Cosmic CO and [C II] backgrounds and the fuelling of star formation over 12 Gyr

Abstract

Molecular gas, modest in mass yet pivotal within the cosmic inventory, regulates baryon cycling as the immediate fuel for star formation. Across most of cosmic history, its reservoir has remained elusive, with only the tip of the iceberg revealed by luminous CO-emitting galaxies. Here we report the detection of the mean cosmic CO background across its rotational ladder at 7σ, together with ionized carbon ([C II]) at 3σ, over 0 < z < 4.2. This uses tomographic clustering of diffuse broadband intensities with reference galaxies, directly probing aggregate emission in the cosmic web. From CO(1–0) we infer the total molecular gas density, ${\varOmega }_{{{\rm{H}}}_{2}}$, finding it about twice that resolved in galaxy surveys. The global depletion time is ~1 Gyr, shorter than the Hubble time, requiring sustained inflow. CO excitation is linked to star-formation surface density and, with depletion time, yields a super-linear Kennicutt–Schmidt law that appears universal. Together these results establish a global picture of galaxy growth fuelled by a larger, short-lived molecular reservoir. The CO and [C II] detections mark a turning point for line-intensity mapping, replacing forecasts with empirical line strengths and defining sensitivity requirements for upcoming three-dimensional experiments poised to open new windows on galaxy formation and cosmology.

Main

Understanding the cosmological growth of galaxies requires probing the gas reservoirs that regulate the baryon cycle within the cosmic ecosystem¹. A defining feature of cosmic history is the rise of the star-formation rate (SFR) density to a peak at z ≈ 2, followed by a tenfold decline to the present^2,3. The driver of this evolution remains uncertain: is it set by the global gas supply, or by changes in how efficiently galaxies consume it? Line emission from CO and [C II] probes key baryon phases in the interstellar medium (ISM) and circumgalactic medium, with CO tracing the bulk molecular hydrogen (H₂) that fuels star formation and [C II] tracing the cooling medium left in its wake. Beyond the local universe, however, galaxy surveys face challenges in sensitivity and efficiency; existing CO and [C II] programmes cover only small areas or preselected targets, yielding a few hundred emitters in total and far fewer at any given redshift^{4,5,6,7,8,9,10}. Such sparse sampling, coupled with surface-brightness limits, leaves the luminosity functions and total light budgets highly uncertain. These challenges, along with opportunities to probe large-scale structure (LSS) and cosmology, motivate line-intensity mapping (LIM), a new approach that bypasses the need to resolve individual galaxies and instead captures the integrated emission of all sources in the cosmic web.

Our CO and [C II] intensity measurements build on a companion study³ of the cosmic infrared background (CIB), where young starlight is absorbed by dust and thermally re-emitted, a process that dominates the energy output of galaxy formation. That analysis used diffuse emission in 11 broadband intensity maps from Planck, Herschel and the Infrared Astronomical Satellite^11,12,13,14, spanning a 50-fold frequency range from 100 GHz to 5 THz. Redshift information was obtained through spatial cross-correlations with reference galaxies and quasars (Methods) over an effective sky of 1π sr, enabling tomographic reconstruction of the CIB spectrum over 0 < z < 4.2 robust against Galactic foregrounds. In the large-scale ‘clustering’ regime, gravity links these bright references to the underlying galaxy population rather than to their own emission or the shot noise from discrete bright sources. This enables direct estimates of the mean backgrounds, free from galaxy incompleteness and astrophysical assumptions. Using this clustering-based CIB tomography, the companion paper³ traced the evolution of dust temperature and the total dust mass budget Ω_dust, recovered the cosmic SFR history with substantially improved precision over the previous benchmark² and showed that star formation has been predominantly dust obscured for over 12 Gyr.

Here we unlock the full potential of intensity tomography to detect CO and [C II] emission and probe the cosmic budgets of molecular and cooling gas. Figure 1a shows the rest-frame bias-weighted mean CIB emissivity spectrum, ϵ_νb(ν, z), from the companion work³, where the emissivity ϵ_ν is the radiative power per unit frequency and comoving volume in erg s⁻¹ Hz⁻¹ Mpc⁻³, and b is the effective CIB bias relative to the underlying matter density fluctuations. The broad thermal dust emission traces the obscured SFR history, while the discrete frequencies of the CO rotational ladder and the [C II] 158-μm line mark where these line contributions enter. Redshift tomography enhances the effective spectral resolution by mapping each observed band ν_obs to ν = ν_obs(1 + z) across 16 redshift bins, yielding super-bandpass sampling for line diagnostics.

Fig. 1: Tomographic CIB spectrum and cosmic CO and [C II] histories.

a, Evolving CIB spectrum shown as ϵ_νb versus rest-frame frequency, colour-coded by redshift. Data points are from 11-band intensity–galaxy cross-correlations in the two-halo clustering regime³. Vertical bands mark nine CO and the [C II] 158-μm lines, with widths set by the mean spectral resolution R ≈ 3.5; data points within these bands receive line excess. Coloured curves show the best-fit continuum-plus-line model, with unresolved CO and [C II] drawn as top-hats at 1% of their frequencies. b, We detect the cosmic CO and [C II] backgrounds by exploiting their characteristic frequency–redshift line-excess patterns in a. Their comoving luminosity densities, with CO summed over nine lines (black and red curves with shaded intervals), peak broadly near cosmic noon. Dashed lines give posterior medians for individual CO transitions, revealing evolving excitation and the dominance of mid-J transitions. Black and red data points are per-redshift refits, supporting our redshift parameterization; the single transparent data point for [C II] is negative due to noise. [C II] is unconstrained at z < 0.4 owing to a gap in spectral coverage. For broader context, yellow data points and upper limits (95%) indicate Lyα intensity-mapping constraints^17,18, and the green dashed line shows predicted hydrogen 21-cm amplitudes¹⁹. [C II] probably rivals Lyα as the brightest line background, while CO remains much brighter than 21 cm. All error bars and shaded bands indicate 1σ (68%) uncertainties.

We analyse the total emissivity ϵ_ν as the sum of continuum and line components (Methods). The continuum follows the ensemble dust spectrum from the companion study³, with parameters describing dust density, temperature and opacity index and their redshift evolution. The line term includes [C II] 158 μm at 1,900.54 GHz, treated as a Dirac delta function, and nine CO transitions from J = 1−0 (115.27 GHz) to J = 9−8. Other lines are fainter and expected to be negligible at our current emissivity precision. The CO ladder is fitted jointly with a spectral line energy distribution (SLED) controlled by an excitation parameter linked to an effective galaxy-scale star-formation surface density Σ_SFR, described by equation (7) in Methods with coefficients given in Extended Data Table 1. We perform Bayesian inference on the tomographic ϵ_νb(ν, z) spectrum in Fig. 1a, supplemented by Far-InfraRed Absolute Spectrophotometer and Planck monopole measurements¹⁵ to break the bias-intensity degeneracy. This yields redshift-resolved luminosity densities for CO and [C II], ρ_{CO, J(J−1)}(z) and ρ_C II(z), together with the mean Σ_SFR(z) that governs CO excitation. A summary of all model parameters and posteriors is provided in Extended Data Table 2.

We detect a statistically significant 6.6σ cosmic CO signal, quantified by ρ_CO summed over nine lines after marginalizing over the continuum and accounting for redshift covariance. [C II] is also detected at 3.0σ. Figure 1b shows the posterior evolution of ρ_CO and ρ_C II (thick lines and bands), both peaking at z ≈ 2. For visualization, we refit the data for per-redshift estimates (circular markers), allowing CO(1–0) and [C II] amplitudes to vary across eight and six bins, respectively. Posterior medians for individual CO transitions are also shown, providing excitation diagnostics and further validation of the CO signals. These detections are robust to foregrounds, coeval continuum, and line confusion, probing the CO and [C II] backgrounds beyond the shot-noise regime. They recover the total cosmic mean intensity, encompassing the full galaxy population down to the faintest objects, with potential minor contributions from molecular outflows and extended circumgalactic medium halos¹⁶. Compared with the CIB continuum³, [C II] accounts for about 0.3% of the total infrared luminosity density, and the nine CO lines contribute about 0.03%. Their redshift evolution tracks the total cosmic SFR more closely than the CIB, implying sensitivity to the full star-forming gas rather than only its dust-obscured component (Methods).

To place CO and [C II] in a broader context of line backgrounds, Fig. 1b also shows Lyα intensity constraints: a 2σ detection at z = 1 and a limit at z = 0.3 (ref. ¹⁷), and another limit at z = 2.55 (ref. ¹⁸). Given current uncertainties, the competition for the brightest line is a tie between Lyα and [C II], with [C II] probably pulling ahead at low redshift as metallicity increases and Lyα escape decreases. The predicted hydrogen hyperfine 21-cm background is much fainter¹⁹. Notably, [C II], though a ‘metal’ line, may outshine all hydrogen lines, making it a premier LSS tracer.

Figure 2 summarizes the sky monopole, that is, the mean brightness temperatures T_b of the CO and [C II] line backgrounds, converted from the luminosity densities in Fig. 1b. An intensity version is shown in Extended Data Fig. 1. Each line appears at its rest frequency at z = 0 and redshifts to lower frequencies, extrapolated to z = 10, with shaded bands indicating the 1σ uncertainties. For comparison, we overlay the jointly constrained dust continuum from this work and the empirical estimate of the thermal Sunyaev–Zeldovich (SZ) effect background²⁰, whose decrement exceeds the dust emission below ~80 GHz. Fainter lines, as well as the radio free–free and synchrotron backgrounds rising at the lowest frequencies, are neglected. Although [C II] is far brighter in absolute terms, CO lines become increasingly prominent at lower frequencies, showing substantially higher equivalent widths (Methods) relative to the CIB and other secondary effects of the cosmic microwave background (CMB).

Fig. 2: Landscape of millimetre line monopoles in brightness temperature.

The mean sky monopoles T_b versus observed frequency for the CO and [C II] backgrounds, measured in this work up to z = 4.2 and extrapolated to z = 10, are shown as curves colour-coded by redshift. For comparison, the black curve indicates the updated CIB dust-continuum monopole, fitted jointly with the lines in this work, with their sum closely matching the previous continuum-only fit³. The spectral distortion of the thermal SZ effect background is shown in red, with the dashed segment indicating the decrement (y ≈ 1.22 × 10⁻⁶)²⁰. The amplitudes of all these components are now empirically constrained, with shaded bands indicating 68% uncertainties. These lines trace the three-dimensional LSS of the universe, while the continua trace it in projection; together they constitute the main extragalactic foreground of the CMB. Filled grey regions mark atmospheric transmission windows at a relatively dry site⁴⁹, highlighting more accessible low-frequency bands from the ground. This figure establishes key benchmarks for the far-infrared to millimetre line backgrounds, replacing previously divergent model forecasts with direct measurements and defining amplitude baselines and sensitivity targets for future LIM experiments.

The CO background reveals the cosmic molecular history shown in Fig. 3, tracing the fuel of cosmic star formation. This gas reservoir is dominated in mass by H₂, which is difficult to observe directly as it lacks a permanent dipole moment, but is traced by readily excited low-J CO lines. Assuming a Milky Way-like CO-to-H₂ conversion factor of α_CO = 3.6M_⊙ (K km s⁻¹ pc²)⁻¹ (ref. ²¹) for J = 1–0, we infer the molecular density parameter ${\varOmega }_{{{\rm{H}}}_{2}}\equiv {\rho }_{{{\rm{H}}}_{2}}/{\rho }_{{\rm{c}}{\rm{r}}{\rm{i}}{\rm{t}}}$, where ${\rho }_{{{\rm{H}}}_{2}}={\alpha }_{{\rm{C}}{\rm{O}}}{\rho }_{{\rm{C}}{\rm{O}},10}$ is the comoving molecular gas density and ρ_crit the present-day critical density. Although the conversion is anchored at low J to extract mass rather than excitation, the constraint uses all transitions (Fig. 4a), with the SLED turnover well sampled across redshift (Methods). We find that ${\varOmega }_{{{\rm{H}}}_{2}}$ peaks at z ≈ 1.5 and declines towards both lower and higher redshifts, with evolution well described by

$${\varOmega }_{{{\rm{H}}}_{2}}(z)\approx 1.9\times 1{0}^{-4}\,\frac{{(1+z)}^{2.3}}{1+{[(1+z)/2.8]}^{6.5}}.$$

(1)

This delivers an empirical ${\varOmega }_{{{\rm{H}}}_{2}}$ from direct ρ_CO constraints, modulo α_CO, bypassing galaxy-survey incompleteness and avoiding the shot-noise-only extrapolation by measuring diffuse CO in the clustering regime.

Fig. 3: History of molecular gas inventory traced by CO.

Expressed as the molecular density parameter ${\varOmega }_{{{\rm{H}}}_{2}}\equiv {\rho }_{{{\rm{H}}}_{2}}/{\rho }_{{\rm{c}}{\rm{r}}{\rm{i}}{\rm{t}}}$ evolution, our intensity-based measurement is from the CO(1–0) background, jointly constrained by all nine transitions over z = 0–4.2. Results are shown as the black line (posterior median) with shaded 68% credible intervals and black data points for per-redshift refits. Modulo α_CO, this provides a direct constraint on the total molecular gas density over 12 Gyr. For comparison, we overlay estimates from CO galaxy surveys at z = 0 (Five College Radio Astronomy Observatory (FCRAO), xCOLD GASS)^22,23 and at higher redshifts (ASPECS, CO Luminosity Density at High z (COLDz), Hubble Deep Field (HDF), Plateau de Bure High-z Blue Sequence Survey 2 (PHIBSS2))^4,5,6,7,8, the latter including only bright detected galaxies and thus shown as lower limits. At z = 1–2.5, our total ${\varOmega }_{{{\rm{H}}}_{2}}$ is about twice that resolved in galaxy surveys, revealing a substantial faint population previously missed and disfavouring shallow faint-end luminosity function slopes. This comparison adopts the same α_CO for all z > 0 sources and intensity estimates, making their ratios independent of α_CO assumptions. Early LIM results for CO from the CO power spectrum survey (COPSS)²⁴ and millimeter-wave intensity mapping experiment (mmIME)²⁵ support high ${\varOmega }_{{{\rm{H}}}_{2}}$, though these detections are limited to small-scale shot power and require model-dependent conversion to mean CO. Two sets of upper limits are currently compatible with our result: a LIM constraint from the CO mapping array project (COMAP)²⁶ and a null detection of CO absorbers in ALMACAL²⁷. Predictions from IllustrisTNG²⁸, EAGLE²⁹ and a semi-empirical model³⁰ are also shown, requiring no CO-to-H₂ conversion, with the latter two more consistent with our results.

Fig. 4: Astrophysical summary of star-forming gas over cosmic history.

Thick solid lines and shaded bands or contours show our constraints from the cosmic CO background, coloured consistently by redshift across all panels. Literature galaxy-based results are overplotted for comparison. All uncertainties are shown at the 1σ (68%) level. a, CO SLED, expressed as the velocity-integrated line-intensity ratios versus J, revealing higher excitation at earlier epochs. b, Σ_SFR rising with redshift as constrained by the CO SLED in a (Methods). c, Global molecular gas depletion time ${t}_{dep}={\rho }_{{H}_{2}}/{\rho }_{SFR}$ derived from the CO-based ${\rho }_{{H}_{2}}$ (Fig. 3) and CIB-based ρ_SFR (ref. ³), resembling the main-sequence galaxy relation³². The inverse of t_dep gives the star-formation efficiency. A short t_dep relative to the Hubble time necessitates sustained inflow to maintain cosmic star formation. d, Star-formation law linking galaxy-scale star-formation and gas surface densities, ${\varSigma }_{{\rm{S}}{\rm{F}}{\rm{R}}}\propto {\varSigma }_{{{\rm{H}}}_{2}}^{ \sim 1.5}$. Diagonal lines indicate constant t_dep, enabling the mapping from Σ_SFR (panel b) and 𝑡_dep (panel c) to $\varSigma_{{\rm{H}}_2}$ for the cosmic background. Coloured ellipses show 1σ uncertainties of the cosmic mean at a given redshift. The superlinear slope matches the local Kennicutt–Schmidt relation³⁴, revealing a universal mode of star formation that persists from galaxies to the aggregate background across 90% of cosmic history.

Figure 3 also overlays literature constraints on ${\varOmega }_{{{\rm{H}}}_{2}}$ from integrating CO emission in detected galaxies, both locally^22,23 and at higher redshifts^4,5,6,7,8. Although the absolute scale of α_CO remains uncertain (Methods), all z > 0 studies adopt the same Milky Way value as used here, enabling direct comparison in their ratios. Beyond the local universe, only about 200 CO emitters of all J have been detected across a total <0.1°² area, sparsely distributed in redshift and leaving the faint end and sometimes even the knee of the evolving luminosity functions poorly constrained. As a result, these studies avoid extrapolation and report ${\varOmega }_{{{\rm{H}}}_{2}}$ only from detected galaxies, which we plot as lower limits. During the peak star-formation epoch at z = 1–2.5, our diffuse, wide-field CO intensity measurement yields a ρ_CO,10 about twice that from individually detected galaxies above survey limits, giving an ${\varOmega }_{{{\rm{H}}}_{2}}$ larger by the same factor under any shared α_CO assumption. This factor-of-two gap most probably reflects incompleteness in galaxy surveys, revealing a substantial population of faint, previously missed galaxies. Combining individually detected bright populations with our total-background estimates would yield constraints that disfavour top-heavy luminosity functions.

CO has been a key LIM target for probing three-dimensional LSS via aggregate line emission. Alongside our results, Fig. 3 also summarizes existing LIM constraints, including the pioneering CO power-spectrum detections from the CO power spectrum survey²⁴ and millimeter-wave intensity mapping experiment²⁵ at 2–4σ, both limited to the shot-noise regime. These small-scale signals probe only the second moment of the luminosity function, making conversion to total CO emission (the first moment) and thus to ${\varOmega }_{{{\rm{H}}}_{2}}$ highly model dependent. In addition, without redshift tomography, the single millimeter-wave intensity mapping experiment band suffers from line confusion, with a degeneracy among four different J transitions and epochs. We also show two upper limits, both compatible with our detections: one from the CO mapping array project²⁶, a CO LIM experiment continuing to advance the constraints, and another from the non-detection of intervening CO absorbers²⁷, whose interpretation is less direct. Finally, we overlay three H₂ model predictions that do not require an α_CO assumption: those obtained by post-processing the IllustrisTNG²⁸ and EAGLE²⁹ hydrodynamic simulations, together with a semi-empirical model³⁰ linking observed galaxy SFR to halo mass in N-body simulations. The latter two predict higher ${\varOmega }_{{{\rm{H}}}_{2}}$ at z = 1–3, in better agreement with our results.

We now exploit astrophysical information in the CO background for diagnostics of star-forming gas in the universe and summarize the results in Fig. 4. Beyond the overall amplitudes, we uncover a clear evolutionary trend in the relative strengths of the CO ladder, or the SLED, with rising excitation with redshift (Fig. 4a, see also Fig. 1b). In studies of individual galaxies, CO SLED varies strongly between normal galaxies, submillimetre-selected starbursts, and quasars³¹. By contrast, our measurement uniquely pinpoints the averaged excitation over cosmic time free from selection effects. The elevated SLED implies systematically higher Σ_SFR at high redshifts (Fig. 4b), consistent with denser molecular gas, stronger turbulence and more intense radiation fields in the ISM^31,32.

The global molecular gas depletion time t_dep can be estimated by dividing our molecular density ${\rho }_{{H}_{2}}$ in Fig. 3 by the cosmic SFR density ρ_SFR. Figure 4c shows the resulting posterior anchored on the recent ρ_SFR(z), derived from the CIB with a minor UV correction following equation (42) of the companion study³, for which we adopt a 10% uncertainty. Evaluated on a dense redshift grid, the evolution is well approximated by

$${t}_{{\rm{d}}{\rm{e}}{\rm{p}}}\equiv {\rho }_{{{\rm{H}}}_{2}}/{\rho }_{{\rm{S}}{\rm{F}}{\rm{R}}}\simeq 3\,{(1+z)}^{-1}\,\mathrm{Gyr},$$

(2)

on the long side but remaining consistent with estimates for main-sequence galaxies³². This timescale of ~1 Gyr is shorter than the Hubble time, so once gas cools into the molecular phase it is converted into stars, requiring ongoing replenishment through gas inflows that naturally occur alongside cosmological structure growth. Meanwhile, t_dep is far longer than the local free-fall time, indicating that some regulating processes, possibly feedback and turbulence, keep star-formation efficiency well below the dynamical limit³³. The redshift dependence of t_dep is mild compared with the strong evolution of both ${\rho }_{{H}_{2}}$ and ρ_SFR, implying that changes in the global fuel supply, rather than large variations in consumption efficiency, drive the cosmic star-formation history.

In the classic picture of star formation in galactic disks, surface densities regulate disk instability and the efficiency of gas collapse. We therefore investigate the star-formation law in Fig. 4d by comparing the characteristic galaxy-scale Σ_SFR from Fig. 4b inferred from CO excitation against the molecular gas surface density ${\varSigma }_{{{\rm{H}}}_{2}}={t}_{{\rm{d}}{\rm{e}}{\rm{p}}}{\varSigma }_{{\rm{S}}{\rm{F}}{\rm{R}}}$ obtained via depletion-time scaling. This assumes that t_dep derived from the volumetric ratios of ${\rho }_{{H}_{2}}/{\rho }_{SFR}$ in Fig. 4c can be applied to galaxy-scale surface quantities. Unlike galaxy-based scaling relations, the mean background yields only one point estimate in the Σ_SFR–${\Sigma }_{{H}_{2}}$ plane at each redshift, with the fully propagated 1σ covariance also shown in Fig. 4d. As the universe evolves, these snapshots trace a cosmic star-formation law of the form ${\varSigma }_{{\rm{S}}{\rm{F}}{\rm{R}}}\propto {\varSigma }_{{{\rm{H}}}_{2}}^{N}$ with N ≈ 1.5, consistent with the local Kennicutt–Schmidt relation (N ≈ 1.4)³⁴. The superlinear slope indicates that higher molecular surface densities yield disproportionately more efficient consumption through enhanced gravitational cloud collapse³³. The persistence of this law, from local galaxies to global backgrounds and across 12 Gyr of cosmic history, suggests that the underlying physics may be fundamentally universal.

We now turn to [C II], the dominant coolant of diffuse ISM, excited by coupling stellar feedback in photodissociation regions, molecular cloud surfaces, and the neutral and ionized media. Each [C II] photon carries away thermal energy from collisions with gas particles, so the luminosity density ρ_C II measures the global cooling rate of star-forming gas after processes such as stellar winds, ionization and photoelectric heating. From ρ_C II in Fig. 1b, we find a cooling-rate evolution

$${\varLambda }_{{\rm{C}}\,\mathrm{II}}(z)\,\equiv \,{\rho }_{{\rm{C}}\,\mathrm{II}}(z)\approx 5.9\times 1{0}^{38}\,\frac{{(1+z)}^{3.2}}{1+{[(1+z)/2.9]}^{6.6}}\,{\rm{e}}{\rm{r}}{\rm{g}}\,{{\rm{s}}}^{-1}\,{\rm{M}}{\rm{p}}{{\rm{c}}}^{-3}.$$

(3)

This provides a global benchmark for ISM energy balance, constraining the net feedback coupling of young stars to diffuse gas as an endpoint of the baryonic energy cascade, and complements the CO-based picture by explaining the cooling and funnelling of cosmological inflows to replenish star-forming gas in cold clouds.

As [C II] responds directly to star formation and is the brightest line background, once calibrated it enables three-dimensional LSS mapping through LIM, complementing CO. Using our [C II] detection at z = 2, where ρ_C II is best constrained, we derive the mean [C II]–SFR relation:

$$\mathrm{SFR}\approx 4.{5}_{-1.6}^{+3.5}\times 1{0}^{-8}\,({L}_{{\rm{C}}\,\mathrm{II}}/{L}_{\odot })\,{M}_{\odot }\,{\rm{y}}{{\rm{r}}}^{-1},$$

(4)

where L_C II is the per-object [C II] luminosity. This is anchored to the precise CIB-based cosmic SFR density³, the same as that used for t_dep, and assumes a Chabrier initial mass function³⁵, providing a global [C II] calibration at the peak star-formation epoch. Our scaling is consistent with local trends for most galaxy types³⁶, supporting a non-evolving [C II]–SFR relation. The ‘[C II] deficit’ seen in ultraluminous infrared galaxies³⁷ is not prominent in the [C II] background, reflecting the dominance of normal star-forming galaxies over rare starbursts in the cumulative light.

Together, these findings establish a coherent picture of galaxy formation: the molecular reservoir is larger than previously charted yet depleted on short timescales, requiring replenishment through cosmological matter assembly. CO excitation traces star-formation intensity under a universal, superlinear law set by gas supply, while [C II] captures the cooling imprints of stellar radiation and feedback coupling. Stellar growth in galaxies is thus sustained by a large, short-lived molecular supply, maintained by the continuous cycling of baryons via inflow, cooling-induced collapse, star formation and feedback. This global, cosmological view anchors and contextualizes the diverse approaches to studying galaxy formation, from individual galaxies to cosmic backgrounds.

In the broader context of LIM, the sky background monopoles in Fig. 2 and Extended Data Fig. 1 serve as anchor points for anisotropy signals, placing our CO and [C II] detections within a cosmological framework. At its core, the LIM approach expands LSS cosmology by using photons rather than galaxies as matter tracers. Unlike galaxy surveys, LIM does not impose surface-brightness thresholds for source detection and requires much lower angular resolution, boosting survey efficiency, sky coverage and completeness³⁸. This enables access to LSS at higher redshifts with more linear modes, opening opportunities to probe reionization, dark energy, primordial non-Gaussianity and beyond^39,40. Yet, before this work, results were limited to shot power^24,25 and galaxy stacking⁴¹, both confined to small scales where line amplitudes do not generalize. The global strength of line backgrounds therefore remained unknown, leaving forecasts, sensitivity requirements and survey strategies uncertain. Our detections of mean CO and [C II] provide an empirical benchmark that directly closes this gap, establishing the line strengths needed to guide forecasts and instrument design across lines, frequencies and redshifts.

A growing set of experiments now targets CO and [C II]^{42,43,44,45,46,47,48}. Alongside our detections, Fig. 2 overlays atmospheric transmission⁴⁹, showing the frequency windows accessible from the ground, while a space mission could open the full parameter space⁵⁰. Our measurements provide empirical CO and [C II] line strengths and their evolution over a wide redshift range, closing the long-standing uncertainty that has limited LIM survey planning. Future experiments can therefore be designed and optimized using empirical targets rather than a broad span of model predictions. This establishes CO and [C II] as calibrated matter tracers, laying the empirical foundation for full three-dimensional LIM measurements and robust cosmological inference, guiding LIM into its next phase of scientific development.

Methods

Tomographic intensity mapping

Our analysis employs clustering tomography, a method initially introduced to analyse the UV background¹⁷ and later applied to the thermal SZ effect²⁰ and the CIB³, from which the intensity measurements used here are derived. The method cross-correlates diffuse intensity maps with spectroscopic reference samples, in this case 11 broadband maps from Planck, Herschel and the Infrared Astronomical Satellite^11,12,13,14 spanning 100 GHz–5 THz, with ~3 million galaxies and quasars from the Sloan Digital Sky Survey (SDSS)^{51,52,53,54,55,56}. On scales larger than individual dark matter halos, the fluctuations in intensity and galaxy density follow δ_I = b_Iδ_m and δ_g = b_gδ_m, with δ_m being the matter overdensity, and b_I (denoted b hereafter) and b_g are the respective bias factors. The tomographic cross-correlation amplitude measured in bins of redshift deprojects the extragalactic intensity I without responding to foregrounds:

$$\langle {\delta }_{I}{\delta }_{{\rm{g}}}\rangle =\frac{{\rm{d}}I}{{\rm{d}}z}b{b}_{{\rm{g}}}\langle {\delta }_{{\rm{m}}}^{2}\rangle ,$$

(5)

where dI/dz is intensity emitted per unit redshift interval and 〈δ_m²〉 is the matter autocorrelation. Dividing by the theoretical 〈δ_m²〉 under a fiducial Λ cold dark matter model cosmology⁵⁷, where Λ is the cosmological constant, and correcting for the independently measured b_g yields empirical estimates of the background as the bias-weighted differential intensity (dI/dz)b. Because the 11 intensity maps have different beam sizes, the minimum physical scale cut ranges from 0.74 to 3.71 Mpc (1.48 Mpc for the most constraining bands), while the maximum is fixed at 14.84 Mpc. In this clustering regime, the reference galaxies’ own emission contributes only at the percentage level or lower, ensuring robust estimates of (dI/dz)b. Adopting the cosmological radiative transfer solution in equation (9) of the formalism³, which is equivalent to a simple unit conversion, yields ϵ_νb in redshift–frequency space, shown in Fig. 1a.

As described in the companion work³, a small thermal SZ distortion from hot gas in the cosmic web, detected jointly at low frequencies²⁰, has already been removed from the multi-band, multi-epoch CIB (dI/dz)b data vector. Radio free–free and synchrotron emission are undetected even in our lowest-frequency band (100 GHz), with their inclusion yielding only upper limits. We therefore do not include them in this work.

Line sampling

The intensity tomography ϵ_νb in Fig. 1a probes the total CIB, including both the dust continuum and spectral lines. Supplementary Fig. 1 shows the coverage matrix for [C II] 158 μm and nine CO transitions (J = 1–0 to 9–8) across 16 redshift bins spanning 0 < z < 4.2, based on instrument bandpasses and precise cross-correlation redshifts from spectroscopic galaxy references. Line coverages by Planck and Herschel are indicated separately. Of the 157 data points used for spectral fitting (below 8-THz rest-frame in Fig. 1a), 17 of them sample [C II], 67 sample CO (31 include two transitions) and the remaining 73 are in line-free regions, providing equally important continuum anchors for line detection.

CO lines from J = 4–3 to 9–8 are fully sampled across 0 < z < 4.2. Limited by the lowest Planck band (100 GHz), lower-J transitions are not directly covered at high redshifts. Nonetheless, the turnover behaviour (Fig. 4a) of the CO excitation or the SLED is well sampled throughout, enabling robust simultaneous constraints on mass (low J) and excitation (high J). For [C II], coverage starts at z ≈ 0.4 and is complete at higher redshifts. This is faithfully reflected in the line luminosity posteriors (Fig. 1b), where the [C II] 68% interval widens towards z = 0.

CO excitation

To link the nine CO transitions, we seek a family of SLEDs, f_SLED, defined as the ratios of velocity-integrated line intensities I or comoving luminosity densities ρ:

$${f}_{SLED}\equiv {I}_{J(J-1)}/{I}_{10}=({\rho }_{CO,J(J-1)}/{\rho }_{CO,10})/J.$$

(6)

To parameterize f_SLED with a single variable controlling the level of excitation, we use results from numerical simulations of disk and merging galaxies with molecular-line radiative transfer⁵⁸, which identify the effective galaxy-scale star-formation surface density Σ_SFR as the primary regulator of CO excitation. Higher Σ_SFR corresponds to increased gas densities and temperatures, enhancing collisions between CO and H₂. Such conditions also entail stronger interstellar radiation fields, elevated cosmic-ray heating and enhanced turbulence, all contributing to higher CO excitation. To capture this behaviour, we adopt a modified version of the simulation-motivated functional form,

$$\log \,{f}_{{\rm{S}}{\rm{L}}{\rm{E}}{\rm{D}}}({\varSigma }_{{\rm{S}}{\rm{F}}{\rm{R}}})=A\,\frac{{\chi }^{B}}{1+{\chi }^{B}}+C,\,\,{\rm{w}}{\rm{i}}{\rm{t}}{\rm{h}}\,\chi =\log \,{\varSigma }_{{\rm{S}}{\rm{F}}{\rm{R}}}-{\chi }_{0},$$

(7)

where I has units of Jy km s⁻¹, Σ_SFR is in M_⊙ yr⁻¹ kpc⁻² with χ₀ = −2 set to the lowest log Σ_SFR considered, and A, B and C are J-dependent coefficients. This form reduces to the original relation⁵⁸ at χ^B ≪ 1 and saturates at χ^B ≫ 1, allowing a fully thermalized ceiling of f_SLED = J² to be imposed with improved numerical stability. We set this thermalized bound at log Σ_SFR = 4, although the exact anchor point is unimportant as long as it lies far above the Σ_SFR for typical starbursts.

The original simulated family of SLEDs was validated against available excitation measurements, especially for high-Σ_SFR systems such as M82, submillimeter-selected dusty galaxies, and quasars, but was not well tested for more typical star-forming galaxies at lower Σ_SFR out to high J (ref. ⁵⁸). For the cosmic CO background, however, the population is expected to be dominated by these more moderate, main-sequence-like systems⁵⁹, as commonly assumed in previous LIM studies^24,25,60, motivating an empirical recalibration. Multi-J observations of normal star-forming galaxies beyond the local universe now exist for three samples at z = 1–3: one ‘BzK’ sample⁶¹ and two selected in CO J = 2–1 and 3–2 (ref. ⁶²). Using their observed SLEDs and measured Σ_SFR values, we refit the coefficients in equations (7) by requiring smooth evolution with both J and Σ_SFR, matching these observed SLEDs within uncertainties, and satisfying the thermalized boundary condition. The best-fit coefficients are listed in Extended Data Table 1.

On the basis of this parameterization, Fig. 4a shows the allowed family of SLEDs across the full range of excitations (short-dashed lines) compared with the observed ones for the galaxy samples at z = 1–3 (refs. ^61,62), with their corresponding Σ_SFR shown in Fig. 4b. Relative to the original simulation output, our recalibration remains consistent at high Σ_SFR but yields a slower high-J decline at low Σ_SFR, in better agreement with the new measurements. This may reflect intrinsic Σ_SFR scatter or additional excitation mechanisms, both expected in the diverse population of galaxies contributing to the cosmic CO background. Unlike previous LIM studies with fixed SLEDs^24,25,60, our new parameterization, combined with the 11-band redshift tomography, enables an empirical estimate of CO excitation alongside the attempt for global CO background detection.

Full spectral model

Following the companion study³, we adopt a spectral model that provides an analytic description of two observables: (1) ϵ_νb = ϵ_ν(ν, z)b(z) (Fig. 1a) and (2) the CIB monopole I_ν(ν_obs) (ref. ¹⁵) (cyan points in Supplementary Fig. 2), which constrains the black curve in Fig. 2. The latter is a derived quantity and provides an integral constraint on ϵ_ν(ν, z) after converting units to intensity. The CIB emissivity comprises thermal dust continuum and line terms

$${\epsilon }_{\nu }(\nu ,z)={\epsilon }_{\nu ,cont}(\nu ,z)+{\epsilon }_{\nu ,line}(\nu ,z)$$

(8)

to be specified shortly. For the evolution of the light-weighted CIB bias factor, we adopt

$$b(z)={b}_{0}+{b}_{1}z+{b}_{2}{z}^{2},$$

(9)

jointly constrained by the two observables ϵ_νb and I_ν, one with bias, one without.

The continuum term ϵ_ν, cont is parameterized as a generalized greybody spectrum³, weighted by a range of dust properties to capture the diversity of galaxies contributing to the CIB:

$${\epsilon }_{\nu ,{\rm{c}}{\rm{o}}{\rm{n}}{\rm{t}}}(\nu ,z)=\frac{4{\rm{\pi }}{\rho }_{{\rm{d}}}(z)}{{\varSigma }_{{\rm{d}}}}\,[1-{{\rm{e}}}^{-\langle \tau \rangle (\nu ,z)}]\,\langle {B}_{\nu }\rangle (\nu ,z).$$

(10)

The key components are the following: ρ_d = ρ_d(a, b, c, d), the cosmic-dust mass density setting the overall normalization with four degrees of freedom (DOFs) for redshift evolution; Σ_d, the effective galaxy-scale dust surface density setting the transition wavelength between optically thin and thick regimes; 〈τ〉 = 〈τ〉(β₀, C_β, σ_β), the effective optical depth weighted by a distribution of dust opacity index β with evolving mean and scatter; 〈B_ν〉 = 〈B_ν〉(μ₀, C_μ, s_T, α_T), the Planck function weighted by a log-normal power-law dust-temperature distribution characterized by an evolving logarithmic mean 𝜇, scatter 𝑠_𝑇, and high-end power-law index 𝛼_𝑇. Full parameterizations for each term are given in equation (20), equations (13) and (14), and equations (17)–(19) of the companion paper³, respectively.

We model the line term ϵ_ν, line from CO and [C II] simply by describing its unique frequency–redshift patterns: the emissivity excess in the ith data point, at redshift zⁱ and rest-frame frequency νⁱ, sampled by a bandwidth $\Delta {\nu }^{i}=\Delta {\nu }_{obs}^{i}(1+{z}^{i})$, is

$$\begin{array}{rcl}{\epsilon }_{\nu ,\,line}^{i} & = & {\epsilon }_{\nu ,\,CII}^{i}+{\epsilon }_{\nu ,\,CO}^{i}\\ & = & \frac{{\rho }_{CII}({z}^{i})}{\Delta {\nu }^{i}}{A}_{CII}^{i}+\mathop{\sum }\limits_{J=1}^{9}\frac{{\rho }_{CO,JJ-1}({z}^{i})}{\Delta {\nu }^{i}}{A}_{CO,\,J}^{i},\end{array}$$

(11)

where A_CO, J and A_C II are binary activation vectors, with elements set to 1 if the line falls within the rest-frame bandwidth and 0 otherwise (Fig. 1a and Supplementary Fig. 1). The nine CO lines are connected by f_SLED, as parameterized in equations (7) with coefficients in Extended Data Table 1. We anchor CO (1–0) and [CII] to their local dust continua:

$$\begin{array}{rcl}{\rho }_{{\rm{C}}{\rm{O}},10}(z) & = & {R}_{{\rm{C}}{\rm{O}}}(z){\nu }_{{\rm{C}}{\rm{O}}}{\epsilon }_{\nu ,{\rm{c}}{\rm{o}}{\rm{n}}{\rm{t}}}({\nu }_{{\rm{C}}{\rm{O}}},z);\\ {\rho }_{{\rm{C}}\,\mathrm{II}}(z) & = & {R}_{{\rm{C}}\,\mathrm{II}}(z){\nu }_{{\rm{C}}\,\mathrm{II}}{\epsilon }_{\nu ,{\rm{c}}{\rm{o}}{\rm{n}}{\rm{t}}}({\nu }_{{\rm{C}}\,\mathrm{II}},z),\end{array}$$

(12)

where ν_CO and ν_C II are the rest frequencies for CO 1–0 and [C II], respectively, and R_CO(z) and R_C II(z) define the dimensionless equivalent widths quantifying the line-to-continuum ratios, to be constrained by the data. We allow R_CO, R_C II and the CO SLED parameter Σ_SFR to evolve with redshift, each with one additional DOF, giving six free parameters in total for ϵ_ν, line.

The full model has 21 free parameters. These include 12 for the CIB thermal dust continuum (a, b, c, d, μ₀, C_μ, s_T, α_T, Σ_d, β₀, C_β and σ_β), three for the bias evolution (b₀, b₁ and b₂) and six for CO and [C II] lines (R_CO, 0, C_CO, log Σ_SFR,2, ${C}_{{\Sigma }_{SFR}}$, ${R}_{CII,0}^{100}$ and ${C}_{CII}^{100}$). For convenience, in our fitting we use ${R}_{CII}^{100}=100\times {R}_{CII}$ to reflect the percentage-level [C II]-to-continuum ratio expected, and adopt logarithmic forms for a, Σ_d and Σ_SFR. Among these parameters, pairs of (X₀, C_X) describe the redshift evolution of each quantity $X\in \{\mu ,\beta ,{R}_{CII}^{100},{R}_{CO},\log \,{\Sigma }_{SFR}\}$ related to the spectral shape:

$$X(z)={X}_{0}+{C}_{X}\,\log (1+z).$$

(13)

Note that, as long as no informative prior is imposed, the anchor redshift is mathematically arbitrary and does not affect the inferred X(z) posteriors. This log–linear form is preferred over a power law for data-driven constraints, as it permits negative values (and sign flips) for noisy quantities such as ${R}_{CII}^{100}$ and R_CO, where line detections would otherwise be prior driven if non-negativity were enforced. All X are anchored at z = 0 via X₀, except for log Σ_SFR, which is anchored at z = 2 with the same log–linear form X(z) = X₂ + C_X[log(1 + z) − log 3], chosen only to test priors informed by observational results of CO excitation^61,62 at cosmic noon. The full parameter set is summarized in Extended Data Table 2.

Priors

Of the 21 model parameters, most could be constrained empirically, with only a few DOFs that are prior driven, as summarized in Extended Data Table 2. For thermal dust and bias parameters, we adopt the same priors as used before³, largely uninformative except for a Gaussian prior on log Σ_d, which sets the optically thin–thick transition at frequencies above CO and [C II] and thus does not affect line detections. The evolving b(z), assumed common to continuum and lines, is jointly constrained by ϵ_νb and I_ν, with additional assumptions: a Gaussian prior b₀ = 1 ± 0.1 for z = 0, corresponding to dark-matter halo masses of M_h = 10^12.5–10¹³ M_⊙ (ref. ⁶³), together with bounds b(z = 2) > 2.4 to exceed the bias of typical Lyman-break galaxies⁶⁴ and b(z = 3) < 4.81 to remain below the extreme M_h^5/3 scaling relevant to the thermal SZ effect²⁰.

For R_CO,0 and ${R}_{CII,0}^{100}$ we impose only broad bounds so that the detections are data driven. A weak Gaussian prior ${C}_{CII}^{100}=0\pm 2$ is applied to exclude extreme redshift evolution of [C II] relative to the continuum, while no such prior is placed on CO. For the CO SLED we adopt a Gaussian prior log Σ_SFR,2 = 0.3 ± 0.25 at z = 2, motivated by the galaxy samples used to calibrate the parameterization^61,62 (Fig. 4b), and require log Σ_SFR > −2 at all redshifts to avoid unphysical line ratios.

Fitting to data

We derive parameter posteriors using Bayesian inference, fitting the full continuum-plus-line model to the ϵ_νb and I_ν data vector and covariance matrix. The exploration of parameter space is performed with Markov chain Monte Carlo (MCMC) using the affine-invariant ensemble sampler implemented in the emcee package⁶⁵. The resulting posteriors are listed in Extended Data Table 2, with the joint covariances between line and nuisance parameters shown in the segment of the full corner plot in Supplementary Fig. 3.

The best-fit model projections in data space are shown as lines in Fig. 1a and Supplementary Fig. 2, which closely follow the data and provide a visual check of fit quality, especially for the continuum. Because CO and [C II] lines are much fainter, the overall goodness of fit is only modestly better than the continuum-only case in the companion study³. Nonetheless, the reduced χ² decreases from 1.49 to 1.40 despite six fewer DOFs, reflecting a measurable improvement, primarily from accounting for the low-frequency CO excess visible below ~500 GHz in Fig. 1a.

Both CO and [C II] are significantly detected (6.6σ and 3.0σ) on top of the CIB continuum. The empirically recovered CO SLED trend in Fig. 4a,b, showing increasing excitation with redshift, further supports the robustness of the CO detection. The 21-parameter posteriors fully describe the evolving CIB-plus-line spectrum and are propagated into derived quantities including the line luminosity densities (Fig. 1b), sky monopole brightness temperatures (Fig. 2), molecular gas history (Fig. 3) and star-forming gas diagnostics (Fig. 4).

Null test

To verify that the CO and [C II] detections are not spurious, we perform a null test by repeating the MCMC fit 50 times after randomly shuffling the line–redshift associations in the activation vectors A_CO, J and A_C II in equation (11). This destroys the physical frequency–redshift patterns while keeping all other inputs unchanged. The recovered line luminosity densities are consistent with zero, as shown in Supplementary Fig. 4, with dispersions smaller than the detections. This confirms that the signals arise only when the correct line placements are used.

Model-level preference

We now assess model selection together with parameter degeneracies. Because line luminosity densities are sub-percent relative to the CIB continuum, commonly used metrics such as the Akaike and Bayesian information criteria (AIC/BIC) were not ideal for testing whether adding lines improved the fit, as they are ‘amplitude weighted’ and therefore strongly driven by the continuum. This means that continuum residuals at frequencies unrelated to the lines—for example, near the CIB peak where the emissivity is highest—could saturate the Akaike and Bayesian information criteria and overshadow improvements from the faint emission lines. Nevertheless, adding lines does yield a modest reduction in χ². Much stronger support for the model-level preference comes from several direct physical arguments discussed next.

Compared with the continuum-only fit³, adding CO and [C II] leaves the posteriors for continuum parameters unchanged except for σ_β. Without lines, the low-frequency excess in Fig. 1a forces a large σ_β ≈ 0.71, broader than observed for diverse galaxy populations, and the 1σ lower tail extends to β_low ≈ 1, in tension with galaxy observations⁶⁶. With lines included, this excess is more correctly attributed to CO, σ_β drops to ~0.24, and the mean β ≈ 2, consistent with physical expectations. The continuum and lines are robustly separated with only a very weak degeneracy (correlation coefficient −0.14 between σ_β and R_CO,0 in Supplementary Fig. 3), thanks to strong leverage from the multiline CO pattern in redshift–frequency space. The continuum-plus-line model is therefore favoured by the data and more physically realistic.

Moreover, the presence of CO and [C II] emission at known rest-frame frequencies is a direct physical expectation from quantum mechanics, so including them defines the more appropriate model space. In this sense, our inference of known lines is not subject to the ‘look-elsewhere effect’ that would apply to blind searches over unknown frequencies. Within this physically preferred model, the data then constrain the line amplitudes, making the detections well grounded.

Monopole

We examine in detail the millimetre background monopoles from multiple LSS components, focusing on the CIB dust continuum and CO and [C II]. Supplementary Fig. 2 presents these monopoles in intensity units, closely corresponding to Extended Data Fig. 1 and the brightness-temperature representation in Fig. 2, and adds direct comparisons with existing measurements and model predictions. As shown there, the 100–545-GHz data points from the Far-InfraRed Absolute Spectrophotometer and Planck enter our spectral fit as integral constraints and are therefore consistent with the total-background posterior by construction. Supplementary Fig. 2 further shows that our empirical CO amplitude, combining nine transitions, agrees with SIDES⁵⁹, a widely used model, but disfavours more extreme scenarios⁶⁷ with suppressed low-J amplitudes and steep high-J decline. For [C II], our posterior is broadly consistent with recent predictions^59,67,68 but lies well below earlier claimed detections^69,70, which were probably contaminated by correlated CIB continuum⁷¹.

The CO background becomes increasingly important near and below 100 GHz and may influence precision studies of CMB spectral distortion and secondary anisotropies^67,72,73. Towards the low-frequency end, the picture becomes more complex. Radio free–free and synchrotron emission rise as modelled in SIDES⁵⁹ but remain negligible for our fit, which uses data above 100 GHz. The thermal SZ decrement, extrapolated from empirical constraints at higher frequencies²⁰, becomes stronger than the other components below ~80 GHz in Supplementary Fig. 2 (see also Fig. 2 and Extended Data Fig. 1). If the radio background catches up only slowly, this could produce a net negative LSS contribution, that is, an ‘absolute LSS decrement’, in front of the CMB spanning several tens of gigahertz.

Line-to-total infrared ratios

We evaluate the contributions of CO and [C II] to the total infrared luminosity density ρ_TIR in Supplementary Fig. 5a, finding that [C II] accounts for ~0.3% of ρ_TIR and CO for ~0.03%, in broad agreement with model expectations apart from some low-CO scenarios⁶⁷. This estimate is anchored to the ρ_TIR reported in equation (39) of the companion study³, for which we adopt a 10% uncertainty. Because the line contributions are small, the ρ_TIR from the continuum-only fit therein is nearly identical to that from the continuum-plus-line fit in this study.

Supplementary Fig. 5a also shows that the line-to-total ratios rise mildly with redshift. Supplementary Fig. 5b demonstrates that this evolution becomes nearly flat when the line luminosity densities are normalized by ρ_TIR/f_obs, where f_obs is the cosmic-dust-obscured fraction, which decreases towards higher redshift as UV emission from unobscured star formation becomes important relative to the CIB³. The resulting constant ratios of line-to-CIB+UV emission suggest that CO and [C II] trace the total SFR density more closely than the dust continuum. Producing CIB emission requires dust to be heated by young stars, but star formation alone does not ensure strong dust emission. In contrast, CO provides the fuel for star formation, and [C II] is excited as a direct consequence of it. These causal relations make the lines potentially more unbiased tracers of the total SFR in galaxy studies, especially in the early, dust-poor universe.

Equivalent widths

An alternative way to characterize CO and [C II] line strengths is through their rest-frame equivalent widths relative to the local CIB continuum at the line frequency, rather than to the integrated energy budget. LIM noise is complex and often foreground limited. If, however, residual CIB-continuum photon noise—an often overlooked component—dominates after standard high-pass filtering, the equivalent widths become especially informative, as they scale directly with the resulting signal-to-noise ratios.

To detect lines through their contrast with the continuum, our spectral framework naturally incorporates the dimensionless equivalent-width parameter R. We generalize R in equation (12), originally defined for CO 1–0 and [C II], to any line with rest frequency ν₀, luminosity density ρ_line and underlying ϵ_ν,cont(ν₀):

$$R\equiv \frac{{\rho }_{{\rm{l}}{\rm{i}}{\rm{n}}{\rm{e}}}}{{\nu }_{0}{\epsilon }_{\nu ,{\rm{c}}{\rm{o}}{\rm{n}}{\rm{t}}}({\nu }_{0})}.$$

(14)

The conventional wavelength and frequency equivalent widths, EW_λ and EW_ν, then follow directly as

$$\begin{array}{l}{\mathrm{EW}}_{\lambda }=R{\lambda }_{0};\\ {\mathrm{EW}}_{\nu }=R{\nu }_{0},\end{array}$$

(15)

where λ₀ = c/ν₀ and c is the speed of light.

Using our empirically constrained 21-parameter posterior, we compute R, EW_λ and EW_ν for the CO and [C II] backgrounds as functions of redshift in Supplementary Fig. 6. This complements the ρ_line in Fig. 1b. For CO, although ρ_line peaks at mid J, the equivalent widths increase monotonically towards low J owing to the declining CIB continuum. Conversely, because [C II] lies near the thermal CIB peak, its high ρ_line still translates into substantially smaller equivalent widths than CO 1–0 by about 3.5 dex in EW_λ and 1 dex in EW_ν, with the midpoint of 2.25 dex (a factor of ~200) in R. This underscores the added challenge for LIM experiments targeting [C II] and suggests that low-J CO may be more accessible targets instead. As an additional note, this multiline comparison also highlights R as a more fundamental equivalent-width definition, free of the floating anchors inherent in wavelength or frequency units and thus well suited for broader adoption in general astronomy applications.

CO-to-H₂ conversion factor

The ${\varOmega }_{{{\rm{H}}}_{2}}$ in Fig. 3 derived from CO 1–0 luminosity density (Fig. 1b) depends directly on α_CO. In the local universe, α_CO is known to increase towards lower metallicity, especially below one-third solar, due to reduced dust and gas shielding of CO over H₂ against far-UV radiation⁷⁴. Even at fixed metallicity, the absolute calibration carries a 30% uncertainty⁷⁵. Following common practice in high-redshift blank-field CO surveys (that is, those that do not preselect dusty starbursts) and in CO LIM studies, we adopt α_CO = 3.6 M_☉ (K km s⁻¹ pc²)⁻¹, consistent with the Milky Way value. Although an evolving α_CO might be expected given the lower metallicities of higher-redshift galaxies, dynamical mass measurements of BzK galaxies at z ≈ 1.5 favour a non-evolving, Milky Way-like value²¹. This suggests either mild metallicity evolution in typical star-forming galaxies or ISM conditions that compensate for reduced relative shielding. Nevertheless, α_CO remains a systematic uncertainty in estimating ${\varOmega }_{{{\rm{H}}}_{2}}$.

Despite these caveats, relative amplitudes of ${\varOmega }_{{{\rm{H}}}_{2}}$ in Fig. 3 are robust, as most data points adopt the same α_CO; the ratios are thus unaffected by this assumption and correspond to ratios of the underlying CO 1–0 luminosity densities. Two exceptions are xCOLD GASS²³ at z = 0, which employs a metallicity-dependent α_CO, and the ALMACAL CO absorber-based limits²⁷, which rely on a column-density version of the conversion factor. With a consistent α_CO applied to all other measurements, our main finding that intensity mapping reveals roughly twice the molecular gas mass already resolved in galaxies holds directly. Any future revision of α_CO would simply rescale all inferred ${\varOmega }_{{{\rm{H}}}_{2}}(z)$ in Fig. 3 but leave their ratios unchanged.

A varying α_CO would, however, affect the inferred t_dep and the star-formation law in Fig. 4c,d. A higher α_CO at early times would imply larger H₂ reservoirs for the same CO intensity, yielding longer t_dep and a shallower cosmic star-formation law deviating from the Kennicutt–Schmidt relation. Conversely, if we assume that cosmic star formation follows the Kennicutt–Schmidt law with a fixed slope near N ≈ 1.4, consistent with our best-fit N ≈ 1.5, then the effective α_CO would evolve only weakly with redshift, in line with our fiducial assumption.