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

Abstract

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 $n=(GM{a}_0{)}^{-1/2}$ 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 ${\nabla }_{\nu }{\mathcal{K}}_{\mu }^{\nu }=8\pi GM{\Psi }^{\ast }{\mathcal{D}}_{\mu }\Psi$, 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 ${\int }_0^{1/a_0}dr4\pi r^2{\Psi }^{\ast }\Psi =n$ 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.

arXiv comment13 pages, 1 figure

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