Friedmann equations have a solution for a mix of matter - even for a sudden jump in pressure

We presented a hypothesis in our blog post on October 20, 2021 that a mixture of slow massive matter and radiation has no solution for both Friedmann equations. The hypothesis is false. All mixtures of matter do satisfy the second Friedmann equation if they satisfy the first one. This is because of energy conservation.

Our October 16, 2021 claim that Friedmann equations do not have a solution for a sudden jump of pressure was incorrect.

https://en.wikipedia.org/wiki/Friedmann_equations

Theorem. Any uniform matter has a solution for both Friedmann equations in the case where the curvature parameter k = 0. We assume energy conservation.

Proof. Let us have any uniform matter. The first Friedmann equation clearly has a solution. Let

        ρ(a)

be the mass-energy density as a function of the scale factor a. If the matter is slow massive matter, then

       ρ(a) ~ 1 / a³,

       D_a ρ(a) = -3 / a * ρ(a),

where we denote the derivative operator against a by D_a

We assume energy conservation. If the derivative D_a ρ(a) has some other value, it has to be because pressure is doing work. If we let a grow to a + da, then the lost mass-energy is

       [-3 / a * ρ(a) - D_a ρ(a)] da * a³ * c²

in a cube whose side is a. The volume growth of the cube is 3 a² da. The pressure has to be

        p(a) = [-ρ(a) - 1/3 a D_a ρ(a)] * c²            (1)

to conserve energy.

We denote the derivative against the time coordinate t with a prime mark '. The first Friedmann equation is

       (a' / a)² = 2/3 κ ρ                                        (F1)

<=>

       a'          = sqrt(2/3 κ) a sqrt(ρ).

We calculate the time derivative of both sides of the equation (we denote C = sqrt(2/3 κ):

=>  a''          = C a' sqrt(ρ)

                       + C a * 1/2 * 1 / sqrt(ρ) * 

                          * (D_a ρ) a'

                    = C² a sqrt(ρ) sqrt(ρ)

                        + 1/2 C² a² sqrt(ρ) / sqrt(ρ)

                           * D_a ρ

                    = 2/3 κ a ρ

                       + 1/3 κ a² D_a ρ

<=>

     a'' / a     = 2/3 κ ρ                                           (2)

                      + 1/3 κ a D_a ρ.

The second Friedmann equation is:

     a'' / a       = -1/3 κ ρ - κ p / c²                      (F2)

Now we subtract (F2) from (2). We assume that we have a solution for (F1), which means that (2) is true. Then (F2) is true if and only if the subtracted equation is true:

    (F2) <=>

     0            = ρ

                      + 1/3 a D_a ρ

                      + p / c².

Our formula for pressure (1) makes the right side of the equation zero, which means that the subtracted equation is true if we assume energy conservation. QED.