  There is renewed interest in direct-drive inertial confinement fusion,
following the milestone December 2022 3.15 MJ ignition result on the National
Ignition Facility. A key obstacle is the control of the two plasmon decay
instability. Here, recent advances in inhomogeneous turbulence theory are
applied to the broadband parametric instability problem for the first time. A
novel dispersion relation is derived for the two plasmon decay in a uniform
plasma valid under broad-bandwidth laser fields with arbitrary power spectra.
The effects of temporal incoherence on the instability are then studied. In the
limit of large bandwidth, the well-known scaling relations for the growth rate
are recovered, but it is shown that the result is more sensitive to the
spectral shape of the laser pulse rather than to its coherence time. The range
of wavenumbers of the excited plasma waves is shown to be substantially
broadened, suggesting that the absolute instability is favoured in regions
further away from the quarter critical density. The intermediate bandwidth
regime is explored numerically -- the growth rate is reduced to half its
monochromatic value for laser intensities of $10^{15} \,
\text{W}/\text{cm}^{2}$ and relatively modest bandwidths of $5 \, \text{THz}$.
The instability-quenching properties of a spectrum of discrete lines spread
over some bandwidth have also been studied. The reduction in the growth rate is
found to be somewhat lower compared to the continuous case but is still
significant, despite the fact that, formally, the coherence time of such a
laser pulse is infinite.
