Could the universe behave like a Bose-Einstein condensate in the distant future? Question:
If the universe keeps expanding and getting cooler, and the temperature asymptotically approaches absolute zero over time...

Could the universe behave like a Bose-Einstein condensate in the distant future?

Density distribution of a Bose-Einstein condensate
 A: Fundamentally, no.
Bose-Einstein condensation is driven by increase in phase-space density, quantified by the degeneracy parameter $D$:
$$ D = n_0 \lambda_{\text{dB}}^3,$$
where $n_0$ is the peak density in real space, and $\lambda_{\text{dB}}$ is the de Broglie (thermal $\propto T^{-1/2}$) wavelength, quantifiying the extent in momentum space.
The degeneracy parameter is related to the entropy $S$ and the 'container' shape $\gamma$ as:
$$ D \propto \exp \left (\gamma - \frac{S}{N k_B} \right ), $$
where $\gamma = 0$ in the absence of a trapping potential, $N$ is the number of particles, and $k_B$ is Boltzmann's constant.
Bose-Einstein condensation kicks in when $D \sim 1$.
To achieve that, you have to remove entropy from your system. The laser and evaporative cooling techniques that have so far achieved BEC of cold atoms, photons, & co. not only reduce the temperature of the sample, but also its entropy. The change in entropy being $\geq 0$ in any process, you are basically extracting entropy from the sample to the environment -- casing point, laser cooling: you send in a directional laser beam (low entropy), and it comes out as spontaneously emitted (more disordered, higher entropy).
In the case of the universe expansion:

*

*slow expansions are usually adiabatic, i.e. at constant entropy. Particles in their ground state stay in their ground state and, sure, get "cooler". But they also increase their spatial profile and hence maintain constant phase space density.

*Assuming the universe has too high an entropy $S$ to Bose-Condense now, if will only have a higher entropy in the future. To reduce it, you'd have to dump it somewhere outside the system (the Universe...) - and if you figure that one out, let me know.


From an engineering point of view, like the other answer & comments therein say, you also have problems. Low densities mean low thermalisation rates. Large particle separation bring more thermalisation issues and even causality and relativistic issues.
A: No, because a BEC requires not only incredibly cold temperatures but high enough densities as well. You can have a gas approach absolute zero just as easily as long as the particles remain non-interacting, which they would do if they remain incredibly far apart relative to their size.
As the universe expands, it not only gets cooler but also more rarified. Eventually even fundamental particles will be outside the range to interact with any other particle even if they moved at lightspeed for infinite time towards each other.
