The swept-back multipolar magnetic field of neutron stars: Application to NICER MSP J0030+0451

NICER observations of millisecond pulsars (MSPs) suggest that non-dipolar magnetic fields are required to explain their surface X-ray hotspots. C. Kalapotharakos et al. (2021) modeled the NICER light curve of MSP J0030+0451 (J0030) using a static vacuum offset dipole-plus-quadrupole field and corresponding force-free (FF) solutions to jointly reproduce the X-ray and Fermi-LAT $\gamma$-ray emission. We substitute their static vacuum field model with a more realistic swept-back configuration that accounts for rotational effects. This field more closely resembles the corresponding FF solutions, making it a more physically motivated choice for future multiwavelength modeling. We adopt a centered swept-back vacuum multipolar magnetic field (SVM2F; J. P\'etri 2015), expressed as a complete expansion in vector spherical harmonics, enabling flexible descriptions of arbitrary magnetic field geometries. We introduce a metric to quantify the complexity among different field prescriptions, illustrated for the static offset vacuum field. To efficiently explore parameter space, we train a neural network surrogate (G. Olmschenk et al. 2025) on SVM2F light curves including components up to the octupole, accelerating Markov chain Monte Carlo sampling by $\sim 10^3$ compared to direct physical model evaluations. Applying this framework to J0030, we constrain the field parameter space and find that a centered swept-back multipolar field including terms up to the octupole adequately reproduces the bolometric thermal X-ray light curve. Our study highlights the importance and inherent complexity of prescribing different multipolar magnetic field models for rotating stars, and can be extended to other MSPs to ultimately constrain the masses and radii of neutron stars, and hence their equation of state.