Validity of Friedmann's equation

Question

Recently, a friend gives me the Friedmann's equations under the following form:

\begin{gathered} \left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3 c^{4}} \rho+\frac{\Lambda}{3}-\frac{k}{3 a^{2}}\quad(1) \\ \frac{\ddot{a}}{a}=-\frac{4 \pi G}{c^{4}} P-\frac{1}{2}\left(\frac{\dot{a}}{a}\right)^{2}-\frac{k}{2 a^{2}}+\frac{\Lambda}{2}\quad(2) \end{gathered}

I have difficulties to convince myself that they are right since from side I know them under the form :

\begin{aligned} &\frac{R^{\prime \prime}}{R}=-\frac{4 \pi G}{3}\left(\rho+\frac{3 p}{c^{2}}\right)+\frac{\Lambda}{3}\quad(3) \\ &\left(\frac{R^{\prime}}{R}\right)^{2}=\frac{8 \pi G \rho}{3}+\frac{\Lambda}{3}-\frac{k}{R^{2}}\quad(4) \end{aligned}

In order to make the link, I think I have to take $a(t)=\dfrac{R(t)}{R_{0}}$ but even with this convention, I can't get to find the equations (1) and (2). There are factors and terms that differs between eq(2) and eq(3), like between eq(1) and eq(4).

EDIT : so finally, Could someone tell me if (1) and (2) are wrong and if (3) and (4) are right ?

Answer 1

(1) and (4) are almost the same, except for factors of $c^4$ and $3$ in the first and third terms on the right hand side.

If you add zero in the form $\frac12(8πGρ/3 + Λ/3 - k/R^2) - \frac12(R'/R)^2$ to the right hand side of (3), you get (2), except for a factor of $c^2$ difference in the first term.

So (1) and (2) are almost right, aside from a wrong factor of $3$ and some misplaced factors of $c$.