Uniqueness of self-similar asymptotically Friedmann-Robertson-Walker spacetime in Brans-Dicke theory

We investigate spherically symmetric self-similar solutions in Brans-Dicke theory. Assuming a perfect fluid with the equation of state p=(gamma-1)mu(1less than or equal togamma<2), we show that there are no nontrivial solutions which approach asymptotically to the flat Friedmann-Robertson-Walker spacetime if the energy density is positive. This result suggests that primordial black holes in Brans-Dicke theory cannot grow at the same rate as the size of the cosmological particle horizon.