  In two recent publications [Kov{\'a}cs, Larsson, and Mesforush, SIAM J.
Numer. Anal. 49(6), 2407-2429, 2011] and [Furihata, et al., SIAM J. Numer.
Anal. 56(2), 708-731, 2018], strong convergence of the semi-discrete and fully
discrete finite element methods is, respectively, proved for the
Cahn-Hilliard-Cook (CHC) equation, but without convergence rates revealed. The
present work aims to fill the left gap, by recovering strong convergence rates
of (fully discrete) finite element methods for the CHC equation. More
accurately, strong convergence rates of a full discretization are obtained,
based on Galerkin finite element methods for the spatial discretization and the
backward Euler method for the temporal discretization. It turns out that the
convergence rates heavily depend on the spatial regularity of the noise
process. Different from the stochastic Allen-Cahn equation, the presence of the
unbounded elliptic operator in front of the cubic nonlinearity in the
underlying model makes the error analysis much more challenging and demanding.
To address such difficulties, several new techniques and error estimates are
developed. Numerical examples are finally provided to confirm the previous
findings.
