Quantum tunneling is not directly related to the Big Bang itself but has been invoked in some theories to address certain aspects of the early universe. It is important to note that our understanding of the early universe and the specific events surrounding the Big Bang is still an active area of research, and different theories exist with varying degrees of consensus.
That said, one concept in cosmology that involves quantum tunneling is called "eternal inflation." Eternal inflation is a hypothesis proposed to explain why the observable universe appears homogeneous and isotropic on large scales, despite the fact that regions of the universe that should have been causally disconnected during the Big Bang could not have equilibrated their temperatures.
According to this hypothesis, during the extremely rapid expansion of the early universe, there are quantum fluctuations in the inflaton field (the field responsible for inflation). These fluctuations can lead to the formation of "bubbles" within the inflationary space. These bubbles represent regions where inflation ends, and the universe transitions to a state of normal expansion, eventually giving rise to observable regions like our own universe.
Quantum tunneling comes into play when considering the transitions from the inflating region to the bubble regions. It is suggested that the transition can occur through a quantum tunneling process, where the inflating region tunnels through a potential barrier to create a bubble universe. The process is analogous to a quantum particle tunneling through a classically forbidden region.
However, it's important to note that eternal inflation and the role of quantum tunneling in this context are still active areas of research, and various theoretical and mathematical challenges need to be addressed. The specific details and mechanisms of how the early universe evolved and the nature of the Big Bang remain subjects of ongoing investigation and theoretical development.