The ALMA Spectroscopic Survey in the HUDF: Constraining Cumulative CO Emission at 1 ≲ z ≲ 4 with Power Spectrum Analysis of ASPECS LP Data from 84 to 115 GHz

The largest physical scale, then, accessible in real space in the line-of-sight dimension, r_∥,max, is determined by the survey's frequency coverage, Δν_BW,

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\parallel ,\max } & = & \displaystyle \frac{c}{{H}_{0}}{\displaystyle \int }_{{z}_{\min }}^{{z}_{\max }}\displaystyle \frac{{dz}}{\sqrt{{{\rm{\Omega }}}_{{\rm{M}}}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}\\ & = & \chi ({z}_{\max })-\chi ({z}_{\min }),\end{array}\end{eqnarray} \tag{ 1 }$$

where χ(z) is the comoving line-of-sight distance to redshift z. In the above expression, z_min and z_max correspond to the minimum and maximum redshifts observed at, respectively, the highest and lowest frequencies, ν_max = 114.750 GHz and ν_min = 84.278 GHz, of the survey bandwidth, so that ${z}_{\min }\,={\nu }_{\mathrm{rest},\mathrm{CO}(2-1)}/{\nu }_{\max }-1=1.009$, ${z}_{\max }={\nu }_{\mathrm{rest},\mathrm{CO}(2-1)}/{\nu }_{\min }-1\,=1.735$, and r_∥,max = 1054.8 Mpc h⁻¹.

Similarly, the channel resolution establishes the smallest physical scale probed in the line of sight, r_∥,min:

$$\begin{eqnarray}&&{r}_{\parallel ,\min }=\chi ({z}_{\mathrm{chn},i+1})-\chi ({z}_{\mathrm{chn},i}).\end{eqnarray} \tag{ 2 }$$

Here z_chn,i and ${z}_{\mathrm{chn},i+1}$ correspond to redshifts of the ith and ith+1 channels at observed frequencies ν_i and ${\nu }_{i+1}={\nu }_{i}+{\rm{\Delta }}{\nu }_{\mathrm{chn}}$, so that ${z}_{\mathrm{chn},i}={\nu }_{\mathrm{rest},\mathrm{CO}(2-1)}/{\nu }_{i}-1$ and ${z}_{\mathrm{chn},i+1}={\nu }_{\mathrm{rest},\mathrm{CO}(2-1)}/({\nu }_{i}+{\rm{\Delta }}{\nu }_{\mathrm{chn}})-1$. Because Δν_chn is a constant across the band, the physical separation between channels increases gradually with redshift. In practice, then, to facilitate computing the Fourier transform, we do not use Equation (2) when converting channel widths from frequency to physical distance. Rather, we define channel separations of equal width in space, dividing the total line-of-sight distance, r_∥,max, by the number of channels, N_chn = 196, in the band, yielding r_∥,min = 5.38 Mpc h⁻¹. This value is equal to the channel width at ν_cen and is a reasonable substitute for the true r_∥,min per channel, due to the modest 12.6% relative change in r_∥,min from either band edge to the band center. Note that the choice of N_chn = 196 reflects the fact that we have imaged the ASPECS data cube while rebinning the native ALMA channel resolution by a factor of 40 (i.e., ${\rm{\Delta }}{\nu }_{\mathrm{chn}}={\rm{\Delta }}{\nu }_{40\mathrm{chn}}$), compared to the factor of 2 rebinning (${\rm{\Delta }}{\nu }_{\mathrm{chn}}={\rm{\Delta }}{\nu }_{2\mathrm{chn}}$) used when imaging the data cube for purposes of CO line searches, etc. The coarser spectral resolution Δν_40chn = 0.156 GHz, with a velocity width ${\rm{\Delta }}{v}_{40\mathrm{chn}}\sim 470$ km s⁻¹ at ν_cen, ensures that most CO emission is spectrally unresolved throughout the data cube; the median FWHM of the Gaussian line flux profiles of the blindly detected lines in ASPECS is 355 km s⁻¹, and the full range of observed FWHMs spans 40.0–617 km s⁻¹ (GL19). We favor the larger channel width to avoid significant contributions to the power spectrum from emission lines with FWHM ≫ Δv_chn, since we do not attempt to characterize the effect of falsely elongating the observed flux density of these lines from the ∼kpc scales of localized emission within the CO-bright galaxy to ∼Mpc scales when converting channel widths to cosmological line-of-sight distances.

In the transverse, or on-sky, dimensions, the largest and smallest physical scales accessible to probe CO fluctuations in real space, r_⊥,max and r_⊥,min, are determined by the survey width and synthesized beam size,

$$\begin{eqnarray}&&{r}_{\perp ,\max }={D}_{{\rm{A}},\mathrm{co}}({z}_{\mathrm{cen},\mathrm{CO}(2-1)}){\rm{\Delta }}{\theta }_{S},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{r}_{\perp ,\min }={D}_{{\rm{A}},\mathrm{co}}({z}_{\mathrm{cen},\mathrm{CO}(2-1)}){\rm{\Delta }}{\theta }_{b},\end{eqnarray} \tag{ 4 }$$

where the units of Δθ_S and Δθ_b are in radians, and ${D}_{{\rm{A}},\mathrm{co}}(z)$, the comoving angular diameter distance at redshift z, is equal to χ(z) for Ω_k = 0.

As the antenna primary beam size grows with observed wavelength, Δθ_b and Δθ_S gradually increase with redshift across the survey bandwidth. As with ${r}_{\parallel ,\min }$, we adopt fixed values for each quantity calculated at ν_cen, where change is modest to either band edge. At this frequency, the synthesized beam—an ellipse described by the FWHMs of its major and minor axes, ${\rm{\Delta }}{\theta }_{b,\mathrm{maj}}$ and ${\rm{\Delta }}{\theta }_{b,\min }$, respectively—for the full data set is ${\rm{\Delta }}{\theta }_{b}={\rm{\Delta }}{\theta }_{b,\mathrm{maj}}\times {\rm{\Delta }}{\theta }_{b,\min }=1\buildrel{\prime\prime}\over{.} 80\,\times \,1\buildrel{\prime\prime}\over{.} 48$, corresponding to comoving transverse distances 0.0250 Mpc h⁻¹ × 0.0205 Mpc h⁻¹. (For reference, ${\rm{\Delta }}{\theta }_{b}=2\buildrel{\prime\prime}\over{.} 11\,\times 1\buildrel{\prime\prime}\over{.} 67$ at ν_min, and ${\rm{\Delta }}{\theta }_{b}\,=1\buildrel{\prime\prime}\over{.} 51\,\times 1\buildrel{\prime\prime}\over{.} 32$ at ν_max.) Expressing the beam area as ${A}_{b}={\rm{\Delta }}{\theta }_{b,\mathrm{maj}}{\rm{\Delta }}{\theta }_{b,\min }\pi /\left(4\mathrm{ln}2\right)$, we let ${r}_{\perp ,\min }=\sqrt{{A}_{b}}=0.0240$ Mpc h⁻¹. Note that the data cube for an interferometric image is gridded with rectangular cell sizes Δθ_cell a factor of a few times smaller than Δθ_b, chosen such that Δθ_cell represents Nyquist sampling of the longest-baseline visibility data (Taylor et al. 1999). Thus, the smallest transverse dimension present in the data set is actually Δθ_cell = 036 (=0.005 Mpc h⁻¹ at ν_cen), though there is no information on CO fluctuations contained within physical scales smaller than Δθ_b.

At ν_cen, the full width of the survey spans roughly ${\rm{\Delta }}{\theta }_{S,\mathrm{tot}}=2\buildrel{\,\prime}\over{.} 83$ at a primary beam response cutoff of 20%. The sensitivity profile of the mosaic primary beam implies, however, that the antenna response drops to 50% at ${\rm{\Delta }}{\theta }_{S,\mathrm{HPBW}}\approx 2\buildrel{\,\prime}\over{.} 15$. Beyond this threshold, the rms noise statistics deteriorate rapidly, as indicated by the noise map²⁰ in Figure 1 for ν_cen. Not only is the overall rms noise higher for the survey field past the half-power point of the mosaic primary beam, the spatial variation of the rms is also significantly greater in this region, compared to the central ∼4 arcmin², e.g., where the rms remains mostly between 0.13 and 0.18 mJy beam⁻¹ channel⁻¹, gradually reaching 0.3 mJy beam⁻¹ channel⁻¹ at the half-power point;²¹ beyond the half-power point, the rms increases from 0.3 to 0.8 mJy beam⁻¹ at the outermost edge of the mosaic, defined by the primary beam cutoff at 20% antenna response. We note as well that results from the blind search for individual CO emitters in ASPECS data (GL19) suggest a lower fidelity (i.e., higher probability of false identification) of line candidates in the survey volume corresponding to <50% antenna response, so other studies (e.g., CO LF measurements presented in D19) within the ASPECS collaboration have excluded data that lie outside ${\rm{\Delta }}{\theta }_{S,\mathrm{HPBW}}$. Thus, we limit our analysis to a square region (shown as the black dotted square in Figure 1) with area ${\rm{\Delta }}{\theta }_{S}^{2}={\left(1.84\mathrm{arcmin}\right)}^{2}={\left(1.53\mathrm{Mpc}{h}^{-1}\right)}^{2}$, chosen to lie within the 50% power threshold at all observed frequencies. For reference, this region encompasses roughly 85% of the volume contained within ${\rm{\Delta }}{\theta }_{S,\mathrm{HPBW}}$ and 55% of the volume within ${\rm{\Delta }}{\theta }_{S,\mathrm{tot}}$.

Zoom In Zoom Out Reset image size

Since we will be characterizing the CO fluctuation field by its power spectrum, we must relate the relevant physical scales probed in real space (Equations (1)–(4)) to the wavevectors with magnitude $k=\sqrt{{k}_{\parallel }^{2}+{k}_{\perp }^{2}}$ that are accessible to ASPECS in Fourier space:

$$\begin{eqnarray}&&{k}_{\parallel ,\min }=\displaystyle \frac{2\pi }{{r}_{\parallel ,\max }}\,\mathrm{and}\,{k}_{\perp ,\min }=\displaystyle \frac{2\pi }{{r}_{\perp ,\max }},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{k}_{\parallel ,\max }=\displaystyle \frac{2\pi }{2{r}_{\parallel ,\min }}\,\mathrm{and}\,{k}_{\perp ,\max }=\displaystyle \frac{2\pi }{2{r}_{\perp ,\min }},\end{eqnarray} \tag{ 6 }$$

where k_∥,min and k_⊥,min represent the lowest k modes available in the line-of-sight and transverse dimensions, respectively; note that these fundamental modes map to the largest scales accessible in real space, with frequencies spanning a single oscillation across Δν_BW and Δθ_S. The highest k modes probed by the survey, k_∥,max and k_⊥,max, correspond to Nyquist frequencies, ${k}_{\parallel ,\mathrm{Nyq}}\propto 1/(2{r}_{\parallel ,\min })$ and ${k}_{\perp ,\mathrm{Nyq}}\propto 1/(2{r}_{\perp ,\min })$, mapping these modes to the smallest physical scales in the survey.

Table 1 summarizes the ASPECS survey parameters adopted for the power spectrum analysis and their mappings to physical dimensions in real and k space. In Figure 2, we indicate the location of k_∥ and k_⊥ values for ASPECS relative to the predicted total CO(2–1) power—including both clustering and shot-noise contributions—at z = 1 from Sun et al. (2018). The ranges of transverse and line-of-sight k modes have important implications to be considered when computing the power spectrum.

Zoom In Zoom Out Reset image size

Table 1.  Mapping ASPECS Survey Parameters to Real- and Fourier-space Dimensions

Survey bandwidth, ΔνBW	84.278–114.750 GHz
Channel resolution, Δνchn	0.156 GHz
Survey width, ΔθS	184
Beam size, Δθb	180 × 148
Central redshift, ${z}_{\mathrm{cen},\mathrm{CO}(2-1)}$	1.315
${r}_{\parallel ,\min }\leqslant {r}_{\parallel }\leqslant {r}_{\parallel ,\max }$	5.38 Mpc ${h}^{-1}\lt {r}_{\parallel }\lt 1054.8$ Mpc h−1
${r}_{\perp ,\min }\leqslant {r}_{\perp }\leqslant {r}_{\perp ,\max }$	0.0240 Mpc ${h}^{-1}\lt {r}_{\perp }\lt 1.53$ Mpc h−1
${k}_{\parallel ,\min }\leqslant {k}_{\parallel }\leqslant {k}_{\parallel ,\max }$	0.00596 h Mpc${}^{-1}\lt {k}_{\parallel }\lt 0.584$ h Mpc−1
${k}_{\perp ,\min }\leqslant {k}_{\perp }\leqslant {k}_{\perp ,\max }$	4.107 h Mpc${}^{-1}\lt {k}_{\perp }\lt 130.900$ h Mpc−1

Download table as:  ASCIITypeset image

For example, the large bandwidth and relatively narrow survey area of ASPECS dictate that the CO brightness fluctuations on physical scales larger than the survey width, ${r}_{\perp ,\max }=1.53$ Mpc h⁻¹ (i.e., for $k\lt {k}_{\perp ,\min }=4.107\,h$ Mpc⁻¹), will be probed exclusively by k_∥ modes. Thus, the power spectrum measured at k < 4.107 h Mpc⁻¹ will be an inherently one-dimensional (1D) measurement, dominated by power from the shorter wavelength, high-k_⊥ modes projected into the line of sight. These physical scales are important, however, for extracting information about large-scale clustering. Based on models (e.g., Pullen et al. 2013; Sun et al. 2018) for the total CO power spectrum at the redshift range relevant to this study, we expect any power from galaxy clustering between dark matter halos, ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{clust}}(k,z)$, to dominate the total CO power spectrum, ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{tot}}(k,z)$, up to k ≲ 1 h Mpc⁻¹ compared to contributions from small-scale clustering of galaxies that share a common host dark matter halo or shot-noise power, ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{shot}}$ (see Figure 2). If the CO surface brightness fluctuations, $\langle {T}_{\mathrm{CO}}\rangle$, trace the large-scale clustering of galaxies with some mean bias factor, $\langle {b}_{\mathrm{CO}}\rangle$, that offsets CO-emitting galaxies from the underlying dark matter distribution, i.e., if

$$\begin{eqnarray}&&{P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{clust}}(k)=\langle {T}_{\mathrm{CO}}(z){\rangle }^{2}\langle {b}_{\mathrm{CO}}(z){\rangle }^{2}{P}_{{\rm{m}},{\rm{m}}}(k,z),\end{eqnarray} \tag{ 7 }$$

where ${P}_{{\rm{m}},{\rm{m}}}(k,z)$ is the linear matter power spectrum (appropriate for k < 0.1 h Mpc⁻¹), then the low-k component of the power spectrum is, in principle, useful for constraining the aggregate CO emission within a given cosmological volume. Note that the units of ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{clust}}(k,z)$ in Equation (7) are in μK² (Mpc h⁻¹)³ for $\langle {T}_{\mathrm{CO}}(z)\rangle$ in μK, ${P}_{{\rm{m}},{\rm{m}}}(k,z)$ in (Mpc h⁻¹)³, and a dimensionless ${b}_{\mathrm{CO}}(z)$.

For physical scales smaller than ${r}_{\perp ,\max }=1.53$ Mpc h⁻¹ (i.e., for $k\geqslant {k}_{\perp ,\min }=4.107$ h Mpc⁻¹), it is clear from Table 1 that k can have contributions from both k_⊥ and k_∥ as long as $k=\sqrt{{k}_{\parallel }^{2}+{k}_{\perp }^{2}}$ is within the range k = 4.107–130.900 h Mpc⁻¹—we are measuring a full 3D power spectrum in this regime—though the k_⊥ modes, with wavelengths on the order of Δθ_b up to Δθ_S, provide most of the information on the power at these scales. At z ∼ 1, we expect any power from galaxy–galaxy clustering at k ≳ 4 h Mpc⁻¹ to be buried under a Poissonian shot-noise component, ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{shot}}(k)$, dominated by bright CO emitters in the survey volume. By restricting our analysis to k ≳ 10 h Mpc⁻¹, where the true power spectrum is expected to be flat, the power spectrum measurement is unaffected by the highly anisotropic ASPECS survey window function.

As with blindly detected individual line candidates, where the redshift of a line candidate without spectroscopically or photometrically confirmed counterparts can be ambiguous, the true redshifts of sources contributing to the intensity fluctuations contained within the ASPECS survey volume are unknown. However, in defining a real- and Fourier-space grid to perform our power spectrum calculations, we have assumed a specific target redshift ${z}_{\mathrm{cen},\mathrm{CO}(2-1)}=1.315$—corresponding to the central redshift of CO(2–1) in the ASPECS bandwidth—for the emission. While CO(2–1) is expected to dominate the mean surface brightness at 99 GHz, based on the number of blind detections in ASPECS relative to other line transitions, for example, it is not the only source of spectral line emission present in the survey volume. Figure 2 of D19 illustrates the number of different spectral lines that are, in principle, observable within the ASPECS frequency coverage, emitted from galaxies within the local universe (such as CO(1–0) at z < 0.37) and at high redshift (such as any CO transition from J > 2 at z > 2). Thus, the measured autopower spectrum of the ASPECS data should be interpreted as a sum of the power from surface brightness fluctuations from all relevant CO transitions,

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{CO},\mathrm{CO}}(k) & = & {P}_{\mathrm{CO}(1-0),\mathrm{CO}(1-0)}({k}_{\mathrm{CO}(1-0)})\\ & & +{P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}({k}_{\mathrm{CO}(2-1)})\\ & & +{P}_{\mathrm{CO}(3-2),\mathrm{CO}(3-2)}({k}_{\mathrm{CO}(3-2)})\\ & & +{P}_{\mathrm{CO}(4-3),\mathrm{CO}(4-3)}({k}_{\mathrm{CO}(4-3)})+...,\end{array}\end{eqnarray} \tag{ 8 }$$

where we truncate the sum, in practice, to include contributions from CO(1–0), CO(2–1), CO(3–2), and CO(4–3), because the ASPECS survey has only resulted in CO blind detections up to J = 4 (GL19). Each term in the right-hand side of the above equation is expressed as a function of wavenumber ${k}_{\mathrm{CO}(J-(J-1))}$, corresponding to the Fourier space defined at the emitted redshift of the respective J transition at the band center. For reference, at ν_cen = 99.572 GHz, CO(1–0), CO(3–2), and CO(4–3) can be emitted from z = 0.157, 2.470, and 3.629, respectively. The k appearing on the left-hand side of Equation (8) is intentionally ambiguous; ultimately, we would like to write ${P}_{\mathrm{CO},\mathrm{CO}}(k)$ as ${P}_{\mathrm{CO},\mathrm{CO}}({k}_{\mathrm{CO}(2-1)})$. We can convert ${k}_{\mathrm{CO}(J-(J-1))}$ to k_CO(2-1), defined at ${z}_{\mathrm{cen},\mathrm{CO}(2-1)}$, using so-called "distortion" factors (as given in, e.g., Lidz & Taylor 2016),

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{\perp }({z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))}) & = & \displaystyle \frac{{k}_{\perp ,\mathrm{CO}(2-1)}}{{k}_{\perp ,\mathrm{CO}(J-(J-1))}}\\ & & =\displaystyle \frac{{D}_{{\rm{A}},\mathrm{co}}({z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))})}{{D}_{{\rm{A}},\mathrm{co}}({z}_{\mathrm{cen},\mathrm{CO}(2-1)})}\end{array}\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{\parallel }({z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))}) & = & \displaystyle \frac{{k}_{\parallel ,\mathrm{CO}(2-1)}}{{k}_{\parallel ,\mathrm{CO}(J-(J-1))}}\\ & = & \displaystyle \frac{H({z}_{\mathrm{cen},\mathrm{CO}(2-1)})}{H({z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))})}\\ & & \times \,\displaystyle \frac{\left(1+{z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))}\right)}{\left(1+{z}_{\mathrm{cen},\mathrm{CO}(2-1)}\right)},\end{array}\end{eqnarray} \tag{ 10 }$$

that relate transverse and line-of-sight comoving distances, respectively, between the true emitted redshift of intensity fluctuations, ${z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))}$, and the adopted redshift ${z}_{\mathrm{cen},\mathrm{CO}(2-1)}$. In Equation (10), the expression H(z) refers to the Hubble parameter at redshift z. Finally, combining Equations (8)–(10), we obtain the total CO power in terms of k_CO(2-1):

$$\begin{eqnarray}&&\begin{array}{l}{P}_{\mathrm{CO},\mathrm{CO}}({k}_{\mathrm{CO}(2-1)})={P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}({k}_{\mathrm{CO}(2-1)})\\ \quad +\displaystyle \sum _{J=1,3,4}\left[\displaystyle \frac{1}{{\alpha }_{\perp }{\left({z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))}\right)}^{2}{\alpha }_{\parallel }({z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))})}\right.\\ \quad \times \ {P}_{\mathrm{CO}(J-(J-1)),\mathrm{CO}(J-(J-1))}\left(\displaystyle \frac{{k}_{\perp ,\mathrm{CO}(2-1)}}{{\alpha }_{\perp }({z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))})},\right.\\ \quad \times \left.\left.\displaystyle \frac{{k}_{\parallel ,\mathrm{CO}(2-1)}}{{\alpha }_{\parallel }({z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))})}\right)\right].\end{array}\end{eqnarray} \tag{ 11 }$$

The multiplicative pre-factor $1/({\alpha }_{\perp }^{2}{\alpha }_{\parallel })$ in the second term on the right-hand side of the above equation represents the ratio of volume probed by the survey in CO(2–1) relative to the other J transitions. The ratio $1/({\alpha }_{\perp }^{2}{\alpha }_{\parallel })\gt 1$ for J = 1, reflecting the fact that the volume probed by CO(1–0) is less than the volume probed by CO(2–1), while the opposite is true for the J = 3 and 4 transitions, where $\tfrac{1}{{\alpha }_{\perp }^{2}{\alpha }_{\parallel }}\lt 1$. Contributions to the total measured power from different CO transitions are correspondingly magnified or demagnified when projected into the CO(2–1) frame (Equation (11)).

Note that Equation (11) is only valid for large separations in redshift between the sources of CO emission; if there is overlap between the redshift ranges of the different transitions in the ASPECS survey volume, then Equation (11) will contain cross-terms that represent the cross-power spectrum between the CO transitions that overlap in redshift. For the ASPECS spectral coverage, we point out that there is a small overlap in redshift for the CO(3–2) and CO(4–3) in the survey at z = 3.011–3.107 (see Table 1 in D19), but the cross-terms here will be negligible given that the mean redshifts ${z}_{\mathrm{cen},\mathrm{CO}(3-2)}=2.470$ and ${z}_{\mathrm{cen},\mathrm{CO}(4-3)}=3.629$ are widely separated.

A search for continuum emission in the ASPECS LP Band 3 data was presented in GL19. This study identified six continuum sources, with the brightest emission on the order of ∼10 μJy, indicating that the continuum level in each channel of the 3 mm cube is negligible for our purposes. To ensure, however, that our power spectrum measurements reflect power from spectral line (CO) fluctuations only and do not contain contributions from the continuum, we perform continuum subtraction on the cube with a linear baseline fit, described in Section 3. This continuum-subtracted cube is used for all power spectrum and cross-power spectrum analyses.

A direct measurement of the mean CO surface brightness, $\langle {T}_{\mathrm{CO}}\rangle$, across observed frequencies ν_obs = 84.3–114.8 GHz provides an empirical point of comparison to model predictions in the context of CO intensity mapping experiments at moderate redshifts, and also places a constraint on foreground emission for cosmic microwave background (CMB) experiments aiming to map spectral distortions at high redshift. We repeat the analysis performed for the ASPECS-Pilot program by Carilli et al. (2016) to place a lower limit on $\langle {T}_{\mathrm{CO}}\rangle$ at ν_obs = 99 GHz, based on blindly detected sources in the ASPECS LP 3 mm survey.

Following Carilli et al. (2016), we consider the aggregate emission from all observed CO transitions that contribute to the mean sky brightness at 99 GHz. We begin by summing the line fluxes of the 16 blindly detected CO emission line candidates reported in GL19 to obtain a total CO flux of 6.91 ± 0.19 Jy km s⁻¹ or, equivalently, $\left(2.28\pm 0.06\right)\times {10}^{6}$ Jy Hz. Dividing this total flux by Δν_BW yields a total mean CO flux density $\langle {S}_{\nu }\rangle =\left(7.53\pm 0.21\right)\times {10}^{-5}$ Jy. Finally, to derive a mean surface brightness in units of μK, we apply the Rayleigh–Jeans approximation, $\langle {T}_{\mathrm{CO}}\rangle \sim 1360\langle {S}_{\nu }\rangle {\lambda }_{\mathrm{obs}}^{2}/{A}_{S}^{\mathrm{blind}}\,=0.55\pm 0.02$ μK, where λ_obs is the observed wavelength (in units of cm) and ${A}_{S}^{\mathrm{blind}}$ is the survey area (in units of arcsec²) utilized in the line search, corresponding to the region of the mosaic where primary beam attenuation is less than 20%: 1.69 × 10⁴ arcsec² at 99 GHz. Following the prescription in Moster et al. (2011), we estimate a 19.5% relative uncertainty on $\langle {T}_{\mathrm{CO}}\rangle$ due to cosmic variance in the pencil beam survey by combining the fractional uncertainties²⁵ calculated for each identified line entering into the above flux sum, given the survey depth in stellar mass, mean redshift, and survey volume probed by the respective J transition. Because Moster et al. (2011) estimated cosmic variance as the product of galaxy bias and the dark matter cosmic variance, their prescription is strictly applicable here in the case where the galaxy bias is identical to the bias b_CO of CO emission with respect to the matter density field.

For ASPECS-Pilot, which consisted of a single ∼1 arcmin² pointing with the same spectral coverage as ASPECS LP, $\langle {T}_{\mathrm{CO}}\rangle$ was found to be 0.94 ± 0.09 μK at 99 GHz (Carilli et al. 2016), which is a factor of 1.72 times greater than reported here for ASPECS LP. Since the time of publication of that analysis, however, four of the 10 line candidates reported by ASPECS-Pilot (namely, 3 mm.4, 3 mm.7, 3 mm.8, and 3 mm.9 in Table 2 of Walter et al. 2016) have been reclassified as "unconfirmed"—i.e., likely spurious, given their narrow line widths—sources based on the improved line search algorithms developed in GL19 and are excluded from the present analysis. Additionally, two of the ASPECS-Pilot line candidates (3 mm.6 and 3 mm.10) are outside the ASPECS LP survey coverage and similarly excluded. Thus, when including emission from only the four remaining confirmed sources from the original 10 sources listed in Walter et al. (2016), one finds that the total observed CO flux scales linearly with the decrease in observed survey area, resulting in a revised $\langle {T}_{\mathrm{CO}}\rangle =0.55\pm 0.05$ μK for ASPECS-Pilot, consistent with our new measurement.

It is important to note that the measurement of $\langle {T}_{\mathrm{CO}}\rangle$ presented here is considered a lower limit because the blind detections represent only a fraction of the total CO emission in the ASPECS LP survey volume; the fraction recovered by blind detections is determined by the sensitivity limit of the survey and the shape of the relevant CO LFs. We compute the mean CO surface brightness based on the observed CO(2–1), CO(3–2), and CO(4–3) LFs for ASPECS LP presented in D19 as follows:

$$\begin{eqnarray}&&\begin{array}{l}\langle {T}_{\mathrm{CO}(J-(J-1))}\rangle =\displaystyle \int d{\mathrm{log}}_{10}{L}_{\mathrm{CO}(J-(J-1))}\\ \quad \times \,{\rm{\Phi }}({L}_{\mathrm{CO}(J-(J-1))})\displaystyle \frac{{L}_{\mathrm{CO}(J-(J-1))}}{4\pi {D}_{L}^{2}}{{yD}}_{{\rm{A}},\mathrm{co}}^{2},\end{array}\end{eqnarray} \tag{ 17 }$$

where D_L, y, and D_A,co refer, respectively, to the luminosity distance, the derivative of the comoving radial distance with respect to the observed frequency (i.e., $y=d\chi /d\nu \,={\lambda }_{\mathrm{rest}}{(1+z)}^{2}/H(z)$), and the comoving angular diameter distance and are all evaluated at ${z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))}$. Here ${\rm{\Phi }}({L}_{\mathrm{CO}(J-(J-1))})$ is originally expressed as a function of the integrated source brightness temperature, ${L}_{\mathrm{CO}(J-(J-1))}^{{\prime} }$ (in units of K km s⁻¹ pc²), ${\rm{\Phi }}({L}_{\mathrm{CO}(J-(J-1))^{\prime} })$, and is given in the logarithmic Schechter form

$$\begin{eqnarray}&&\begin{array}{r}{\mathrm{log}}_{10}{\rm{\Phi }}({L}_{\mathrm{CO}(J-(J-1))}^{{\prime} })={\mathrm{log}}_{10}{{\rm{\Phi }}}_{* }+\alpha {\mathrm{log}}_{10}\left(\displaystyle \frac{{L}_{\mathrm{CO}(J-(J-1))}}{{L}_{\mathrm{CO}(J-(J-1))* }^{{\prime} }}\right)\\ \,\times \displaystyle \frac{1}{\mathrm{ln}10}\displaystyle \frac{{L}_{\mathrm{CO}(J-(J-1))}^{\prime} }{{L}_{\mathrm{CO}(J-(J-1))* }^{{\prime} }}+{\mathrm{log}}_{10}\left(\mathrm{ln}10\right).\end{array}\end{eqnarray} \tag{ 18 }$$

We convert from ${L}_{\mathrm{CO}(J-(J-1))}^{{\prime} }$ to ${L}_{\mathrm{CO}(J-(J-1))}$ (in units of solar luminosity) via

$$\begin{eqnarray}&&{L}_{\mathrm{CO}(J-(J-1))}=3\times {10}^{-11}{\nu }_{\mathrm{rest},\mathrm{CO}(J-(J-1))}^{3}{L}_{\mathrm{CO}(J-(J-1))}^{{\prime} }\end{eqnarray} \tag{ 19 }$$

from Carilli & Walter (2013). Fits to the LF data have yielded Schechter parameters α, Φ_*, and ${L}_{\mathrm{CO}(J-(J-1))* }^{{\prime} }$, with uncertainties summarized in Table 3. Note that the faint-end slope, α, has been fixed at α = −0.2 for all LFs.

Table 3.  CO LF Schechter Parameters from D19

Line	Redshift	α	${\mathrm{log}}_{10}{{\rm{\Phi }}}_{* }$	${\mathrm{log}}_{10}{L}_{* }^{{\prime} }$
			$[{\mathrm{log}}_{10}$ (Mpc−3 dex−1)]	$[{\mathrm{log}}_{10}$ (K km s−1 pc2)]
(1)	(2)	(3)	(4)	(5)
CO(2–1)	1.43	−0.2 (fixed)	$-{2.79}_{-0.09}^{+0.09}$	${10.09}_{-0.09}^{+0.10}$
CO(3–2)	2.61	−0.2 (fixed)	$-{3.83}_{-0.12}^{+0.13}$	${10.60}_{-0.15}^{+0.20}$
CO(4–3)	3.80	−0.2 (fixed)	$-{3.43}_{-0.22}^{+0.19}$	${9.98}_{-0.14}^{+0.22}$

Note. (1) Line transition. (2) Mean redshift of LF redshift bin. (3) Faint-end slope parameter in Equation (18). (4) Normalization parameter in Equation (18). (5) Characteristic luminosity parameter in Equation (18).

Download table as:  ASCIITypeset image

Integrating the LFs (Equation (17)) from an upper luminosity limit ${L}_{\mathrm{upp}}^{{\prime} }={10}^{12}$ K km s⁻¹ pc² down to the mean 7σ line sensitivity²⁶ ${L}_{\min ,7\sigma }^{{\prime} }$ in the respective redshift interval covered by each CO transition, which reflects the ASPECS LP detection threshold,²⁷ yields a mean total surface brightness $\langle {T}_{\mathrm{CO}}{\rangle }_{\mathrm{LF},7\sigma }=0.49$–1.78 μK, where the quoted range reflects the uncertainty in the LF parameters; please see Table 4 for a breakdown of the inferred $\langle {T}_{\mathrm{CO}}\rangle$ by J transition. Extending the lower limit of integration down to L'_min = 10⁸ K km s⁻¹ pc² at all redshifts implies a total mean surface brightness of $\langle {T}_{\mathrm{CO}}{\rangle }_{\mathrm{LF}}=0.72$–2.24 μK. Therefore, we estimate that our blind detections represent $\langle {T}_{\mathrm{CO}}{\rangle }_{\mathrm{LF},7\sigma }/\langle {T}_{\mathrm{CO}}{\rangle }_{\mathrm{LF}}=68.1 \%$–79.5% of the total CO surface brightness at this observed frequency.

Table 4.  Mean CO Surface Brightness Inferred from Schechter-form LFs

Line	$\langle {T}_{\mathrm{CO}(J-(J-1))}\rangle$ ((${L}_{\min }^{{\prime} }={L}_{\min ,7\sigma }^{{\prime} }$)	$\langle {T}_{\mathrm{CO}(J-(J-1))}\rangle$ ((${L}_{\min }^{{\prime} }={10}^{8}$ K km s−1 pc2)
	(μK)	(μK)
(1)	(2)	(3)
CO(2–1)	${0.53}_{-0.21}^{+0.32}$	${0.72}_{-0.25}^{+0.38}$
CO(3–2)	${0.25}_{-0.12}^{+0.31}$	${0.29}_{-0.13}^{+0.33}$
CO(4–3)	${0.12}_{-0.08}^{+0.25}$	${0.20}_{-0.11}^{+0.32}$

Notes. (1) Line transition. (2) Mean CO surface brightness calculated by integrating Equation (17) with lower and upper limits of integration ${L}_{\min }^{{\prime} }={L}_{\min ,7\sigma }^{{\prime} }$ and ${L}_{\mathrm{upp}}^{{\prime} }={10}^{12}$ K km s⁻¹. (3) Same as column (2) but for ${L}_{\min }^{{\prime} }={10}^{8}$ K km s⁻¹ pc².

Download table as:  ASCIITypeset image

As with the limit on mean CO surface brightness, the blindly detected sources in Table 2 can also be used to place a lower limit on the expected CO shot-noise power.

The total CO shot-noise power from only the detected sources, ${\left[{P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det }$, will contain contributions from galaxies emitting in the observed transitions J = 2, 3, and 4,

$$\begin{eqnarray}\begin{array}{rcl}{\left[{P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det } & = & {\left[{P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det }\\ & & +{\left[{P}_{\mathrm{CO}(3-2),\mathrm{CO}(3-2)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det }\\ & & +{\left[{P}_{\mathrm{CO}(4-3),\mathrm{CO}(4-3)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det },\end{array}\end{eqnarray} \tag{ 20 }$$

where ${\left[{P}_{\mathrm{CO}(3-2),\mathrm{CO}(3-2)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det }$ and ${\left[{P}_{\mathrm{CO}(4-3),\mathrm{CO}(4-3)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det }$ have been converted to the CO(2–1) frame using Equation (11). Each term on the right-hand side of Equation (20) can be determined analytically by summing the N individual line fluxes per the expression

$$\begin{eqnarray}&&\displaystyle \sum _{i=1}^{N}\displaystyle \frac{1}{{V}_{S}}{\left(\displaystyle \frac{{L}_{\mathrm{CO}(J-(J-1))}}{4\pi {D}_{L}^{2}}{{yD}}_{{\rm{A}},\mathrm{co}}^{2}\right)}^{2},\end{eqnarray} \tag{ 21 }$$

where V_S refers to the survey volume at ${z}_{\mathrm{cen},\mathrm{CO}(J-(J-1))}$. Note that the above expression for shot-noise power has units of surface brightness squared times volume (μK² (Mpc h⁻¹)³) and is equal to the same value at all k, appropriate for a Poisson sampling of galaxies.

Starting with the CO(2–1), CO(3–2), and CO(4–3) source fluxes from GL19, reported in Table 2, we find, for the entire ASPECS LP 3 mm survey volume used in the blind search, lower limits on the expected shot-noise power of ${\left[{P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det }\,=\,63.64$, ${\left[{P}_{\mathrm{CO}(3-2),\mathrm{CO}(3-2)}^{\mathrm{shot}}({k}_{\mathrm{CO}(3-2)})\right]}_{\det }=\,98.49$, and ${\left[{P}_{\mathrm{CO}(4-3),\mathrm{CO}(4-3)}^{\mathrm{shot}}({k}_{\mathrm{CO}(4-3)})\right]}_{\det }=1.05$ μK² (Mpc h⁻¹)³, respectively. For the cropped region (black dotted square in Figure 1) corresponding to the volume used in the power spectrum analysis, we find that the CO(2–1), CO(3–2), and CO(4–3) line emitters each give rise to a respective shot-noise power of 73.99, 71.06, and 1.21 μK² (Mpc h⁻¹)³. The slightly higher shot-noise power predicted in the cropped region for CO(2–1) and CO(4–3) is due to the decrease in volume after the crop; the CO(3–2) shot-noise power decreases due to the fact that two of the five detected sources are located outside of the boundary of the cropped region. After converting the CO(3–2) and CO(4–3) shot-noise power into the CO(2–1) frame, we sum each contribution to arrive at a total shot-noise power at z_cen = 1.315 arising from the blind detections: ${\left[{P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})\right]}_{\det }=118.45$ and 113.24 μK² (Mpc h⁻¹)³ for the full survey and cropped region, respectively.

Finally, we estimate the expected shot-noise power based on the D19 CO LFs,

$$\begin{eqnarray}&&\begin{array}{l}{P}_{\mathrm{CO}(J-(J-1)),\mathrm{CO}(J-(J-1))}^{\mathrm{shot}}=\displaystyle \int d{\mathrm{log}}_{10}{L}_{\mathrm{CO}(J-(J-1))}\\ \quad \times \,{\rm{\Phi }}({L}_{\mathrm{CO}(J-(J-1))}){\left[\displaystyle \frac{{L}_{\mathrm{CO}(J-(J-1))}}{4\pi {D}_{L}^{2}}{{yD}}_{{\rm{A}},\mathrm{co}}^{2}\right]}^{2},\end{array}\end{eqnarray} \tag{ 22 }$$

and find that the detected sources (i.e., integrating Equation (22) down to the relevant 7σ line sensitivity limit) recover 95.2%–97.7% of ${P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})$, 98.6%–99.7% of ${P}_{\mathrm{CO}(3-2),\mathrm{CO}(3-2)}^{\mathrm{shot}}({k}_{\mathrm{CO}(3-2)})$, and 84.9%–96.0% of ${P}_{\mathrm{CO}(4-3),\mathrm{CO}(4-3)}^{\mathrm{shot}}({k}_{\mathrm{CO}(4-3)})$. In total, the recovered fraction is 96.0%–98.6% of ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})$.

Integrating the LFs measured by ASPECS LP (with lower and upper limits of integration equal to 10⁸ and 10¹² K km s⁻¹ pc²) per Equation (22) yields the following estimates of shot-noise power for CO(2–1), CO(3–2), and CO(4–3), respectively: ${80}_{-38}^{+71}$, ${130}_{-82}^{+320}$, and ${27}_{-19}^{+90}$ μK² (Mpc h⁻¹)³. In order to compare to our measurement of the total CO shot-noise power, we convert the individual shot-noise powers estimated for each transition into the CO(2–1) frame and sum, finding ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})={170}_{-92}^{+290}$ μK² (Mpc h⁻¹)³. The 3σ upper limit on ${P}_{\mathrm{CO},\mathrm{CO}}({k}_{\mathrm{CO}(2-1)})\leqslant 190$ μK² (Mpc h⁻¹)³ determined via the power spectrum places a more stringent constraint on the total CO shot-noise power; it rules out a significant fraction of the allowable range, ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)})=78$ to 460 μK² (Mpc h⁻¹)³, obtained from the LF fits. The low sensitivity, however, on the power spectrum measurement precludes us from determining the amplitude of individual contributions from the different CO transitions to the aggregate value, so translating our limit on ${P}_{\mathrm{CO},\mathrm{CO}}({k}_{\mathrm{CO}(2-1)})$ to constraints on individual LFs is highly speculative—as would be attempting to constrain the Schechter parameters of the individual CO LFs. We note, however, that the fixed faint-end slope α = −0.2 in D19 implies that the shot-noise power spectrum is relatively insensitive to the low-luminosity systems in the CO LF. In this case, the shot-noise power is more sensitive to the Schechter parameters Φ_* and L_*, and our measurement suggests either lower normalizations Φ_* or a knee in the Schechter function that occurs at lower luminosities ${L}_{* }^{{\prime} }$. This may be particularly applicable to the observed CO(3–2) and CO(4–3) LFs at z > 2, where the empirical constraints span a limited range in luminosity compared to the CO(2–1) LF at z ∼ 1, and the resulting uncertainties on the Schechter parameters are large.

In this section, we compare our measured ${P}_{\mathrm{CO},\mathrm{CO}}({k}_{\mathrm{CO}(2-1)})$ to independent observational constraints on CO(1–0) shot-noise power at higher redshift.

First, we consider the CO(1–0) LFs determined at z = 2–3 by the VLA COLDz program (Riechers et al. 2019), which targeted survey volumes in COSMOS (∼9 arcmin²) and GOODS-N (∼51 arcmin²) across 8 GHz of bandwidth in the Ka band (ν_obs ≈ 30–38 GHz) with typical synthesized beam sizes ∼3''. We compute the probability distribution of ${P}_{\mathrm{CO}(1-0),\mathrm{CO}(1-0)}({k}_{\mathrm{CO}(1-0)},{z}_{\mathrm{cen}}=2.4).$ using Equation (22) and the Schechter function parameter samples from the posterior distributions obtained with the approximate Bayesian computation method for the merged COSMOS and GOODS-N data set (see Figure 6 in Riechers et al. 2019). We find that the distribution has a median ${P}_{\mathrm{CO}(1-0),\mathrm{CO}(1-0)}({k}_{\mathrm{CO}(1-0)},{z}_{\mathrm{cen}}=2.4)=276.58$ μK² (Mpc h⁻¹)³, with a probable range of ${P}_{\mathrm{CO}(1-0),\mathrm{CO}(1-0)}({k}_{\mathrm{CO}(1-0)},{z}_{\mathrm{cen}}=2.4)=74.75$ μK² (Mpc h⁻¹)³ (5th percentile) to 1119.35 μK² (Mpc h⁻¹)³ (95th percentile).

The COLDz constraints on the CO(1–0) shot-noise power spectrum at ${z}_{\mathrm{cen},\mathrm{CO}(1-0)}=2.4$ can be converted to a constraint on the CO(2–1) luminosity density at z ∼ 1 (and, thus, the CO(2–1) shot-noise power spectrum at the same redshift) assuming that (1) the CO(2–1) line is thermalized and has the same brightness temperature as CO(1–0) at z_cen = 2.4 and (2) there is no evolution in the CO(2–1) luminosity density from ${z}_{\mathrm{cen},\mathrm{CO}(1-0)}=2.4$ to ${z}_{\mathrm{cen},\mathrm{CO}(2-1)}=1.3$. The first assumption is reasonable for the low J transitions relevant here and is supported by observations of a variety of high-z systems (see, e.g., Table 2 in Carilli & Walter 2013, which shows ${L}_{\mathrm{CO}(2-1)}^{{\prime} }/{L}_{\mathrm{CO}(1-0)}^{{\prime} }\sim 0.9$ for submillimeter galaxies, color-selected star-forming galaxies, etc.). The latter assumption is likely false in detail, but, given that the cosmic SFR density is relatively flat between z = 1 and 3, it is a reasonable first approximation; note also that the cosmic molecular gas densities at z ∼ 1 and 3 are indistinguishable by current empirical standards. In any case, assumption (2) results in an underestimation of ${P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}$ if the CO(2–1) luminosity density at z = 2.4 is, in reality, lower than the luminosity density at z = 1, and vice versa. So, assuming that items (1) and (2) are valid, we have ${P}_{\mathrm{CO}(1-0),\mathrm{CO}(1-0)}({k}_{\mathrm{CO}(1-0)},{z}_{\mathrm{cen}}\,=2.4)={P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}({k}_{\mathrm{CO}(2-1)},{z}_{\mathrm{cen}}=2.4)$, which implies a CO(2–1) luminosity density ${\rho }_{\mathrm{CO}(2-1)}({z}_{\mathrm{cen}}=2.4)\,={\rho }_{\mathrm{CO}(2-1)}({z}_{\mathrm{cen}}=1.3)=5.63\times {10}^{10}$ ${L}_{\odot }$ Mpc⁻³, yielding ${P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)},{z}_{\mathrm{cen}}=1.3)=167.03$ μK² (Mpc h⁻¹)³; the 5th and 95th percentiles similarly give lower and upper bounds on ${P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)},{z}_{\mathrm{cen}}\,=1.3)=45.14$ and 675.98 μK² (Mpc h⁻¹)³, respectively. This range is consistent with both our upper limit on the measured ${P}_{\mathrm{CO},\mathrm{CO}}({k}_{\mathrm{CO}(2-1)},{z}_{\mathrm{cen}}=1.3)$ and the ${P}_{\mathrm{CO}(2-1),\mathrm{CO}(2-1)}^{\mathrm{shot}}({k}_{\mathrm{CO}(2-1)},{z}_{\mathrm{cen}}=1.3)$ estimated from the CO(2–1) LF fit derived from ASPECS LP data (see Section 4.2.1).²⁸ A direct comparison of the CO(1–0) LF inferred²⁹ by ASPECS at z_cen = 2.6 and that measured by COLDz at nearby redshifts also reveals an excellent agreement across the overlapping CO(1–0) luminosity range of ${L}_{\mathrm{CO}(1-0)}^{{\prime} }=(0.1$–4) × 10¹¹ K km s⁻¹ pc², which suggests that our assumptions (1) and (2) are reasonable. Importantly, the apparent agreement at the bright end of the CO(1–0) LFs also suggests that the impact of cosmic variance on the shot-noise power spectrum presented in this study is modest, since the shot-noise measurement is inherently more sensitive to high-luminosity systems (because of the ∝L² dependence in Equation (22)).

We also compare the measured CO power spectrum of ASPECS LP data to the power spectrum measurement of CO(1–0) at wavenumbers k_CO(1-0) ∼ 1–10 h Mpc⁻¹ at z ∼ 2−3 from COPSS II (Keating et al. 2016). COPSS II was a dedicated intensity mapping experiment carried out with the Sunyaev–Zel'dovich Array to observe noncontiguous fields (totaling a survey area of ∼0.7 deg²) on the sky at ∼2' spatial resolution. Keating et al. (2016) presented a marginal 2σ detection of ${P}_{\mathrm{CO}(1-0),\mathrm{CO}(1-0)}^{\mathrm{shot}}({k}_{\mathrm{CO}(1-0)})=3000\pm 1300$ μK² (Mpc h⁻¹)³ at z_cen = 2.8, which is at least a factor of 6 larger than the COLDz measurement at similar redshift. Following the same procedure as outlined above, we estimate the CO(2–1) shot noise at z_cen = 1.3 based on the COPSS II measurement, finding 1600 ± 700 μK² (Mpc h⁻¹)³, which is 4.8–12 times higher than our current 3σ upper limit, though the ASPECS measurement probes CO power at higher wavenumbers (>10 h Mpc⁻¹) compared to COPSS II.

We obtain ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{gal}}({k}_{\mathrm{CO}(2-1)})$ by adopting the same methods to measure the noise-bias-free autopower spectrum, ${P}_{\mathrm{CO},\mathrm{CO}}({k}_{\mathrm{CO}(2-1)})$, with one key difference: masking was performed on τ₁, τ₂, τ₃, and τ₄ to remove flux densities from voxels beyond a spatial radius of ∼1'' and spectral width of 0.165 GHz (or 1 channel) from the center of known positions of MUSE galaxies listed in Inami et al. (2017), prior to estimating the power spectrum. The ∼1'' radius was chosen so that any enclosed source flux would be encompassed by a full beamwidth; the true source flux is not recovered in the case of extended emission (see Table 2 for sources identified as extended and corresponding fluxes extracted per GL19). Explicitly, we are evaluating a statistic defined as

$$\begin{eqnarray}&&\langle {T}_{\mathrm{CO}}^{\mathrm{gal}* }({\boldsymbol{k}}){T}_{\mathrm{CO}}^{\mathrm{gal}}({\boldsymbol{k}}^{\prime} )\rangle \equiv {\left(2\pi \right)}^{3}{\delta }_{{\rm{D}}}({\boldsymbol{k}}-{\boldsymbol{k}}^{\prime} ){P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{gal}}(k).\end{eqnarray} \tag{ 23 }$$

In the above equation, ${T}_{\mathrm{CO}}^{\mathrm{gal}}$ represents the masked ASPECS data cube, which has been weighted at every ith voxel according to

$$\begin{eqnarray}&&{({T}_{\mathrm{CO}}^{\mathrm{gal}})}_{i}={w}_{i}{({T}_{\mathrm{CO}})}_{i},\end{eqnarray} \tag{ 24 }$$

where w_i = 1 for voxels containing a MUSE galaxy, and w_i = 0 otherwise. Thus, in the same way that the noise-bias-free autopower spectrum measurement at k = 10–100 h Mpc⁻¹ yielded constraints on the second moments of the CO LFs (per Equation (22)), the masked noise-bias-free autopower spectrum measurement at the same k range constrains the second moments of the LFs of CO-emitting MUSE galaxies.

Errors on the masked noise-bias-free autopower spectrum, $\delta {P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{gal}}({k}_{\mathrm{CO}(2-1)})$, were obtained from Equation (16) and include an additional contribution estimated from 100 simulations of random MUSE source positions. This Poisson term was deemed necessary to prevent underestimating the error based only on Equation (16); after removing significant source flux from the cubes in the masking step, the pre-factor γ = 0.25 that appears in this equation might no longer accurately describe the relation of noise properties in τ₁, τ₂, τ₃, and τ₄ to those in T_I and T_II.

Figures 5 and 6 show the measured masked noise-bias-free CO autopower spectrum, ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{MUSE}}({k}_{\mathrm{CO}(2-1)})$, between CO fluctuations in the ASPECS LP data cube and 3D positions from the MUSE spec-z catalog. In these figures, we compare the power measured when including (1) the MUSE positions of the CO blind detections (solid red curves), (2) all available MUSE positions (solid black curves), and (3) all available MUSE positions, excluding those corresponding to the positions of the CO blind detections (dashed red curves). To facilitate comparison of the relative contributions of each sample described by items (1)–(3), we have scaled the y-axis in each plot by a factor $1/\langle {P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{MUSE}}({k}_{\mathrm{CO}(2-1)}){\rangle }^{\mathrm{all}}$, where the denominator represents the total noise-bias-free cross-power using all MUSE positions and including potential CO emission from J = 1–4 transitions. In the case where item (3) yields ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{MUSE}}=0$, we can be confident that there is no measured "excess" power from MUSE galaxies with previously undetected CO emission. Note that, although our power spectrum measurements cannot probe fluctuations on scales of ${k}_{\mathrm{CO}(2-1)}\gtrsim 100$ h Mpc⁻¹, which correspond to the sub-beam size pixel gridding, we include these wavenumbers in the plot to show the effect of the beam size on the measured power, which follows an exponential drop-off as expected; the sensitivity on the autopower spectrum measurement was insufficient to "detect" the beam roll-off, so we truncated the spectrum in Figure 4 at k_CO(2-1) ≈ 100 h Mpc⁻¹.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We find that the noise-bias-free cross-power spectrum is detected at high significance with an amplitude suggesting that the majority of CO emission contained within the survey volume is accounted for by the ASPECS blind detections and that the rest-frame optical/UV galaxies in the same field closely trace the observed CO emission (Figure 5). The latter result is unsurprising, given that 12/16 of CO blind detections had known MUSE counterparts. Only a small level of excess power—$\langle {P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{MUSE}}({k}_{\mathrm{CO}(2-1)}{\rangle }^{\mathrm{blind}\mathrm{removed}}\,/\langle {P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{MUSE}}({k}_{\mathrm{CO}(2-1)}{\rangle }^{\mathrm{all}}\,=7.05\pm 3.2 \%$, where we have averaged over wavenumber bins from k_CO(2-1) = 9.55 to 54.98 h Mpc⁻¹ to avoid the effects of the exponential beam roll-off³³ seen at higher k_CO(2-1)—is observed to originate from the positions of MUSE galaxies in the same field that do not have previously detected ASPECS counterparts, amounting to 3.8%–10.7% of the total masked CO autopower measured when including all MUSE positions with potential CO $J=1,2,3,$ or 4 emission. In other words, the percentage of power recovered by the MUSE sources with previously detected ASPECS counterparts is 89%–96%.

The high S/N on ${P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{MUSE}}({k}_{\mathrm{CO}(2-1)})$ enables further decomposition of the total masked CO autopower spectrum into the individual J transitions contributing to the aggregate signal (Figure 6). Here we identify emission from the CO(3–2) transition as the principal source of residual power after removing the ASPECS blind detections in the total masked CO autopower spectrum in Figure 6, while the power derived from all other J transitions is zero. The level of residual power detected in the masked CO(3–2) power spectrum,³⁴ ${P}_{\mathrm{CO}(3-2),\mathrm{CO}(3-2)}^{\mathrm{MUSE}}({k}_{\mathrm{CO}(2-1)})=1.8\pm 0.56$ μK² (Mpc h⁻¹)³, represents 5.2% ± 1.6% of the total masked CO autopower $\langle {P}_{\mathrm{CO},\mathrm{CO}}^{\mathrm{MUSE}}({k}_{\mathrm{CO}(2-1)}{\rangle }^{\mathrm{all}}$ (including all J transitions), or 14%–22% of the total masked CO(3–2) power spectrum amplitude.

Next we consider the nature of the sources contributing to the excess in CO(3–2) emission relative to the expected power from the CO(3–2) blind detections only. The level of observed power could be attributed to a single bright source, e.g., with a CO(3–2) flux of the order of ∼0.10 Jy km s⁻¹, or multiple fainter sources. Since a CO(3–2) source flux of ∼0.10 Jy km s⁻¹ at 2 ≤ z ≤ 3 implies a line luminosity of 4.2 × 10⁶ L_⊙ that is just below the mean sensitivity limit for the survey for that redshift range, it is possible that a single bright source with this flux would have been previously undetected in the ASPECS line search. The scenario where the excess power originates from CO(3–2) emitters below the individual detection threshold is also plausible, however, given that the MUSE catalog contains galaxies with lower stellar mass M_* and SFRs than probed by the CO blind detections. Specifically, at 2 ≤ z ≤ 3, MUSE sources probe down to M_* ∼ 10⁹ M_⊙ and SFR ∼ 0.3 M_⊙ yr⁻¹, while the ASPECS-detected galaxies have M_* ≥ 10¹⁰ M_⊙ and SFR ≥ 10 M_⊙ yr⁻¹ (B19). We can test these scenarios by masking the ASPECS data cube down to progressively lower flux thresholds and determine the flux level where the excess power vanishes. Figure 7 shows the results of this analysis, where voxels with flux densities with $| {F}_{\nu }| \gt 1.0,$ 0.75, 0.50, 0.25, and 0.1 mJy have been blanked so that the remaining emission in the data cubes is due to sources (or noise) fainter than the masking threshold; note that we consider the absolute value of the flux densities because we are working with dirty image cubes that contain F_ν < 0. From Figure 7, it is clear that roughly 75% of the excess power originates from voxels with flux densities greater than 0.25–0.50 mJy, which implies fluxes ${F}_{\nu }{\rm{\Delta }}{\nu }_{40\mathrm{chn}}\gt 0.12$–0.24 Jy km s⁻¹, suggesting that a single relatively bright emitter is responsible for the majority of the observed power from galaxies with previously undetected CO sources. We note that this flux level is fainter than any of the previous blind CO(3–2) detections in Table 2 and comparable to a 0.17 Jy km s⁻¹ ASPECS CO(3–2) detection identified with a MUSE spec-z prior at z = 2.028; this source cannot be responsible for the excess power discussed here, however, as it lies outside the region of sky used in the power spectrum analysis. Nonzero power due to voxels with F_ν < 0.25 mJy implies that very faint sources with fluxes less than 0.10 Jy km s⁻¹ may exist in the data cube and contribute to the observed excess in the CO-galaxy power spectrum, but their overall contribution is small (i.e., <25%).

Zoom In Zoom Out Reset image size

Since the masking analysis suggests that the observed excess CO(3–2) power could be due to a single source, we can try to identify this source by masking (i.e., setting to zero) one by one each MUSE source position; note that only one MUSE position is masked at a time. If the excess power is due to a single source, then the power will remain unchanged (within the measured uncertainties) until the MUSE position corresponding to the CO(3–2) emission is masked and the power goes to zero. Following this procedure, we observe the power drop to zero (magenta curve in Figure 6) upon masking source MUSE ID 24. Masking of all other MUSE sources resulted in negligible changes to the power spectrum. Examination of the data cube T₀ reveals no significant flux at the source position corresponding to MUSE ID 24 (R.A. = 53160088, decl. = −27776356, ν_obs = 97.57 GHz). However, this location is within a beam's width of a known blind detected source, ASPECS LP-3mm.01 or MUSE ID 35 (at R.A. = 53160587, decl. = −27.776120, ν_obs = 97.58), so the ∼1'' radius used to extract emission from MUSE positions when computing the masked autopower spectrum encompasses flux from the blindly detected source. Thus, we conclude that the observed excess in CO(3–2) power is not due to any MUSE sources with previously undetected CO(3–2) emission.

We thus recompute the masked autopower spectrum using all MUSE positions with potential CO J = 1, 2, 3, or 4 emission, now excluding MUSE ID 24, in order to revise the estimate of the fraction of total masked autopower recovered by MUSE positions corresponding to ASPECS blind detections. We find that the MUSE positions corresponding to ASPECS blind detections recover 106% ± 6.5% of the total CO shot-noise power.

Additionally, we measure the cross-power spectrum between the ASPECS data and MUSE position field and refer to this quantity as the cross-shot-noise power spectrum, ${P}_{\mathrm{CO},\mathrm{gal}}({k}_{\mathrm{CO}(2-1)})$. In this case, we work directly with the ASPECS data cube T₀, which has the lowest rms compared to the subsets T_I, T_II, etc. Since the noise in T₀ has not been quantified, the error on ${P}_{\mathrm{CO},\mathrm{gal}}({k}_{\mathrm{CO}(2-1)})$ is derived via simulation of random MUSE positions only and is a lower estimate of the true error.

We normalize the grid of MUSE galaxy positions to have units corresponding to the dimensionless density fluctuation field ${\delta }_{\mathrm{gal}}({{\boldsymbol{x}}}_{i})=\left({n}_{\mathrm{gal}}({{\boldsymbol{x}}}_{i})-\langle {n}_{\mathrm{gal}}\rangle \right)/\langle {n}_{\mathrm{gal}}\rangle$, where ${n}_{\mathrm{gal}}({{\boldsymbol{x}}}_{i})$ refers to the number density of galaxies at position ${{\boldsymbol{x}}}_{i}$ in the cube, and $\langle {n}_{\mathrm{gal}}\rangle$ is the mean number density of galaxies in the full volume. The cross-shot-noise power spectrum between T₀ and the dimensionless density fluctuation cube, G, is then

$$\begin{eqnarray}&&\langle {T}_{0}^{* }({\boldsymbol{k}})G({\boldsymbol{k}}^{\prime} )\rangle \equiv {\left(2\pi \right)}^{3}{\delta }_{{\rm{D}}}({\boldsymbol{k}}-{\boldsymbol{k}}^{\prime} ){P}_{\mathrm{CO},\mathrm{gal}}(k),\end{eqnarray} \tag{ 25 }$$

with units of μK (Mpc h⁻¹)³.

As derived in, e.g., Wolz et al. (2017) and Breysse & Alexandroff (2019), ${P}_{\mathrm{CO},\mathrm{gal}}(k)$ is proportional to the mean CO surface brightness of CO-emitting MUSE galaxies, $\langle {T}_{\mathrm{CO},\mathrm{gal}}\rangle$,

$$\begin{eqnarray}&&{P}_{\mathrm{CO},\mathrm{gal}}(k)=\displaystyle \frac{\langle {T}_{\mathrm{CO},\mathrm{gal}}\rangle }{\langle {n}_{\mathrm{gal}}\rangle }.\end{eqnarray} \tag{ 26 }$$

Since the factor $1/\langle {n}_{\mathrm{gal}}\rangle$ is equal to the amplitude of the shot-noise term of the galaxy autopower spectrum, ${P}_{\mathrm{gal},\mathrm{gal}}^{\mathrm{shot}}$ (units of (Mpc h⁻¹)³), we rearrange Equation (26) to write

$$\begin{eqnarray}&&\langle {T}_{\mathrm{CO},\mathrm{gal}}\rangle =\displaystyle \frac{{P}_{\mathrm{CO},\mathrm{gal}}(k)}{{P}_{\mathrm{gal},\mathrm{gal}}^{\mathrm{shot}}},\end{eqnarray} \tag{ 27 }$$

which has units of μK.

Constraints on $\langle {T}_{\mathrm{CO},\mathrm{gal}}\rangle$ are obtained from the cross-shot-noise power spectrum, ${P}_{\mathrm{CO},\mathrm{gal}}({k}_{\mathrm{CO}(2-1)})$, according to Equation (27). As illustrated in Figure 8, we strongly detect the cross-shot-noise power spectrum in both considered J = 3 and J = 2 transitions, finding no significant contribution to the observed mean CO surface brightness from MUSE sources with previously undetected CO(2–1) or CO(3–2) emission. It is interesting to note, however, that the average value of $\langle {T}_{\mathrm{CO}(2-1),\mathrm{gal}}\rangle$ is slightly nonzero after removing the blindly detected sources from the sample (red dashed curve, top panel). After we discard the 18 MUSE positions with potential CO(2–1) emission that have poorly characterized MUSE spectra (or CONFID = 1, i.e., a nonsecure redshift based on a singly unidentified line), then there is marginal (2.5σ) detection of excess CO(2–1) surface brightness from MUSE emitters without previous ASPECS CO detections (orange dashed curve) that can contribute 0.07 ± 0.02 μK, or 19% ± 7.8% of the total observed CO(2–1) emission.³⁵ Not shown in Figure 8, we measure nondetections of CO(1–0) and CO(4–3) surface brightnesses (−0.0020 ± 0.0082 and −0.022 ± 0.028 μK, respectively) in MUSE galaxies.

Zoom In Zoom Out Reset image size

Implications for CO LFs. If there is a one-to-one correlation between CO-emitting galaxies and MUSE-selected galaxies, then $\langle {T}_{\mathrm{CO},\mathrm{gal}}\rangle =\langle {T}_{\mathrm{CO}}\rangle$ (and ${P}_{\mathrm{CO},\mathrm{gal}}^{\mathrm{NBF}}({k}_{\mathrm{CO}(2-1)})={P}_{\mathrm{CO},\mathrm{CO}}({k}_{\mathrm{CO}(2-1)})$), and one can use the above measurements on $\langle {T}_{\mathrm{CO},\mathrm{gal}}\rangle$ (and ${P}_{\mathrm{CO},\mathrm{gal}}^{\mathrm{NBF}}({k}_{\mathrm{CO}(2-1)})$) to constrain the CO LF via Equations (17) and (22). For example, if the observed ∼20% excess surface brightness in CO(2–1) from galaxies without ASPECS detections is real, and the MUSE galaxies represent the complete population of total CO(2–1) emitters, then one can deduce that the ASPECS survey recovers ≳80% of the CO(2–1) surface brightness at its sensitivity threshold. Keeping other Schechter parameters in Table 3 fixed, this suggests a relatively flat faint-end slope α ≤ −0.1 for the CO(2–1) LF at z ∼ 1.³⁶ Of course, if there are CO(2–1) emitters that do not have MUSE spec-z, then $\langle {T}_{\mathrm{CO},\mathrm{gal}}\rangle \ne \langle {T}_{\mathrm{CO}}\rangle$, and we cannot reliably infer constraints on the CO(2–1) LF. Given the high percentage (100%) of ASPECS CO(2–1) detections with MUSE spec-z counterparts, it is possible that $\langle {T}_{\mathrm{CO}(2-1),\mathrm{gal}}\rangle$ is not dramatically different from $\langle {T}_{\mathrm{CO}(2-1)}\rangle$. For CO(3–2), however, the percentage is much lower (40%), so we do not attempt to draw conclusions about the CO(3–2) LF based on $\langle {T}_{\mathrm{CO}(3-2),\mathrm{gal}}\rangle$.