Microwave background temperature at a redshift of 6.34 from H2O absorption

NOEMA observations

The target was observed in the 3-mm wavelength band 1 (rest-frame 400 μm) with NOEMA as part of project S20DA (Principal Investigators: D. A. Riechers, F. Walter). Three partially overlapping spectral setups were observed under good weather conditions between 26 July 2020 and 25 August 2020 with ten antennas in the most compact D configuration, using a bandwidth of 7.7 GHz (dual polarization) at 2-MHz spectral resolution per sideband. We also included previously published5 observations between 6 February 2012 and 31 May 2012 in the A and D configurations tuned to 110.128 and 113.819 GHz, respectively, and previously unpublished observations between 1 June 2012 and 4 June 2012 and on 10 July 2017 in the D configuration tuned to 78.544 and 101.819 GHz taken as part of projects V0BD, W058, and S17CC (Principal Investigator: D. A. Riechers), all using 3.6 GHz of bandwidth (dual polarization), yielding 21 observing runs in total. Nearby radio quasars were used for complex gain, bandpass and absolute flux calibration. The target was also observed in the 0.87-mm wavelength band 4 (rest-frame 122 μm) with NOEMA as part of project X0CC (Principal Investigator: D. A. Riechers). Observations were carried out during three observing runs with six antennas in the A and C configurations under good weather conditions between 4 December 2013 and 12 March 2015, with the band 4 receivers tuned to 335.5 GHz and using a bandwidth of 3.6 GHz (dual polarization). Nearby radio quasars were used for complex gain, bandpass and absolute flux calibration. The GILDAS package was used for data calibration and imaging. All 3-mm data were combined to a single visibility cube before imaging. Imaging was carried out with natural baseline weighting. The band 4 data were also imaged with Briggs robust weighting to increase the spatial resolution. A map of the continuum emission at the frequency of the H2O line was created by averaging the visibility data over a bandwidth of 2.04 GHz centred on the line. This range was chosen to avoid other lines in the bandpass. Continuum emission was subtracted from the H2O line cube in the visibility plane. Moment 0 images of the line absorption were created before and after continuum subtraction by integrating the signal over a bandwidth of 100 MHz, corresponding to 395 km s−1. The resulting r.m.s. noise levels are provided in Extended Data Fig. 2. We also make use of previously published5 rest-frame 158-μm NOEMA data, which were adopted without further modification.

Line and continuum parameters

The flux of the H2O(110–101) line was extracted by simultaneous Gaussian fitting of the line and continuum emission (including a linear term for the continuum) in the one-dimensional spectrum shown in Fig. 1, which was extracted from the image cube. The source is unresolved at the frequency of the H2O(110–101) line, such that the main uncertainties are due to the slope of the continuum emission and the appropriate fitting of other nearby lines, in particular, CO(5–4). The uncertainties in these parameters are part of the quoted uncertainties. We find a line peak flux of −818 ± 145 μJy at a line full width half maximum (FWHM) of 507 ± 111 km s−1, centred at a frequency of 75.8948 GHz (±46 km s−1; the calibration uncertainties on the line FWHM and centre frequency are negligible and that on the line peak flux is <10%—that is, minor compared with the measurement uncertainty). Given the rest frequency of the line of 556.9359877 GHz, this corresponds to a redshift of 6.3383, which is consistent with the systemic redshift of HFLS3 (z = 6.3335 and 6.3427 with uncertainties of ±14 and ±54 km s−1 at Gaussian FWHM of 243 ± 39 and 760 ± 152 km s−1, respectively, for the two velocity components detected in the 158-μm [CII] line)5. For comparison, the H2O(202–111) and H2O(211–202) emission lines in HFLS3 have FWHM of 805 ± 129 and 927 ± 330 km s−1, respectively5—that is, only marginally broader than the 110–101 line at the current measurement uncertainties. The continuum flux at the line frequency is 396 ± 15 μJy, corresponding to 48% ± 9% of the absorption-line flux (the relative flux calibration uncertainty between the line and continuum emission is negligible). We also measured the 335.5-GHz continuum flux by two-dimensional fitting to the continuum emission in the visibility plane. We find a flux of 33.9 ± 1.1 mJy, which agrees with previous lower-resolution observations at the same wavelength5. The major (minor) axis FWHM diameter of the source is 0.617 ± 0.074 arcsec (0.37 ± 0.20 arcsec). This yields the physical source size quoted in the main text at the redshift of HFLS3.

Brightness temperature contrast

The H2O(110–101) line leads to a decrement in continuum photons from the starburst and, as such, is observed as a lack of continuum emission at its frequency at the position of the starburst. It therefore appears as negative flux in an image where starburst continuum emission has been subtracted. In addition, (sub)millimetre-wavelength interferometric images reveal structure against a flat sky background defined by the large-scale CMB surface brightness, which the interferometer does not detect itself due to its limited spatial sampling. Therefore the fraction of the signal due to the decrement in CMB photons at the position of the starburst not only appears as negative flux without subtracting any further signal but it also corresponds to a lack of continuum emission at the line frequency in practice. As the mere presence of an absorption-line signal stronger than the measured continuum emission implies absorption against the CMB, this interpretation is not limited by uncertainties in the galaxy continuum flux or uncertainties in the absolute flux calibration.

Line-excitation modelling

RADEX is a radiative transfer program to analyse interstellar line spectra by calculating the intensities of atomic and molecular lines, assuming statistical equilibrium and considering collisional and radiative processes, as well as radiation from background sources. Optical depth effects are treated with an escape probability method8. Studies of nearby star-forming galaxies show that the observed absorption strengths of the ground-state H2O and H2O+ transitions are due to cooler gas that is located in front of, and irradiated by, a warmer background source that is emitting the infrared continuum light that also excites the higher-level H2O emission lines11,29. We therefore adopt the same geometry for the modelling in this work, which is adequately treated within RADEX (that is, treating the dust continuum plus the CMB as background fields for the absorbing material)8. The dust continuum emission is modelled as a grey body with treating Tdust, βIR and the wavelength where the dust optical depth reaches unity as free-fitting parameters for each dust continuum size and TCMB sampled by the models. The observed spectral energy distribution of HFLS3, including all literature5 photometry and the measurements presented in this work, is then treated as the contrast between the dust continuum and CMB background fields, such that the resulting fit parameters for the dust continuum source change with TCMB in a self-consistent manner. In the RADEX models, we derive the H2O peak absorption depth into the CMB. We then multiply the best matching peak absorption depth found by RADEX with a Gaussian matched to the fitted line centroid and line width obtained from the observed line profile in Fig. 2 to determine the model line profile. In this approach, the shallower absorption in the line wings either corresponds to a lower filling factor of the H2O layer at the corresponding velocities or to lower H2O column densities. Although collisions of H2O molecules with H2 is another mechanism that can modify the level populations especially at very high gas densities (which is an important mechanism for the cooling of low-excitation-temperature transitions of molecules like H2CO to below TCMB)12,30, the RADEX models show that they do not affect our findings (see Fig. 3c). We therefore adopt models with essentially no collisions by assuming a very low gas density of n(H2) = 10 cm−3. We then compare our findings to those obtained when adopting conditions that are similar to those found in local starburst galaxies11 and to those found for high-density environments with n(H2) > 105 cm−3. The cross sections for collisions out of the 101 level are always larger than those out of the 110 level, independent of the collision partner and the temperature at which the collisions take place31,32,33. Therefore collisions cannot be responsible for an over-proportional de-population of the 110 level relative to the 101 ground state, and the net effect of including collisions is a decrease in the absorption depth into the CMB by reducing the TCMB − Tex temperature difference at very high gas densities compared with cases without collisions. For reference, the effect of collisions on the determination of TCMB is negligible for the typical conditions found in local starbursts (that is, n(H2) ~ 104 cm−3; Tkin = 20–180 K)11 and only starts to have an impact for very high densities n(H2) > 105 cm−3. For a given continuum source size, the constraints on TCMB would therefore be tighter (that is, would more quickly become inconsistent with the observations) for the high-density case than for the case without collisions, such that the latter approach is more conservative (see Fig. 3b). The overall impact of collisional excitation would therefore be more stringent requirements on the source size, covering fraction and water column, such that their inclusion would only further strengthen our conclusions. We note that this is the opposite effect to the case of the studies of ultraviolet lines3,17,19,20,21,22,23,24,25,26,27,28, where neglecting collisional excitation results in less conservative constraints on TCMB. If we were to assume that the H2O absorption were to emerge from within the infrared continuum-emitting region, a larger source size would probably be required to obtain the same absorption-line strength due to a reduced effective radiation field strength from the starburst. Previous modelling attempts of nearby galaxies assuming such geometries have not been able to produce H2O(110–101) line absorption on the scales necessary to explain the observations of HFLS3, which may indicate that even more complex assumptions would be required11. Thus, the resulting constraints would, once again, be less conservative, perhaps acting in a similar manner as the high-density case. Excluding both of these effects from the models leads to a maximally conservative estimate of TCMB and its uncertainties. Assuming a plane-parallel or similar geometry instead of a spherical geometry would only have a minor impact on our findings8. The models shown in Fig. 3 assume a filling factor of unity, which is the most conservative possible assumption. A more clumpy geometry with a lower covering fraction remains possible for all TCMB values for which the predicted absorption strength exceeds the observed value (see shaded regions in Fig. 3b). For reference, the minimum covering fractions consistent with the continuum size at the observed signal strength are shown for the different cases considered in Fig. 3. The line absorption is also found to be optically thick, with an optical depth of τH2O = 21.1 for the solution shown in Fig. 2b. To determine the redshift above which the effect becomes observable (Fig. 3c), we fixed r108μm, Tdust, βIR and Mdust to the observed values and the H2O column density to the value corresponding to the model spectrum. H2O line absorption into the dust continuum of HFLS3 would already become visible at z > 2.9, but absorption into the CMB only becomes observable at z > 4.5 (or higher for H2 densities of >105 cm−3). These values account for changes in the shape of the dust grey-body spectrum (that is, changes in the relative availability of 538-μm and 108-μm photons) due to changes in TCMB with redshift. To better quantify the impact of different modelling parameters, we have varied Tdust and βIR beyond their previously estimated uncertainties (nominal reference values without considering variations in TCMB from the literature are Tdust = ({63.3}_{-5.8}^{+5.4}) K and βIR = ({1.94}_{-0.09}^{+0.07}))5,6. This is necessary because both parameters are dependent on the varying TCMB in our models (and therefore are changing parameters in Fig. 3b, c), such that their true uncertainties need to be re-evaluated. We independently varied βIR in the 1.6–2.4 range and Tdust in the ±20 K range as functions of TCMB around the best-fit values. This shows that βIR > 2.0 and Tdust lower by more than 10 K from the best fits yield very poor fits to the spectral energy distribution data, whereas ΔβIR > −0.1 below the best-fit value would require a larger continuum size than the measured r108μm + 1σ and therefore are disfavoured by the size constraint. Excluding these ranges, the extrema across this entire range would extend the uncertainty range in the predicted TCMB by only −1.7 and +5.4 K and −0.8 and +4.4 K for the r108μm + 1σ and r108μm + 2σ cases, respectively. For comparison, the difference between the +1σ and +2σ uncertainty ranges is −3.6 and +3.8 K). This shows that the impact of the uncertainties in the dust spectral energy distribution fitting parameters on those in TCMB are subdominant to those in the continuum size measurement. Conversely, we have studied the impact of changes in TCMB on the best-fit Tdust and βIR. For the values corresponding to r108μm + 1σ and r108μm + 2σ ranges, Tdust typically changes by <0.5 K and βIR typically changes by <0.1–0.2 when varying the parameters independently. These changes are larger than the actual uncertainties, because the fit to the dust spectral energy distribution becomes increasingly poorer with these single-parameter variations. At the same time, these changes are subdominant to those induced by changes in dust continuum size within the +1σ and +2σ uncertainty ranges, which is consistent with our other findings.

Other H2O transitions in HFLS3

Five H2O lines were previously detected towards HFLS3 (202–111, 211–202, 312–221, 312–303 and 321–312) and two additional lines were tentatively detected (413–404 and 422–413)5. The Jup = 3 transitions are due to ortho-H2O and all other transitions are due to para-H2O. All of these transitions appear in emission. Given the high critical densities of these transitions, our RADEX models cannot reproduce the strength of these lines as the same time as the observed ortho-H2O(110–101) absorption strength, which suggests that they emerge from different gas components. For reference, to reproduce the strength of the H2O(211–202) in Fig. 1 alone with collisional excitation, n(H2) = 2 × 107 cm−3 and Tkin = 200 K would be required, but the H2O(110–101) would no longer appear in absorption against the CMB if it were to emerge from the same gas component. This is consistent with the picture that the H2O absorption is due to a cold gas component along the line of sight to the warm gas that gives rise to the emission lines11. Observations of the para-H2O(111–000) ground state do not currently exist for HFLS3, but our models do not show this line in absorption towards the CMB.

Origin of the lower and upper limits on T

Our models show that the lower limit on TCMB at a given redshift based on the observed H2O absorption is due to the minimum ‘seed’ level population due to the CMB black-body radiation field. To determine a conservative lower limit, we have calculated models with continuum sizes up to r108μm = 5 kpc (see Fig. 3b), corresponding to a +7.5σ deviation from the observed continuum size, and recorded the temperatures at which such weakly constrained models turn into absorption. We find that this results in a lower limit of TCMB > 7–8 K, independent of the model assumptions. This finding alone does not explain the existence of an upper limit in Fig. 3b. For a given size of the dust continuum emission, an increase in TCMB also requires an increase in Mdust to still reproduce the observed dust spectral energy distribution, which leads to an effective increase in the dust optical depth at a given wavelength. The result of a rising optical depth is that the grey-body spectrum between 538 and 108 μm increasingly resembles a black-body spectrum and, hence, a decrease in the H2O absorption against the CMB. This effect is responsible for the upper limit in allowed TCMB for a given dust continuum size and absorption strength.

Uncertainties of T
CMB measurements

The uncertainties shown for the literature data in Fig. 4 are adopted from the literature sources without modification, and they typically represent the statistical uncertainties from the individual measurements or sample averages. Individual cluster measurements of the thermal SZ effect may be affected by dust associated with foreground galaxies or the Milky Way, the galaxy clusters or background galaxies that may be amplified by gravitational lensing, uncertainties in the reconstruction of the Compton-y parameter maps due to flux uncertainties, radio emission due to active galactic nuclei and/or relics, the kinetic and relativistic SZ effects, and general bandpass and calibration uncertainties17. Furthermore, uncertainties on the cluster geometry—and therefore line-of-sight travel distance of the CMB photons through the cluster—and on the temperature of the intra-cluster gas limit the precision of individual SZ measurements. Sample averages may also be affected by systematics in the stacking procedures. Individual data points deviate by up to at least two standard deviations from the trend, which may indicate residual uncertainties beyond the statistical error bars provided, such that the error bars shown in Fig. 4 are underestimated. The main source of uncertainty for the ultraviolet absorption-line-based measurements are due to the assumption of no collisional excitation, which is not taken into account in the statistical uncertainties shown in Fig. 4. Attempts to take this effect into account appear to suggest substantially larger uncertainties than indicated by individual error bars27 (Fig. 4). To expand on earlier estimates21, we have calculated RADEX models for typical Tkin, n(H) and column densities found from [CI] measurements in the diffuse interstellar medium34, which suggests that collisional excitation contributes to the predicted Tex of the lower fine-structure transition. Although we show the original unmodified data, the ultraviolet-based measurements are therefore subject to uncertainties due to model-dependent excitation corrections in addition to the statistical uncertainties. Furthermore, the fine-structure levels of tracers like the [CI] lines can be excited by ultraviolet excitation and following cascades. To constrain TCMB based on these measurements, the kinetic temperature, particle density and local ultraviolet radiation field must be known, and are typically determined based on tracers other than the species used to constrain TCMB. Also, some measurements are based on spectrally unresolved lines, which limits the precision of kinetic temperature measurements based on thermal broadening21. Owing to these uncertainties, the ultraviolet absorption-line-based measurements are probably consistent with the standard ΛCDM value, but they do not constitute a direct measurement of TCMB without notable further assumptions. For reference, the median TCMB/(1 + z) estimate based on the [CI] measurements alone (excluding upper limits) is 3.07 K, with a median absolute deviation of 0.09 K and a standard deviation of 0.31 K. Therefore the current sample median deviates from the ΛCDM value by about one standard deviation. A combination of the (uncorrected) measurements based on CO, [CI] and [CII] provides a median value of 2.84 K, with a median absolute deviation of 0.15 K and a standard deviation of 0.25 K. This highlights the importance of the corrections discussed above and in the literature and the value of measurements with systematic uncertainties that differ from this method to obtain a more complete picture. The main source of uncertainty of the H2O-based measurements, beyond the caveats stated in the line-excitation-modelling section, are the statistical uncertainties on the source size, the lack of a direct measurement of the absorbing H2O column density, variations in the dust mass absorption coefficient and the filling factor. Given the high metallicity suggested by other molecular line detections, the limitation to high filling factors due to the source size and the constraint on the gas mass from dynamical mass measurements, the main source of uncertainty resides in the source size due to limited spatial resolution in the current data. As such, major improvements should be possible by obtaining higher, (sub-)kpc resolution (that is, <0.2”) imaging with the Atacama Large Millimeter/submillimeter Array (ALMA; for other targets) and planned upgrades to NOEMA, and, in the future, with the Next Generation Very Large Array (ngVLA). Statistical uncertainties will also be greatly reduced by observing larger samples of massive star-forming galaxies over the entire redshift range where measurements are possible, closing the gap to SZ-based studies, which are currently limited to z < 1. The resulting improvement in precision will provide the constraints that are necessary to confirm or challenge the evolution of the CMB temperature with redshift predicted by standard cosmological models.

Accessibility of the line signal

The frequency range currently covered by NOEMA is 70.4–119.3, 127.0–182.9 and 196.1–276.0 GHz (with greatly reduced sensitivity above about 115 and 180 GHz in the first two frequency ranges). ALMA covers the 84–500-GHz range with gaps at 116–125 and 373–385 GHz, with a future extension down to 65 GHz (with greatly reduced sensitivity below about 67 GHz). The ngVLA is envisioned to cover the 70–116-GHz range. Excluding regions of poor atmospheric transparency, the H2O(110–101) line is therefore observable in these frequency ranges at redshifts of z = 0.1–0.4, 0.5–2.0, 2.1–3.4 and 3.8–6.9 in principle, but the detectability of the line in absorption against the CMB is probably limited to the z ~ 4.5–6.9 range if the spectral energy distribution shape of HFLS3 is representative. At lower frequencies, the Karl G. Jansky Very Large Array and, in the future, ALMA and the ngVLA also provide access to the <52-GHz range, such that the signal also becomes observable at z > 9.7 in principle. In conclusion, the absorption of the ground-state H2O transition against the CMB identified here could be traced from the ground towards star-forming galaxies across most of the first approximately 1.5 billion years of cosmic history.

Detectability of the line signal for different spectral energy distribution shapes

To investigate whether the effect is expected to be detectable for different galaxy populations, we have applied our modelling to the z = 3.9 quasar APM 08279+5255, for which the dust spectral energy distribution is composed of a dominant 220-K dust component and a weaker 65-K dust component, contributing only 10–15% to the far-infrared luminosity35,36,37,38,39,40,41,42,43,44,45,46. The models suggest that the line is expected to occur in emission and that it would not be expected to be detectable in absorption at any redshift out to at least z = 12 in galaxies with similar dust spectral energy distributions. Other far-infrared-luminous, high-redshift, active galactic nucleus host galaxies typically show a stronger relative contribution of their lower-temperature dust components, such that the effect may remain detectable in less extreme cases. For galaxies with lower dust temperatures than HFLS3, the effect may be present even at lower redshifts, but is typically expected to be weaker in general and to disappear at redshifts where TCMB approaches their Tdust. For a dust spectral energy distribution shape resembling the central region of the Milky Way but otherwise similar properties, the effect is expected to be reduced by more than two orders of magnitude at its redshift peak, and to become virtually unobservable at the redshift of HFLS3. Thus, dusty starburst galaxies appear to be some of the best environments to detect the effect.

Derivation of equation of state parameters

To determine the adiabatic index, we assume a standard Friedmann–Lemaitre–Robertson–Walker cosmology with zero curvature and a matter-radiation fluid that follows the standard adiabatic equation of state quoted in the main text. This would correspond to a redshift scaling TCMB(z) = TCMB(z = 0)*(1 + z)3(γ − 1) in the presence of a dark energy density that does not scale with redshift. The dark energy density is parameterized to scale with a power law (1 + z)m, where m = 0 corresponds to a cosmological constant. With standard assumptions, this yields a redshift scaling of TCMB (ref. 15):

$${T}_{CMB}(z)={T}_{CMB}(z=0){(1+z)}^{3(gamma -1)}{left[frac{(m-3{varOmega }_{m,0})+m{(1+z)}^{(m-3)}({varOmega }_{m,0}-1)}{(m-3){varOmega }_{m,0}}right]}^{(gamma -1)}$$

and an effective dark energy equation of state Pde = weffρde, where the effective equation of state parameter weff = (m/3) − 1. This fitting function is used here with a canonical value of Ωm,0 = 0.315 (ref. 4). The uncertainty of Ωm,0 is small compared with all other sources of uncertainty and, hence, is neglected. All data used in the fitting are provided in Extended Data Table 1 (refs. 36,37,38,39,40,41,42,43,44,45,46).

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