Veff(a)=a2+a4eff()V_{\text{eff}}(a) = - a^2 + a^4 \Lambda_{\text{eff}}(\phi)
where:
The term a2-a^2 represents spatial curvature (assuming k=+1k = +1),
eff()=8G3V()\Lambda_{\text{eff}}(\phi) = \frac{8\pi G}{3} V(\phi) plays the role of an effective vacuum energy.
The potential has a barrier-like structure, with a turning point a0a_0 below which classical evolution is prohibited:
Veff(a0)+Q(a0)=0V_{\text{eff}}(a_0) + Q(a_0) = 0
Thus, universe creation is understood as a tunneling through the potential barrier from a=0a = 0 (non-existence) to a=a0a = a_0 (emergence).
4. Tunneling Action and Probability
The Euclidean tunneling action SES_E across the potential barrier is given by:
SE=0a02Veff(a)+Q(a)daS_E = \int_0^{a_0} \sqrt{2 |V_{\text{eff}}(a) + Q(a)|} \, da
In the semi-classical limit, the tunneling probability P\mathcal{P} is approximated by:
Pe2SE/\mathcal{P} \sim e^{-2 S_E / \hbar}
The finiteness of this action in our model, due to the regularized quantum potential Q(a)Q(a), avoids the initial singularity and permits a non-zero probability for spontaneous universe genesis.
