paths. We have $a(t_i=0)=0 $ and take $a(t=t_f)=a_f$.

For a given $a_f=real,$ we find from (18) $$\beta \cosh{(\nu t_R)}\sin{(\nu t_I)}=\nu a_f$$ $$\cos{(\nu t_I)}=0$$ where $t_f=t_R+i~t_I$. Corresponding to different $t_i$ given by (25) we have the same $t_R$ for a given real $a_f$. In otherwords, there are many different end points $t_f$ which give the same $a_f$ and many different contours to each of these end points. A numerical plot of the paths in the complex scale factor $(a)$ space of solution for various complex time $t$ will be found in [@unru:gr-qc]. In the cited work, they deal with complex solutions to study the classical evoluation without showing the construction of the wavefunction but in our work we construct the wavefunction taking the contribution of complex paths thereby exposing the connection between the small $a$ and large $a$ boundary conditions on the wavefunction.

Let us calculate the zeroth order action $$\begin{aligned}
S_0(a_f,\,\phi_f) &=& \int^{t_0}_{0}P_{a}^{0} \,a'_0\, dt\nonumber \\
&=& -M\int^{t_0}_{0} a_0(a'_0)^2\,dt\nonumber \\
&=& -M\int^{a_f}_{0} a_0 a'_0\,da_0\end{aligned}$$ and consider $$\Psi_0\,( a_f,\,\phi_f)\propto e^{iS_0},$$ as the wavefunction of the universe (assuming that this is possible in the approach we follow). According to the wormhole dominance proposal, the wavefunction $\Psi$ given in (27) must be multiplied 