Early Initial Conditions?313Using these observations, the pump lds of the scalar and tensor modes of the geometry can be expressed solely in terms of the slow-roll parameters. In particular, in the case of the tensor modes it is easy to derive the following chain of equality on the basis of the relation between cosmic and conformal time, and using Eq. (10.32): 藱 H a = H2 + H = a2 H 2 (2 + 2 ) = a2 H 2 (2 �), (10.36) a H Inserting Eq. (10.35) into Eq. (10.36) we will also have, quite simply a 2�= 2 . a (1 �)2where the second equality follows after integration by parts assuming that is constant (as it happens in the case when the potential, at least locally, can be approximated with a monomial in 蠒). Since dt da = , (10.34) a a2 H Eq. (10.33) allows us to express aH in terms of : 1 aH = �. (10.35) (1 �)(10.37)The evolution equation for the tensor mode functions is fk + k 2 �whose solution is N �(1) fk ( ) = � H ( ), 2k(1)a fk = 0, a i(2+1)/4 e , 2(10.38)N =(10.39)which implieswhere H ( ) are the Hankel functions [215, 216] already encountered in chapter 6. In Eq. (10.39) the relation of tp is determined from the relation 2�1 , (10.40) 2 �= 4 (1 �)2 = 3�. 2(1 �) (10.41)The same algebra allows us to determine the relation of the scalar pump ld with the slow-roll parameters. In particular, the scalar pump ld is z = z z z2+z z.(10.42)
