Structure formation with scalar field dark matter: the field approach
  We study the formation of structure in the Universe assuming that dark matter
can be described by a scalar field $\tilde{\Phi}$ with a potential
$V(\Phi)=-\mathfrak{m}^{2}\tilde{\Phi}^{2}/2+\lambda\tilde{\Phi}^4/4$. We
derive the evolution equations of the scalar field in the linear regime of
perturbations. We investigate the symmetry breaking and possibly a phase
transition of this scalar field in the early Universe. At low temperatures, the
scalar perturbations have an oscillating growing mode and therefore, this kind
of dark matter could lead to the formation of gravitational structures. In
order to study the nonlinear regime, we use the spherical collapse model and
show that, in the quadratic potential limit, this kind of dark matter can form
virialized structures. The main difference with the traditional Cold Dark
Matter paradigm is that the formation of structure in the scalar field model
can occur at earlier times. Thus, if the dark matter is of scalar field nature
we expect to have large galaxies at high redshifts.
