Effects of the treatment of the mass quadrupole moment on ray-tracing applications for rapidly rotating neutron stars

The Neutron Star Interior Composition Explorer (NICER) mission has provided a unique opportunity to constrain the equation of state of neutron stars by using the technique of pulse-profile modelling. This technique requires accurate and efficient ray-tracing, that in turn requires a robust representation of the space-time around a neutron star. Several exact and approximate metrics have been proposed, and used, to perform ray-tracing around neutron stars, with both moderate and fast rotation. In this paper, we perform a comparison between several of these metrics, when used for ray-tracing. We calculate the shape of the neutron star as seen by a distant observer using two different surface formulae, the thermal spectrum and pulse profiles from circular and crescent-shaped hotspots, for four configurations of pulsars with rotation rates ranging from 622 to 1000 Hz, and using both a moderate and a stiff equation of state to include realistic and extreme cases. We find small differences between the metrics for rotation frequencies starting at ${\sim}700\, \mathrm{Hz}$ that could theoretically be used for constraining the quadrupole moment or the space-time models. We also determine the practicality of use of each metric in larger-scale applications such as pulse-profile modelling.