Friedmann Equation & World Models


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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.



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