I am driving the paper (http://arxiv.org/abs/gr-qc/0412089) as part of my research project, i have some problems to obtain the effective Lagrangian for Einstein Hilbert action the eq.(4). If any one knows about this, please help me.
They begin with evaluating the Ricci scalar $R$ (2) using the metric given by the equation (1). Then they substitute it in the Einstein-Hilbert action with cosmological constant (3). The final step is integration over space coordinates $r^i$ and $\rho$ to obtain the Lagrangian analogous to the one in classical mechanics of two particles with coordinates $R(t)$ and $a(t)$ and highly non-trivial interaction between them. Is it OK?