Reproducing the asymptotic behaviour of galaxy rotation curves by a novel constraint in general relativity

The cold dark matter paradigm has been posited as the standard explanation for the non-Keplerian behavior of galaxy rotation curves, where for galaxies satisfying the Tully-Fisher relation, the mass of the dark matter halo from a large class of universal dark matter profiles ought to roughly increase linearly with radial distance at large distances, $m(r) \sim r/nG$ ($G$ is the gravitational constant and $n$ is a dimensionless parameter which depends on the amount of baryonic matter $M$ within the galaxy). Despite numerous advances in modeling galaxy formation and evolution, a scientific consensus on the origin of the observed dependence of the dimensionless parameter $n = (GMa_{0})^{-1/2}$ on the mass of baryonic matter $M$ within the galaxy (the Tully-Fisher relation), and the connection of the cosmological constant $\Lambda$ to the parameter $a_{0} \sim (\Lambda/3)^{1/2}$ remains elusive. Here, we show that Einstein Field Equations can be remolded into $\nabla_{\nu}\mathcal{K}^{\nu}_{\,\,\mu} = 8\pi GM\Psi^{*}\mathcal{D}_{\mu}\Psi$, where $\mathcal{K}_{\mu\nu}$ is a complex Hermitian tensor, $\mathcal{D}_{\mu}$ is a covariant derivative and $\Psi$ is a complex-valued function. This avails a novel constraint, $\nabla_{\mu}\nabla_{\nu}\mathcal{K}^{\mu\nu} = 0$ not necessarily available in Einstein's General Relativity. In the weak-field regime, we can readily reproduce the Tully-Fisher relation using the usual charge-less pressure-less fluid. Moreover, our approach is equivalent to a Ginzburg-Landau theory of $n$ bosons, where the order parameter is normalized as $\int_{0}^{1/a_{0}} dr\,4\pi r^2\Psi^*\Psi = n$ and $1/a_{0} \sim (\Lambda/3)^{-1/2}$ is the cut-off length scale comparable to the size of the de Sitter universe. Our investigations provide a framework that reproduces the mass-asymptotic speed relation in galaxies within the cold dark matter paradigm.