# Friedmann Equation & World Models

In this chapter, we will understand what the Friedmann Equation is and study in detail regarding the World Models for different curvature constants.

## Friedmann Equation

This equation tells us about the expansion of space in homogeneous and isotropic models of the universe.

$$\left ( \frac{\dot{a}}{a} \right )^2 = \frac{8\pi G}{3}\rho + \frac{2U}{mr_c^2a^2}$$

This was modified in context of General Relativity (GR) and Robertson-Walker Metric as follows.

Using GR equations −

$$\frac{2U}{mr_c^2} = -kc^2$$

Where k is the curvature constant. Therefore,

$$\left ( \frac{\dot{a}}{a} \right )^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2}$$

Also, $\rho$ is replaced by energy density which includes matter, radiation and any other form of energy. But for representational purposes, it is written as $\rho$.

## World Models for Different Curvature Constants

Let us now look at the various possibilities depending on the curvature constant values.

### Case 1: k=1, or Closed Universe

For an expanding universe, $da/dt > 0$. As expansion continues, the first term on the RHS of the above equation goes as $a^{-3}$, whereas the second term goes as $a^{-2}$. When the two terms become equal the universe halts expansion. Then −

$$\frac{8 \pi G}{3}\rho = \frac{kc^2}{a^2}$$

Here, k=1, therefore,

$$a = \left [ \frac{3c^2}{8 \pi G\rho} \right ]^{\frac{1}{2}}$$

Such a universe is finite and has finite volume. This is called a Closed Universe.

### Case 2: k=-1, or Open Universe

If k < 0, the expansion would never halt. After some point, the first term on the RHS can be neglected in comparison with the second term.

Here, k = -1. Therefore, $da/dt ∼ c$.

In this case, the universe is coasting. Such a universe has infinite space and time. This is called an Open Universe.

### Case 3: k=0, or Flat Universe

In this case, the universe is expanding at a diminishing rate. Here, k = 0. Therefore,

$$\left ( \frac{\dot{a}}{a} \right )^2 = \frac{8\pi G}{3}\rho$$

Such a universe has infinite space and time. This is called a Flat Universe.

### Points to Remember

• The Friedmann equation tells us about the expansion of space in homogeneous and isotropic models of the universe.

• Depending on different curvature constant values, we can have a Closed, Open or Flat Universe.