Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results

Several recent astrophysical observations of distant type Ia supernovae have revealed that the content of the universe is made of about 70% of dark energy, 25% of dark matter and 5% of baryonic (visible) matter [1]. Thus, the overwhelming preponderance of matter and energy in the universe is believed to be dark i.e. unobservable by telescopes. The dark energy is responsible for the accelerated expansion of the universe. Its origin is mysterious and presumably related to the cosmological constant. Dark energy is usually interpreted as a vacuum energy and it behaves like a fluid with negative pressure. Dark matter also is mysterious. The suggestion that dark matter may constitute a large part of the universe was raised by Zwicky [2] in 1933. Using the virial theorem to infer the average mass of galaxies within the Coma cluster, he obtained a value much larger than the mass of luminous material. He realized therefore that some mass was “missing” in order to account for observations. This missing mass problem was confirmed later by accurate measurements of rotation curves of disc galaxies [3, 4]. The rotation curves of neutral hydrogen clouds in spiral galaxies measured from the Doppler effect are found to be roughly flat (instead of Keplerian) with a typical rotational velocity v∞ ∼ 200km/s up to the maximum observed radius of about 50 kpc. This mass profile is much more extended than the distribution of starlight which typically converges within ∼ 10 kpc. This implies that galaxies are surrounded by an extended halo of dark matter whose mass M(r) = rv 2/G increases linearly with radius [56]. This can be conveniently modeled by an isothermal self-gravitating gas the density of which scales asymptotically as r −2 [6].