  We analyse the possibility that the dark matter candidate is from the
approximate scale symmetry theory of the hidden scalar sector. The study
includes the warm dark matter scenario and the Bose-Einstein condensation which
may lead to the scalar boson stars (BS) giving rise to direct detection through
the observation of the primary (direct) photons. The dynamical system of the
scalar particles, the dilatons, at finite temperature and chemical potential is
considered. The fluctuation of the particle density increases sharply within
the increasing of the temperature. When the phase transition approaches, the
fluctuation of the particle density has the non-monotonous rising when the
ground state of the relative chemical potential tends to the critical value
equal to one accompanying by the infinite number of particles. Our results
suggest that the phase transition in the BS may be identified through the
fluctuation in yield of primary photons induced directly by the conformal
anomaly. The fluctuation rate of the photons grows up intensively in the
infra-red to become very large at the phase transition.
