with a ≫ a$_{1}$ or a ≪ a$_{2}$ . 

The Horizon Problem arises when we consider how the horizon of various places in the observable Universe has varied over the lifetime of the Universe and then compare this with observation. Our observable Universe is statisti- cally homogeneous and isotropic on cosmological scales. One manifestation of this is that of the large isotropy that exists in the CMB. Over the entire sky, the temperature of the CMB is the same to within 1 part in ∼ 10 $^{5}$. Therefore, it is very safe to assume that all parts of the CMB were in ther- mal equilibrium with each other at the time of last scattering and thus were in casual contact with each other. However, when we consider two parts of the CMB that are separated by more than a few degrees across the sky, we find that the particle horizon’s of these two parts do not overlap. Therefore, they have never been in casual contact with each other, as there has not been enough time for light (information) to travel between the two parts given the age of the Universe. The Horizon Problem is thus to explain how each part of the observable Universe is so extremely statistically similar to every other part, given that the vast majority of parts have never been in casual contact with each other. 

## 3.1.2 Flatness Problem 

From looking at Eq. (2.29) regarding the Density Parameter, we see that as time evolves from the birth of the Universe, i.e. as ˙ a 2 decreases rapidly, Ω grows rapidly, away from a value of 1. We observe for the Density Parameter the current value Ω$_{0}$ =1 . 000 ± 0 . 005 (Eq. (2.30)), which is very close to 1, i.e. 

