Modelling the Future of Gaia Neutron Star-Main Sequence Binaries: From Eccentric Orbits to Millisecond Pulsar-White Dwarfs

We model the evolution of 21 Gaia neutron star (NS)-main-sequence binaries (orbital period $P_{\mathrm{orb}}\sim200$--$1000$ days, eccentricity $e\gtrsim0.2$) using binary evolution with \texttt{MESA}. We examine eccentric mass transfer and models assuming prior circularization. All systems end as NS-white dwarf (WD) binaries, but transfer modes yield distinct outcomes. Under eccentric transfer, binaries are driven to higher $e$, forming orbits with $e\gtrsim0.6$ and $P_{\rm orb}\sim1000$-$4000$ days. Periastron bursts are brief ($\lesssim10^6$ yr), transfer only a few $\times10^{-2}$ M$_\odot$, and produce mildly recycled pulsars ($P_{\mathrm{spin}}\gtrsim50$ ms) with low-mass He WDs. Artificially circularized transfer gives $P_{\rm orb}\sim200$-$2000$ days, lasts $\sim10^7$ yr, and allows NSs to accrete $\sim0.1$ M$_\odot$, forming fully recycled MSPs ($P_{\mathrm{spin}}\sim$ few-30 ms) with CO WDs. Allowing super-Eddington accretion up to $100\times$ the canonical rate makes even eccentric systems efficient MSP producers, though torque coupling remains uncertain. Using an adaptive, field-dependent magnetic-field decay timescale, we find MSPs stay radio-active over Gyr spans. Gaia systems undergoing stable mass transfer remain wide and fail to match the Galactic MSP-WD population, where most, nearly circular systems have $P_{\rm orb}\lesssim100$ days. Binaries with different mass ratios and initial configurations -- likely leading to unstable mass transfer -- are needed to reproduce the observed MSP-WD distribution.