A Formaldehyde Deep Field

Formaldehyde (H$_2$CO) is often observed at centimeter wavelengths as an absorption line against the cosmic microwave background (CMB). This is possible when energy level populations are anti-inverted to the point where line excitation temperatures fall below the local CMB temperature. Collisions with molecular hydrogen "pump" this anti-maser excitation, and the cm line ratios of H$_2$CO provide a measurement of the local H$_2$ density. H$_2$CO absorption of CMB light provides all of the benefits of absorption lines (no distance dimming) but none of the drawbacks: the CMB provides uniform illumination of all molecular gas in galaxies (no pencil beam sampling), and all galaxies lie in front of the CMB (no fortuitous alignments with background light sources are needed). A formaldehyde deep field (FDF) would therefore provide a blind, mass-limited survey of molecular gas across the history of star formation and galaxy evolution. Moreover, the combination of column density and number density measurements may provide geometric distances in large galaxy samples and at higher redshifts than can be done using the Sunyaev Zel'dovich effect in galaxy clusters. We present a possible ngVLA FDF that would span redshifts $z=0-7$ and provide H$_2$CO line ratios over $z=1.36-3.14$.