We likely totally agree on this: generically we can take an expanding universe and set a boundary condition at the small end defining minimal entropy there. More specifically we can use an expanding dust solution that is dense at one end of time and sparse at the other end of time has higher entropy where there is a lot more empty space. This is just saying "initial conditions" to the question of the arrow-of-time, and whether the low entropy end is "initial" or "final" is a matter of choice: always calling it initial is simply temporal chauvinism.

The gauging here is simply in picking out an appropriate slicing or threading. As you say, with mild assumptions, we seem to be offered a highly useful and effective one on a plate.

Now we can focus on what I think we haven't agreed.

Let's take a page out of perturbative cosmology and think of a second (explicit) gauging, wherein each hypersurface in the observed(observable)/perturbed universe is matched uniquely to a hypersurface in the model/background. Let's also ignore uniqueness problems that might arise in the deep dS era or before the very early universe (we could evolve to a static equilibrium and lose observational means of checking the hypersurface matching) and pretend that in introducing inhomogeneities perturbatively we don't run into any problems selecting the families of hypersurfaces to associate with t=const.

Let's also not argue about whether we run into problems if we define a background and do observations in the perturbed spacetime rather than determine the background from observations; for easiness, let's take the former approach, and talk about model vs observed rather than background vs perturbed.

The question, in a nutshell, is whether the observed systems at the scale of galaxies and smaller should have their entropy grow or shrink in the same way as the model. Informally, when the model is collapsing, do we expect the observed to have many more instances of e.g. blocks sitting still on a flat but not perfectly smooth surface accelerate along the surface; do we expect to see a mug milky coffee separate and eject the milk; do we expect to see omelettes evolve into whole eggs -- observations these dominating any observations of dissipative (classical) processes like friction slowing a sliding block to a standstill, and so on?

That is, is the observed matter's thermodynamic arrow of time determined by whether the model is expanding or collapsing? If there are no perturbations at all (we observe only the model), then we can agree that the answer is "yes", right? But if the observed has evolving structure, and the perturbations' entropy decreases or increases exactly like the model's, is the answer still "yes"? If so, then is the question of whether we're in an expanding or contracting spacetime merely "make your own choice, the physics doesn't care"?

Let's consider unaligning the arrows: in a collapsing background, can we have observed systems in which Boltzmann entropy is increasing? If there is interacting content in the background, what happens in the presence of these incompatible arrows of time?

In the collapsing dust background we have an initial configuration with a large coarse-grained entropy, and marching "forward" in time we see the Boltzmann entropy decreasing, with the momenta of the matter content of the background finely tuned so that it arrives in a small region of phase space. Our problem is when we define that small region as having lower entropy. Perturbations of the collapsing dust check the decrease in entropy in general; as an example we can add in matter that collapses into a black hole.

Conversely, in an expanding FLRW background, can we find theoretical support for perturbations which evolve to lower Boltzmann entropy? How common can systems like that really be? Where in spacetime can they safely interact with the content of the background, given the latter has more degrees of freedom?

By implicitly choosing to keep the Boltzmann arrows of time compatible in the background/model and in the perturbed spacetime/observed universe, we sweep these questions under the rug.

This is the implicit (and here, third) gauging, wherein we pick out and totally order hypersurfaces by their coarsest-grained Boltzmann entropy.

> Or are you saying that those other solutions also exist, they're just in different universes from the one we live in?

I'm not sure we should trust that the RW metric should be always-expanding or always-contracting. Here I'll note that you omitted Lemaître's initial. :-)

What evidence conflicts with us being in one region of a reproducing cosmology where we have possibly many (unobserved but not necessarily in principle unobservable) hot dense phases and many cold sparse phases, where the one that we can see (back to the surface of last scattering) is only relatively low entropy? Where's the beginning of the universe in that? Maybe there isn't one.

Things get even messier if we take a Hartle-Hawking or Page-Wooters view of time as non-fundamental and use that as a basis for considering that the universe may not have had a beginning.

> Isn't this just a restatement of the problem?

Possibly, but even if it's "just" that, I think it probably helps to consider the statement of the problem from different angles.