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 $x approx 2times 10^{-3}$. Thus our new results could be used to further constrain the gluon polarization for $x< 0.05$.
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