content_image/303970.jpg content_image/303969.jpg content_image/303968.jpg content_image/303967.jpg content_image/303966.jpg
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 

<img src='content_image/303970.jpg'>

<img src='content_image/303969.jpg'>

<img src='content_image/303968.jpg'>

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 

<img src='content_image/303967.jpg'>

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 

<img src='content_image/303966.jpg'>

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 

