LMC-induced Perturbations in the Milky Way Halo:I. HaloDance Simulation Suite and Observational Forecasts

The gravitational interaction between the Milky Way (MW) and the Large Magellanic Cloud (LMC) perturbs the MW halo's density and kinematics, encoding information about both galaxies' masses and structures. We present a suite of 2,848 high-resolution ($10^7$ particles) N-body simulations that systematically vary the mass and shape of both galaxies' haloes. We model how the mean velocities and velocity dispersions of halo stars (30--120 kpc) depend on system parameters, and forecast constraints achievable with current and future observations. Assuming Gaia DR3-level astrometry, 20 km/s radial velocity precision, 10% distance precision, and a sample of $\sim$4,000 RR Lyrae stars, we achieve 1$σ$ uncertainties of $0.11 \times 10^{12} M_\odot$ in MW mass, $2.33 \times 10^{10} M_\odot$ in LMC mass, 2.38 in halo concentration ($c$), and 0.06 in halo flattening ($q$). These correspond to fractional uncertainties of 11%, 16%, 25%, and 6% respectively, relative to fiducial values. Improved Gaia proper motions (DR5) yield modest gains (up to 14%), while adding radial velocities improves constraints by up to 60% relative to using Gaia astrometry alone. Doubling the sample size to $\sim$8,000 stars yields an additional 30% improvement, whereas reducing distance uncertainties has minimal impact ($\le$10%). Mean velocities trace LMC-induced perturbations, while velocity dispersions constrain MW halo properties, jointly breaking degeneracies. Our results demonstrate that combining Gaia astrometry with large spectroscopic surveys will enable precise characterization of the MW-LMC system. This methodology paper establishes the framework for interpreting observations; future work will apply these tools to existing spectroscopic datasets. The full simulation suite, HaloDance, will be made publicly available at: https://github.com/Yanjun-Sheng/HaloDance.