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The first binary neutron star merger, GW170817, was accompanied by aradioactivity-powered optical/infrared transient called a kilonova. To date, nocompelling kilonova has been found during optical surveys of the sky,independent of gravitational-wave triggers. In this work, we searched the first23 months of the Zwicky Transient Facility (ZTF) data stream for candidatekilonovae in the form of rapidly evolving transients. We combined ZTF alertqueries with forced point-spread-function photometry and nightly flux stackingto increase our sensitivity to faint and fast transients. Automatic queriesyielded $>11,200$ candidates, 24 of which passed quality checks and strictselection criteria based on a grid of kilonova models tailored for both binaryneutron star and neutron star-black hole mergers. None of the candidates in oursample was deemed a possible kilonova after thorough vetting, catalogcross-matching, and study of their color evolution. The sources that passed ourselection criteria are dominated by Galactic cataclysmic variables. Inaddition, we identified two fast transients at high Galactic latitude, one ofwhich is the confirmed afterglow of long-duration GRB190106A, and the other isa possible cosmological afterglow. Using a survey simulation code, weconstrained the kilonova rate for a range of models including top-hat andlinearly decaying light curves and synthetic light curves obtained withradiative transfer simulations. For prototypical GW170817-like kilonovae, weconstrain the rate to be $R < 1775$ Gpc$^{-3}$ yr$^{-1}$ at 95% confidencelevel by requiring at least 2 high-significance detections. By assuming apopulation of kilonovae with the same geometry and composition of GW170817observed under a uniform viewing angle distribution, we obtained a constrainton the rate of $R < 4029$ Gpc$^{-3}$ yr$^{-1}$.

Constraining the Kilonova Rate with Zwicky Transient Facility Searches Independent of Gravitational Wave and Short Gamma-ray Burst Triggers