Are Friedmann equations linear or nonlinear?

Question

I'm trying to improve my understanding of cosmology, and these 2 equations are basic . You can find them here:

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

Also, if you could tell why they are or aren't linear, it would help me a lot.

Answer 1

A differential equation is linear if it is a linear equation with respect the unknown function and its derivatives. I.e. if it has the form: $$ f_0(x)y^{(n)}+ f_1(x) y^{(n-1)}+ \cdots + f_n(x)y= g(x) $$

Look at the first Friedmann equation. We can write it as ( assume $c=1$): $$ \dot a(t)^2-\left[\frac{8\pi G}{3}\rho +\frac{\Lambda}{3}\right]a(t)^2=k $$ where $a(t)$ is the unknown function and $\dot a(t)$ is its first derivative with respect to the independent variable $t$. So it is obviously not linear since $a(t)$ and $\dot a(t)$ are at square.

Another complication comes from the fact that the density $\rho$ can be, in general, a function of $a$.