"The Schrodinger equation does not reproduce the Born rule." - This is a strange statement. Schrodinger's equation is not supposed to do this. The fact that $|\psi|^2$ represents probabilities is what relates QM to observations in the physical world. This remains true whether you apply QM to the whole universe or not.

@J.Delaney the Schrodinger eqn is not supposed to derive Born rule in conventional Quantum mechanics (and most interpretations), yes. But interpretations like MWI and De Broglie Bohm theory aim to derive the Born rule from Schrodinger equation and few other postulates

@sashoalm I am not familiar with the no-hair theorem. "Entanglement with the measurement device + decoherence" is supposed to play a role in measurements. See this answer physics.stackexchange.com/a/152916 The role that decoherence plays may be interpretation dependent. MWI tries to use these ideas to define what a world is

@RyderRude Whether or not the Born rule can be derived from MWI is a separate issue. Either way, it is an essential part of QM, so "applying QM to the whole universe", means also applying the Born rule. Your answer seem to suggest that applying QM to the whole universe somehow eliminates the Born rule. I don't see any basis for this claim.

@J.Delaney The basis for the claim is that the Born rule is supposed to be applied when an external measurement device measures the system. If we take the universe as our system, then all measurement devices are already part of the system. It's not that I am actively getting rid of the Born rule. It's just that it never gets applied in the situation we have considered.

@RyderRude This is not true. Take for example protons colliding in the center of the sun. You can calculate - using Born's rule - the probability that they will fuse into Helium. From this you can deduce characteristics (such as brightness or lifetime) of typical stars in a universe governed by QM. There is no external measurement device looking at those protons. (Of course you can always consider the rest of universe as an "external" device, but that also leads to applying Born's rule)

@J.Delaney I think we are disagreeing on something subtle here. The way OP intended their question is what happens when one considers the whole universe as a "quantum system". Now, any quantum system behaves deterministically when not being measured and probabilistically when measured. So, if consider the whole universe as a "quantum system", then indeed the process where the protons probabilistically fuse into Helium is ruled out (unless we do MWI)

@J.Delaney but the phrase "a universe governed by quantum mechanics" is vague enough to allow for Copenhagen type universes where we haven't made a measurement well defined. In this use of the terminology, you are correct. My answer is based on the idea "what happens if we consider the whole universe as a quantum system". The behavior of a quantum system is precise and it is what leads to the conclusions

@RyderRude "any quantum system behaves deterministically when not being measured" - It is the wavefunction that evolves deterministically, but the wavefunction encodes probabilities. Applying QM to a system means using the wavefunction to calculate the probabilities of different outcomes. You can apply it to an atom and find the probability that it will decay, or apply it to a star and find the probability that it will explode, or apply it to the whole universe and find the probability that it will end up in whatever state you want.

@J.Delaney i understand that sentiment and I have even expressed this in my answer. I write that, if we take the whole universe as our system and model it by a wavefunction, not only is the evolution deterministic but the wavefunction is also meaningless, as in conventional QM, the meaning of the wavefunction comes from measurements. See the section before the summary. Again, if there is any disagreement, it is extremely subtle and maybe about terminology (e.g. how to define a quantum system or a quantum universe)

@J.Delaney I also find OP's line of inquiry to be natural. On the surface, it seems there is nothing inherently different about modeling the entire universe as wavefunction. But it leads to absurdities like we saw. And MWI needs additional assumptions to remove these absurdities

@RyderRude Applying QM to the whole universe is not equivalent to MWI in the interpretational sense. For example you could just simulate the wavefunction of the universe on a computer and use it to study various observable characteristic (such as the average lifetime of stars as I mentioned before). There is absolutely nothing wrong or "absurd" about this.

@RyderRude In fact , people actually do treat the universe as a quantum system when modeling, for example, primordial quantum fluctuations that led to the the observed power spectrum of the CMB.

@J.Delaney but you just said one comment ago that "it's the wavefunction that evolves deterministically". Now you're saying that the state of the universe probabilistically transitions into other states. This explicitly violates the postulates of quantum mechanics. Can you tell me which postulate of quantum mechanics allows the state to probabilistically collapse by itself without an external measurement device?

@RyderRudeThat's not what I said. I said that the wavefunction of the universe could be simulated deterministically (by Schrodinger's eq.). If you'll run such a simulation, you'll find a superposition of many states corresponding, for example. to different star densities. So you get a probability distribution $|\langle \ n | \psi \rangle|^2$ over $n$, where $|n\rangle$ is a state with definite number of stars and $ | \psi \rangle$ is the wavefunction of the universe. This is just an application of Born's rule.

@RyderRude You could then compare this distribution to the number of stars we actually observe in the universe, and deduce whether our observation is consistent (statistically) with the QM prediction.

@J.Delaney If you do that, you are just using QM the way everyone else uses QM. You are modeling a sub-system of the universe as a wavefunction and treating measurements as a black box. If you model literally everything as a wavefunction, then the postulates say that the state will never probabilistically transition to another state. In the application you are talking about, the state does probabilistically transition to the post-measurement state, so it is decidedly not modeling the entire universe as a wavefunction and applying QM to it.

@J.Delaney in particular, in any of these experiments, you are at least not modeling yourself as a wavefunction. In practice, you are always studying a tiny fraction of the universe. As soon as you model yourself as wavefunction (which is implied by modeling the entire universe as a wavefunction), you run into the absurdities

Well, even in classical physics you can't create a simulation that includes yourself (because it will need to simulate the computer that runs the simulation and so on...) so those absurdities are not unique QM. When we are talking about modeling the entire universe we are obviously talking about a hypothetical universe that does not include ourselves, so that's beside the point. But we can definitely model a universe that operates by the same physical laws as ours, and use that to predict properties of our own universe.

@J.Delaney so we agree that there is an absurdity. i would disagree that this particular absurdity also arises in classical mechanics. The absurdity doesn't necessarily have to do with simulations.

but if you want to think in terms of simulations, then in classical mechanics, one can trap a human in an isolated room and simulate the room on a computer (the computer is only simulating the room. it is not simulating itself). there are no absurdities here

but if one does a similar experiment in QM, the simulation would make the human go deterministically in a superposition of multiple observations, while the human would report that the evolution was non deterministic

but the problem isnt really about simulations. it is more that conceptually, you can think of an entire classical universe modeled by classical mechanics. but if u think of an entire quantum universe (including yourself) modeled by quantum mechanics, then it leads to completely deterministic evolution of the state