COPSS II: The Molecular Gas Content of Ten Million Cubic Megaparsecs at $z\sim3$

We present a measurement of the abundance of carbon monoxide in the early Universe, utilizing the final results from the CO Power Spectrum Survey (COPSS). Between 2013 and 2015, we performed observations with the Sunyaev-Zel'dovich Array (SZA) to measure the aggregate CO emission from $z\sim3$ galaxies with the intensity mapping technique. Data were collected on 19 fields, covering an area of 0.7 square degrees, over a frequency range of 27 to 35 GHz. With these data, along with data analyzed in COPSS I, we are able to observe the CO(1-0) transition in the redshift range of $z=2.3-3.3$ for spatial frequencies between $k=0.5-10\ h\,\textrm{Mpc}^{-1}$, spanning a comoving volume of $4.9\times10^{6}\ h^{-3}\,\textrm{Mpc}^{3}$. We present estimates of contributions from continuum sources and ground illumination within our measurement. We constrain the amplitude of the CO power spectrum to be $P_{\textrm{CO}}=3.0_{-1.3}^{+1.3}\times10^{3}\ \mu\textrm{K}^{2} (h^{-1}\,\textrm{Mpc})^{3}$, or $\Delta^{2}_{\textrm{CO}}(k\!=\!1\ h\,\textrm{Mpc}^{-1})=1.5^{+0.7}_{-0.7} \times10^{3}\ \mu\textrm{K}^{2}$, at 68% confidence, and $P_{\textrm{CO}}>0$ at 98.9% confidence. These results are a factor of 10 improvement in sensitivity compared to the previous limits set in COPSS I. We use this measurement to place constraints on the CO(1-0) galaxy luminosity function at $z\sim3$. We constrain the ratio of CO(1-0) luminosity to host halo mass to $A_{\textrm{CO}}=6.3_{-2.1}^{+1.4}\times10^{-7}\ L_{\odot}\ M_{\odot}^{-1}$, and estimate a mass fraction of molecular gas of $f_{\textrm{H}_{2}}=5.5^{+3.4}_{-2.2}\times10^{-2}$ for halos with masses of order $10^{12}M_{\odot}$. Using simple theoretical estimates for the scaling of molecular gas mass fraction and halo mass, we estimate the global density of molecular gas to be $\rho_{z\sim3}(\textrm{H}_{2})=1.1_{-0.4}^{+0.7}\times10^{8}\ M_{\odot}\ \textrm{Mpc}^{-3}$.