Deriving Friedmann Equations without General Relativity

Question

Can we derive the analytic Friedmann Equations without general relativity, starting from completely classical/nonrelativistic arguments? (If we consider sufficiently small volumes.)

Answer 1

A Newtonian approach is possible but of of course it is not rigorous.

We're in a newtonian approximation now, and we want to describe the motion of a unit mass at a point P on the surface of a sphere. So let's take this spherical distribution of matter, take a point at a distance $l$ from the origin. The equation of motion of that point is

$$ \frac{d^2l}{dt^2} = - \frac{GM}{l^2} \tag{1} $$

$$ \Rightarrow \, \, \, \ddot{l}\, \dot{l} = \frac{d}{dt} \frac{\dot{l}^2}{2}= - \frac{GM}{l^2} \dot{l} = \frac{d}{dt} \frac{GM}{l} \tag{2} $$ from which you can find

$$ \frac{d}{dt} \Big(\frac{\dot{l}^2}{2} - \frac{GM}{l} \Big) =0 \tag {3} $$

$$ \Rightarrow \, \, \dot{l}^2 = \frac{2GM}{l} + constant \tag{4} $$

This is the equation of conservation of energy per unit mass.

Now in cosmology you have $$l= d_c \frac{a}{a_0} = \tilde{D} a \tag {5}$$

where $d_c$ is the comoving distance and $a$ is the usual scale factor. We put this in $(4)$ and remembering that $\tilde{D}$ is not affected by the time derivative and $M = \frac{4\pi}{3} \rho (\tilde{D} a \, ) ^3$, we get

$$ ( \,\tilde{D} \dot{a} \, )^2 = \frac{2G \frac {4\pi}{3} \rho (\tilde{D} a \, ) ^3}{\tilde{D}a} \tag{6} $$

From which we have

$$ \dot{a} = \frac{8\pi G}{3} \rho a^2 + constant \tag{F2} $$

that is the second Friedmann equation when $constant = -kc^2$.

Now we derive the first Friedmann equation simply replacing $\rho$ with $\rho_{eff} = \rho + \frac{3p}{c^2}$ that is we put a relativistic term in the density by hand. It goes like this, starting from $(1)$

$$ \frac{d^2l}{dt^2} = - \frac{GM}{l^2} = -\frac{G}{l^2} \frac{4\pi}{3} \rho_{eff} l^3 = - \frac{4 \pi G}{3} l \big( \, \rho + \frac{3p}{c^2} \, \big) $$

Using again $(5)$ one obtains the first Friedmann equation

$$ \ddot{a} = - \frac{4 \pi}{3} G \big( \, \rho + \frac{3p}{c^2} \, \big) a \tag{F1} $$

You can find this kind of treatment in Coles-Luccin: Cosmology-The origin and evolution of cosmic structure. I don't know in which chapter cause I don't have the book and I used my notes for this answer, but I'm pretty sure my notes were taken from that book.