  Results are presented from data recorded in 2009 by the PHENIX experiment at
the Relativistic Heavy Ion Collider for the double-longitudinal spin asymmetry,
$A_{LL}$, for $\pi^0$ and $\eta$ production in $\sqrt{s} = 200$ GeV polarized
$p$$+$$p$ collisions. Comparison of the $\pi^0$ results with different theory
expectations based on fits of other published data showed a preference for
small positive values of gluon polarization, $\Delta G$, in the proton in the
probed Bjorken $x$ range. The effect of adding the new 2009 \pz data to a
recent global analysis of polarized scattering data is also shown, resulting in
a best fit value $\Delta G^{[0.05,0.2]}_{\mbox{DSSV}} = 0.06^{+0.11}_{-0.15}$
in the range $0.05<x<0.2$, with the uncertainty at $\Delta \chi^2 = 9$ when
considering only statistical experimental uncertainties. Shifting the PHENIX
data points by their systematic uncertainty leads to a variation of the
best-fit value of $\Delta G^{[0.05,0.2]}_{\mbox{DSSV}}$ between $0.02$ and
$0.12$, demonstrating the need for full treatment of the experimental
systematic uncertainties in future global analyses.
