Spatially Resolved Water Emission from Gravitationally Lensed Dusty Star Forming Galaxies at z $sim$ 3

Water ($rm H_{2}O$), one of the most ubiquitous molecules in the universe, has bright millimeter-wave emission lines easily observed at high-redshift with the current generation of instruments. The low excitation transition of $rm H_{2}O$, p$-$$rm H_{2}O$(202 $-$ 111) ($ u_{rest}$ = 987.927 GHz) is known to trace the far-infrared (FIR) radiation field independent of the presence of active galactic nuclei (AGN) over many orders-of-magnitude in FIR luminosity (L$_{rm FIR}$). This indicates that this transition arises mainly due to star formation. In this paper, we present spatially ($sim$0.5 arcsec corresponding to $sim$1 kiloparsec) and spectrally resolved ($sim$100 kms$^{-1}$) observations of p$-$$rm H_{2}O$(202 $-$ 111) in a sample of four strong gravitationally lensed high-redshift galaxies with the Atacama Large Millimeter/submillimeter Array (ALMA). In addition to increasing the sample of luminous ($ > $ $10^{12}$L$_{odot}$) galaxies observed with $rm H_{2}O$, this paper examines the L$_{rm H_{2}O}$/L$_{rm FIR}$ relation on resolved scales for the first time at high-redshift. We find that L$_{rm H_{2}O}$ is correlated with L$_{rm FIR}$ on both global and resolved kiloparsec scales within the galaxy in starbursts and AGN with average L$_{rm H_{2}O}$/L$_{rm FIR}$ =$2.76^{+2.15}_{-1.21}times10^{-5}$. We find that the scatter in the observed L$_{rm H_{2}O}$/L$_{rm FIR}$ relation does not obviously correlate with the effective temperature of the dust spectral energy distribution (SED) or the molecular gas surface density. This is a first step in developing p$-$$rm H_{2}O$(202 $-$ 111) as a resolved star formation rate (SFR) calibrator.