Quantum Vortices in Superfluids and Cosmological Applications

or problems. 1.2 Turbulence in superfluids and quantized vortices Due to the absence of viscosity, a vortex that forms in a superfluid continues permanently its vortical motion without dispersion. If we consider the circulation along a loop path around a vortex in a superfluid, given that the velocity of the superfluid depends on the gradient of the phase $\phi$ according to equation 1.7, we obtain that: $$\oint ds \cdot v_s = \frac{\hbar}{m} \oint ds \cdot \nabla \phi \quad (1.12)$$ This implies that, for the wave function to be continuous, the circulation must be an integer multiple of $2\pi\hbar/m$, i.e.: $$\oint ds \cdot v_s = \frac{2\pi n\hbar}{m}; \quad n = 0, 1, 2, \ldots \quad (1.13)$$ meaning that the circulation of the vortex in the superfluid is quantized. The inner tube of the vortex cannot terminate inside the fluid but can reconnect to itself and form a ring. In the absence of external potentials, vortices have a tubular interior of dimension $\xi$ function of $\hbar/\sqrt{mg_0}$: $$\xi = \frac{\hbar}{\sqrt{mg_0}} \quad (1.14)$$ Two vortices can reconnect with each other when they reach the critical distance: $$\delta = \frac{\xi}{2R} \ln \frac{R}{c_0 R_0} \quad (1.15)$$ where $c_0$ and $R_0$ are constants. **Figure 1.1:** Quantum turbulence. Simulation of quantum turbulence in a superfluid liquid operated by means of a surface layer. In a quantum turbulence process in a superfluid liquid, two antagonistic phenomena balance between the production of new vortices and the reconnections of vortices that are at distances less than the critical distance. Indicating with $l$ the density of vortex lines (the average per unit volume), we have that the evolution of turbulence is governed by Vinen's empirical law: $$\frac{dl}{dt} = Al^{3/2} - Bl^2 \quad (1.16)$$ where $A$ and $B$ are constants dependent on temperature. 1.3 Relativistic quantized vortices and string theory It is important to note how the equation governing the evolution of a non-relativistic quantized vortex line is given by the nonlinear Schrödinger equation (NLSE), while that of a relativistic quantized vortex is naturally governed by the nonlinear Klein-Gordon equation (NLKG): $$\nabla^2 \psi - \frac{\partial^2 \psi}{\partial t^2} - \frac{dV}{d|\psi|^2} \psi = 0 \quad (1.17)$$ and therefore the dynamical evolution of a quantized relativistic vortex line is the same evolution as a relativistic string. This allows many of the results found in String Theory to be traced back to quantized vortices in a superfluid [17]. **Figure 1.2:** Dynamics of quantum vortices. (a) reconnection of two vortex lines; (b) scattering between a ring and a line; (c) leap of two annular vortices [30]. 1.4 Origins of spacetime superfluidity Despite the evident analogy, the reasons that lead spacetime to behave like a superfluid are currently not clear. In particular, it is of interest to understand what the particles of the spacetime fluid consist of, if this is what we are talking about, or what the scalar field is that transforms spacetime into a superfluid. Huang [4] proposes the Universe pervaded by a Bose-Einstein condensate with Higgs-type bosons. Clearly in this proposal, the reason why spacetime would remain superfluid at high temperatures would need to be clarified. Among the alternative hypotheses is Verlinde [20], for whom spacetime is a perfect conductor of information. In Verlinde's vision, every loss of information given by the interaction of spacetime with matter elements inside a volume transforms - through a sort of holographic principle - into an area defect in the surface that encloses such volume. This area defect is then perceived as the curvature of spacetime formulated in General Relativity. In view of some identities between energy, matter and information, we believe it is not impossible that Verlinde's vision could be traced back to a mathematically equivalent formulation to that of a superfluid spacetime. **Figure 2.1:** Lyman series. The Lyman series represents the spectral emission produced by a hydrogen atom when its electron transitions from an excited state ($n \geq 2$) to the ground state ($n = 1$). The numbering follows the Greek alphabet so that Ly-α corresponds to the emission that occurs when the electron transitions from state $n = 2$ to $n = 1$, Ly-β from $n = 3$ to $n = 1$, etc. 2 Applications to contemporary cosmology 2.1 Two problems of cosmology **Red shift, Hubble's law and Dark energy.** When the electron of a hydrogen atom jumps from one orbital to another, it produces a light emission according to a wavelength governed by the Rydberg formula for hydrogen, i.e.: $$\frac{1}{\lambda} = R \left( \frac{1}{n^2} - \frac{1}{m^2} \right) \quad (2.1)$$ where $R$ is the Rydberg constant, $m$ and $n$ represent the principal quantum number of the orbital before and after the quantum jump. In particular, when the final state of the electron is the lowest possible ($n = 1$), we have the Lyman series represented in Fig. 2.1. Since the beginning of the 20th century