## Ew RlKim

mi m

The first summation dominates the second in the above equation; the first grows like log2 m, while the second grows like m log m.

The same pattern emerges at all ordersâ€”the dominant contribution can be isolated as:

The sum of the dominant contributions remains finite:

and suggests that for large ot, ym will decay as a power-law with exponent QR.

Motivated by this observation, and recalling that for large m, Am - l/mR*2 (from equation (8)), we suggest the following approximation for Bm, valid for all values of m, not just when m is large:

where C is a constant that is independent of m. The above expression for Bm is derived by replacing R by R + QR in the denominator of the product that defines Am (equation (8)) This is really nothing more than informed guesswork; this is the simplest expression for Bm that recovers a power-law with exponent R + QR for large m and reduces to Am when Q = 0.

In order to determine F(t), the total number of folds at time t, equation (11) has to be solved using the approximate solution (62). First, the a choice has to be made for the constant Câ€”since the equation is an approximation, there is freedom in the choice. One way is to enforce the consistency of equation (53) for m = 1:

As F{t) is direcdy affected by B]t it is natural to focus on m= 1. Note that for small Q, C~ 1 + 2/(R + 2)(R + 3).

Equation (11) can be integrated to give an approximation for F(t):

Using the identity of Appendix H, the normalized coefficients are given by:

0 0