Some miscellaneous equations on the Friedmann cosmology.

I investigated the phase transition of the energy in Friedmann cosmology arising as a power equation:

Non-conservation it seemed, just appeared from a time derivative of the Friedmann equation and from there the continuity equation needs a new definition. The continuity has relationships to the first law of thermodynamics:

Variations in density can be achieved by density perturbations in early cosmology symbolized by

Moving on, the variations in density we will use will be how they vary with time including the energy content of the universe, we will explore possible ways (solutions) to the non-conserving model. The curvature part of the Friedmann equation can be written as:

In which the Friedmann equation itself is given as

replacing we get (by using

Multiplying through by we can see how the Friedmann equation at the very core is just a statements about the energy of the universe:

To achieve the non-conserved form, we just take the derivative on both sides:

Divide through by retrieves a form close to the original Freidmann equation, except it is for non-conservation

It's helpful to keep in mind the following relationships:

Another version, differentiating:

We can give it as

With variations in the density term and the energy term in the last expression will both meaure how density and energy varies in the evolution of the universe. Going back to this form now

inverting through and using the following fluid equation (which describes how density changes in the universe)

we have

(Note in this form, it seems to be the last term that contributes to the variation of energy in the universe).

The energy may also be seen as

and

Where epsilon is just some other density term related to the energy. Using the previous form

It is strange though, that the non-conserved form can be achieved very easily using a form of the fluid equation (which itself is a statement about conservation),

Using

We obtain after substitution

Which is the non-conserved form if and only if:

The fluid equation does indeed describe the evolution of the density parameters in the universe but is only a statement about conservation if it satisfies

The first term in the fluid equation describes how density dilutes over time. Second term describes loss of energy because the pressure has done work as the universes volume increases. The equation is either incomplete, or incompatible as a conservation equation satisfying only the -solution.