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 orders—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 C—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

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