  Direct laser acceleration (DLA) of electrons in a plasma of near critical
electron density (NCD) and associated synchrotron-like radiation are discussed
for moderate relativistic laser intensity (the normalized laser amplitude $a_0$
$\leq$ 4.3) and ps-long pulse. This regime is typical for kJ PW-class laser
facilities designed for high energy density research. Currently, in experiments
at the PHELX laser it was demonstrated that interaction of 10$^{19}$ W/cm$^{2}$
sub-ps laser pulse with sub-mm long NCD plasma results in generation of
high-current well-directed super-ponderomotive electrons with effective
temperature that is 10$\times$ higher than the ponderomotive potential [O.
Rosmej et al., PPCF 62, 115024 (2020)]. Three-dimensional Particle-In-Cell
simulations provided a good agreement with the measured electron energy
distribution and were used in the current work to study synchrotron radiation
of the DLA accelerated electrons. The resulting x-ray spectrum with a critical
energy of 5 keV reveals an ultra-high photon number of 7$\times$10$^{11}$ in
the 1-30 keV photon energy range at the focused laser energy of 20 J. Numerical
simulations of a betatron x-ray phasecontrast imaging based on the DLA process
for the parameters of a PHELIX laser is presented. The results are of interest
for applications in high energy density (HED) experiments, which require a
picosecond x-ray pulse and a high photon flux.
