3.2 Cosmic Inflation 31 As this inequality becomes stronger, H becomes more and more constant during the period of inflation, with the expansion becoming more and more exponential, i.e. a ∝ e $^{Ht}$. It should be noted that Eqs. (3.12)–(3.14) are all equivalent. If we now include GR in our discussion, we have one final definition of inflation. Using the (Friedmann) Acceleration Equation, Eq. (2.20), we have This implies that we require negative enough pressure (“repulsive gravity”), P < − ρ $_{3}$, in order to achieve a period of inflation. Now we will discuss how a period of inflation can solve the five problems of an isolated Big Bang Cosmology mentioned above. Firstly, the Horizon Problem. The solution to this problem is to say that the entire observable Universe used to be inside the horizon, prior to the end of inflation. Dur- ing inflation however, different scales of relevance to our observable Universe were “stretched” to outside of the horizon at different times, remembering that one of the definitions of inflation is that of a decreasing comoving Hubble length, Eq. (3.13). At later times, after inflation, these scales then started to re-enter the horizon, again at different times for different scales. Now, the Flatness Problem. From looking at Eq. (2.29) regarding the Density Parameter, we can see that the RHS will tend towards 0 during inflation. The reason for this, is that during inflation ¨ a > 0 by definition. Therefore, ˙ a 2 will grow rapidly, which will have the affect of rapidly driving the value of Ω towards 1 throughout the entire period of inflation. However, ρ + 3 P < 0 (3.15)