+ c_{52} (\tau-\tau_B)^{q_2}  .\end{aligned}$$

Note that in the above, the time $\tau_H$ depends on $k$, $\tau_H\equiv \tau_H(k)$. To simplify the notation we do not write the k-dependence explicitly. Note also that $q_2$ (the power of the scale factor in conformal time) and $p_2$ (the power of the scale factor in cosmological time) are the indices of the scale factor during the initial contracting and final expanding phase, and that $q_1$ and $p_1$ are the indices in the intervening periods. In particular, the index 1 in these quantities should not be confused with the index in Region I. Note also that the $q_i$ are related to the $p_i$ through the relation

$$q_i = \frac{p_i}{1-p_i},\;\;\;i=1,2.
\label{qipi}$$

Matching conditions
-------------------

Let us now discuss the process of matching between the different regions discussed above. The matching conditions for the metric across a space-like hypersurface were derived in Hwang-Vishniac [@HV] and Deruelle-Mukhanov [@DM] and are the generalization of the Israel matching conditions [@Israel]. For cosmological fluctuations, the matching conditions say the solutions in different regions can be connected by enforcing two conditions, namely the continuity of both $v$ and its derivative across the boundary surface.

As mentioned earlier, for the two non-singular bouncing points 