Here are some basic questions about how thing evolve with time (we are looking for \( n \sim 1/ a^3\) type of answers). These things are very useful to know (and something you should know) when trying to understand things in cosmology and especially when we just want to estimate roughly how things evolve which comes up quite often so its good to review them and make sure you know it by heart as they come up again and again:
[1] What is the definition of the equation of state \(w\)? What is the equation of state for all the components (baryons, CDM, photons, neutrinos, dark energy (cosmological constant) and curvature)?
[2] How does the energy density \(\rho(a)\) of the different components vary with the scale-factor? How can we explain the result for matter and radiation physically?
[3] What equation of state (for the total matter budget) do we need to get accelerated expansion \(\ddot{a} > 0\)
[4] How does the Hubble factor evolve with \(a\) (i.e. \( H \sim a^n\)) in the radiation and matter eras? What is \(H\) when dark energy dominates? What is \(a(t)\) when \(H\) is constant?
[5] How does the temperature evolve with scalefactor \(T(a)\) ?
[6] How does the energy density \(\rho\) and numberdensity \(n\) evolve with temperature \(T\) for a relativistic gas?
[7] What is the energy density \(\rho\) in terms of the number density \(n\) for non-relativistic matter?
[8] How does the conformal time \(\eta \) evolve with both \(\mathcal{H}\) and \(a\) in the radiation dominated era?
[9] What are some different measures of time in cosmology (name atleast 3) and how are they related?
[10] What is the definition of matter-radiation equality and at roughly what redshift is this in our Universe? At (roughly) what redshift, time and temperature is the CMB released? At what redshift does dark energy start to dominate?
[11] We often have to change between different times. What is \(\frac{d}{dt}\) in terms of \(\frac{d}{dx}\) (\(x=\log a\)) ? What is \(\frac{d}{d\eta}\) in terms of \(\frac{d}{dt}\) ? (Chain rule)
See also
https://cmb.wintherscoming.no/problems.php#b8