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In this article, we study the hyperbolic Anderson model driven by aspace-time \emph{colored} Gaussian homogeneous noise with spatial dimension$d=1,2$. Under mild assumptions, we provide $L^p$-estimates of the iteratedMalliavin derivative of the solution in terms of the fundamental solution ofthe wave solution. To achieve this goal, we rely heavily on the \emph{Wienerchaos expansion} of the solution.  Our first application are \emph{quantitative central limit theorems} forspatial averages of the solution to the hyperbolic Anderson model, where therates of convergence are described by the total variation distance. Thesequantitative results have been elusive so far due to the temporal correlationof the noise blocking us from using the It\^o calculus. A \emph{novel}ingredient to overcome this difficulty is the \emph{second-order GaussianPoincar\'e inequality} coupled with the application of the aforementioned$L^p$-estimates of the first two Malliavin derivatives. Besides, we provide thecorresponding functional central limit theorems.  As a second application, we establish the absolute continuity of the law forthe hyperbolic Anderson model. The $L^p$-estimates of Malliavin derivatives arecrucial ingredients to verify a local version of Bouleau-Hirsch criterion forabsolute continuity. Our approach substantially simplifies the arguments forthe one-dimensional case, which has been studied in the recent work by Balan,Quer-Sardanyons and Song (2019).

The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications