  We report the double helicity asymmetry, $A_{LL}^{J/\psi}$, in inclusive
$J/\psi$ production at forward rapidity as a function of transverse momentum
$p_T$ and rapidity $|y|$. The data analyzed were taken during $\sqrt{s}=510$
GeV longitudinally polarized $p$$+$$p$ collisions at the Relativistic Heavy Ion
Collider (RHIC) in the 2013 run using the PHENIX detector. At this collision
energy, $J/\psi$ particles are predominantly produced through gluon-gluon
scatterings, thus $A_{LL}^{J/\psi}$ is sensitive to the gluon polarization
inside the proton. We measured $A_{LL}^{J/\psi}$ by detecting the decay
daughter muon pairs $\mu^+ \mu^-$ within the PHENIX muon spectrometers in the
rapidity range $1.2<|y|<2.2$. In this kinematic range, we measured the
$A_{LL}^{J/\psi}$ to be $0.012 \pm 0.010$~(stat)~$\pm$~$0.003$(syst). The
$A_{LL}^{J/\psi}$ can be expressed to be proportional to the product of the
gluon polarization distributions at two distinct ranges of Bjorken $x$: one at
moderate range $x \approx 0.05$ where recent RHIC data of jet and $\pi^0$
double helicity spin asymmetries have shown evidence for significant gluon
polarization, and the other one covering the poorly known small-$x$ region $x
\approx 2\times 10^{-3}$. Thus our new results could be used to further
constrain the gluon polarization for $x< 0.05$.
