  We consider a system of $d$ non-linear stochastic heat equations driven by an
$m$-dimensional space-time white noise on $\mathbb{R}_+\times \mathbb{R}$. In
this paper we study the asymptotic behavior of spatial averages over large
intervals $[-R,R]$. We establish a rate of convergence to a multivariate normal
distribution in the Wasserstein distance and a functional central limit
theorem.
