Forecasting [C ii] Line-intensity Mapping Measurements between the End of Reionization and the Epoch of Galaxy Assembly

We consider three experiments designed to probe [C ii] at high redshift:

• 1.  
  The Epoch of Reionization Spectrometer (EoR-Spec) on CCAT-prime (or CCAT-p) is designed for [C ii] line-intensity mapping, covering observing frequencies of ν_obs = 210–420 GHz in two bands, altogether covering z = 3.5–8.1. For this paper, we assume a Phase I instrument with a single dichroic TES bolometer array over 1004 spatial positions occupying one-third of the image plane of one instrument module (of up to seven possible), and a modest resolving power of R = 100 or a frequency resolution of δ_ν ≈ ν_obs/100 throughout the observing bands. The nominal survey program covers 2 deg² over 4000 hr.⁶
• 2.  
  CONCERTO (the CarbON C ii line in post-rEionization and ReionizaTiOn epoch project) is expected to deploy two arrays of spectrometer pixels with channels of δ_ν = 1.5 GHz, with one array observing in 125–300 GHz and the other in 200–360 GHz (or redshift ranges of 5.3–14 and 4.3–8.5). The nominal program is a survey of 1.4 deg² over 1200 hr.
• 3.  
  TIME (the Tomographic Ionized-carbon Mapping Experiment) is a R ∼ 150 grating spectrometer planning to survey a one-beam-wide 78' × 05 (or 1.3 × 0.0083 deg²) slice of sky over 1000 hr (Crites et al. 2014; see also the TIME subsection of Kovetz et al. 2017, whose parameters in part supercede that of Crites et al. 2014). The spectrometer operates over two bands spanning 183–230 and 230–326 GHz (or redshift ranges spanning 7.3–9.4 and 4.8–7.3). We simplify the resolving power into an optimistic, constant figure of δ_ν = 1.5 GHz for the lower-frequency band and 1.9 GHz for the higher-frequency band.

These experiments represent a wide array of state-of-the-art but proven technologies. EoR-Spec on CCAT-p (which we will often refer to simply as CCAT-p in this work) is an evolution of previous spectrometers using Fabry–Perot interferometers (FPI) like SPIFI (Bradford et al. 2002; Oberst et al. 2011). CONCERTO will use arrays of kinetic inductance detectors evolved from NIKA (Adam et al. 2014) and NIKA2 (Adam et al. 2018). TIME inherits the novel waveguide grating spectrometer architecture of Z-Spec (Bradford et al. 2004). Each of these technology sets represents a different approach to enabling a compact, broadband, background-limited, low- to medium-resolution (R ≳ 100) direct-detection spectrometer.

The experiments also represent a range of survey strategies, ranging from TIME's deep line-scan strategy spanning a volume only one beam wide, to CCAT-p's wide-field survey enabled by a degree-scale field of view. Far from being redundant, these three experiments will complement each other in their scope and analysis techniques. However, we are interested in the fundamental ability of each one to achieve a detection of the [C ii] power spectrum.

The N-body (dark matter only) cosmological simulation at the base of the UniverseMachine EDR is the Bolshoi–Planck simulation (Klypin et al. 2016; Rodríguez-Puebla et al. 2016), a periodic box 250h⁻¹ Mpc on each side with 2048³ particles of mass 1.6 × 10⁸h⁻¹ M_⊙. The mass resolution is good enough to resolve a complete sample of halos down to ∼10¹⁰ M_⊙; the halo catalogs are incomplete below this mass, which has implications for signal amplitude that we discuss at the end of this section.

We generate sets of 42 lightcones at four different redshift ranges, populated with Bolshoi–Planck dark matter halos with star formation rates derived from UniverseMachine, and fully covering the extents outlined in Table 1. (We generate 58 extra lightcones at the highest redshift of z ∼ 7.4 for a total of 100 lightcones.) CCAT-p has the widest expected spectral coverage of all the instruments we consider, so its coverage influences our choice of redshifts; the CONCERTO and TIME spectral coverages reach the three and two lowest-frequency simulated bands. We use the 250h⁻¹ Mpc size of Bolshoi–Planck as an approximate limit for the angular sizes of our lightcones, which are indicated in Table 1. In calculating sensitivities for each experiment, on the other hand, we assume the full expected survey area (but still only the simulated range of frequencies indicated in Table 1).

Table 1.  Simulation and Experimental Parameters Used for Signal and Sensitivity Forecasts in This Work

z	Frequencies	Lightcone	σpix			Npix			${\sigma }_{\mathrm{pix}}{t}_{\mathrm{pix}}^{-1/2}$ per beam
		Size	CCAT-p	CONCERTO	TIME	CCAT-pa	CONCERTO	TIME	CCAT-p	CONCERTO	TIME
	(GHz)	(arcsec)	(MJy sr−1 s1/2)						(Jy sr−1)
3.7	428–388	180	2.8	⋯	⋯	1004	⋯	⋯	2.2 × 104	⋯	⋯
4.5	365–325	165	1.7	18.	⋯	1004	1500	⋯	1.2 × 104	4.7 × 104	⋯
6.0	290–250	150	0.86	11.	11.	1004	3000	32	6.2 × 103	1.8 × 104	1.6 × 104
7.4	240–212	135	0.70	7.5	5.2	1004	3000	32	3.9 × 103	8.0 × 103	5.7 × 103

Notes. Lightcone sizes are not field sizes (2 deg², 1.4 deg², and 78' × 05 for CCAT-p, CONCERTO, and TIME, respectively), and the latter are used for sensitivity calculations. CCAT-p ${\sigma }_{\mathrm{pix}}{t}_{\mathrm{pix}}^{-1/2}$ values assume 45° elevation and first-quartile weather with 0.28 mm precipitable water vapor over a 4000 hr survey. For CONCERTO and TIME, we use Equation (4) to calculate t_pix from N_pix as well as survey times (1200 hr for CONCERTO and 1000 hr for TIME) and field sizes (mentioned previously), setting ${{\rm{\Omega }}}_{\mathrm{pix}}={{\rm{\Omega }}}_{\mathrm{beam}}$.

^aCCAT-prime instrumental details mean that the number of spatial pixels alone does not inform t_pix as in the other surveys—see main text (Section 2.3).

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We assign each halo in the lightcones a luminosity based solely on its star formation rate⁷ and cosmological redshift (we ignore peculiar velocities and redshift-space distortions throughout). We impose a minimum emitter halo mass of 10¹⁰ M_⊙, and for halos above this mass, we use a power-law SFR–[C ii] luminosity relation:

$$\begin{eqnarray}&&\mathrm{log}\left(\displaystyle \frac{{L}_{\mathrm{line}}}{{L}_{\odot }}\right)=\alpha \mathrm{log}\left(\displaystyle \frac{\mathrm{SFR}}{{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right)+\beta .\end{eqnarray} \tag{ 1 }$$

Lagache et al. (2018) find a redshift-dependent mean relation with α = 1.4–0.07z and β = 7.1–0.07z. We assume 0.5 dex scatter on the SFR–[C ii] relation, based on the 0.5–0.6 dex dispersion that Lagache et al. (2018) find. The middle panels of Figure 1 illustrate the mean halo mass–SFR relation and the mean halo mass–[C ii] luminosity relation at the simulated redshifts, including the quenched fractions prescribed by Behroozi et al. (2019).⁸ As the figure shows, our mean relation compares favorably to the best-fit model from Padmanabhan (2019), which comes from relying on abundance matching at z ∼ 0 against the Hemmati et al. (2017) luminosity function and inferring redshift evolution based on constraints from Pullen et al. (2018) on integrated [C ii] emission at z ∼ 2.6. Note that Padmanabhan (2019) finds that current observations only allow constraints down to M_vir = 10¹¹ M_⊙, and Lagache et al. (2018) state that their average relation describes [C ii] luminosities down to 10⁷ L_⊙.

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Note again that the mean SFR–[C ii] relation in Lagache et al. (2018) derives from simulations of high-redshift galaxies using semi-analytic modeling (G.A.S.; updated from Cousin et al. 2015a, 2015b, 2016) and a photoionization code (CLOUDY; Ferland et al. 2017), rather than observations of local galaxy samples as in Spinoglio et al. (2012) (a synthesis of data from Brauher et al. 2008), De Looze et al. (2014), and Herrera-Camus et al. (2015). As Lagache et al. (2018) note, analysis of line observations plus heating and attenuation effects from the cosmic microwave background suggest that high-z [C ii] emission is dominated by ionized carbon in photodominated regions (PDR), rather than neutral gas as appears to be the case in local star-forming galaxies. By combining G.A.S. modeling of galaxy formation history with CLOUDY modeling of the PDR in each galaxy, Lagache et al. (2018) should represent a state-of-the-art understanding of [C ii] emission at high redshift.

At z ∼ 6, the Lagache et al. (2018) SFR–[C ii] relation takes α = 0.98 and β = 6.68. The local calibrations, by comparison, would assign luminosities 2–10 times higher at a given SFR. For Herrera-Camus et al. (2015), α = 0.967 and β = 7.65; for De Looze et al. (2014), α = 0.99 and β = 6.92 (based on the entire literature sample); and α = 0.89 and β = 7.27 for Spinoglio et al. (2012) (converting the original L_IR–[C ii] relation to a SFR–[C ii] one by taking L_IR/L_⊙ =10¹⁰ SFR/(M_⊙ yr⁻¹)).⁹

We use limlam_mocker¹⁰ to generate line-intensity cubes and power spectra for each lightcone using this model. Doing this requires defining a grid of volume elements (or voxels ), each taking up a solid angle and frequency interval within the mocked line-intensity cube. We use the frequencies and lightcone sizes in Table 1 and create an intensity cube of 450³ voxels.¹¹ All halo luminosities are binned per voxel, and the [C ii] luminosity per voxel L_vox is converted into an intensity ${I}_{\nu }={L}_{\mathrm{vox}}/(4\pi {D}_{L}^{2}{{\rm{\Omega }}}_{\mathrm{pix}}{\delta }_{\nu })$ by dividing by the voxel frequency interval δ_ν, voxel solid angle Ω_pix, and $4\pi {D}_{L}^{2}$ given the luminosity distance D_L from the observer to the voxel. We can then calculate both the mean map intensity $\left\langle {I}_{\nu }\right\rangle$ from the simulated intensity cube, presented in Section 3.1, and the spherically averaged, comoving 3D power spectrum P(k), which we present as the main signal in Section 3.2. The latter is obtained from the full 3D power spectrum P(k) of the intensity cube in comoving space, averaged in spherical k-shells.

Note that, while $\left\langle {I}_{\nu }\right\rangle$ is not to be measured directly by any of the experiments we consider, it is nonetheless a useful and potentially important statistic. Unlike P(k), it can be analytically calculated purely from the halo mass function (HMF) with the model outlined above. Thus, analytic expectations for $\left\langle {I}_{\nu }\right\rangle$ will act as a sanity check on our simulation results.

Given that analysis of recent observations points to star formation arising in z ≳ 6 galaxies with halo masses as low as several 10⁹ M_⊙ (Finlator et al. 2017), the minimum emitter halo mass of 10¹⁰ M_⊙ imposed in these simulations is potentially too high, but is forced by the Bolshoi–Planck mass resolution, which results in halo incompleteness below ∼10¹⁰ M_⊙. However, analytic calculations of $\left\langle {I}_{\nu }\right\rangle$ are not subject to the same limits. To gauge the effects of the HMF and the minimum emitter halo mass on the signal, we make analytic estimates of $\left\langle {I}_{\nu }\right\rangle$ at each redshift using the above model with an HMF fit at each redshift that should predict the correct abundances of halos with masses ≲10¹⁰ M_⊙. In the process, we also calculate the expected contribution to the mean intensity from different mass bins. We present these results alongside the simulation results (for total $\left\langle {I}_{\nu }\right\rangle$ only) in Section 3.1.

We follow the formalism of Li et al. (2016) and quantify the uncertainty in P(k) as

$$\begin{eqnarray}&&{\sigma }_{P}(k)=\displaystyle \frac{P(k)+{P}_{n}}{\sqrt{{N}_{m}(k)}},\end{eqnarray} \tag{ 2 }$$

which is the total observed power spectrum—signal P(k) plus noise P_n—divided by the number of Fourier modes N_m(k) available for averaging near that given k. We detail the calculation of N_m(k) in Appendix A.

We calculate P_n from instrumental noise only. If our survey volume is observed uniformly so that each volume element (or voxel) of some comoving volume V_vox has been observed for some integration time t_pix, then

$$\begin{eqnarray}&&{P}_{n}=\displaystyle \frac{{\sigma }_{\mathrm{pix}}^{2}}{{t}_{\mathrm{pix}}}{V}_{\mathrm{vox}},\end{eqnarray} \tag{ 3 }$$

where σ_pix is the sensitivity per instrumental pixel per spectral element, and ${\sigma }_{\mathrm{pix}}{t}_{\mathrm{pix}}^{-1/2}$ the final survey sensitivity per voxel (the equivalent of σ_n in Appendix C of Li et al. 2016). We show σ_pix in Table 1 for each survey, quantified as noise-equivalent intensity/input (NEI), as well as the number of instrumental pixels expected (per band). We explain the figures for each survey in more detail below.

• 1.  
  For CCAT-p, we use figures for ${\sigma }_{\mathrm{pix}}{t}_{\mathrm{pix}}^{-1/2}$ per beam for a single EoR-Spec array given by the collaboration (G. Stacey 2018, private communication). The figures are calculated at the central frequencies of each simulated frequency band, assuming first-quartile weather (0.4 mm precipitable water vapor) and an observing elevation of 45°. Each array will have 1004 spatial beams sensitive to two polarizations (which are folded into the σ_pix given), but instrumental details mean that only a fraction of the total spectral coverage can be instantaneously observed and the FPI must step across multiple settings to sample the full bandwidth, which slightly complicates the calculation of t_pix. However, broadly speaking, the results are equivalent to taking N_pix to be 1004 spatial beams times a factor ≪1 that depends on the observing frequency, for an effective count of around 20.
• 2.  
  CONCERTO will have an array of 1500 pixels for each band, for a total of 3000 pixels in the 200–300 GHz overlap between the two bands (G. Lagache 2018, private communication). The overlap excludes z = 4.5 but includes our two highest simulated redshifts. We use the noise-equivalent flux density (NEFD) given in Serra et al. (2016) of 155 mJy s^1/2, and divide by the beam solid angle to obtain the NEI.We wish to clarify a few caveats around this NEFD value. The most important caveat is that only one value is given with no frequency dependence. In reality, atmospheric opacity is far greater at 325–365 GHz compared to below 300 GHz, so we expect higher NEFD near 345 GHz versus at lower frequencies. Another caveat worth noting is that Serra et al. (2016) actually consider the NEFD divided by $\sqrt{{N}_{\mathrm{pix}}};$ however, this is still used as if it were merely the NEFD per pixel, since their expression for t_pix also includes a factor of N_pix. That said, we show in Appendix B that the 155 mJy s^1/2 figure is not unreasonable, in principle, as an NEFD per pixel for CONCERTO. Note, though, that while Dumitru et al. (2019) use the same NEFD when referring to the NIKA2 sensitivities demonstrated in Adam et al. (2018), our calculations in Appendix B make far more optimistic assumptions about system efficiency and emissivity than NIKA2 operating conditions would indicate. (We also still end up with somewhat higher estimated noise, but within a factor of order unity of the figures provided by the CONCERTO team.)Ultimately, for the purposes of this work, we will use the 155 mJy s^1/2 figure as-is at all frequencies, and we leave a re-examination of CONCERTO sensitivities for future work from the CONCERTO team or others.
• 3.  
  For TIME, we use the median of the NEI ranges for the low- and high-frequency bands quoted for operation at the Arizona Radio Observatory (ARO) assuming 3 mm precipitable water vapor (TIME Collaboration 2018, private communication). Crites et al. (2014) also indicate that the TIME experiment will have 32 spectrometers (16 per polarization).

Converting between NEFD and NEI requires knowledge of the beamwidth, quantified as the full width at half maximum (FWHM). Since TIME and CONCERTO are both to operate on 12 m telescopes, we assume that the beams for both instruments have a diffraction-limited FWHM of 1.22λ/(12 m), ranging from 17'' to 31'' throughout the full spectral range; Sun et al. (2018) and Dumitru et al. (2019) assume similarly for TIME and CONCERTO. For the CCAT-p beam FWHM, we use figures provided by the collaboration of (37'', 39'', 46'', 53'') at (408, 345, 280, 214) GHz.

For CONCERTO and TIME, which should have simultaneous uniform coverage of all frequency channels within each band by virtue of their architectures, we take t_pix to be simply the integration time per pixel:

$$\begin{eqnarray}&&{t}_{\mathrm{pix}}=\displaystyle \frac{{N}_{\mathrm{pix}}{t}_{\mathrm{surv}}}{{{\rm{\Omega }}}_{\mathrm{surv}}/{{\rm{\Omega }}}_{\mathrm{pix}}},\end{eqnarray} \tag{ 4 }$$

which is to say the total survey time multiplied by the number of instrumental pixels, divided by the number of map pixels (the ratio of the survey solid angle to the pixel solid angle). For CCAT-p, as noted above, the instantaneous spectral coverage and thus the calculation of t_pix is more complex, and ${\sigma }_{\mathrm{pix}}{t}_{\mathrm{pix}}^{-1/2}$ is presented per beam. We present the same in Table 1 for CONCERTO and TIME, using Ω_pix = Ω_beam.

The only input left for P_n is the comoving volume per voxel. Because ${V}_{\mathrm{vox}}\propto {{\rm{\Omega }}}_{\mathrm{pix}}{\delta }_{\nu }$, this cancels out the Ω_pix dependence of t_pix and the ${\delta }_{\nu }^{-1/2}$ dependence of σ_pix (see Appendix B for a detailed explanation) when calculating P_n, which thus does not depend on the voxel extent in any dimension. The ${\sigma }_{\mathrm{pix}}{t}_{\mathrm{pix}}^{-1/2}$ obtained for all experiments is per beam, so we take V_vox to be the comoving volume within a solid angle of Ω_beam and a frequency interval of δ_ν.

Having calculated P_n and thus σ_P(k) for each experiment, we must finally consider attenuation of the observed power spectrum at high wavenumber k due to the finite beam size of each telescope. This attenuation W(k) of the signal results in an effective sensitivity limit of σ_P(k)/W(k). Appendix C.3 of Li et al. (2016) details an analytic calculation of W(k), but we make an analogous numerical calculation in this work based on the expected voxel grid for each experiment. For this calculation only, we assume map pixel widths of 15'' for CCAT-p and 5'' for the other experiments. Once the pixel width is finer than the standard deviation of the Gaussian beam profile (FWHM/2.355), the degree of angular oversampling makes little difference in the numerical calculations and resulting W(k).

Table 2 shows the mean map intensity both from our simulated survey volumes and from the expected contributions from halos based on analytic calculations. The latter uses the best-fit model from Behroozi et al. (2019) with the HMF of Sheth et al. (2001), modified to fit the HMF values provided in the Behroozi et al. (2019) EDR at each redshift. For the HMF, setting A = 0.62–0.071z+0.0039z² and a = 0.96–0.072z +0.0058z²—instead of the original redshift-independent values of A ≈ 0.322 and a = 0.707 from Sheth et al. (2001)—provides an adequate description of the actual HMF down to M_vir = 10¹⁰ M_⊙ at the redshifts considered here (as we illustrate in the uppermost panel of Figure 1). Since the best-fit model prescribes average star formation rates and quenched fractions as functions of the halo maximum circular velocity at peak historical virial mass ${v}_{{M}_{\mathrm{peak}}}$, rather than of virial mass, we use the relation given in Appendix E2 of Behroozi et al. (2019) to convert the model relations into functions of virial mass.¹² This mass–${v}_{{M}_{\mathrm{peak}}}$ relation is inexact, with Behroozi et al. (2019) indicating scatter of ∼0.1 dex, meaning ∼10% discrepancies between our numerical results and analytic results for $\left\langle {I}_{\nu }\right\rangle$ should not be surprising. Indeed, any discrepancies between analytic and simulated results in Table 2 are within this expectation.

Table 2.  Mean Map Intensities from Simulations and from an Analytic Fit with Different Minimum Emitting Halo Masses

Redshift	$\left\langle {I}_{\nu }\right\rangle$ (Jy sr−1)
	Median and 90% Interval	Analytic, ${M}_{\mathrm{vir},\min }=\cdots$
	from Simulations	${10}^{10}\,{M}_{\odot }$	109 M⊙
3.7	${924.2}_{-54.1}^{+35.1}$	865.0	870.3
4.5	${339.2}_{-9.7}^{+14.5}$	308.4	315.6
6.0	${64.74}_{-2.33}^{+3.08}$	63.48	73.42
7.4	${16.58}_{-0.69}^{+0.66}$	17.55	27.81

Note. The HMF-based analytic calculation uses an approximation to the Behroozi et al. (2019) mean halo mass–SFR relation, and includes no scatter in SFR or line luminosity. The analytic HMF used is a modification of the fit in Sheth et al. (2001) described in the main text.

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The lowermost panel of Figure 1 shows the expected relative contribution of different halo mass ranges to $\left\langle {I}_{\nu }\right\rangle$. At z = 5.8 and below, the slope of the halo mass–[C ii] relation is steep enough compared to the slope of the HMF that a majority of $\left\langle {I}_{\nu }\right\rangle$ comes from halos of masses ≳10¹¹ M_⊙. Lowering the minimum emitting halo mass to 10⁹ M_⊙ (with a simple extrapolation below 10¹⁰ M_⊙) thus has little effect on the analytically estimated $\left\langle {I}_{\nu }\right\rangle$ below z ∼ 6, increasing only by 0.6% at z = 3.7, 2.3% at z = 4.5, and 15.6% at z = 6.0.

By z = 7.4, however, halos with M_vir ≲ 10¹¹ M_⊙ contribute a majority of the average signal, and lowering the cut to 10⁹ M_⊙ would allow more low-mass halos to contribute, increasing the analytically calculated mean [C ii] intensity by a factor of 1.6. This roughly translates to an increase in the power spectrum ($P(k)\sim {\left\langle {I}_{\nu }\right\rangle }^{2}$) by a factor of three. The analytic $\left\langle {I}_{\nu }\right\rangle$ with ${M}_{\mathrm{vir},\min }={10}^{10}\,{M}_{\odot }$ is within 10% of the numerically simulated mean intensity, so our P(k) forecast may indeed be too low by a factor of three, due to an overly high cutoff mass.

Whether lowering the cutoff halo mass below 10¹⁰ M_⊙ is well-motivated is unclear. As noted above, Finlator et al. (2017) do find evidence for unsuppressed star formation activity in z ≳ 6 halos with masses below 10¹⁰ M_⊙, but the evidence is not strong at z = 8 and not very strong at z = 7, and z ∼ 7.4 is the only redshift at which a lower cutoff significantly affects $\left\langle {I}_{\nu }\right\rangle$ in Table 2. We discuss the cutoff further in the next section, although we find that even a factor-of-three increase in the signal at z ∼ 7.4 will require greatly upgraded surveys to distinguish or detect.

We show the P(k) values calculated from the simulations in Figure 2. Our forecast signal level appears to drop an order of magnitude or so with each increase in observed redshift. However, in light of the analytic checks of Section 3.1, we consider the possibility that the decline in P(k) between z = 6.0 and z = 7.4 depends on the choice of cutoff halo mass. If halos below our chosen cutoff of 10¹⁰ M_⊙ emit in [C ii], our simulated P(k) at z = 7.4 may be an underestimate by a factor of several. Therefore, we also show an "optimistic" forecast at z = 7.4 alongside the fiducial one in Figure 2 by multiplying the fiducial P(k) by a factor of three (suggested by the comparison to analytic calculations in Table 2).

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We put our predictions in the context of previous work by plotting them together with P(k) from Serra et al. (2016) (which claims agreement with Gong et al. 2012) and Dumitru et al. (2019),¹³ interpolated or extrapolated in redshift as necessary (as the approximate evolution of P(k) with redshift at each k is apparent in both works). Serra et al. (2016) use measurements of the cosmic infrared background anisotropies with a halo model to constrain a halo mass–infrared luminosity relation, and combine this with the local L_IR–[C ii] relation of Spinoglio et al. (2012) to relate halo mass to [C ii] luminosity and thus enable an analytic calculation of P(k). Given our discussion about the difference between local SFR–[C ii] calibrations and the relation from Lagache et al. (2018) being as large as 1 dex at z ∼ 6, we find it unsurprising that the P(k) values of Serra et al. (2016) are almost 2 dex higher than our predictions.

The work of Dumitru et al. (2019) is more similar to ours in comparison, assigning [C ii] luminosities to halos identified in cosmological simulations to directly simulate cubes of [C ii] intensity, and using Lagache et al. (2018) for the SFR–[C ii] scaling relation in their model. Given these similarities, the discrepancy between our prediction and theirs is more surprising at first glance. However, the model of Dumitru et al. (2019) prescribes star formation rates that are simply proportional to halo mass, does not model a quiescent galaxy population, and results in a somewhat higher cosmic star formation rate density at z ∼ 6 compared to UniverseMachine. These differences between the halo mass–SFR relations of the two models help explain the factor-of-four difference we see between our forecast P(k) and the extrapolation from the results of Dumitru et al. (2019) at z ∼ 6.

At z ∼ 7–8, the two models result in a more similar cosmic SFR density, but still prescribe SFR in halos in substantively different ways. Additionally, the minimum halo mass of 2.3 × 10⁸h⁻¹ M_⊙ = 3.4 × 10⁸ M_⊙ in the simulations that Dumitru et al. (2019) use (with no additional cutoff imposed for [C ii] emission) becomes a more significant source of discrepancy at these highest redshifts. We have stated above that a lower cutoff halo mass in our simulations could increase our forecast P(k) by a factor of three, but this is merely a zeroth-order estimate. A perfectly fair comparison at these highest redshifts against the results of Dumitru et al. (2019) would require deploying the UniverseMachine framework on a simulation fine enough to allow resolution of halos with M_vir = 10⁹ M_⊙. This would enable [C ii] simulations that incorporate a more complete halo population, although it is an entirely open question as to how well-justified it would be either to set a lower cutoff mass or to extrapolate the M_vir–[C ii] relation so far below 10¹¹ M_⊙ in halo mass or 10⁷ L_⊙ in line luminosity. We leave this for possible future work.

Finally, while we do not explicitly plot P(k) from Padmanabhan (2019) to compare, there is broad agreement here with that work, with our P(k) below the best-fit model but still within the associated uncertainties. This is to be expected, given the level of agreement in the [C ii] luminosity prescription between our model and that of Padmanabhan (2019) already shown in Figure 1.

Between our work and previous work considered in Section 3.2, predictions for the [C ii] signal span a range of several orders of magnitude, unconstrained by any observational data. An improved understanding of [C ii] emission and its connection to star formation activity at high redshift would be made possible with a P(k) detection or even an upper limit that could exclude the more optimistic models. To consider the ability of near-future surveys to do this, we return to the sensitivities considered in Section 2.3.

We plot the expected sensitivities of CCAT-p, TIME, and CONCERTO in Figure 2 given the P(k) obtained in this work, accounting for the expected signal attenuation W(k) due to beam smoothing; we plot the noise in Figure 2 as ${\sigma }_{P}(k)/W(k)$. Table 3 reports the expected total S/N over all modes up to k = 1 Mpc⁻¹, calculated as

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}={\left[\sum _{i}{\left(\displaystyle \frac{P({k}_{i})W({k}_{i})}{{\sigma }_{P}({k}_{i})}\right)}^{2}\right]}^{1/2},\end{eqnarray} \tag{ 5 }$$

summing over all k-bins with central values k_i < 1 Mpc⁻¹.

Note the slope of the sensitivity curves at low k for CCAT-p and CONCERTO. For the lowest redshifts, the signal is large enough for sample variance to be a significant—if not dominant—component of σ_P(k). Therefore, at the low-k end, where P(k) ∼ k⁻¹ and $1/\sqrt{{N}_{m}(k)}\sim {k}^{-1}$, ${\sigma }_{P}(k)\sim {k}^{-2}$. (Figure D1 from Appendix D shows the sensitivity curves if only instrumental noise is considered.) At higher redshifts, where instrumental noise dominates σ_P, P_n ∼ k⁰ but $1/\sqrt{{N}_{m}(k)}\sim {k}^{-1}$ still, so ${\sigma }_{P}(k)\sim {k}^{-1}$. A similar argument holds for TIME at the redshifts it observes, except $1/\sqrt{{N}_{m}(k)}\sim {k}^{-1/2}$ at low k (see Appendix A).

More importantly, there is a discontinuity in the slope of all sensitivity curves, at k ∼ 0.1–0.3 Mpc⁻¹. This fact stems from the limited spectral resolution of all instruments, which significantly affects the growth of N_m(k) beyond a specific k as line-of-sight modes become inaccessible (again, see Appendix A). The effect is particularly severe for TIME's line-scan survey strategy, as it targets only one angular dimension.

Nonetheless, a detection of the [C ii] signal at z ∼ 6 as predicted here is within reach. TIME is at a disadvantage due to the relatively limited N_m(k) it probes, but as it is expected to deploy first out of the three surveys, it will at minimum set bleeding-edge upper limits on the z ∼ 6 [C ii] auto spectrum, and an extended campaign with deeper mapping could yield a tentative detection. For instance, if deployed on a high-quality Atacama site, TIME would be capable of approximately twice the mapping speed it could achieve at the ARO, which translates to half the noise power spectrum after the same survey time (TIME Collaboration 2019, private communication). Such a deployment may be realized, for example, on the proposed Leighton Chajnantor Telescope (LCT), a refurbishment and move of the Caltech Submillimeter Observatory (CSO) from Maunakea to the Chajnantor plateau.¹⁴ Either in isolation or in combination with the ARO deployment, a 1000 hr survey with TIME at the LCT is extremely competitive against the other surveys presented here.

To be more exact about necessary extensions to survey times for a detection of P(k) at small k, Table 3 also shows how much time would be required to achieve an S/N of unity at a k-bin centered at k = 0.026 Mpc⁻¹ of width Δk = 0.035 Mpc⁻¹. (The corresponding total all-k S/N varies, and is especially lower for TIME due to its shallower sensitivity limit curve.) Optimization of survey areas can also increase t_pix and improve sensitivity, but depending on the criteria, the optimal survey areas at z ∼ 6 are too small for the instruments considered (see Appendix C).

Table 3 also shows that surveys would need to be unrealistically lengthy to detect the expected signal at z ∼ 8. This comes with the caveat from the end of Section 3.1, i.e., that the predicted P(k) at z ∼ 8 is likely too low. However, even if P(k) here is too conservative by an order of magnitude, none of the surveys above would be sensitive enough to even reach an S/N of 1 with their fiducial survey programmes, and all would require 5–10 times greater time on sky for an all-k S/N of 2–4. TIME at ARO is more competitive at this redshift range than at z ∼ 6, with map noise expected to be several times better than either CCAT-p or CONCERTO. However, the line-scan strategy limits relative detectability of P(k) for k ≳ 10⁻¹ Mpc⁻¹ and thus total S/N across the scales considered here.

A second generation of [C ii] line-intensity surveys might attain a fully three-dimensional, wide-field detection (i.e., over ≳1 deg²), potentially even through a significant upgrade to an existing instrument or extension of an existing survey. Sensitivities must improve over the immediate generation by at least an order of magnitude, however. This would enable a more confident detection of our forecast signal at z ∼ 6 in addition to a tentative detection at z ∼ 8.

In view of this, we note the significant upgrade potential for EoR-Spec on CCAT-p. The Phase I instrument assumed here only occupies one-third of one instrument module, when in fact the overall Prime-Cam design (i.e., the first-light instrument design for CCAT-p) can accommodate up to seven instrument modules (Vavagiakis et al. 2018). An upgraded EoR-Spec configuration—or a more aggressive first-light configuration—would use two fully occupied instrument modules for six times the field of view and six times the mapping speed. Note also that the seven-module Prime-Cam design does not entirely fill the CCAT-p telescope's field of view, which could accommodate as many as 19 modules (Stacey et al. 2018). Such "Phase II" extensions of the EoR-Spec array by factors of 6–18 would provide the generational leap that may enable a detection of the [C ii] at z ∼ 7.4 (given the more optimistic version of our prediction).

TIME also has a clear path forward from what we present here. In addition to potential deployment at another telescope like the LCT, one way to increase sensitivity may be to make use of more space by moving from a grating spectrometer to a more compact architecture. The on-chip spectrometer technology to be deployed in SuperSpec (Shirokoff et al. 2014) provides a potential technology path. Once SuperSpec successfully demonstrates this technology in the field, future work should consider what may be possible with hundreds or thousands of SuperSpec-style pixels instead of the 16 considered here for TIME.

The NEFD is effectively the noise per beam, and is given by system efficiencies, instrumental bandwidth, telescope aperture, and the total noise-equivalent power (NEP). Equation (A8) of Gong et al. (2012) gives a dimensionally incorrect expression for the NEFD, so we refer to Equation (7.41) from Walker (2015), presented below with a minor change in notation:

$$\begin{eqnarray}&&\mathrm{NEFD}=\displaystyle \frac{\mathrm{NEP}}{{\eta }_{c}{\eta }_{t}{A}_{e}{e}^{-{\tau }_{\nu }A}{\delta }_{\nu }}.\end{eqnarray} \tag{ B1 }$$

Here, A_e is the effective aperture of the telescope, while δ_ν is the bandwidth of the observation (the spectrometer channel bandwidth in this case). The NEP itself depends on frequency and detector bandwidth, and in background-limited operation at the frequencies considered in this work, we adapt Equation (46) of Zmuidzinas (2003), which provides an approximate expression for background-limited power uncertainty if the Rayleigh–Jeans temperature T₀ of the background radiation is known:

$$\begin{eqnarray}&&{\sigma }_{P}\approx \displaystyle \frac{{{kT}}_{0}}{\sqrt{\mathrm{BW}\cdot \tau }}\cdot \mathrm{BW},\end{eqnarray} \tag{ B2 }$$

where ${\mathrm{BW}}_{\det }$ is the bandwidth seen by the detector, and τ the integration time. By convention, this quantity is the NEP when we refer this uncertainty to a 1 Hz post-detection bandwidth, or τ = 0.5 s. Then we find

$$\begin{eqnarray}&&\mathrm{NEP}\approx 0.62\ \mathrm{aW}\ {\mathrm{Hz}}^{-1/2}\left(\displaystyle \frac{{T}_{0}}{{\rm{K}}}\right)\sqrt{\displaystyle \frac{{\mathrm{BW}}_{\det }}{\mathrm{GHz}}}.\end{eqnarray} \tag{ B3 }$$

Note that ${\mathrm{BW}}_{\det }$ is not necessarily equal to the bandwidth used in Equation (B1): a grating spectrometer or Fabry–Perot interferometer will only expose each detector to the bandwidth per channel (so that the NEFD scales as ${\delta }_{\nu }^{-1/2}$), but detectors behind a Fourier-transform spectrometer must see the entire instrumental bandwidth (so that the NEFD scales as ${\delta }_{\nu }^{-1}$).

We define the aperture efficiency ${\eta }_{A}\equiv {A}_{e}/(\pi {D}^{2}/4)$, where D is the dish diameter, and then $\eta \equiv {\eta }_{c}{\eta }_{t}{\eta }_{A}{e}^{-{\tau }_{\nu }A}$, to encapsulate the end-to-end optical coupling efficiency, including atmospheric attenuation.

The NEP is usually taken to be the power incident across the solid angle projected from the detector to the sky; see Equation (7.38) of Walker (2015), for instance. Therefore, to get the sensitivity per pixel σ_pix as noise-equivalent intensity (NEI), we divide the NEFD by the beam solid angle ${{\rm{\Omega }}}_{\mathrm{beam}}\,={\mathrm{FWHM}}^{2}\cdot \pi /(4\mathrm{ln}2)$ (which the dependence of the NEFD on A_e broadly cancels out):

$$\begin{eqnarray}&&{\sigma }_{\mathrm{pix}}=\displaystyle \frac{\mathrm{NEFD}}{{{\rm{\Omega }}}_{\mathrm{beam}}}=\displaystyle \frac{\mathrm{NEP}}{\eta (\pi {D}^{2}/4){\delta }_{\nu }{{\rm{\Omega }}}_{\mathrm{beam}}}.\end{eqnarray} \tag{ B4 }$$

This expression, together with Equation (B4), allow us in principle to rederive the sensitivities considered in Section 2.3 from scratch assuming background-limited operation.

We now recap the basic design of each experiment considered in the main text, considering various trade-offs involved in each.

• 1.  
  TIME uses a grating spectrometer. The size of the grating limits the number of beams on sky, but all pixels integrate simultaneously on all frequencies, and each detector only sees the light corresponding to the width of the spectral bin.
• 2.  
  CCAT-p uses an FPI, which effectively acts as a narrow-band filter in front of the detectors. This means that the detectors only see the light corresponding to the width of the spectral bin, as in a grating spectrometer, but the FPI must scan through the different spectral bins. Therefore, at some point, there is a penalty proportional to N_chan (the number of spectral bins) on the final map noise, as each detector spends only $\sim 1/{N}_{\mathrm{chan}}$ of its time integrating in a given spectral bin.
• 3.  
  CONCERTO uses a Fourier-transform spectrometer (FTS), and specifically a Martin–Puplett interferometer (MPI). The relevant detail is that an interferometer in front of the camera scans through different time delays, so that each detector records an interferogram that can then be Fourier-transformed into a spectrum. As a result, each detector must see the entire bandwidth of the instrument rather than the bandwidth of the spectral bin. Unlike the FPI, despite the fact that there is a scan in time delay, the pixels do effectively integrate down on all frequency channels as the scan proceeds, so no N_chan penalty applies.

For the purposes of this discussion, we consider all experiments only at 250 GHz, summarizing the relevant parameters in the first several columns of Table B1. We assume a diffraction-limited beam for all experiments, with an FWHM of 50'' for D = 6 m and 25'' for D = 12 m at 250 GHz. (Note that this assumption is acceptable even for CCAT-p, as its beam FWHM approaches the diffraction limit at lower frequencies.)

Table B1.  Instrumental Parameters and Approximate Sensitivities for All Experiments at 250 GHz

Experiment	Spectrometer	Npix	${N}_{\mathrm{spec},\mathrm{eff}}$	D	δν	Δν	BWdet for NEP	Approx. NEP	Approx. NEFD	Approx. σpix
	Architecture			(m)	(GHz)	(GHz)	(GHz)	(aW Hz−1/2)	(mJy s1/2)	(MJy sr−1 s1/2)
CCAT-p	Fabry–Perot	1004	20	6	2.5	210	2.5	15	69	1.0
CONCERTO	Martin–Puplett	3000	3000	12	1.5	160	160	120	230	14
TIME	Grating	32	32	12	1.9	96	1.9	51	160	9.5

Note. Calculations of approximate NEP, NEFD, and σ_pix assume a background brightness temperature of 15 K (60 K for TIME), corresponding to a ground-based 300 K antenna with 5% system emissivity (20% for TIME). The effective count of spectrometers ${N}_{\mathrm{spec},\mathrm{eff}}$ integrating simultaneously across the full band is given by N_pix divided by the number of channels in the band multiplied by the number of channels simultaneously observed at any given instant. For CONCERTO, we are considering a region of overlap between the 125–300 GHz and 200–360 GHz bands; Δν differs between the two by 10%, and we take the smaller Δν here. The TIME parameters are for the high-frequency 230–326 GHz band. The actual numbers provided by the collaborations are ${\sigma }_{\mathrm{pix}}=0.86$ MJy sr⁻¹ s^1/2 for CCAT-p, $\mathrm{NEFD}=155$ mJy s^1/2 (yielding σ_pix = 11 MJy sr⁻¹ s^1/2) for CONCERTO, and σ_pix = 11 MJy sr⁻¹ s^1/2 for TIME.

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Calculating the NEP and NEFD requires not only the instrument parameters, but also assumed values for the background Rayleigh–Jeans temperature T₀ and the end-to-end optical efficiency of the system η. We assume that, for all experiments, a detector couples to 30% of the source emission, so that η = 0.3 for dual-polarization pixels (as in CCAT-p and CONCERTO) and η = 0.15 for single-polarization spectrometers (which are what are counted up for TIME). We also assume that T₀ is given by 300 K times a system emissivity of 5% for CCAT-p and CONCERTO, or 20% for TIME (as it is at a significantly less dry site than the others, with much lower atmospheric transmission). Then T₀ = 15 K for CCAT-p and CONCERTO, and 60 K for TIME.

We show the resulting approximate NEP and NEFD, as well as σ_pix (the NEI), in the final several columns of Table B1. The results match the figures provided by each collaboration to within a factor of 2. Although these approximate calculations are highly dependent on our assumptions and ignore many complexities surrounding direct-detection spectrometer noise, they do show that all claimed figures for the sensitivity per pixel are facially at sensible ranges.

We do caution that our assumptions are at times highly optimistic—in particular, an assumption of 5% system emissivity represents an extreme best-case scenario for both CONCERTO and CCAT-p, but especially so for CONCERTO (given the optical configuration, antenna design, site altitude, and other considerations). Compare also to Dumitru et al. (2019), where the 155 mJy ${{\rm{s}}}^{1/2}$ NEFD figure is extrapolated from NIKA2 sensitivities demonstrated on sky with 2 mm precipitable water vapor and an observing elevation of 60 degrees; 5% emissivity is unachievable under such high water vapor content along the line of sight. However, as in the main text, we leave a re-examination of CONCERTO sensitivities for future work by others.

Furthermore, given the work that the individual collaborations have put into their own publicly and privately communicated sensitivity figures, which likely incorporate greater detail than we are able to, we do not replace the figures of Table 1 with the figures of Table B1; nor should we expect the two sets of figures to match perfectly, for similar reasons.

In any case, based on σ_pix alone, CONCERTO and TIME appear to be at a significant disadvantage relative to CCAT-p, respectively due to the increased NEP from the FTS architecture (which sets ${\mathrm{BW}}_{\det }$ equal to Δν rather than δ_ν) and the atmospheric conditions at ARO (reflected in our estimates in the higher background temperature). However, recall that CCAT-p uses a scanning spectrometer, with the FPI transmitting two colors at once to the dichroic TES array. Therefore, with ≈100 spectral bins in the entire band and only two bins seen by the spectrometer at any given time, there is a fifty-fold penalty incurred in the integration time per pixel. We encapsulate this in Table B1 by borrowing a notation from Padmanabhan (2019) and noting the effective count of spectrometers ${N}_{\mathrm{spec},\mathrm{eff}}$ with full simultaneous frequency coverage across the entire instrument band, which is simply equal to N_pix for CONCERTO and TIME, but is N_pix/50 for CCAT-p. Thus, ${N}_{\mathrm{spec},\mathrm{eff}}$ takes the place of N_pix in Equation (4) for CCAT-p alone. Since the final survey sensitivity is given by ${\sigma }_{\mathrm{pix}}{t}_{\mathrm{pix}}^{-1/2}\propto {\sigma }_{\mathrm{pix}}{N}_{\mathrm{spec},\mathrm{eff}}^{-1/2}/{{\rm{\Omega }}}_{\mathrm{surv}}$, the large spectrometer count of CONCERTO and the small survey area of TIME compensate for their relatively high σ_pix, which is how the final map noise ends up around the same order of magnitude for all experiments (as shown in Table 1 in the main text).