3.2 Cosmic Inflation 34 Now let us consider a generic coordinate gauge, in which we still have a comoving threading but a generic slicing, as opposed to one in which each slice has a uniform energy density. We have where and At any given value of t , the scale factors a ( x , t ) and ˜ a ( x , t ) differ only because of the difference in the time coordinate of the two spacetime slices. Therefore, in order to maintain generality, we will drop the ~ on the scale factor. We can define a quantity called the e-folding number as which is the number of exponential expansions of the Universe between when the scale factor is a$_{1}$ and when it is a$_{2}$ . As H ≡ ˙ a $_{a}$, this can also be expressed as and is thus sometimes called the number of Hubble times. The difference in the number of e-foldings between any two generic spacetime slices is given N = ∫ t$_{2}$ t$_{1}$ H d t (3.24) N ≡ ln ( a$_{2}$ a$_{1}$ ) (3.23) ˜ γ$_{ij}$ ( x ) ≡ ( I e 2 ˜ h ( x ) ) ij (3.22) ˜ a ( x , t ) ≡ a ( t ) e ψ ( x ,t ) (3.21) ˜ g$_{ij}$ = ˜ a $^{2}$( x , t )˜ γ$_{ij}$ ( x ) (3.20)