  We study the stochastic gravitational wave background (SGWB) produced by
freely decaying vortical turbulence in the early Universe. We thoroughly
investigate the time correlation of the velocity field, and hence of the
anisotropic stresses producing the gravitational waves. With hydrodynamical
simulations, we show that the unequal time correlation function (UETC) of the
Fourier components of the velocity field is Gaussian in the time difference, as
predicted by the "sweeping" decorrelation model. We introduce a decorrelation
model that can be extended to wavelengths around the integral scale of the
flow. Supplemented with the evolution laws of the kinetic energy and of the
integral scale, this provides a new model UETC of the turbulent velocity field
consistent with the simulations. We discuss the UETC as a positive definite
kernel, and propose to use the Gibbs kernel for the velocity UETC as a natural
way to ensure positive definiteness of the SGWB. The SGWB is given by a
4-dimensional integration of the resulting anisotropic stress UETC with the
gravitational wave Green's function. We perform this integration using a Monte
Carlo algorithm based on importance sampling, and find that the result matches
that of the simulations. Furthermore, the SGWB obtained from the numerical
integration and from the simulations show close agreement with a model in which
the source is constant in time and abruptly turns off after a few eddy turnover
times. Based on this assumption, we provide an approximate analytical form for
the SGWB spectrum and its scaling with the initial kinetic energy and integral
scale. Finally, we use our model and numerical integration algorithm to show
that including an initial growth phase for the turbulent flow heavily
influences the spectral shape of the SGWB. This highlights the importance of a
complete understanding of the turbulence generation mechanism.
