content_image/307788.jpg content_image/307787.jpg content_image/307786.jpg content_image/307785.jpg content_image/307784.jpg
and 

<img src='content_image/307788.jpg'>

We assume that the vacuum state is that of the Bunch-Davies vacuum. This vacuum state is the ground state of the system within a curved spacetime background. For very early times, as η → −∞ , the Bunch-Davies vacuum gives the initial condition 

<img src='content_image/307787.jpg'>

As this is for very early times, it also corresponds to very small wavelengths and so corresponds to the Minkowski vacuum, i.e. the vacuum of the system within a flat spacetime background. The solution for ϕ$_{k}$ ( η ) is 

<img src='content_image/307786.jpg'>

which is the mode function for the Bunch-Davies vacuum. The spectrum is given by 

<img src='content_image/307785.jpg'>

Substituting Eq. (3.46) and the commutation relations Eqs. (3.47) and (3.48) into Eq. (3.51) yields 

<img src='content_image/307784.jpg'>

Substituting Eq. (3.50) into this, dividing by a 2 (to return back to the δφ perturbations) and evaluating it a few Hubble times after horizon exit gives 

