3.1 Problems of Big Bang Cosmology 25 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 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 a ( t ) x$_{PH}$ ( t ) ≡ a ( t ) ∫ t 0 1 a ( t ′ ) d t ′ (3.4) a ( t ) x$_{PH}$ ( a ) = a ( t ) ∫ a 0 1 a $^{2}$H d a (3.5) x$_{PH}$ ( t ) ≡ ∫ t 0 1 a ( t ) d t (3.1) = η ( t ) − η (0) (3.2) x$_{PH}$ ( a ) = ∫ a 0 1 a $^{2}$H d a (3.3)