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 $z=1.36-3.14$.
