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A system with two massive bodies has potential energy proportional to their separation. Since the universe is expanding, is the potential (and total?) energy of such a system slowly increasing? What is the scale of this effect if it exists?

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Yes, in an expanding universe there is an increase in potential energy (with same caveats).

The (Friedmann) equations of a homogeneous and isotropic universe with no spatial curvature or cosmological constant can, in fact, be derived very easily from nonrelativistic Newtonian gravity. (This derivation is shown, e.g., in Mukhanov's Physical Foundations of Cosmology.) So it is actually valid to think of such a cosmos in Newtonian terms, using randomly but approximately homogeneously distributed point masses. At early times, the point masses are flying apart at high speed but the average distance between them is small. At late times, they will have slowed down, but the average distance between them will have increased. All that kinetic energy is converted into gravitational potential energy. This also remains true if we take into account Newtonian perturbation theory (the growth of overdense and underdense regions from initial perturbations).

As for the caveats: First, as I mentioned above, when spatial curvature is present, the Newtonian solution is no longer valid. Second, there is a century of literature with a variety of attempts to define a meaningful stress-energy tensor for the gravitational field in general relativity, none completely satisfying. Last but not least, a cosmological constant/dark energy term adds its own interesting twist to the story, as its energy density remains constant in an expanding universe, causing the expansion to accelerate: in this case, both the kinetic and the potential energy increase, thanks to the role played by dark energy's negative pressure.

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Universe is picking up kinetic energy due to the expansion of space:

A photon will gain energy (blueshift) when it heads into a supercluster on its way to the Earth. This is an effect of general relativity. And as it leaves the other side of the supercluster as it continues its journey, it will lose energy (redshift) as it climbs out of the gravitational potential well. But while it is passing through the supercluster, that structure is spreading out due to the Big Bang overall expansion, and its gravitational potential is weakening. So the redshift or energy loss is smaller than the original energy gain or blueshift. So net-net, photons gain energy passing through a supercluster.

Potential energy on that scale, not so much.

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Not within general relativity.

Except in special circumstances, such as an asymptotically flat spacetime, there is no definable total energy in general relativity. The Friedmann–Roberston–Walker family of spacetimes that describe a homogeneous and isotropic universe are not an exception, although there's a separate argument that can be made for the case of spatially compact universes towards their total energy being exactly zero, no matter how they expand or contract.

For the matter distributions, gravitational potential can be useful in the weak-field limit of general relativity when the matter covers a finite region of space, but in general it's not present in general relativity. Or rather, the metric itself can be thought of as taking the role of a 'gravitational potential', but the way it does so is very different from the Newtonian theory.

A Newtonian cosmology having formally analogous behavior to the pressureless FRW cosmology with kinetic and potential terms can be done: $$E = \frac{1}{2}\sum_k m_k\dot{r}_k^2 - G\sum_{j\neq k}\frac{m_jm_k}{|\mathbf{r}_j-\mathbf{r}_k|} - \frac{1}{6}\Lambda\sum_k m_kr_k^2\text{.}$$ However, this is incompatible with an infinite universe, and is neither homogeneous nor isotropic (being rather spherically symmetric with a distinguished center), and therefore violates the cosmological principle. That issue can be fixed by introducing a potential for the scale factor itself, but in that case interpreting it as the gravitational potential of the individual gravitating masses is problematic.

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