Question

Prove this.(a˙a)2=8πG3ρ−k∗c2a2(a˙a)2=8πG3ρ−k∗c2a2

Answer 1

Answer:

Friedmann Equation

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

(a˙a)2=8πG3ρ+2Umr2ca2(a˙a)2=8πG3ρ+2Umrc2a2

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

Using GR equations −

2Umr2c=−kc22Umrc2=−kc2

Where k is the curvature constant. Therefore,

(a˙a)2=8πG3ρ−kc2a2(a˙a)2=8πG3ρ−kc2a2

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

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

8πG3ρ=kc2a28πG3ρ=kc2a2

Here, k=1, therefore,

a=[3c28πGρ]12a=[3c28πGρ]12

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∼cda/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,

(a˙a)2=8πG3ρ(a˙a)2=8πG3ρ

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