  We study the sharp interface limit of the stochastic Cahn-Hilliard equation
with cubic double-well potential and additive space-time white noise
$\epsilon^{\sigma}\dot{W}$ where $\epsilon>0$ is an interfacial width
parameter. We prove that, for sufficiently large scaling constant $\sigma >0$,
the stochastic Cahn-Hilliard equation converges to the deterministic
Mullins-Sekerka/Hele-Shaw problem for $\epsilon\rightarrow 0$. The convergence
is shown in suitable fractional Sobolev norms as well as in the $L^p$-norm for
$p\in (2, 4]$ in spatial dimension $d=2,3$. This generalizes the existing
result for the space-time white noise to dimension $d=3$ and improves the
existing results for smooth noise, which were so far limited to $p\in \left(2,
\frac{2d+8}{d+2}\right]$ in spatial dimension $d=2,3$. As a byproduct of the
analysis of the stochastic problem with space-time white noise, we identify
minimal regularity requirements on the noise which allow convergence to the
sharp interface limit in the $\mathbb{H}^1$-norm and also provide improved
convergence estimates for the sharp interface limit of the deterministic
problem.
