Abstract
The mismatch between the locally measured expansion rate of the universe and the one inferred from the cosmic microwave background measurements by Planck in the context of the standard ΛCDM, known as the Hubble tension, has become one of the most pressing problems in cosmology. A large number of amendments to the ΛCDM model have been proposed in order to solve this tension. Many of them introduce new physics, such as early dark energy, modifications of the standard model neutrino sector, extra radiation, primordial magnetic fields or varying fundamental constants, with the aim of reducing the sound horizon at recombination r_{⋆}. We demonstrate here that any model which only reduces r_{⋆} can never fully resolve the Hubble tension while remaining consistent with other cosmological datasets. We show explicitly that models which achieve a higher Hubble constant with lower values of matter density Ω_{m}h^{2} run into tension with the observations of baryon acoustic oscillations, while models with larger Ω_{m}h^{2} develop tension with galaxy weak lensing data.
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Introduction
Decades of progress in observational and theoretical cosmology have led to the consensus that our universe is well described by a flat Friedman–Robertson–Lemaitre metric and is currently comprised of around 5% baryons, 25% cold dark matter (CDM), and 70% dark energy in its simplest form—the cosmological constant Λ. Although this ΛCDM model fits many observations exquisitely well, its prediction for the presentday cosmic expansion rate, H_{0} = 67.36 ± 0.54 km/s/Mpc^{1}, based on precise cosmic microwave background (CMB) radiation observations by the Planck satellite, do not compare well with direct measurements of the Hubble constant. In particular, the Supernovae H0 for the Equation of State (SH0ES) collaboration^{2}, using Cepheid calibrated supernovae Type Ia, finds a much higher value of H_{0} = 73.5 ± 1.4 km/s/Mpc. This 4.2σ disagreement, known as the “Hubble tension”, has spurred much interest in modifications of the ΛCDM model capable of resolving it (cf.^{3} for a comprehensive list of references). Several other determinations of H_{0}, using different methods, are also in some degree of tension with Planck, such as the Megamaser Cosmology Project^{4} finding 73.9 ± 3.0 km/s/Mpc or H0LiCOW^{5} finding \(73.{3}_{1.8}^{+1.7}\) km/s/Mpc. It is worth noting that a somewhat lower value of 69.8 ± 2.5 km/s/Mpc was obtained using an alternative method for calibrating SNIa^{6}.
Among the most precisely measured quantities in cosmology are the locations of the acoustic peaks in the CMB temperature and polarization anisotropy spectra. They determine the angular size of the sound horizon at recombination,
with an accuracy of 0.03%^{1}. The sound horizon r_{⋆} is the comoving distance a sound wave could travel from the beginning of the universe to recombination, a standard ruler in any given model, and D(z_{⋆}) is the comoving distance from a presentday observer to the last scattering surface, i.e., to the epoch of recombination. D(z_{⋆}) is determined by the redshiftdependent expansion rate H(z) = h(z) × 100 km/s/Mpc which, in the flat ΛCDM model, depends only on two parameters (see Methods for details): Ω_{m}h^{2} and h, where Ω_{m} is the fractional matter energy density today and h = h(0) = H_{0}/100 km/s/Mpc. Thus, given r_{⋆} and an estimate of Ω_{m}h^{2}, one can infer h from the measurement of θ_{⋆}. Using the Planck best fit values of Ω_{m}h^{2} = 0.143 ± 0.001 and r_{⋆} = 144.44 ± 0.27 Mpc, obtained within the ΛCDM model^{1}, yields a Hubble constant significantly lower than the more direct local measurements.
If the value of the Hubble constant was the one measured locally, i.e., h ≈ 0.735, it would yield a much larger value of θ_{⋆} unless something else in Eq. (1) was modified to preserve the observed CMB acoustic peak positions. There are two broad classes of models attempting to resolve this tension by introducing new physics. One introduces modifications at late times (i.e., lower redshifts), e.g., by introducing a dynamical dark energy or new interactions among the dark components that alter the Hubble expansion to make it approach a higher value today, while still preserving the integrated distance D in Eq. (1). In the second class of models, the new physics aims to reduce the numerator in Eq. (1), i.e., modify the sound horizon at recombination.
Late time modifications based on simple phenomenological parameterizations tend to fall short of fully resolving the tension^{7}. This is largely because the baryon acoustic oscillation (BAO) and and supernovae (SN) data, probing the expansion in the 0 ≲ z ≲ 1 range, are generally consistent with a constant dark energy density. One can accommodate a higher value of H_{0} by making parameterizations more flexible, as e.g., in^{8,9}, that allow for a nonmonotonically evolving effective dark energy fluid. Such nonmonotonicity tends to imply instabilities within the context of simple dark energy and modified gravity theories^{10} but can, in principle, be accommodated within the general Horndeski class of scalartensor theories^{11}.
Earlytime solutions aim to reduce r_{⋆} with essentially two possibilities: (i) a coincidental increase of the Hubble expansion around recombination or (ii) new physics that alters the rate of recombination. Proposals in class (i) include the presence of early dark energy^{12,13,14,15,16,17}, extra radiation in either neutrinos^{18,19,20,21} or some other dark sector^{22,23,24,25,26,27}, and dark energy–dark matter interactions^{28}. Proposals in class (ii) include primordial magnetic fields^{29}, nonstandard recombination^{30}, or varying fundamental constants^{31,32}. In this work we show that any earlytime solution which only changes r_{⋆} can never fully resolve the Hubble tension without being in significant tension with either the weak lensing (WL) surveys^{33,34} or BAO^{35} observations.
Results and discussion
The acoustic peaks, prominently seen in the CMB anisotropy spectra, are also seen as BAO peaks in the galaxy power spectra and carry the imprint of a slightly different, albeit intimately related, standard ruler—the sound horizon at the “cosmic drag” epoch (or the epoch of baryon decoupling), r_{d}, when the photon drag on baryons becomes unimportant. As the latter takes place at a slightly lower redshift than recombination, we have r_{d} ≈ 1.02r_{⋆} with the proportionality factor being essentially the same in all proposed modified recombination scenarios. More importantly for our discussion, the BAO feature corresponds to the angular size of the standard ruler at z ≪ z_{⋆}, i.e., in the range 0 ≲ z ≲ 2.5 accessible by galaxy redshift surveys. For the BAO feature measured using galaxy correlations in the transverse direction to the line of sight, the observable is
where z_{obs} is the redshift at which a given BAO measurement is made. For simplicity, we do not discuss the line of sight and the “isotropic” BAO measurements^{36} here, but our arguments apply to them as well. It is well known that BAO measurements at multiple redshifts provide a constraint on r_{d}h and Ω_{m}.
In any particular model, r_{⋆} (and r_{d}) is a derived quantity that depends on Ω_{m}h^{2}, the baryon density and other parameters. However, in this work, for the purpose of illustrating trends that are common to all models, we treat r_{⋆} as an independent parameter and assume that no new physics affects the evolution of the universe after recombination.
Without going into specific models, we now consider modifications of ΛCDM which decrease r_{⋆}, treating the latter as a free parameter and taking r_{d} = 1.0184r_{⋆}. The relation between r_{⋆} and r_{d} in different models that reduce the sound horizon is largely the same as the one in ΛCDM, hence we fix it at the Planck best fit ΛCDM value. For a given Ω_{m}h^{2}, Eq. (1) defines a line in the r_{d}–H_{0} plane, and since Eqs. (1) and (2) are the same in essence, a BAO measurement at each different redshift also defines a respective line in the r_{d}–H_{0} plane. However, the significant difference between z_{⋆} and z_{obs} results in different slopes of the respective r_{d}(h) lines (see Methods for details), as illustrated in Fig. 1. The latter shows the r_{d}(h) lines from two different BAO observations, one at redshift z = 0.5 and another at z = 1.5, at Ω_{m}h^{2} fixed to the Planck best fit ΛCDM value of 0.143, and the analogous lines defined by the CMB acoustic scale plotted for three values of Ω_{m}h^{2}: 0.143, 0.155, and 0.167. Both lines correspond to transverse BAO measurements. Slopes derived from the line of sight and isotropic BAO at the same redshift would be different, but the trend with increasing redshift is the same. The lines are derived from the central observational values and do not account for the uncertainties in \({\theta }_{\perp }^{{\rm{BAO}}}\) and θ_{⋆} (although the uncertainty in θ_{⋆} is so tiny that it would be difficult to see by eye on this plot). As anticipated, the slope of the r_{d}(h) lines becomes steeper with increased redshift.
Also shown in Fig. 1 are the marginalized 68% and 95% confidence levels (CL) derived from the combination of all presently available BAO observations in a recombinationmodelindependent way, namely, while treating r_{d} as an independent parameter (see^{37} and Methods for details). The red contours show the ΛCDM based constraint from Planck, in good agreement with BAO at H_{0} ≈ 67 km/s/Mpc, but in tension with the SH0ES value shown with the gray band. In order to reconcile Planck with SH0ES solely by reducing r_{d}, one would have to move along one of the CMB lines. Doing it along the line at Ω_{m}h^{2} = 0.143 would quickly move the values of r_{d} and H_{0} out of the purple band, creating a tension with BAO. Full consistency between the observed CMB peaks, BAO and the SH0ES Hubble constant could only be achieved at a higher value of Ω_{m}h^{2} ≈ 0.167. However, unless one supplements the reduction in r_{d} by yet another modification of the model, such high values of Ω_{m}h^{2} would cause tension with galaxy WL surveys such as the Dark Energy Survey (DES)^{33} and the KiloDegree Survey (KiDS)^{34}, which we illustrate next.
DES and KiDS derived strong constraints on the quantity \({S}_{8}\,\equiv\, {\sigma }_{8}{({{{\Omega }}}_{m}/0.3)}^{0.5}\), where σ_{8} is the matter clustering amplitude on the scale of 8 h^{−1} Mpc, as well as Ω_{m}. The value of S_{8} depends on the amplitude and the spectral index of the spectrum of primordial fluctuations, which are welldetermined by CMB and have similar best fit values in all modified recombination models. S_{8} also depends on the net growth of matter perturbations which increases with more matter, i.e., a larger Ω_{m}h^{2}.
The values of S_{8} and Ω_{m} obtained by DES and KiDS are already in slight tension with the Planck best fit ΛCDM model, and the tension between KiDS and Planck is notably stronger than that between DES and Planck. Increasing the matter density aggravates this tension – a trend that can be seen in Fig. 2. The figure shows the 68% and 95% CL joint constraints on S_{8}Ω_{m} by DES supplemented by the Pantheon SN sample^{38} (which helps by providing an independent constraint on Ω_{m}), along with those by Planck within the ΛCDM model. The purple contours (Model 2) correspond to the model that can simultaneously fit BAO and CMB acoustic peaks at Ω_{m}h^{2} = 0.155, i.e., the model defined by the overlap between the BAO band and the \({\theta }_{\star }^{(2)}\) (blue dashed) line in Fig. 1. The green contours (Model 3) are derived from the model with Ω_{m}h^{2} = 0.167 corresponding to the overlap region between the \({\theta }_{\star }^{(3)}\) (green dotted) line and the BAO and SH0ES bands in Fig. 1 (see Methods for details). The figure shows that when attempting to find a full resolution of the Hubble tension, with CMB, BAO, and SH0ES in agreement with each other, one exacerbates the tension with DES and KiDS.
We note that there is much more information in the CMB than just the positions of the acoustic peaks. It is generally not trivial to introduce new physics that reduces r_{⋆} and r_{d} without also worsening the fit to other features of the temperature and polarization spectra^{39,40}. Our argument is that, even if one managed to solve the Hubble tension by reducing r_{⋆} while maintaining a perfect fit to all CMB data, one would still necessarily run into problems with either the BAO or WL.
Surveying the abundant literature of the proposed earlytime solutions to the Hubble tension, one finds that the above trends are always confirmed. Figure 3 shows the best fit values of r_{d}h, H_{0}, and S_{8} in models from Refs. ^{13,14,18,23,24,28,29,30,32}. Note that there are other proposed earlytime solutions to the Hubble tension. Figure 3 only shows the models for which explicit estimates of H_{0}, Ω_{m}h^{2}, S_{8}, and possibly r_{d}h were provided. One can see that, except for the model represented by the red dot at the very right of the plot, corresponding to the strongly interacting neutrino model of^{18}, solutions requiring low Ω_{m}h^{2} are in tension with BAO, whereas solutions with higher Ω_{m}h^{2} are in tension with DES and KiDS. This latter tension was previously observed and extensively discussed in the context of the early dark energy models^{41,42,43,44,45,46}. As we have shown in this paper, it is part of a broader problem faced by all proposals aimed at reducing the Hubble tension in which the main change amounts to a reduction of r_{d}.
In most of the models represented in Fig. 3, the effect of introducing new physics only amounts to a reduction in r_{d}. We note that, in any specific model of a reduced r_{d}, the best fit values of other cosmological parameters also change, which can affect the quality of the fit to various datasets. However, such changes, e.g., in the best fit value of the spectral index n_{s} which affects S_{8}, tend to be small for the models studied in the literature and have a minor impact compared to the effect of reducing r_{d}, which is a prerequisite for reconciling CMB with SH0ES. As we have argued, this will necessarily limit their ability to address the Hubble tension while staying consistent with the large scale structure data. Resolving the Hubble tension by new earlytime physics without creating other observational tensions requires more than just a reduction of the sound horizon. This is exemplified by the interacting dark matterdark radiation model^{25} and the neutrino model^{18} proposed as solutions. Here, extra tensions are avoided by supplementing the reduction in the sound horizon due to extra radiation by additional exotic physics: dark matterdark radiation interactions in the first case and neutrino selfinteractions and nonnegligible neutrino masses in the second case. Consequently, with so many parameters, the posteriori probabilities for cosmological parameters are highly inflated over those for ΛCDM. It is not clear how theoretically appealing such scenarios are, and the model in^{18} seems to be disfavored by the CMB polarization data.
In conclusion, we have argued that any model which tries to reconcile the CMB inferred value of H_{0} with that measured by SH0ES by only reducing the sound horizon automatically runs into tension with either the BAO or the galaxy WL data. While we do not expect our findings to be surprising for the majority of the community, the novelty of our result is in isolating and clearly stating the essence of the problem—that the slopes of the r_{⋆}–H_{0} degeneracy lines for BAO and CMB are vastly different, thus making it impossible to reconcile CMB with SH0ES by reducing r_{⋆} without violating BAO. We believe this very simple fact has not been stated before in this context in a modelindependent way. With just a reduction of r_{⋆}, the highest value of the Hubble constants one can get, while remaining in a reasonable agreement with BAO and DES/KiDS, is around 70 km/s/Mpc. Thus, a full resolution of the Hubble tension will require either multiple modifications of the ΛCDM model or discovering systematic effects in one or more of the datasets.
Methods
The acoustic scale measurements from the CMB and BAO
The CMB temperature and polarization anisotropy spectra provide a very accurate measurement of the angular size of the sound horizon at recombination,
where r_{⋆} is the sound horizon at recombination, or the comoving distance a sound wave could travel from the beginning of the universe to recombination, and D(z_{⋆}) is the comoving distance from a presentday observer to the last scattering surface, i.e., to the epoch of recombination. In a given model, r_{⋆} and D(z_{⋆}) can be determined from \({r}_{\star }\,=\,\mathop{\int}\nolimits_{{z}_{\star }}^{\infty }{c}_{s}(z){\rm{d}}z/H(z)\) and \(D({z}_{\star })\,=\,\mathop{\int}\nolimits_{0}^{{z}_{\star }}c\,{\rm{d}}z/H(z)\), where c_{s}(z) is the sound speed of the photon–baryon fluid, H(z) is the redshiftdependent cosmological expansion rate and c is the speed of light. To complete the prescription, one also needs to determine z_{⋆} using a model of recombination.
The redshift dependence of the Hubble parameter in the ΛCDM model can be written as
where h(z) is simply H(z) in units of 100 km/s/Mpc, and h is the value at redshift z = 0. Here, Ω_{r}, Ω_{m}, and Ω_{Λ} are the presentday density fractions of radiation, matter (baryons and CDM) and dark energy. From the precise measurement of the presentday CMB temperature T_{0} = 2.7255 K (however, also see^{47}), and adopting the standard models of particle physics and cosmology, one knows the density of photons and neutrinos Ω_{r}h^{2}. Using the theoretically well motivated criticality condition on the sum of the fractional densities, i.e., Ω_{r} + Ω_{m} + Ω_{Λ} = 1, one finds that h(z) is dependent only on two remaining quantities: Ω_{m}h^{2} and h. The photon–baryon sound speed c_{s} in Eq. (1) is determined by the ratio of the baryon and photon densities and is wellconstrained by both Big Bang nucleosynthesis and the CMB. Fitting the ΛCDM model to CMB spectra also provides a tight constraint on Ω_{m}h^{2}, making it possible to measure h.
In alternative models, a smaller r_{⋆} is achieved by introducing new physics that reduces z_{⋆} through a modification of the recombination process or by modifying h(z) before and/or during recombination, or a combination of the two. In our analysis, we consider Eq. (3) while remaining agnostic about the particular model that determines the sound horizon. Namely, we treat r_{⋆} as an independent parameter. We assume, however, that after the recombination, the expansion of the universe is well described by Eq. (4), which is the case in many alternative models. Thus, our independent parameters are r_{⋆}, Ω_{m}h^{2}, and h, with the latter two determining D(z_{⋆}). The dependence of D(z_{⋆}) on the precise value of z_{⋆} is very weak, so that the differences in z_{⋆} in different models do not play a role.
The same acoustic scale is also imprinted in the distribution of baryons. There are three types of BAO observables corresponding to the three ways of extracting the acoustic scale from galaxy surveys^{36}: using correlations in the direction perpendicular to the line of sight, using correlations in the direction parallel to the line of sight, and the angleaveraged or “isotropic” measurement. While our MCMC analysis includes all three types of the BAO data, for the purpose of our discussion it suffices to consider just the first type, which is the closest to CMB in its essence, but our conclusions apply to all three. Namely, we consider
where \({r}_{{\rm{d}}}\,=\,\mathop{\int}\nolimits_{{z}_{{\rm{d}}}}^{\infty }{c}_{s}(z){\rm{d}}z/H(z)\) is the sound horizon at the epoch of baryon decoupling, closely related to r_{⋆}, and z_{obs} is the redshift at which a given BAO measurement is made. We adopt a fixed relation r_{d} = 1.0184r_{⋆} that holds for the Planck best fit ΛCDM model and is largely unchanged in the alternative models.
As the distance integrals D(z_{⋆}) and D(z_{obs}) in the denominators of Eqs. (3) and (5) are dominated by the matter density at low redshifts, one can safely neglect Ω_{r}h^{2} and write
where ω_{m} = Ω_{m}h^{2} and 2998 Mpc = c/100km/s/Mpc, and an analogous equation for BAO with the replacement \(({r}_{\star },{\theta }_{\star },{z}_{\star })\,\to\, ({r}_{{\rm{d}}},{\theta }_{\perp }^{{\rm{BAO}}},{z}_{{\rm{obs}}})\). For a given Ω_{m}h^{2}, Eq. (6) defines a line in the r_{d}–H_{0} plane. Similarly, a BAO measurement at each different redshift also defines a respective line in the r_{d}–H_{0} plane. Taking the derivative of r_{⋆} with respect to h one finds
and a completely analogous equation for BAO. It is important to realize that the derivative is very different for CMB and BAO due to the vast difference in redshifts at which the standard ruler is observed, z_{⋆} ≈ 1100 for CMB vs. z_{obs} ~ 1 for BAO, resulting in different values of the integral in Eq. (7). This results in different slopes of the respective r_{d}(h) lines. Note that the slopes of the r_{d}(h) lines differ for the transverse, parallel and volume averaged BAO measured at the same redshift. While important for constraining cosmological parameters^{48}, these differences are small compared to that caused by the big difference between the BAO and CMB redshifts.
Obtaining the contours and the r _{d}(h) lines in Fig. 1
The marginalized joint r_{d}–H_{0} constraints from BAO were obtained using CosmoMC^{49} modified to work with r_{d} as an independent parameter. The cosmological parameters we vary are r_{d}, Ω_{m}h^{2}, and h, and the shown constraint is obtained after marginalizing over Ω_{m}h^{2}. The BAO data included the recently released Date Release (DR) 16 of the extended Baryon Oscillation Spectroscopic Survey (eBOSS)^{50} that includes BAO and redshift space distortions measurements at multiple redshifts from the samples of Luminous Red Galaxies (LRGs), Emission Line Galaxies (ELGs), clustering quasars (QSOs), and the Lymanα forest. We use the BAO measurement from the fullshape auto and crosspower spectrum of the eBOSS, LRGs, and ELGs^{51,52}, the BAO measurement from the QSO sample^{53}, and from the Lymanα forest sample^{54}. We combine these with the lowz BAO measurements by 6dF^{55} and the SDSS DR7 main Galaxy sample^{56}.
The CMB and BAO lines shown in Fig. 1 were obtained by talking the measured value of θ_{⋆} or θ_{⊥}(z_{obs}), fixing Ω_{m}h^{2} at a certain value (provided for each line in the legend), varying h and deriving r_{d} from Eqs. (3) and (5). We do not show the uncertainties around the individual lines because they are only meant to demonstrate the differences in slopes and the effect of different Ω_{m}h^{2}. The marginalized BAO and the Planck CMB contours provide a more accurate representation of the uncertainties involved.
The dependence of the CMB r_{d}(h) lines on Ω_{m}h^{2} may appear contradictory to the Ω_{m}h^{2} dependence shown in Fig. 1 of a wellknow paper by Knox and Millea^{40}. There, increasing Ω_{m}h^{2} moves the CMB best fit (r_{d}, h) point in a direction orthogonal to where our CMB lines move. The reason for the difference is that their r_{d} is a derived parameter obtained from the standard recombination model and, hence, depends on Ω_{m}h^{2}. In our derivation of the CMB lines, on the other hand, the Ω_{m}h^{2} dependence only appears in D(z_{⋆}) and D(z_{obs}).
Obtaining the S _{8} constraints in Fig. 2
The joint DES + SN contours in Fig. 2 are obtained using the default version of CosmoMC and marginalizing over all relevant ΛCDM and nuisance parameters. To derive the Model 2 and Model 3 contours in Fig. 2, we fit the ΛCDM model to the BAO data using r_{d}, Ω_{m}h^{2}, and h as a free parameters, supplemented by Gaussian priors on Ω_{m}h^{2} and h, and with the primordial spectrum amplitude A_{s} and the spectral index n_{s} fixed to their best fit ΛCDM values. The fit then generates constraints on S_{8} and Ω_{m} as derived parameters. For Model 2, the Gaussian priors were Ω_{m}h^{2} = 0.155 ± 0.0012, where we assumed the same relative uncertainty in Ω_{m}h^{2} as for the Planck best fit ΛCDM model, and h = 0.71 ± 0.01, corresponding to the central value and the 1σ overlap between the CMB2 line and the BAO band. For Model 3, the priors were Ω_{m}h^{2} = 0.167 ± 0.0013 and h = 0.735 ± 0.14.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
The numerical codes used in this paper are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Eiichiro Komatsu and Joulien Lesgourgues for helpful comments on the draft of the paper and Kanhaiya Pandy and Toyokazu Sekiguchi for kindly providing us with data of their models. We gratefully acknowledge using CosmoMC^{49} and GetDist^{57}. This research was enabled in part by support provided by WestGrid (www.westgrid.ca) and Compute Canada Calcul Canada (www.computecanada.ca). L.P. is supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada, and by the Chinese Academy of Sciences President’s International Fellowship Initiative, Grant No. 2020VMA0020. G.B.Z. is supported by the National Key Basic Research and Development Program of China (No. 2018YFA0404503), a grant of CAS Interdisciplinary Innovation Team, and NSFC Grants 11925303, 11720101004, 11673025, and 11890691.
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K.J. proposed the idea, drafted the paper, and participated in all stages of the project; L.P. codeveloped the idea, ran the numerical simulations and contributed to the text; G.B.Z. developed the numerical code used in this work, and contributed to the text.
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Jedamzik, K., Pogosian, L. & Zhao, GB. Why reducing the cosmic sound horizon alone can not fully resolve the Hubble tension. Commun Phys 4, 123 (2021). https://doi.org/10.1038/s4200502100628x
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DOI: https://doi.org/10.1038/s4200502100628x
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