or rather a peculiar type of fluid was initially reproposed by Dirac[9] in the 1950s, by Sinha et al.[10] in the late 1970s, to arrive at the present day with Barceló, Liberati and Visser[11] and more completely and organically by Volovik[5, 7] and Huang et al.[4]. The analogies between the behavior of spacetime and fluid dynamics are so recurrent and evident that it is difficult to consider them coincidental. The superfluidity of spacetime allows us to derive, without further artificial hypotheses, the answer to two observational anomalies that contemporary cosmology calls "dark energy" and "dark matter" (cf. section 2). However, the hypothesis of a superfluid spacetime, however it is intended to be reformulated, would imply a violation of Lorentz symmetry for very high energies that has not yet been observed. This violation of Lorentz symmetry - which should still be located at energies between 2 and 3 TeV[4, 5] - constitutes the main theoretical obstacle to this hypothesis.
1.1 Superfluid Equations
The first superfluids encountered experimentally in the laboratory were formed from some isotopes of Helium ${}^4$He and ${}^3$He which respectively at temperatures below 2.17K and 2.5 mK manifest a transition towards a superfluid state. Subsequently, Ketterle's[25] experiments on ${}^6$Li at nano-Kelvin temperatures led to the realization of many other forms of superfluids obtained as Bose-Einstein Condensates of gases at low temperatures.
Bose-Einstein Condensates
Most of the superfluids currently known are obtained as Bose-Einstein condensates. Bosons are integer spin particles that, like photons, can be found in the same quantum state, i.e., they are not subject to the Pauli exclusion principle. This implies that at very low temperatures a large number of particles, virtually all of them, can be found in the lowest quantum state and act in phase as if they were a single wave. In Bose-Einstein statistics, the distribution of bosons for a specific energy level $E$ is given by:
$$f(E) = \frac{1}{e^{(E-\mu)/kT} - 1}$$ (1.1)
where $T$ is the Temperature, $k$ is the Boltzmann constant and $\mu$ is the chemical potential. When temperatures are low, the chemical potential is considered close to zero. We thus obtain that the number of bosons $N_0$ in the lowest quantum state, compared to the total number of bosons $N$, is given by:
$$\frac{N_0}{N} \approx 1 - C(kT)^{3/2}$$ (1.2)
where $C$ is a constant${}^1$. Consequently, for a hypothetical zero temperature, all atoms would be in the same lowest state. Let us call $\psi$ a complex scalar field that represents the average wave function of the system, or also the order parameter of the system, then indicating with $g$ the boson coupling constant proportional to the scattering distance $a$, i.e.:
$$g = \frac{4\pi\hbar^2 a}{m}$$ (1.4)
we have that the parameter $\psi$ satisfies a nonlinear Schrödinger equation (NLSE) called the Gross-Pitaevskii equation (GPE), i.e.:
$$-\frac{\hbar^2}{2m}\nabla^2\psi + g|\psi|^2\psi - \mu_0\psi = i\hbar\frac{\partial\psi}{\partial t}$$ (1.5)
By operating the Madelung transformation:
$$\psi = \sqrt{\rho}e^{i\phi}$$ (1.6)
where we identify with $\rho$ the density of the superfluid${}^2$ and with $\phi$ the phase, we obtain that the velocity of the superfluid $v_s$ is given by a gradient of the phase:
$$v_s = \frac{\hbar}{m}\nabla\phi$$ (1.7)
and the hydrodynamic equations of the fluid are formed by a continuity equation of the superfluid:
$$\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho v_s) = 0$$ (1.8)
and the equivalent of the Euler equation which turns out to be:
$$m\frac{\partial v_s}{\partial t} + v_s \cdot \nabla v_s = -\nabla\left[\mu_0 - g\rho + \frac{\hbar^2}{2m}\frac{1}{\sqrt{\rho}}\nabla^2\sqrt{\rho}\right]$$ (1.9)
where the last term in brackets is called the quantum pressure. The limits of the Gross-Pitaevskii equation should be underlined. This equation is indeed ideally valid when the temperature is at 0°K and the model is less valid the higher the interaction between bosons. In general, the model can be considered valid when the following conditions on the scattering length are valid:
$$a^3 \ll \frac{V}{N_0}$$ (1.10)
$$a \ll \sqrt{\frac{2\pi\hbar^2}{mkT}}$$ (1.11)
In case of temperatures different from zero, as could be in the case of spacetime which due to background radiation is at a temperature of about 3°K, it might be necessary to modify these equations by inserting dynamics formed by two distinct fluids, $\rho = \rho_s + \rho_n$, of which one is a superfluid, $\rho_s$, while the other, $\rho_n$, is an ordinary fluid. A two-fluid hydrodynamics has been described by Khalatnikov[13] and is anyway a standard model in treating this type...