My question is that how did the Big Bang determine the initial distribution of the particles and their wavefunctions? In theories like Bohmian mechanics or Many worlds, how was the universal wavefunction determined by the Big Bang?
What forces/causes were responsible for the initial conditions that the Big Bang created (for example electrostatic force, gravity, etc.)? Did the Big Bang determine the initial conditions randomly, or such forces led to unique and deterministic initial conditions?
At present, nobody knows how to write down the exact form of the wave function of the entire universe at the moment of the Big Bang, so we'll leave that question for another time.
The known physics of particle and photon interactions and the energies of formation of composite particles like hadrons and of atoms like hydrogen, helium, and their isotopes allow physicists to walk the dynamics of an expanding universe incrementally backwards in time up to the point where the energies shared by the particles and the photons are so great that the laws of physics are known to break down and become invalid.
Conversely, those known physics allow the dynamics of an expanding universe to be walked incrementally forward in time from the point where those laws become valid all the way out to the present time.
These exercises place explicit bounds on exactly how much of the different primary constituents of the universe we inhabit today should be in evidence right now, and the match between the Big Bang predictions and the actual measurements is good enough to validate the Big Bang model.
In this sense, the initial outcome of the Big Bang does indeed furnish valid initial conditions for the subsequent evolution of the universe.
A detailed and readable account of this process, with descriptions of which fundamental forces and particles were significant players at each stage of the process, is given in Weinberg's book The First Three Minutes.
