Black holes and a scalar field in expanding universe

We consider a model of inhomogeneous universe with the presence of a massless scalar field, where the inhomogeneity is assumed to consist of many black holes. Such model can be constructed by following Lindquist and Wheeler, which has already been investigated without the presence of scalar field to show that an averaged scale factor coincides with that of Friedmann model. In this paper we construct the inhomogeneous universe with an massless scalar field, where it is assumed that the averaged scale factor and scalar field are given by those of the Friedmann model with the presence of the scalar field. All of our calculations are carried out in the frame of Brans-Dicke gravity. In constructing the model of inhomogeneous universe, we define a mass of black hole in Brans-Dicke expanding universe which is equivalent to ADM mass in the epoch of the adiabatic time evolution of the mass, and obtain an equation relating our mass with the averaged scalar field and scale factor. It is not a priori obvious whether or not our mass depends on time during cosmological time evolution. As the results we find that the mass has adiabatic time dependence in a sufficiently late stage of the expansion of the universe, and that its time dependence is qualitatively diffenrent according to the sign of the curvature of the universe: the mass increases decelerately in closed universe case, is constant in flat case, and decreases decelerately in open case. It is also noticed that the mass in the Einstein frame depends on time. Our results that the mass has time dependence should be retained even in the general scalar-tensor gravities and with a potential of the scalar field. PACS: 04.70.-s, 04.70.Bw, 98.80.Hw