Still More on Metric Perturbations
Recall that we have our Boltzmann equation, which describes metric perturbations. To gain physical intuition into this equation, started taking velocity moments of this equation (in the non-relativistic limit).
and we defined . We can iterate to calculate any moment. We then used these moments inside the Boltzmann equation.
moment of Boltzmann (the continuity equation):
moment of Boltzmann:
In the Newtonian limit (), dropping the ’s, and using (where is the non-diagonal piece), we have:
Which is just Euler’s Equation for a perfect fluid.
moment of Boltzmann. T his is messy, so we’ll use that for a perfect gas, , which gives us:
where is the heat flux.
moment: Recall the linearized continuity and Euler’s equations (in comoving coordinates):
Defining (and so ):
So compare the Boltzmann equation:
with the full theory (the coupled Boltzmann and Einstein equations in a conformal Newtonian gauge):
This gives us the Geodesic equation (which is the GR equation of motion):
where the term is specific to GR, , and . In GR, the Boltzmann equation is written using the energy-momentum tensor:
where is the phase space distribution. The first-order part of the energy-momentum conservation equation () gives us the following moments:
where . There are, of course, higher moments as well. Compare the above result to the result we got using just the linearized fluid equations. That simplistic result was almost right, except for the factor of . We can look at different components of this energy-momentum conservation equation:
- CDM: here we are pressure-less, , , so we have:
- Baryons: we have a non-relativistic fluid, , and :
- Photons: , :
- Neutrinos: no Thomson scattering, collision-less, etc.
Coupled Baryon-Photon Fluid
Before photon decoupling, baryons and photons are tightly coupled via Thomson scattering (it behaves like a single fluid). Recall that Thomson scattering is the low-energy limit of Compton scattering, which is the general mechanism for photon-electron scattering. We are in this low-energy regime because after about 1 second. The cross-section for Thomson scattering is , and
The optical depth to Thomson scattering is given by:
where is the electron density, and is distance along the line of sight. is optically thick (opaque), and is optically thin (transparent).
We can calculate when photons decouple (when from Thomson scattering ):
where integral over distance is written in terms of redshift. We can use the Friedmann equation to replace , and we’ll use that the free electron density is given by:
where is the # density of hydrogen and is the ionization fraction. We can use the Y parameter to calculate :
which tells us that . Thus:
We’ll actually do this out to get as a function of and cosmological parameters on PS#8. Next time we’ll see if we can calculate the sound speed .