian field (as predicted by many theories), then completely specifies its statistical properties ().
- quantifies the amount of gravitational clustering for each mode.
- The Fourier Transform of is the two-point correlation function(see PS#9):
where . A two-point correlation function describes the excess (above Poisson) probability of finding pairs of points at separation . The Sloan Digital Sky Survey has been finding that the galaxy-galaxy correlation , which is to say, you are more likely to find galaxies close together than far apart. describes clustering.
One more note on : it has units of . We can define a dimensionless version:
Angular Power Spectrum (with CMB as an example)
We then define the Angular Power Spectrum :
where ’s are Legendre Polynomials, and is what everyone typically plots for galaxy surveys. In general, we can make the substitution:
So doing out the math:
Where the last step was taken using the addition theorum of spherical harmonics.
For non-Gaussian fields, the lowest order statistic is the 3-point correlation function, which is equivalent to a bi-spectrum. In space, this says that
where describe a triangle in space.