  The optimal error estimate that depending only on the polynomial degree of $
\varepsilon^{-1}$ is established for the temporal semi-discrete scheme of the
Cahn-Hilliard equation, which is based on the scalar auxiliary variable (SAV)
formulation. The key to our analysis is to convert the structure of the SAV
time-stepping scheme back to a form compatible with the original format of the
Cahn-Hilliard equation, which makes it feasible to use spectral estimates to
handle the nonlinear term. Based on the transformation of the SAV numerical
scheme, the optimal error estimate for the temporal semi-discrete scheme which
depends only on the low polynomial order of $\varepsilon^{-1}$ instead of the
exponential order, is derived by using mathematical induction, spectral
arguments, and the superconvergence properties of some nonlinear terms.
Numerical examples are provided to illustrate the discrete energy decay
property and validate our theoretical convergence analysis.
