Volume 251, number 4 PHYSICS LETTERS B 29 November 1990 The cosmological formation of boson stars Mark S. Madsen and Andrew R. Liddle Astronomy Centre, Universityof Sussex, Falmer,Brighton BN1 9QH, UK Received 10 August 1990 The formation of boson stars by gravitational collapse in the early universe is considered. It is demonstrated that while the simplest models are unfavourable for boson star formation, the inclusion ofa boson self-interaction and consideration ofa boson- antiboson asymmetry allow substantial possibilities for the cosmologicalformation ofboson stars. Dark matter is a necessary component of modern cosmology, in order to reconcile both the prediction of inflationary models that the present mass density is very close to the critical closure density and the re- quirement that galaxies should have had sufficient time to form. There are many candidates for such dark matter, ranging from massive neutrinos to weakly interacting massive particles, or WIMPs [ 1 ]. The latter have also been invoked, under the name of cosmions, as a possible solution to the long-standing solar neutrino problem [ 2 ]. Clearly the question of the distribution of the dark matter is of great interest, and in this letter we wish to consider the possibility that, if the dark matter consists of bosonic particles, some or all of the dark matter may reside in boson stars. Boson stars are gravitationally bound macroscopic quantum states of scalar bosons. They are similar in many respects to neutron stars, differing in that their pressure support derives from the uncertainty rela- tion rather than the exclusion principle. Since the possibility of their existence was first theoretically demonstrated [3-5 ], it has been shown that boson stars are stable to small radial perturbations (pro- vided that their central density does not exceed a critical value which also corresponds to the configu- ration with the maximum possible mass) [ 6,7 ], and that they possess excited states analogous to those of atomic wavefunctions [8,9], which may decay to their ground state by the emission of scalar radiation. States with nonzero angular momentum may also de- cay by radiation of gravitational waves [ 9 ]. While up until now work has concentrated on scalar particles, it seems likely that similar constructs should be pos- sible for massive bosons of any spin. The obvious re- maining unanswered question is whether boson stars can actually form at some stage of the evolution of the cosmos. We shall see below that a large class of models do permit boson stars to form. The structure of stationary boson stars has been ex- tensively investigated using numerical methods [4,5 ], but can also be well understood from simple dimen- sional arguments [ 10]. The gravitationally bound states of a free scalar field ~ with mass m will have a scale radius r given by GM/r~ 1, where the gravita- tional constant G= 1/m 2 with mp the Planck mass, and M is the total mass of the bound state. An indi- vidual boson will have momentum of order m, cor- responding to a typical length r~ 1/m, so that M~m2p/m. In this state, the energy density is p~M/r3~m2m~,, and since /9~m2~ 2, the bound state must have ~~ mp. Numerical results [4] show that in these configurations the gravitational binding energy is only a small fraction of the mass of the star. Bound states with finite mass exist only for complex ~, since in this case there is a conserved current [ 6] which is not present when the field is real. This dis- tinction, however, makes no difference to the consid- erations addressed in this article. The presence of a self-interaction in the scalar po- tential is known to have significant effects on the structure of the boson star [ 5 ]. This is seen by con- sidering the bound states found with a potential term like 2~ 4 with 2<< 1. so that the effective mass of the 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) 507