content_image/314036.jpg content_image/314038.jpg content_image/314037.jpg content_image/314034.jpg content_image/314035.jpg
## 3.1.1 Horizon Problem 

With the definition of the Hubble parameter being H ≡ ˙ a $_{a}$, we define the hori- zon as the distance that light (information) can travel within one Hubble Time H − $^{1}$. 

We define the comoving Particle Horizon as 

<img src='content_image/314036.jpg'>

<img src='content_image/314038.jpg'>

<img src='content_image/314037.jpg'>

with the third line coming from the fact that we take ˙ a ( t ) > 0 (i.e. the Uni- verse is expanding). This is the maximum distance that light (information) could have traveled since the birth of the Universe at t, a =0. Any two events that are separated by a distance of more than twice the particle horizon are out of casual contact. The Particle Horizon in physical coordinates is given by 

<img src='content_image/314034.jpg'>

<img src='content_image/314035.jpg'>

