  We study the parameter estimation for excitations of Dirac fields in the
expanding Robertson-Walker universe. We employ quantum metrology techniques to
demonstrate the possibility for high precision estimation for the volume rate
of the expanding universe. We show that the optimal precision of the estimation
depends sensitively on the dimensionless mass $\tilde{m}$ and dimensionless
momentum $\tilde{k}$ of the Dirac particles. The optimal precision for the
ratio estimation peaks at some finite dimensionless mass $\tilde{m}$ and
momentum $\tilde{k}$. We find that the precision of the estimation can be
improved by choosing the probe state as an eigenvector of the hamiltonian. This
occurs because the largest quantum Fisher information is obtained by performing
projective measurements implemented by the projectors onto the eigenvectors of
specific probe states.
