  In this work a weak-turbulence closure is used to determine the structure of
the two-time power spectrum of weak magnetohydrodynamic (MHD) turbulence from
the nonlinear equations describing the dynamics. The two-time energy spectrum
is a fundamental quantity in turbulence theory from which most statistical
properties of a homogeneous turbulent system can be derived. A closely related
quantity, obtained via a spatial Fourier transform, is the two-point two-time
correlation function describing the space-time correlations arising from the
underlying dynamics of the turbulent fluctuations. Both quantities are central
in fundamental turbulence theories as well as in the analysis of turbulence
experiments and simulations. However, a first-principles derivation of these
quantities has remained elusive due to the statistical closure problem, in
which dynamical equations for correlations at order $n$ depend on correlations
of order $n+1$. The recent launch of the Parker Solar Probe (PSP), which will
explore the near-Sun region where the solar wind is born, has renewed the
interest in the scientific community to understand the structure, and possible
universal properties of space-time correlations. The weak MHD turbulence regime
that we consider in this work allows for a natural asymptotic closure of the
two-time spectrum, which may be applicable to other weak turbulence regimes
found in fluids and plasmas. An integro-differential equation for the
scale-dependent temporal correlation function is derived for incompressible
Alfv\'enic fluctuations whose nonlinear dynamics is described by the reduced
MHD equations.
