In the many-worlds theory and Bohmian mechanics, the universe has an "initial" "universal" wavefunction which then evolves according to the Schrodinger's equation and determines the future of the universe. My question is that what the shape of this initial wavefunction is more likely to be? Can it be any random integrable/differentiable function over the configuration space with equal probabilities, or it is supposed to have anisotropies?
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The universal wave function is a term introduced by Hugh Everettin in his Phd. thesis The theory of the universal wave function.
The wave function of the universe is given by The Hartle–Hawking state is the wave function of the Universe—a notion meant to figure out how the Universe started—that is calculated from Feynman's path integral. More precisely, it is a hypothetical vector in the Hilbert space of a theory of quantum gravity that describes this wave function.
There have been criticism associated with its mathematical validity and what application does it serve. So, in my opinion we cannot give a shape to the universal wave function.
For more info read this dissertation where the word Universal wave function was used -dissertation
