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The cold dark matter paradigm has been posited as the standard explanationfor the non-Keplerian behavior of galaxy rotation curves, where for galaxiessatisfying the Tully-Fisher relation, the mass of the dark matter halo from alarge class of universal dark matter profiles ought to roughly increaselinearly with radial distance at large distances, $m(r) \sim r/nG$ ($G$ is thegravitational constant and $n$ is a dimensionless parameter which depends onthe amount of baryonic matter $M$ within the galaxy). Despite numerous advancesin modeling galaxy formation and evolution, a scientific consensus on theorigin of the observed dependence of the dimensionless parameter $n =(GMa_{0})^{-1/2}$ on the mass of baryonic matter $M$ within the galaxy (theTully-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$\nabla_{\nu}\mathcal{K}^{\nu}_{\,\,\mu} = 8\piGM\Psi^{*}\mathcal{D}_{\mu}\Psi$, where $\mathcal{K}_{\mu\nu}$ is a complexHermitian tensor, $\mathcal{D}_{\mu}$ is a covariant derivative and $\Psi$ is acomplex-valued function. This avails a novel constraint,$\nabla_{\mu}\nabla_{\nu}\mathcal{K}^{\mu\nu} = 0$ not necessarily available inEinstein's General Relativity. In the weak-field regime, we can readilyreproduce the Tully-Fisher relation using the usual charge-less pressure-lessfluid. Moreover, our approach is equivalent to a Ginzburg-Landau theory of $n$bosons, where the order parameter is normalized as $\int_{0}^{1/a_{0}} dr\,4\pir^2\Psi^*\Psi = n$ and $1/a_{0} \sim (\Lambda/3)^{-1/2}$ is the cut-off lengthscale comparable to the size of the de Sitter universe. Our investigationsprovide a framework that reproduces the mass-asymptotic speed relation ingalaxies within the cold dark matter paradigm.

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