  Let $\{u(t,x)\}_{t>0,x\in{{\mathbb R}^{d}}}$ denote the solution to the
linear (fractional) stochastic heat equation. We establish rates of convergence
with respect to the uniform distance between the density of spatial averages of
solution and the density of the standard normal distribution in some different
scenarios. We first consider the case that $u_0\equiv1$, and the stochastic
fractional heat equation is driven by a space-time white noise. When $\alpha=2$
(parabolic Anderson model, PAM for short) and the stochastic heat equation is
driven by colored noise in space, we present the rates of convergence
respectively in the case that $u_0\equiv1$, $d\geq1$ and $u_0=\delta_0$, $d=1$
under an additional condition $\hat f(\mathbb{R}^d)<\infty$. Our results are
obtained by using a combination of the Malliavin calculus and Stein's method
for normal approximations.
