The trouble beyond H_0 and the new cosmic triangles
José Luis Bernal, Licia Verde, Raul Jimenez, Marc Kamionkowski, David Valcin, Benjamin D. Wandelt
TThe trouble beyond H and the new cosmic triangles Jos´e Luis Bernal, Licia Verde,
2, 3
Raul Jimenez,
2, 3
MarcKamionkowski, David Valcin, and Benjamin D. Wandelt
4, 5, 6 Department of Physics and Astronomy, Johns Hopkins University,3400 North Charles Street, Baltimore, Maryland 21218, USA ICC, University of Barcelona, Mart´ı i Franqu`es, 1, E-08028 Barcelona, Spain ICREA, Pg. Lluis Companys 23, Barcelona, 08010, Spain. Sorbonne Universit´e, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France. Sorbonne Universit´e, Institut Lagrange de Paris (ILP), 98 bis bd Arago, 75014 Paris, France. Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA.
The distance ladder using supernovae yields higher values of the Hubble constant H than thoseinferred from measurements of the cosmic microwave background (CMB) and galaxy surveys, adiscrepancy that has come to be known as the ‘Hubble tension’. This has motivated the explorationof extensions to the standard cosmological model in which higher values of H can be obtained fromCMB measurements and galaxy surveys. The trouble, however, goes beyond H ; such modificationsaffect other quantities, too. In particular, their effects on cosmic times are usually neglected. Weexplore here the implications that measurements of the age t U of the Universe, such as a recentinference from the age of the oldest globular clusters, can have for potential solutions to the H tension. The value of H inferred from the CMB and galaxy surveys is related to the sound horizon atCMB decoupling (or at radiation drag), but it is also related to the matter density and to t U . Giventhis observation, we show how model-independent measurements may support or disfavor proposednew-physics solutions to the Hubble tension. Finally, we argue that cosmological measurementstoday provide constraints that, within a given cosmological model, represent an over-constrainedsystem, offering a powerful diagnostic tool of consistency. We propose the use of ternary plots tosimultaneously visualize independent constraints on key quantities related to H like t U , the soundhorizon at radiation drag, and the matter density parameter. We envision that this representationwill help find a solution to the trouble of and beyond H . I. INTRODUCTION
The standard, ΛCDM, cosmological model, has suc-cessfully passed increased scrutiny, as observations ofthe cosmic microwave background (CMB) [1–3], type-Ia supernovae (SNeIa) [4] and large-scale structure [5–8]have improved drastically over recent years. Nonethe-less, tensions have arisen for specific parameters whentheir values are inferred, within the ΛCDM, from differ-ent probes and observables. The biggest tension is relatedto determinations of the Hubble constant H ≡ h km/s/Mpc, and has increased in the last decade to be inthe 4 − σ [9, 10].The current state of the H tension is illustratedin Fig. 1, where we show marginalized posteriors formeasurements depending on early-times physics (like Planck [1] or baryon acoustic oscillations with a bigbang nucleosynthesis prior on the physical density ofbaryons [11, 12]), late-time expansion history (usingstrong lensing time delays from TDCOSMO [13–17] andcosmic chronometers [19, 20]), and local measurements,independent of cosmology, from SH0ES [21] and CCHP[22]. Except for cosmic chronometers, all competitive H There are ongoing efforts to relax the dependence of strong lens-ing time delays H inference on the assumed expansion rate [18]. constraints rely on distance measurements. The two determinations yielding the largest tension areobtained from the CMB power spectra and the SH0ESdistance ladders using SNeIa calibrated by Cepheids.CCHP calibrates the SNeIa instead with the tip of the redgiant branch (TRGB) and finds a lower value of H [22](see also [25–27]).Given the strong constraints imposed by available dataon the product of the sound horizon r d at radiation dragand h , r d has been targeted as the critical quantity tobe modified in order to solve the H tension. In lightof current constraints, the modifications of ΛCDM bestpoised to reduce the H tension involve altering pre-recombination physics [29], as to lower the value of r d .There is a plethora of proposed models to do so and thoseshowing more promise involve boosts of the expansionhistory between matter-radiation equality and recombi-nation (see e.g., [30–44]).Despite the fact that most of the attention has beenfocused on modifying distance scales across cosmic his-tory, the expansion rate, thus H , also determines the Some H constraints related with large-scale structure do notdepend on the sound horizon, but still depend on distance scales,such as the size of the horizon at matter-radiation equality [23,24]. a r X i v : . [ a s t r o - ph . C O ] F e b
64 66 68 70 72 74 76 78 H (Mpc km/s) ( H ) P18 CCHP TDCOSMOSH0ESBAO+BBNCC -Early: P18, BAO+BBN-Late: CC, TDCOSMO-Local: CCHP, SH0ES
FIG. 1. Summary of constraints on H from cosmicchronometers (CC) [20], Planck (P18) [1], baryon acous-tic oscillations with a BBN prior on the baryon abundance(BAO+BBN) [12], CCHP [22], SH0ES [28], and strong-lensing time delays (TDCOSMO) [17]. Note that the resultsshown in this figure are subject to different model assump-tions. age-redshift relation. Measuring cosmic ages can pro-vide a constraint on H completely independent from r d ,other standard scales, or distance measurements. Cos-mic chronometers measure directly the expansion rateusing differential ages [19]; this approach is limited torelatively low redshifts, covering a range that overlapswith distance measurements. On the other hand, sincerelative changes in the expansion history at early timesdo not significantly modify the age of the Universe, in-dependent inferences of absolute lookback times, such asthe age of the Universe, may weigh in on the H tension.In this work, we discuss how the age of the Universe in-ferred from a recent determination of the age of the oldestglobular clusters [45–47] can offer an additional perspec-tive on the H controversy. Our results suggest that anaccurate and precise measurement of the age of the Uni-verse provides an important test of the hypothesis thatthe H tension suggests new early-Universe physics butstandard late-Universe physics. In the process, we alsoupdate constraints on the low-redshift expansion rate us-ing recent relative distance redshift measurements.In the same way as the H tension was reframed asthe inconsistency between r d , h and their product r d h (inferred independently in a model-agnostic way fromlow redshifts observations) [48–50], the same can be saidabout other sets of quantities that can be constrained in-dependently, albeit assuming a cosmological model. Oneis the combination of the matter density parameter todayΩ M and h , and the physical matter density parametertoday, which is the product of the two, Ω M h . The otherset is the age t U of the Universe and h , and their combi-nation t U h , which is completely determined by the shape of the expansion history and measured independently.This is reminiscent of the ‘cosmic triangle’ proposed inRef. [51] two decades ago, where the matter, cosmologicalconstant and curvature density parameters are related toone another because they sum to unity. The original cos-mic triangle is a ternary plot which served to visualizecosmological constraints that led to favor the (now stan-dard) flat ΛCDM model. Here, in full analogy, we pro-pose the use of ternary plots as diagnosis diagrams to ex-amine the tension between cosmological quantities inde-pendently measured from different observations. Ternaryplots are specially suited for this purpose, as we show forthe cases of r d , Ω M and t U listed above.This article is organized as follows. We present up-dated constraints on the late-Universe expansion rate asa function of redshift in Sec. II; discuss the role cosmicages play in the H tension in Sec. III; present the newcosmic triangles in Sec. IV; and finally conclude in Sec. V. II. UPDATED EXPANSION RATECONSTRAINTS
We begin by presenting updated model-agnostic con-straints on the expansion rate as a function of redshift, E ( z ) ≡ H ( z ) /H , using the latest, state-of-the-art data.These constraints on E ( z ) are a key input for the resultsof sections III, IV and our conclusions.We use SNeIa observations from Pantheon [4] andBAO measurements from 6dFGRS [52], SDSS DR7 [53],BOSS [5], WiggleZ [54], and eBOSS, including galaxies,quasars and Lyman- α forest [55–59] as relative distanceindicators. Note that, although BAO-only analyses as-sume a fiducial cosmology, their results are robust to beapplied to other cosmologies (see e.g., [60, 61]).Two models for E ( z ) are examined: ΛCDM, and aparametrization using natural cubic splines, the nodes ofwhich have a varying position, without imposing flat-ness, which we refer to as ‘generic’ expansion and assuch falls under what we here refer to as “model agnos-tic” approach. Given its flexibility, the generic expansionshall be understood as a marginalization over cosmologi-cal models predicting a smooth E ( z ). Other uses of thisparametrization, known as flexknot, can be found in e.g.,Refs. [62, 63]. Standard BAO analyses adopt a prior on r d to break the r d h degeneracy and calibrate the distance measurements, followingthe approach known as inverse cosmic distance ladder. Notusing that prior and marginalizing over r d removes any de-pendence on pre-recombination physics, since the BAO mea-surements are robust to modifications of the pre-recombinationphysics of ΛCDM [60].We use measurements from BAO-only analyses, following theeBOSS likelihoods and criterion to combine with BOSS mea-surements from https://svn.sdss.org/public/data/eboss/DR16cosmo/tags/v1_0_0/likelihoods/BAO-only/ . The free parameters for the ΛCDM case are { Ω M , r d , h, M SN } , where M SN is the absolute magnitudeof SNeIa; on the other hand, the generic expansion needs (cid:110) z (1 ,N − , E (1 ,N )knot , Ω k , r d , h, M SN (cid:111) as free parameters,where E knot are the values of E ( z ) at the knots of thesplines, located at z knot , and Ω k is the density parameterassociated to curvature. The first and last knot are fixedat z = 0 and z = 2 .
4, respectively, and E (0) = 1 by defi-nition. Although our results do not significantly dependon the number of knots used, we find N = 4 providesthe best performance, allowing for as much freedom aspossible but avoiding over-fitting, and report the resultsobtained under this choice. We use uniform priors in allcases.We use the public code MABEL [64], to run run MonteCarlo Markov chains with the sampler zeus [65, 66] toconstrain the shape of the expansion rate in the late-time Universe ( z ≤ .
4) and the quantity r d h with un-calibrated distance measurements from BAO and SNeIameasurements. Note that, with the data included in theanalysis, h and r d individually are completely uncon-strained; only their product is constrained.The new BAO and SNeIa data allow the constraints onthe generic E ( z ) to be extended up to z = 2 .
4, as shownin Fig. 2. The generic reconstruction yields an E ( z )which is consistent with the prediction of a ΛCDM modelfrom Planck and BAO+SNeIa. Allowed deviations from
Planck ’s ΛCDM best fit are (cid:46) −
4% at z (cid:46) .
8; thisbound weakens slightly (cid:46)
10% at 0 . (cid:46) z (cid:46) .
4, due tothe degradation in the constraining power of SNeIa ob-servations. While still being consistent with the ΛCDMprediction, the reconstructed posterior allows for a boostof the expansion rate ( ∼
15% larger than
Planck ’s ΛCDMbest fit) at 1 . (cid:46) z (cid:46) .
4, this can be seen as an “excesswiggle” in the plot; however it is not significant and weshould remark that there are no measurements in thatredshift range corresponding to the gap between the red-shift covered by Supernovae data/eBOSS quasars and theLyman- α forest data. Note also that those expansion his-tories showing an excess expansion rate at these redshiftsneed a lower E ( z ) than ΛCDM at low redshifts. Theseresults extend and improve previous constraints from ag-nostic reconstructions of E ( z ) (see e.g., Ref. [48], wherereported 68% confidence level limits of the deviations are5% at z (cid:46) . k = − . ± .
10 and r d h =100 . ± . E ( z ) is different, so this comparison is more qual-itative than strictly quantitative; the improvement is The code will be publicly available once the corresponding workis submitted to the journal. https://zeus-mcmc.readthedocs.io/ z E ( z ) / E P l a n c k ( z ) CDM (P18)CDM (BAO+SNeIa)Generic (BAO+SNeIa)
FIG. 2. Best fit evolution of the expansion rate with red-shift (thick lines) normalized by
Planck ’s ΛCDM best fit( E ( z ) /E Planck ( z )) and 68% confidence level uncertainties(shaded regions, thin lines). Planck ’s ΛCDM results are re-ported in red and BAO+SNeIa constraints assuming ΛCDMare in blue. In purple, the reconstruction from BAO+SNeIaassuming a generic expansion; thin lines are a sample of 500flexknot splines reconstruction from the 68% cases with high-est posterior. driven by the new data gathered over the past five years).These constraints can be compared to those obtained alsofrom BAO+SNeIa when assuming a flat ΛCDM model: r d h = 100 . ± . M = 0 . ± . r d h constraints. Furthermore, it returns con-straints on r d h comparable to Planck results assumingΛCDM ( r d h = 99 . ± . III. COSMIC AGES AND H In addition to cosmic distances, the expansion rate ofthe Universe determines the look-back time. This opensup the possibility to use time (or age) measurements toweigh in on the H tension. The cosmic chronometersmethod uses relative ages to determine H ( z ), but agescan also be used in a complementary way. The look-backtime t as function of redshift is given by t ( z ) = 977 . H (cid:90) z d z (cid:48) (1 + z (cid:48) ) E ( z (cid:48) ) Gyr , (1)with H ( z ) in km s − Mpc − . Following Eq. (1), the age ofthe Universe is t U ≡ t ( ∞ ). We show the dependence of t U on H , Ω M and a constant equation of state parameter w for dark energy in a w CDM model in Fig. 3. It is evident M H [ M p c k m / s ] . . . . . t U [ G y r s ] w H [ M p c k m / s ] . . . . . t U [ G y r s ] w M [ M p c k m / s ] . . . t U [ G y r s ] FIG. 3. Age of the Universe (in Gyr) as function of H and Ω M for w = − H and w for Ω M = 0 . M and w for h = 0 . Planck
ΛCDM best-fit value.White lines mark contours with constant value of t U . that the strongest dependence is on H , while Ω M and w have less influence.The integral in Eq. (1) is dominated by contributionsfrom redshifts below few tens, decreasing as z grows.Therefore, any exotic pre-recombination physics does notsignificantly affect the age of the Universe. On the otherhand, E ( z ) is bound to be very close to that of a CMB-calibrated ΛCDM model at z (cid:46) .
4, as shown in the pre-vious section. Hence, a precise and robust determinationof t U which does not significantly rely on a cosmologicalmodel, in combination with BAO and SNeIa, may weighin on proposed solutions to the H tension. If an inde-pendent (and model-agnostic) determination of t U wereto coincide with Planck ’s inferred value assuming ΛCDM,alternative models involving exotic physics relevant onlyin the early Universe would need to invoke additionalmodifications also of the late-Universe expansion historyto reproduce all observations as their prediction for t U would be too low.Recently, a value of the age of Universe, t U = 13 . ± .
15 (stat . ) ± .
23 (syst . ) ( ± .
27 when adding statisti-cal and systematic uncertainties in quadrature) was in-ferred from a sample of old globular clusters (GCs) inRefs. [46, 47]. This study involves a Bayesian analysisof the properties of 38 GCs, including their age, distance,metallicity, reddening and abundance of α -enhanced ele-ments. t U is inferred from the age of the oldest of theseGCs (marginalized over all other parameters and includ-ing systematic errors) estimating and correcting for theage of the Universe at the moment of GCs formation, andgenerously marginalizing over the small residual depen-dence on cosmology.We can confront local H measurements with the t U inferred from GCs, since they are related by H t U , which This systematic uncertainty was determined using externalmetallicity spectroscopic measurements of the GCs. We referthe interested reader to Ref. [47] for more details and an alter-native estimate based only on the color-magnitude diagramas ofthe globular clusters. can be obtained using Eq. (1) and a constraint on E ( z )for all the redshifts that contribute significantly to theintegral. Redshifts below 2.4 (where the generic E ( z )reconstruction is available) only cover about 75% of theage of the Universe. If we assume that deviations from aΛCDM expansion history are driven by the poorly knowndark energy component, then E ( z ) at z > E ( z ) is perfectly consistent with ΛCDM andonly relatively small deviations are allowed. If we con-sider more extreme deviations from ΛCDM, additionaldata probing the expansion history at higher redshiftswould be needed to extend the constraints on the generic E ( z ) to cover a larger fraction of t U .Hence, we assume for this study a ΛCDM expansionrate E ( z ), using the value of Ω M inferred from BAO andSNeIa and its error. Note that exotic models modifyingonly pre-recombination cosmology do not affect directlythe late-time E ( z ) (which remains that of a ΛCDM,model) hence our inferred H t U also applies to thesemodels. As an example, we consider early dark energy(EDE) models. In particular, we use the EDE model pos-terior obtained in Refs. [68, 69] for the Planck data; themodel features three additional cosmological parameterscompared to ΛCDM.We show 68% confidence level marginalized con-straints on the H - t U plane from SH0ES, CCHP,GCs, BAO+SNeIa, and Planck in Fig. 4. We find H t U = 945 ±
11 Gyr Mpc − km / s from BAO+SNeIaassuming ΛCDM, while H t U = 928 ± ± − km / s from Planck assuming ΛCDMand EDE, respectively. As a reference, combiningBAO+SNeIa with SH0ES and TRGB returns t U =12 . ± .
29 and t U = 13 . ± .
42 Gyr, respectively, while
Planck ’s inferred values are 13 . ± .
02 Gyr (ΛCDM) The expected effect of adopting the reconstructed E ( z ) whereavailable and a ΛCDM one at higher z is a possible increase ofthe error-bars on t U H of (cid:46)
60 65 70 75 80 H [Mpc km/s] t U [ G y r] SH0ESCCHPBAO+SNeIa Glob. Clust.Planck ( CDM)Planck (EDE)
FIG. 4. 68% confidence level marginalized constraints in the H - t U plane, from independent measurements, as indicated inthe legend. Dashed cyan lines denote the size of the statistical1 σ errors from globular clusters, while the shaded region alsoinclude systematic uncertainties. BAO+SNeIa constraints as-sume a ΛCDM cosmology. We show Planck results assumingΛCDM (red) and EDE (orange). and 13 . +0 . − . Gyr (EDE).These results show that for SH0ES to be compati-ble with BAO+SNeIa the Universe must be significantlyyounger than inferred by
Planck , no matter whetherΛCDM or EDE are assumed; this statement is robustto early-time physics assumptions. The age of the Uni-verse inferred from GCs weakly favors older Universesthan SH0ES combined with BAO+SNeIa, but the cur-rent systematic error budget is too large to firmly distin-guish. There are ongoing efforts to reduce the impact ofsystematic errors (see e.g., [47]), so that GCs constraintson t U have the potential to discriminate among differentscenarios proposed to solve the H tension (statisticalerrors are indicated with dashed lines). IV. THE NEW COSMIC TRIANGLES
The H tension was reframed as a consistency test be-tween r d (an early-time quantity) and H (a late-timequantity), which can be done using a model-agnostic ap-proach, in Ref [48]. Similarly, assuming a cosmologicalmodel, allows for a similar consistency test between Ω M and H to be performed, as proposed in Ref. [70]. Withthe updated constraints on E ( z ), r d h and Ω M obtainedin Sec. II, we can revisit these consistency checks. More-over, the H , t U and H t U constraints obtained with theΩ M values inferred from BAO+SNeIa, adds a third con-sistency test related with H .These three cases are three triads of two cosmolog-ical quantities and their product determined indepen- dently. These triads are { t U , H , H t U } , { r d , h, r d h } , (cid:8) Ω M , h , Ω M h (cid:9) . Within a given cosmological model(although some of the constraints can be obtained modelindependently), and in the absence of systematic errors,a generic triad { a, b, ab } of parameters determined byindependent experiments i , j and k , respectively, is anover-constrained system which must fulfill a i × b j = ( ab ) k within statistical uncertainty. This is what makes thesetriads a powerful diagnostic tool of consistency, especiallyin the context of the H tension. Therefore, the cosmo-logical model(s) yielding agreement of all these triads arefavored by the data.Cosmology faced a similar situation in 1999, when in-formation from CMB anisotropies, SNeIa and clusters ob-servations was combined to determine whether the Uni-verse is flat and if there was evidence for a non-zerocosmological constant [51]. In that case, the triad was { Ω M , Ω k , Ω Λ } , where Ω Λ = 1 − Ω M − Ω k is the densityparameter associated to the cosmological constant today.These triads may be represented in a plane (as donee.g., in Fig. 4), but due to the relation between theircomponents, they can be more efficiently representedin a ternary plot. Taking the logarithm of each quan-tity in the triads of the form { a, b, ab } (which fulfillslog ( a ) + log ( b ) − log ( ab ) = 0), we can build ternaryplots; every point on these ternary plots sum up to 0.This representation provides an intuitive and illustra-tive simultaneous look at independent cosmological con-straints. We use them to illustrate the state of the H tension in each of the three complementary frames thathave been discussed. We refer to these ternary plots asthe new cosmic triangles.Each of the triads discussed in this work involves quan-tities directly related to H and provide different anglesto study the H tension: in terms of times, distances andthe abundance of matter. In interpreting the observa-tional constraints, we can distinguish between early-time,late-time and local observations, which in turn may de-pend on early-time (pre-recombination), late-time (lowredshift) or fully local physics. In all cases, we can useBAO+SNeIa results to link local and early-Universe mea-surements. Note that the triad corresponding to h and r d is the only one that is agnostic with respect to the choiceof a cosmological model for the low-redshift expansionhistory. We show the new cosmic triangles in Fig. 5; the inter-pretation of the ternary plots can be eased by comparingthis figure with Fig. 4. We can appreciate the tensionin the triangles corresponding to r d h and Ω M h . As ex-pected, considering the region favored by BAO+SNeIa, Planck constraints obtained within ΛCDM are consis-tent with CCHP, but show some tension with SH0ES. r d inferred values from Planck are largely independent of stan-dard post-recombination physics, as we can see comparing resultsfrom standard analyses [1] with those using only early-Universeinformation [67]. -3.00 -2.98 -2.96 -2.94 -2.92 -2.90 1.091.111.131.151.171.19 1.81 1.83 1.85 1.87 1.89 1.91
SH0ESCCHPBAO+SNeIaGlobular ClustersPlanck ( CDM)Planck (EDE) -2.05 -2.03 -2.01 -1.99 -1.97 -1.95 -0.18-0.16-0.14-0.12-0.10-0.08 2.13 2.15 2.17 2.19 2.21 2.23 0.75 0.79 0.83 0.87 0.91 0.95 -0.35-0.31-0.27-0.23-0.19-0.15 -0.60 -0.56 -0.52 -0.48 -0.44 -0.40
FIG. 5. 68% confidence level marginalized constraints on the new cosmic triangles: we show the triad corresponding to the ageof the Universe and the Hubble constant (upper left), to the sound horizon at radiation drag and the reduced Hubble constant(bottom left), and to the total matter density parameter today and the square of the reduced Hubble constant (bottom right).Note that all points in each figure sum up to 0, while the ticks in the axes determine the direction of equal values for each axis.
The tensions are always smaller in the case of EDE, butnot enough for this model to be preferred over ΛCDM.Figure 5 clearly shows the synergies of considering thethree triads at the same time. The most studied so farhas been the one involving r d and h , since it was arguedthat the most promising way to solve the H tension wasto reduce the value r d while keeping a standard evolu-tion of the low-redshift expansion rate [29, 48]. We canalso see that this triangle is the one showing the largesttension between Planck assuming ΛCDM, SH0ES andBAO+SNeIa, and the one for which models like EDEshow promise. The triangle including Ω M shows a smallertension: combining BAO+SNeIa with SH0ES (CCHP) we find Ω M = 0 . ± .
009 (Ω M = 0 . ± . σ (0.1 σ ) tension with Planck ’s constraint assum-ing ΛCDM. The tension reduces to 1.5 σ when comparedto the Planck results assuming EDE. Since BAO+SNeIaconstrain E ( z ) at low redshift to be very similar to (andfully consistent with) the best fit of Planck assumingΛCDM, this tension is fully sourced by the H tension,no matter the cosmological model under consideration.However, the situation for the triad involving the age ofthe Universe is different. As argued above, modificationsof the early-Universe cosmology do not directly changethe age of the Universe. This is why Planck
EDE posteri-ors overlap with those assuming ΛCDM (extending alongthe direction of constant Ω M , i.e., the constraint on H t U from BAO+SNeIa). In this representation, the region ofoverlap of Planck , BAO+SNeIa and GCs posteriors is inlarge tension with SH0ES. However, current determina-tions of t U alone are not precise enough to definitivelydisfavor the combination of SH0ES with BAO+SNeIa.Finally, Fig. 5 clearly indicate that if GCs were to stillreturn a high value of t U but with reduced error-bars, de-viations from ΛCDM that only affect pre-recombinationphysics will not be enough to reconcile all the measure-ments. If this will turn out to be the case, a combina-tion of both, high and low redshift modifications to theΛCDM model may be required to solve the H tension.Alternatively one would have to look into much morelocal effects, such as those affecting the distance laddercalibration and in particular effects or processes whichmay be responsible for the mis-match between CCHPand SH0ES. V. CONCLUSIONS
The discrepancies between model-independent mea-surements and model-dependent inferred values of H from different experiments (each of them sensitive to dif-ferent physics and systematic errors) might be a hint forthe need of modifying the standard ΛCDM model. Themost promising deviations from ΛCDM proposed to solvesuch tensions involve a boost in the expansion rate beforerecombination, as to lower the value of r d and reconcilethe direct and the inverse distance ladder. However, weargue in this work, there is a more varied phenomenol-ogy, that goes well beyond r d , to be matched by any newphysics put forward to solve the H tension, especiallyregarding cosmic ages: the trouble goes beyond H .We update agnostic reconstructions of the evolutionof the expansion rate of the late-time Universe with re-cent BAO and SNeIa measurements, extending the re-construction up to z ∼ .
4. We find that BAO andSNeIa constrain the evolution of H ( z ) to be fully con-sistent with the one from ΛCDM Planck ’s best-fit pre-diction: any possible deviation must be well below the5%(10%) level at z < . z < . z ∼ −
20 (covering effectively > H and r d can be addressed with modifications ofthe early-time physics, t U is dominated by the expansionrate at z (cid:46)
30, hence insensitive to high-redshift cos-mology. The t U determination is also insensitive to ef- fects such as cosmological dimming (e.g., violations of theEtherington relation), cosmological screening, deviationsfrom general relativity at large scales affecting growth ofstructures and any phenomenology affecting cosmologi-cal distance measures. Therefore, if a high t U were to bemeasured reliably and with small enough error-bars, itwould disfavor models with high H and standard low-redshift physics. In this case then both, pre- and post-recombination modifications to ΛCDM, may be requiredto reconcile all measurements. Alternatively one wouldhave to invoke much more local effects (be these cosmo-logical, see e.g., [78–81], or astrophysical, in particulareffects or processes which may be responsible for the mis-match between CCHP and SH0ES) affecting the local H determination only, while leaving all other cosmologicalobservations unchanged.In such case, viable solutions to the H trouble willfall in either of two classes of very different nature: lo-cal and global. Global solutions, would have to invokenew physics beyond ΛCDM which affect the entire Uni-verse history from before recombination all the way to thelow-redshift, late-time Universe. Modifying only early-time physics will not be enough. Because of their globalnature, such solutions affect quantities well beyond H ,but would be highly constrained by the wealth of high-precision cosmological observations available. Local so-lutions on the other hand, leave unaffected the globalproperties of cosmology; as such either do not requirenew physics beyond ΛCDM (and thus fall in the realmof astrophysics), or include new physics which only affectvery local observations.A program to improve the inference of t U and reducethe systematic uncertainties, may give this measurementenough power to discriminate between these two differentkinds of viable solutions for the H tension.Finally we identify three triads of independently-measured quantities, relating H with t U , r d , Ω M , re-spectively. Each of these triads is an over-constrainedsystem, hence we propose the use of ternary figures (thenew cosmic triangles) to report and visualize the con-straints. These new cosmic triangles allow for a simulta-neous and easy-to-interpret visual representation of con-straints on different yet related quantities. We hope thatthis representation will help to guide further efforts tofind a solution to the trouble of (and beyond) H . ACKNOWLEDGMENTS
The authors thank Tristan L. Smith, Vivian Poulinand Geoff C.-F. Chen for comments on last versionsof this manuscript. JLB is supported by the Allan C.and Dorothy H. Davis Fellowship. This work is sup-ported in part by MINECO grant PGC2018-098866-B-I00 FEDER, UE. LV acknowledges support by EuropeanUnion’s Horizon 2020 research and innovation programERC (BePreSySe, grant agreement 725327). ICC re-searchers acknowledge “Center of Excellence Maria deMaeztu 2020-2023” award to the ICCUB (CEX2019-000918-M). This work was supported at Johns Hopkinsby NSF Grant No. 1818899 and the Simons Founda-tion. The work of BDW is supported by the LabexILP (reference ANR-10-LABX-63) part of the Idex SU- PER, received financial state aid managed by the AgenceNationale de la Recherche, as part of the programmeInvestissements d’avenir under the reference ANR-11-IDEX-0004-02; and by the ANR BIG4 project, grantANR-16-CE23-0002 of the French Agence Nationale de laRecherche. The Center for Computational Astrophysicsis supported by the Simons Foundation. [1] Planck Collaboration, N. Aghanim, et al., “Planck 2018results. VI. Cosmological parameters,” ArXiv e-prints(July, 2018) , arXiv:1807.06209 .[2] J. W. Henning et al., “Measurements of theTemperature and E-mode Polarization of the CMBfrom 500 Square Degrees of SPTpol Data,” Astrophys.J. no. 2, (Jan., 2018) 97, arXiv:1707.09353[astro-ph.CO] .[3] S. Aiola, E. Calabrese, L. Maurin, S. Naess, B. L.Schmitt, et al., “The Atacama Cosmology Telescope:DR4 Maps and Cosmological Parameters,” arXive-prints (July, 2020) arXiv:2007.07288, arXiv:2007.07288 [astro-ph.CO] .[4] D. M. Scolnic, D. O. Jones, A. Rest, Y. C. Pan, et al.,“The Complete Light-curve Sample of SpectroscopicallyConfirmed SNe Ia from Pan-STARRS1 andCosmological Constraints from the Combined PantheonSample,” Astrophys. J. no. 2, (Jun, 2018) 101, arXiv:1710.00845 [astro-ph.CO] .[5]
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