  We present a comprehensive statistical study of free decay of the quantized
vortex tangle in superfluid $^4$He at low and ultra-low temperatures,
$0\leqslant T \leqslant 1.1\,$K. Using high resolution vortex filament
simulations with full Biot-Savart vortex dynamics, we show that for ultra-low
temperatures $T\lesssim 0.5 \,$K, when the mutual friction parameters
$\alpha\simeq \alpha' < 10^{-5}$, the vortex reconnections excite Kelvin waves
with wave lengths $\lambda$ of the order of the inter-vortex distance $\ell$.
These excitations cascade down to the resolution scale $\Delta\xi$ which in our
simulations is of the order $\Delta \xi\sim \ell/100$. At this scale the Kelvin
waves are numerically damped by a line-smoothing procedure, that is supposed to
mimic the dissipation of Kelvin waves by phonon and roton emission at the scale
of the vortex core. We show that the Kelvin waves cascade is statistically
important: the shortest available Kelvin waves at the end of the cascade
determine the mean vortex line curvature $S$, giving $S \gtrsim 30 /\ell$ and
play major role in the tangle decay at ultra-low temperatures below $0.6\,$K.
The found dependence of $\ell S$ on the resolution scale $\Delta \xi$ agrees
with the L'vov-Nazarenko energy spectrum of weakly-interacting Kelvin waves,
$E\Sb{LN}\propto k^{-5/3}$ rather than the spectrum $E\Sb{LN}\propto k^{-1}$,
suggested by Vinen for turbulence of Kelvin waves with large amplitudes. We
also show that already at $T=0.8\,$K, when $\alpha$ and $\alpha'$ are still
very low, $\alpha\simeq \alpha'<10^{-3}$, the Kelvin wave cascade is fully
damped, vortex lines are very smooth, $S \simeq 2 /\ell$ and the tangle decay
is predominantly caused by the mutual friction.
