Answer to Question #90287 in Astronomy | Astrophysics for Shivam Nishad

Question

Write down Friedman equation and sketch its solutions. Which one of these solutions correspond to the contracting universe ?

Answer 1

The Friedmann equations describe the expansion of space in two models of the universe (homogeneous and isotropic) with use of general relativity. Consider the equations for the matter-dominated universe.

In the flat universe the first Friedmann equation look like:

$$\frac{\dot{a}^2}{a^2}=\frac{8\pi G}{3}\rho_0\Big(\frac{a_0}{a}\Big)^3.$$

The curvature $k$ of such universe is zero, and the deceleration parameter is $q_0=1/2$.

In the open universe with the curvature $k=-1$ and the deceleration parameter $q_0<1/2$ the first Friedmann equations becomes

$$\frac{\dot{a}^2}{a^2}=\frac{8\pi G}{3}\rho_0\Big(\frac{a_0}{a}\Big)^3+\frac{1}{a^2}.$$

In the closed universe, which contracts to singularity, i.e. with $k=1,\space q_0>1/2$, the first Friedmann equation looks as follows:

$$\frac{\dot{a}^2}{a^2}=\frac{8\pi G}{3}\rho_0\Big(\frac{a_0}{a}\Big)^3-\frac{1}{a^2}.$$

The evolution of the scale factor $a$ for these models of the universe can be seen in the following figure presented in lectures on Astrophysics from NICADD:

We can see that the closed universe size (or scale factor) increases from the moment of Big Bang, then this universe decelerates and starts accelerating to the Big Crunch, where it collapses.