  Binary neutron star mergers and neutron star-black hole mergers are
multi-messenger sources that can be detected in gravitational waves and in
electromagnetic radiation. The low electron fraction of neutron star merger
ejecta favors the production of heavy elements such as lanthanides and
actinides via rapid neutron capture (r-process). The decay of these unstable
nuclei powers an infrared-bright transient called a "kilonova". The discovery
of a population of kilonovae will allow us to determine if neutron star mergers
are the dominant sites for r-process element nucleosynthesis, constrain the
equation of state of nuclear matter, and make independent measurements of the
Hubble constant. The Nancy Grace Roman Space Telescope (Roman) will have a
unique combination of depth, near-infrared sensitivity, and wide field of view.
These characteristics will enable Roman's discovery of GW counterparts that
will be missed by optical telescopes, such as kilonova that are associated with
large distances, high lanthanide fractions, high binary mass-ratios, large dust
extinction in the line of sight, or that are observed from equatorial viewing
angles. Our analysis suggests to (i) make available a rapid (about 1 week)
Target of Opportunity mode for GW follow-up; (ii) include observations of the
High Latitude Time-Domain survey footprint in at least two filters (preferably
the F158 and F213 filters) with a cadence of < 8 days; (iii) operate in synergy
with Rubin Observatory. Following these recommendations, we expect that 1-6
kilonovae can be identified by Roman via ToO observations of well localized (A
< 10 sq. deg., 90% C.I.) neutron star mergers during 1.5 years of the
LIGO-Virgo-KAGRA fifth (or about 4-21 in during the sixth) observing run. A
sample of 5-40 serendipitously discovered kilonovae can be collected in a
5-year high latitude survey.
