Spherically symmetric problem on the brane and galactic rotation curves

Abstract

We investigate the braneworld model with induced gravity to clarify the role of the crossover length scale $\ensuremath{\ell}$ in the possible explanation of the dark-matter phenomenon in astrophysics and in cosmology. Observations of the 21 cm line from neutral hydrogen clouds in spiral galaxies reveal that the rotational velocities remain nearly constant at a value ${\ensuremath{\upsilon}}_{c}\ensuremath{\sim}{10}^{\ensuremath{-}3}--{10}^{\ensuremath{-}4}$ in the units of the speed of light in the region of the galactic halo. Using the smallness of ${\ensuremath{\upsilon}}_{c}$, we develop a perturbative scheme for reconstructing the metric in a galactic halo. In the leading order of expansion in ${\ensuremath{\upsilon}}_{c}$, at the distances $r\ensuremath{\gtrsim}{\ensuremath{\upsilon}}_{c}\ensuremath{\ell}$, our result reproduces that obtained in the Randall-Sundrum braneworld model. This inequality is satisfied in a real spiral galaxy such as our Milky Way for distances $r\ensuremath{\sim}3\text{ }\text{ }\mathrm{kpc}$, at which the rotational velocity curve becomes flat, ${\ensuremath{\upsilon}}_{c}\ensuremath{\sim}7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, if $\ensuremath{\ell}\ensuremath{\lesssim}2\text{ }\text{ }\mathrm{Mpc}$. The gravitational situation in this case can be approximately described by the Einstein equations with the so-called Weyl fluid playing the role of dark matter. In the region near the gravitating body, we derive a closed system of equations for the static spherically symmetric situation under the approximation of zero anisotropic stress of the Weyl fluid. We find the Schwarzschild metric to be an approximate vacuum solution of these equations at distances $r\ensuremath{\lesssim}\sqrt[3]{{r}_{g}{\ensuremath{\ell}}^{2}}$. The value $\ensuremath{\ell}\ensuremath{\lesssim}2\text{ }\text{ }\mathrm{Mpc}$ complies well with the solar system tests. At the same time, in cosmology, a low-density braneworld with $\ensuremath{\ell}$ of this order of magnitude can mimic the expansion properties of the high-density $\ensuremath{\Lambda}+\mathrm{\text{cold dark matter}}$ (LCDM) universe at late times. Combined observations of galactic rotation curves and gravitational lensing can possibly discriminate between the higher-dimensional effects and dark matter.