Using the projection operator formalism we explore the decay form of the time correlation function $U_n\left(t\right)\equiv ⟨{\widehat{u}}_n\left(t\right){\widehat{u}}_n^{\ast }\left(0\right)⟩$ of the state variable ${\widehat{u}}_n\left(t\right)$ in the chaotic Kuramoto-Sivashinsky equation. The decay form turns out to be the algebraic decay $1∕\left[1+{\left({\gamma }_{na}t\right)}^2\right]$ in the initial regime $t<1∕{\gamma }_{ne}$ and the exponential decay $\text{exp}\left(-{\gamma }_{ne}t\right)$ in the final regime $t>1∕{\gamma }_{ne}$. The memory function ${\Gamma }_n\left(t\right)$ that represents the chaos-induced transport is found to obey the Gaussian decay $\text{exp}\left[-{\left({\beta }_{ng}t\right)}^2\right]$ in the case of large wave numbers, but the 3/2 power decay $\text{exp}\left[-{\left({\beta }_{n3}t\right)}^{3∕2}\right]$ in the case of small wave numbers. The power spectrum of ${\widehat{u}}_n\left(t\right)$ is given by the real part $U_n^{\prime }\left(\omega \right)$ of the Fourier-Laplace transform of $U_n\left(t\right)$ and has a dominant peak at $\omega =0$. This peak within the linewidth ${\overline{\gamma }}_{ne}\left(\approx {\gamma }_{ne}\right)$ is given by the Lorentzian spectrum ${\overline{\gamma }}_{ne}^2∕\left({\omega }^2+{\overline{\gamma }}_{ne}^2\right)$. However, the wings of the peak outside the width ${\overline{\gamma }}_{ne}$ turn out to take the exponential spectrum $\text{exp}\left(-\omega ∕{\gamma }_{na}\right)$. Thus it is found that the exponential decay $\text{exp}\left(-{\gamma }_{ne}t\right)$ appears to lead to the universal Lorentzian peak, while the algebraic decay $1∕\left[1+{\left({\gamma }_{na}t\right)}^2\right]$ arises to bring about the exponential wing.

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