Age and metallicity of the Milky Way's nuclear star cluster studied at 3 pc from Sagittarius A*

The Milky Way's nuclear star cluster (NSC) is a unique laboratory to study the formation and evolution of dense stellar systems around a supermassive black hole. Previous work suggests that most stars in the NSC are old; however, the detailed age and metallicity distributions remain uncertain. We constrain the star formation history (SFH) and metallicity of a poorly explored region located $\sim$3 pc from SagittariusA*. We analyse VLT/NACO imaging in an intermediate-band filter centred at 2.24 $μ$m, complemented by $H$-band data. We construct completeness-corrected $K$-band luminosity functions (LFs), clearly identifying the Red Clump and Red Giant Branch Bumps. The SFH is derived by fitting cumulative LFs with MIST, PARSEC, and BaSTI models spanning a wide range of ages and metallicities, using Monte Carlo sampling to estimate uncertainties. Metallicity constraints are refined using spectroscopic measurements from the literature. The stellar population is predominantly old and metal-rich: $75.6 \pm 9.5$% of the stellar mass formed $\gtrsim 10$ Gyr ago, with median [M/H] $\sim +0.35$. An intermediate-age component at 2-3 Gyr contributes $20.8 \pm 8.7$%, while minor populations are present at $\sim$400 Myr ($0.9 \pm 0.8$%) and 20 Myr ($3.6 \pm 1.4$%), the latter representing a small but non-negligible young population. Systematic uncertainties from stellar models, binning, photometric range, unresolved binaries, and filter choice are assessed. These results indicate early dominant formation, a significant 2-3 Gyr episode, and minor recent activity, consistent with spectroscopic measurements and with properties of the inner NSC and nuclear stellar disc.