Gravitational potentials
I got some questions regarding Phi and Psi and which one is really the gravitational potential? Both of these are informally referred to as “Gravitational Potentials” in the literature and we cosmologist often mix up what we call them.
In most derivations of the Newtonian limit of GR we find that its the g00 component (i.e. “Psi”) that gives us the Newtonian equations (see e.g. this):
x’’=-DPsi
D^2 Psi = 4piG rho
This is why “Psi” is perhaps most often called the Newtonian potential in the literature (see e.g. the discussion in Bertschinger and Ma).
If we look at the perturbations we are implementing in the Newtonian gauge (in the sub-horizon limit k >> Hp where we can ignore small terms like Psi and Phi’ for big terms like k^2 Phi and “delta”’s) then they can be written:
D^2 Phi = -4piG rho
D^2 (Phi + Psi) = (anisotropic stress)
From the first equation it is -Phi that plays the role of the good old Newtonian potential so it’s also natural to call this the Newtonian potential. This interpretation is the one used on Wikipedia which call -Phi the Newtonian gravitational potential (note its the space-space component which they call Psi which is our -Phi).
The reason both makes sense is that in the absence of anisotropic stress (i.e. Theta2=Nu2=...=0), then the two potentials are exactly the same: Psi = -Phi from the last equation. For example the whole treatment in Baumann for simplicity only works with one single potential.
The distinction between them becomes more interesting when we consider other gravity theories. In that case we no longer need to have Psi = -Phi when Theta2 = Nu2 = 0 and people try to constrain Phi/Psi using different probes to test gravity.
More important than what they are called is how the different potentials enters different physical effects and observables in cosmology:
* Gravitational redshift is determined by the time-time component of the metric, i.e. Psi (see e.g. the line of sight integral in Milestone 4).
* The ISW effect is determined by Psi’ - Phi’ (see e.g. the line of sight integral in Milestone 4).
* When we look at gravitational lensing then it’s the combination (Psi - Phi)/2 which tells us how much light gets bent.
When we ask you to plot the gravitational potential we simply mean to plot any of the potentials (Phi or Psi) and the sum of the potentials (the anisotropic stress Psi+Phi). You can call them what you feel is most natural as long as you are consistent within.