3.2 Cosmic Inflation 40 and 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 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 which is the mode function for the Bunch-Davies vacuum. The spectrum is given by Substituting Eq. (3.46) and the commutation relations Eqs. (3.47) and (3.48) into Eq. (3.51) yields 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 P $_{ϕ}$( k, η ) = k 3 2 π 2 | ϕ$_{k}$ ( η ) | 2 (3.52) 〈 ˆ ϕ k ˆ ϕ k ′ 〉 = 1 4 πk 3 P $_{ϕ}$( k ) δ $^{3}$( k + $_{k}$′$_{)}$ (3.51) ϕ$_{k}$ ( η ) = ( kη − i ) kη e − ikη √$_{2}$$_{k}$ (3.50) ϕ$_{k}$ ( η ) = 1 √$_{2}$$_{k}$ e − ikη (3.49) [ˆ a ( k ) , ˆ a ( $_{k}$′$_{)] = 0}$ (3.48)