Local Determination of the Hubble Constant and the Deceleration Parameter

PHYSICAL REVIEW RESEARCH 2, 013028 (2020)

Local determination of the Hubble constant and the deceleration parameter

David Camarena 1 and Valerio Marra 2 1PPGCosmo, Universidade Federal do Espírito Santo, 29075-910 Vitória, Espírito Santo, Brazil 2Núcleo Cosmo-ufes and Departamento de Física, Universidade Federal do Espírito Santo, 29075-910 Vitória, Espírito Santo, Brazil

(Received 1 July 2019; published 9 January 2020)

The determination of the Hubble constant H0 from the cosmic microwave background by the Planck Collaboration (N. Aghanim et al., arXiv:1807.06209) is in tension at 4.2σ with respect to the local determination

of H0 by the SH0ES collaboration [M. J. Reid et al., Astrophys. J. Lett. 886, L27 (2019)].Hereweimprove upon the local determination, which ﬁxes the deceleration parameter to the standard CDM model value of

q0 =−0.55, that is, uses information from observations beyond the local universe. First, we derive the effective calibration prior on the absolute magnitude MB of type Ia supernovae, which can be used in cosmological analyses in order to avoid the double counting of low-redshift supernovae. We ﬁnd MB =−19.2334 ± 0.0404 mag. Then we use the above MB prior in order to obtain a determination of the local H0 which uses only local observations and assumes only the cosmological principle, that is, large-scale homogeneity and isotropy. This is

achieved by adopting an uninformative ﬂat prior for q0 in the cosmographic expansion of the luminosity distance. −1 −1 We use the latest Pantheon sample and ﬁnd H0 = 75.35 ± 1.68 km s Mpc , which features a 2.2% uncertainty, close to the 1.9% error obtained by the SH0ES Collaboration. Our determination is at the higher tension of 4.5σ with the latest results from the Planck Collaboration that assume the CDM model. Furthermore, we also

constrain the deceleration parameter to q0 =−1.08 ± 0.29, which disagrees with the Planck Collaboration at the 1.9σ level. These estimations only use supernovae in the redshift range 0.023 z 0.15.

DOI: 10.1103/PhysRevResearch.2.013028

I. INTRODUCTION sample of local supernovae increased and the anchors used to calibrate them considerably improved [5–7]. The present The CDM model, the standard model of cosmology, has situation is that the two determinations of the Hubble constant, been extremely successful. Assuming only general relativity one by the Planck Collaboration in 2018 [8] and the other by and well-understood linear perturbations about a homoge- the SH0ES Collaboration in 2019 [7], are now in tension at neous and isotropic background model, with just six param- the considerable 4.2σ level. (See [9] for a historical overview eters it accounts for basically all cosmological observations and [10] for the present status.) on a vast range of scales in space and time. Its key ingredients Much work has been done trying to understand the impli- are the cosmological constant, a constant of nature, and dark cations of this tension: It is indeed, by far, the most severe matter, yet-undetected particles which are predicted by, e.g., problem the CDM model is facing. On one hand, the effect supersymmetric extensions of the standard model of particle of the local structure, the so-called cosmic variance on H , has physics. Although the theoretical basis for both the cosmolog- 0 been thoroughly studied (see [11] and references therein), as ical constant and dark matter may be rightfully questioned, the well as possible reassessments of the error budget [12–14]. On standard model has pragmatically maintained its supremacy, the other hand, physics beyond the standard model has been due to its performance and simplicity. investigated, in the hope that this tension could reveal possible However, since the first release of the cosmic microwave alternatives to the highly tuned cosmological constant and the background (CMB) observations by the Planck Collaboration yet-undetected dark matter (see, e.g., [15–19]). However, at in 2013 [1], the determination of the Hubble constant H based 0 the moment, it is not clear which kind of physics beyond on the standard model of cosmology started to be in tension CDM could solve this crisis [20]. with the model-independent determination via calibrated local Here we wish to improve upon the local determination by Supernovae Ia (SN) by the SH0ES Collaboration in 2011 [2]. the SH0ES Collaboration, which adopts a Dirac δ prior on The initial tension of 2.4σ worsened over the past six years. the deceleration parameter, centered at the standard CDM On one side, systematic uncertainties were better understood model value of q =−0.55. In [14] it was shown that using and CMB data accumulated [3,4]. On the other side, the 0 the broad (truncated) Gaussian prior q0 =−0.5 ± 1, it is indeed possible to obtain a competitive constraint on the Hub- ble constant. In what follows, first we derive the calibration prior on the absolute magnitude MB of supernovae Ia that Published by the American Physical Society under the terms of the was effectively used in the H0 determination by the SH0ES Creative Commons Attribution 4.0 International license. Further Collaboration. Then, using the MB prior and an uninformative distribution of this work must maintain attribution to the author(s) flat prior on q0, we obtain a competitive determination of the and the published article’s title, journal citation, and DOI. H0. Indeed, our determination has an uncertainty comparable

2643-1564/2020/2(1)/013028(8) 013028-1 Published by the American Physical Society DAVID CAMARENA AND VALERIO MARRA PHYSICAL REVIEW RESEARCH 2, 013028 (2020) to the one by the SH0ES Collaboration and it is at the even higher tension of 4.5σ with the latest results from the Planck Collaboration that assume the CDM model. We stress that our determination only uses local observations and only assumes the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, that is, the cosmological principle, according to which the universe is homogeneous and isotropic at large scales. This paper is organized as follows. After introducing low- redshift cosmography in Sec. II, we will obtain the effective calibration prior in Sec. III and the determination of the Hub- ble constant in Sec. IV using the Pantheon data set. We discuss our results in Sec. V and summarize in Sec. VI. Appendix A presents details regarding the derivation of the effective calibration prior, Appendix B reports the results relative to the Supercal supernova data set, and Appendix C reports the FIG. 1. Shown with a solid line is the percentage difference results relative to the older determination of H0 in [6]. between the distance modulus with and without the second-order correction in the expansion of the luminosity distance of Eq. (2). II. COSMOGRAPHY The difference is always smaller than 0.7%. The histogram shows the distribution W of the supernovae in the range 0.023 z The apparent magnitude mt of a supernova at redshift z is SN B 0.15 that are used for local determinations of H0. The filled green given by histogram corresponds to the Pantheon sample (237 supernovae) [25] d (z) and the empty black one to the Supercal sample (217 supernovae) mt (z) = 5log L + 25 + M , (1) B 10 1Mpc B [26]. Using these distributions, the weighted error from neglecting the second-order correction is only 0.2%. where dL (z) is the luminosity distance and MB the absolute magnitude. Using a cosmographic approach within an FLRW redshift supernovae. Indeed, the latest local determinations metric, which only assumes large-scale homogeneity and of H0 use supernovae in the range 0.023 z 0.15, where isotropy, one has the minimum redshift is large enough in order to reduce the − impact of cosmic variance [11,21] and the maximum redshift cz (1 q0 )z 2 dL (z) = 1 + + O(z ) , (2) is small enough in order to reduce the impact of cosmology H0 2 in the determination of H0. We have to bear in mind that the where the Hubble constant and the deceleration parameter are cosmographic expansion could be problematic if extended to defined, respectively, according to high redshifts (z > 1). Indeed, it fails to converge [22] and also its accuracy depends on the order adopted [23]. Here a˙(t ) −a¨(t ) H0 = , q0 = . (3) we avoid these possible issues as the maximum redshift is a(t ) H(t )2a(t ) t0 t0 z = 0.15 [24]. Figure 1 shows the percentage difference with Cosmography is a model-independent approach in the sense respect to the computation that considers the second-order that it does not assume a specific model as it is based on the correction, together with the distribution of the supernovae Taylor expansion of the scale factor. However, this does not Ia that are used to determine H0. The weighted error is mean that its parameters do not contain cosmological infor- 0.2%, which is negligible compared to the error budget to be mation. For example, the deceleration parameter is connected discussed below. Furthermore, the fact that it is safe to neglect to the parameters of wCDM cosmologies according to the second-order correction implies that this analysis is valid also for spatially curved models. m0 1 + 3w 1 + 3wde0 q0 = + de0 = 2 2 2 III. EFFECTIVE CALIBRATION PRIOR 1 − 30 = −0.55, (4) The determination of H by the SH0ES Collaboration 2 0 is a two-step process. On one hand, the anchors, Cepheid where the second equality in the first line assumes spatial hosts, and calibrators are combined to produce a constraint on + = flatness ( m0 de0 1), the first equality in the second line M . This step depends on the astrophysical properties of the =− B assumes the CDM model (w 1), and the second one in sources and is independent of cosmology. On the other hand, . the second line assumes the concordance value 0 0 7. the Hubble-Flow supernovae of the Supercal sample [26] Combining now Eqs. (1) and (2), one has are used to probe the luminosity-distance–redshift relation in (1 − q )z order to determine H0. In this section, using two different mt = 5log 1 + 0 + O(z2 ) B 10 2 methods, we determine the calibration prior f (MB ) which was effectively used in the analysis in [6]. −1 czH0 + 5log + 25 + MB. (5) 10 1Mpc A. Demarginalization

We are neglecting the second-order... correction that contains Within a Bayesian framework, the determination of H0 = a | the jerk parameter j0 H 3a t0 because we consider only low- by the SH0ES Collaboration can be effectively summarized

013028-2 LOCAL DETERMINATION OF THE HUBBLE CONSTANT … PHYSICAL REVIEW RESEARCH 2, 013028 (2020) according to the analysis f (H ) f (M )L(SN|H , q , M ) f H , M | = 0 B 0 0 B , ( 0 B SN) E (6)

f (H0|SN) = dMB f (H0, MB|SN), (7) where the posterior on H0 was obtained by marginalizing over MB. In the equations above, f (H0 ) is an improper ﬂat prior on H0, L is the likelihood, and E is the evidence. Here SN stands for Supercal supernovae in the range 0.023 z 0.15 [26]. In addition, f (MB ) is the informative prior on the supernova FIG. 2. Reconstruction of the determination by Reid et al. [7] absolute magnitude and is the result of the complicated cali- of Eq. (10) using the calibration prior of Eq. (16). Also shown is bration of the local supernovae via the cosmic distance ladder a Gaussian probability distribution function (PDF) with same mean (see [24] for details). and dispersion. It is evident that the deviation from Gaussianity is The likelihood is given by negligible.

1 2 −1/2 − χ (H0,q0,MB ) L(SN|H0, q0, MB ) =|2π| e 2 , (8) One can then match first and second moments of the log- where the χ 2 function is normal distribution with Eq. (10) so that one has χ 2 = − t −1 − t , mB,i mB(zi ) ij mB, j mB(z j ) (9) μRe19 . ,σRe19 . , ln 4 2971 ln 0 019 046 (13) m , where is the supernova covariance matrix and B i are the whereweusedEqs.(A8) and (A9). The calibration prior is z observed apparent magnitudes at the redshifts i. Following then given by the methodology of [24], we fix the nuisance parameters that α = . β = . 5 S ln 10 1 control stretch and color corrections to 0 14 and 3 1, dm = μRe19 + 1 − σ 2 + , MB ln M (14) correct the apparent supernova magnitudes with hosts above ln 10 S0 5 S0 ∼ and below log10 Mstellar 10 by 0.03 mag fainter and brighter, 2 25 2 Re19 1 respectively, and include an intrinsic dispersion of σint = 0.1 σ = σ − . Mdm 2 ln (15) mag together with a peculiar velocity error of 250 km/s. B ln 10 S0 =− . The analysis of [7]fixesq0 0 55 in Eq. (6) and obtains The quantities S0 and S1 are defined in Appendix A, where the Re19 = . ± . −1 −1. full derivation can be found. Using the previous equations, one H0 73 5 1 4kms Mpc (10) obtains the calibration prior f (MB ): Reference [6] states that the sensitivity of H0 to knowledge of Mdm =−19.2334 ± 0.0404 mag. (16) q0 is very low as the mean SN redshift is only 0.07. However, B =− . q0 0 55 is justified only if one uses information beyond In the computation above we considered the second-order the local universe that can constrain the properties of dark correction in dL/z and fixed j0 = 1 because [6] does so [see matter and dark energy [see Eq. (4)], such as data that include Eq. (19)]. The second-order correction shifts the prior by supernovae at z > 0.15 [25,27]. This was necessary as there asmall∼0.1σ . The log-normal distribution on H0 is very were not enough local supernovae and/or their calibration close to a Gaussian as shown in Fig. 2, where the log-normal was less precise so it was not possible to constrain well H 0 f (H0|SN) is compared with a Gaussian, both with mean and if q0 was not fixed. However, as we will show below (see dispersion as in Eq. (10). also [14]), the latest supernova catalog and anchors allow us to perform an analysis which is only based in the z 0.15 B. Deconvolution universe. It is important to stress that the actual analysis of [6] is more complicated than the one of Eq. (6)asitinvolves One may obtain the calibration prior also via the following many intermediate steps. approach. The Hubble constant and the supernova absolute Here, in order to get the calibration prior f (MB )wesolve magnitude are connected through the equation [24] the integral equation obtained by demanding that Eq. (7)gives MB + 5aB(q0, j0 ) + 25 the constraint of Eq. (10). Assuming a Gaussian distribution log H0(q0, j0 ) = , (17) 10 5 for MB with mean M¯ B and dispersion σM , which is also where the “intercept” a is obtained via the Hubble-Flow justified a posteriori by the fact that MB is tightly constrained B by data, it is possible to marginalize analytically the two- supernovae, dimensional (2D) posterior in Eq. (7). The result is that H 0 a (z, m , q , j ) = log [cz f (z, q , j )] − 1 m , (18) is distributed according to a log-normal distribution with the B B 0 0 10 0 0 5 B parameters 1 − q 1 − q − 3q2 + j f = 1 + 0 z − 0 0 0 z2 + O(z3), (19) ln 10 ln 10 1 S μdm = ¯ + σ 2 + − 1 , 2 6 ln MB M (11) 5 5 S0 S0 where z and m are the observed redshift and apparent mag- B nitude of the supernovae. In (17) the absolute magnitude MB ln 10 1 σ dm = σ 2 + . and the intercept aB are independent as the former is obtained ln M (12) 5 S0 via the astrophysical observations of anchors, Cepheid hosts,

013028-3 DAVID CAMARENA AND VALERIO MARRA PHYSICAL REVIEW RESEARCH 2, 013028 (2020) and calibrators, while the latter is obtained via Hubble-Flow C. Robustness supernovae that probe the cosmological luminosity-distance– The computation of the calibration prior with the methods =− . = redshift relation. In particular, fixing q0 0 55 and j0 1 of the previous sections is possible because its determination affects aB but not MB. This shows clearly that it is possible to is independent of the cosmological Hubble flow that is used derive the constraint on MB by “subtracting” (or deconvolv- to determine H0. In particular, MB is independent of the value ing) the contribution of aB from the constraint on H0. adopted for the cosmographic parameters q0 and j0 as their The constraint on aB using the Supercal sample [26]is[24] effect is to alter the luminosity distance and not MB.This aB(q0 =−0.55, j0 = 1) = 0.712 73 ± 0.001 76, (20) would not be the case if SH0ES constrained Hubble flow and MB together as, in this case, MB would be correlated with the which can be obtained via [28] Hubble-flow supernovae. −1 , (C )ij aB,i 1 Even though the methodology is well posed, there could a = i j ,σ2 = , (21) B −1 aB −1 be astrophysical or cosmological effects that could bias the i, j (C )ij i, j (C )ij determination of MB. A calibration bias could be produced where aB,i = aB(zi, mB,i, q0, j0 ) are the intercept measure- by the coherent flow of the local supernovae, akin to cosmic ments with covariance matrix C. After propagating the errors variance [11]. However, both the determination of Eq. (10) on z (mostly peculiar velocity) on mB,itisC = /25, where and the Supercal data set have already been corrected for is the covariance matrix for mB. Regarding mB and ,we such effects using velocity fields derived from galaxy den- follow the methodology of [24] as described in the preceding sity fields [26]. Another bias could be due to correlations section. between supernovae and environmental properties such as One can then proceed in two ways. The simplest approach star formation rate and metallicity. Reference [6] concluded is just to perform first-order error propagation that these correlations are not significant, so they should not

e = Re19 − − =− . , impact Eq. (16) either (see, however, [29,30]). Therefore, we MB 5 log10 H0 aB 5 19 2322 conclude that our determination of the MB prior is robust and σ 2 correctly encodes the supernova calibration via the distance 5 H Re19 σ e = 0 − 25σ 2 = 0.0404. (22) ladder. M Re19 aB ln 10 H0 Alternatively, one can make a change of variable so that the D. Calibration prior in cosmological analyses higher moments are included. Adopting Gaussian distribu- The calibration prior on MB of Eq. (16) can be meaning- tions for MB and aB, it follows that H0 is distributed according fully used in cosmological analyses instead of the correspond- to the log-normal distribution of Eq. (A5) with the parameters ing H0 determination. Indeed, there are 175 supernovae in ln 10 common between the Supercal and Pantheon data sets in the μdc = (M¯ + 5a + 25), (23) ln 5 B B range 0.023 z 0.15 and, in the standard analysis, these supernovae are used twice: once for the H determination and ln 10 0 σ dc = σ 2 + 25σ 2 , (24) once when constraining the cosmological parameters. This ln 5 M aB induces a covariance between H0 and the other parameters so the calibration prior is given by which could bias cosmological inference. The advantage of ln 10 1 adopting the calibration prior is that these 175 supernovae are dc = dm + σ 2 + , MB MB M (25) used only once, avoiding any potential bias. 5 S0

2 2 σ dc = σ dm , (26) IV. DETERMINATION OF LOCAL H0 MB MB where we used the fact that Here we wish to obtain a determination of H0 that is . S 1 based solely on the local universe (z 0 15) and that is truly 5a + 25 =− 1 , 25σ 2 = . (27) B aB independent of any cosmological assumptions besides the cos- S0 S0 mological principle and the astrophysics of anchors, Cepheid Using the previous equations, one obtains the calibration prior hosts, and calibrators. We achieve the latter by considering an f (MB ), uninformative flat prior on q0 and marginalizing the posterior dc =− . ± . , also over q0, that is, MB 19 2326 0 0404 mag (28) f (H ) f (q ) f (M )L which confirms the validity of the first-order estimate of f H , q , M | = 0 0 B , ( 0 0 B SN) E (30) Eq. (22). We see that the demarginalization and deconvolution approaches give very similar results: The variance of the f (H0|SN) = dMBdq0 f (H0, q0, MB|SN), (31) prior is actually identical while the central value differs by a negligible amount: where f (q ) is the flat ignorance prior on q . The method 0 0 σ 2 of [6]isrecoveredifweset f (q ) = δ(q + 0.55). In this 5 H Re19 − 0 0 Mdc − Mdm 0 8 × 10 4. (29) analysis we adopt the more recent Pantheon supernova sample B B ln 10 H Re19 0 [25], which features 237 supernovae in the redshift range We will adopt the demarginalization result as the analysis of 0.023 z 0.15 (see Fig. 1). In this case the covariance the next section generalizes its methodology. matrix and the apparent supernova magnitudes do not depend

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This work Finally, it is interesting to compare directly the marginalized 2D posterior on H0 and q0 from CMB and low-z super- Planck 2018 nova observations. We obtained the CMB posterior using the MCMC chain from the CMB-only Planck analysis available from [33]. The tension in the H0-q0 plane is also presented in Fig. 3. In order to quantity the tension we adopt the index of inconsistency (IOI) [34], which directly generalizes the esti- matorofEq.(35). We ﬁnd √ ≡ δT + −1δ . , 0 2IOI (Cloc CPlanck ) 4 54 δ = loc − Planck, loc − Planck , −0.5 H0 H0 q0 q0 (36)

where C are the covariance matrices on H0 and q0 from the 0 −1 q analysis of Fig. 3 and δ is the difference vector. Note that these −1.5 estimators assume Gaussianity and that the posteriors on H0 and q0 are very close to Gaussian. The tension in the H0-q0 −2 plane is again very strong: 4.5σ .

68 72 76 80 −2.0 −1.5 −1.0 −0.5 0.0 V. DISCUSSION H0 q0 Our determination of the Hubble constant of Eq. (32)is

FIG. 3. Determination of the Hubble constant H0 and the de- based on the calibration prior of Eq. (16) and on the low- celeration parameter q0 from the Pantheon supernovae in the redshift cosmographic expansion of Eq. (2). While the validity redshift range 0.023 z 0.15 [25]. The constraints have been of the calibration prior depends on the standardizable nature marginalized over the absolute magnitude MB. Also shown are the of supernovae Ia, a process that includes corrections due marginalized 2D constraints on H0 and q0 from the CMB-only Planck to color, stretch, and host-galaxy mass, the validity of the Collaboration analysis that assumes the standard flat CDM model cosmographic analysis is solely based on the approximation [8]. The tension between the two determinations is at the 4.5σ level. that the FLRW metric provides a good description of our universe at large scales. While it is possible to test directly on the nuisance parameters that control stretch and color this hypothesis [35–38], the validity of the FLRW metric is corrections, and the correction for the apparent magnitudes a direct consequence of assuming the cosmological principle, ∼ according to which the universe is homogeneous and isotropic with hosts above and below log10 Mstellar 10 is already applied. Also, the covariance matrix includes all the uncertain- at large scales. Therefore, as far as cosmological assumptions ties. The results relative to the Supercal sample are given in are concerned, the determination of Eq. (32) only assumes the Appendix B. cosmological principle. We have used EMCEE [31], an open-source sampler for While our goal here was to improve the local determination Markov chain Monte Carlo (MCMC), to sample the posterior of H0 by considering a flat prior on q0,itisalsopossibleto and GETDIST [32] for analyzing the chains. The result of this perform a more thorough reanalysis of the cosmic distance analysis is presented in Fig. 3. We obtain ladder and consider the impact of non-Gaussianities [14], hyperparameters [12], and a blinded pipeline [13]. In partic- loc = . ± . −1 −1, H0 75 35 1 68 km s Mpc (32) ular, our results are in agreement with the findings of [14], which, using a Bayesian hierarchical model, also found that loc =− . ± . . q0 1 08 0 29 (33) data prefer a lower value of q0 ≈−1 and so, owing to their The latest CMB-only constraint from the Planck Collabora- anticorrelation, a higher value of H0. In particular, Ref. [14] =− . ± tion (see Table 2 in [8]) is adopted a broad (truncated) Gaussian prior q0 0 5 1, similar to our improper flat prior, and obtained a value of Planck = . ± . −1 −1 −1 −1 H0 67 36 0 54 km s Mpc (34) H0 which is ∼1kms Mpc higher than their baseline H0 . σ value. This is in good agreement with our results relative to and the tension with our determination now reaches 4 5 : the Supercal sample, the one adopted in [14]. Indeed, the H loc − H Planck ∼ −1 −1 0 0 4.52. (35) determination of Eq. (B1)is 1kms Mpc rather than the 2 2 Re19 σ + σ H0 of Eq. (10). H0 H Planck 0 It is also possible to extend the cosmic ladder to baryon As it is clear from Fig. 3, low-redshift supernovae are acoustic oscillation (BAO) and CMB observations. Refer- able to constrain not only H0 but also q0, whose distribution ence [39] constrained H0 via a cosmographic inverse lad- is peaked at values lower than the standard model one of der approach, using BAO measurements to break the de- q0 =−0.55. However, the deceleration parameter is not yet generacy between MB and H0 and finding a value of the M18 = tightly constrained and the tension between our determination Hubble constant very close to the Planck one: H0 and the one by Planck is at the 1.9σ level. Note that the 67.8 ± 1.3kms−1 Mpc−1. The cosmographic inverse ladder CDM model cannot give values of q0 below −1. approach propagates the prior on the sound horizon from

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CMB to the BAO scale. However, the calibration of MB where ‘const’ consists of all terms that do not depend on H0 depends on the luminosity-distance–redshift relation of super- and MB and we defined novae both at low and high redshift. This introduces correla- t Wi = mB,i − m˜ (zi ), Vi = 1, tions between MB and the properties of dark energy, which can B −1 T explain why our cosmology-independent determination of q0 S0 = V · · V , M18 =− . ± . differs from the constraint q0 0 37 0 15 obtained in −1 T S1 = W · · V . (A3) [39]. Finally, a low local value of q0 ≈−1 was also obtained in [40], where the cosmic ladder was extended to BAO and The unnormalized posterior of Eq. (6) is then CMB observations using the distance-duality relation instead − 2 (M M¯ B ) of a high-redshift cosmographic expansion. − 1 2 2σ 2 − χ (H0,MB ) f (H0, MB|SN) ∝ e M e 2 , (A4) where an improper flat prior on H was adopted. Equation VI. CONCLUSION 0 (A4) can then be integrated over MB so that the normalized We obtained the effective local calibration prior on the posterior on H0 is distributed according to the log-normal absolute magnitude MB of Supernovae Ia, which we used distribution to obtain a determination of the Hubble constant from the local universe that assumes only large-scale homogeneity and f (H0|SN) = dMB f (H0, MB|SN) −1 −1 isotropy: H0 = 75.35 ± 1.68 km s Mpc . Our determination uses the latest Pantheon sample. As shown in Fig. 3,the 1 (ln H − μ )2 = √ exp − 0 ln , (A5) allowed values are in stronger tension with what the CMB σ 2 H0 2πσln 2 predicts if the standard model of cosmology is valid. The ln H crisis is approaching the 5σ level. We stress that our with the parameters 0 determination relies only on supernovae in the redshift range ln 10 ln 10 1 S . . μdm = ¯ + σ 2 + − 1 , 0 023 z 0 15. If, on one hand, this limits the available ln MB M (A6) 5 5 S0 S0 information as there are many more supernovae at higher redshifts, on the other hand, this restricted redshift range makes ln 10 1 this analysis local. We would like to stress that the calibration σ dm = σ 2 + . (A7) ln 5 M S prior on MB =−19.2334 ± 0.0404 mag can be meaningfully 0 used in cosmological analyses, instead of the corresponding Finally, the mean and variance of the log-normal distribution H determination, in order to avoid the double counting of 0 are low-redshift supernovae. 1 2 The fact that both Hubble constant and deceleration pa- μln+ σ H¯0 = e 2 ln , (A8) rameter differ seems to suggest that a new phenomenon is 2 σ 2 2μ +σ 2 affecting both the sound horizon and the recent expansion σ = (e ln − 1)e ln ln . (A9) rate of the universe. Early dark energy may be a promising H0 candidate [16]. The increased statistical significance of our APPENDIX B: ANALYSIS WITH THE SUPERCAL SAMPLE determination of H0 reinforces this suggestion. Here we present the results using the Supercal supernova catalog [26]thatwasusedin[6,24]. It features 217 supernovae ACKNOWLEDGMENTS in the redshift range 0.023 z 0.15 (see Fig. 1). Figure 4 D.C. thanks CAPES for financial support. V.M. thanks shows the results of this analysis, which are compared with CNPq and FAPES for partial financial support. We thank the ones of Sec. IV that use the more recent Pantheon sample. the referee for suggesting the alternative method used in We obtain Sec. III B. loc = . ± . −1 −1, H0 74 62 1 54 km s Mpc (B1) loc =− . ± . , APPENDIX A: DERIVATION OF THE q0 0 90 0 22 (B2) DEMARGINALIZATION which are in tension with the determinations of H0 and q0 Here we explain how we obtained Eqs. (11) and (12). First, by the Planck Collaboration at the 4.5σ and 1.7σ levels, we define respectively. The tension in the H0 − q0 plane is 4.5σ . There is good agreement between the results that use the Supercal t ≡ t − + , m˜ B mB MB logh (A1) and Pantheon data sets. The previous results were obtained using the methodology = where we used the shorthand notation logh 5log10 H0.We of [24], that is, fixing the nuisance parameters that control can then rewrite Eq. (9)as stretch and color corrections to α = 0.14 and β = 3.1, cor- recting the apparent supernova magnitudes with hosts above χ 2 ={ + − }−1{ + − } ∼ Wi logh MB ij Wi logh MB and below log10 Mstellar 10 by 0.03 mag fainter and brighter, respectively, and including an intrinsic dispersion of σ = = − − / 2 + , int S0(MB logh S1 S0 ) const (A2) 0.1 mag together with a peculiar velocity error of 250 km/s.

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α β Pantheon If instead we adopt a ﬂat prior on and we obtain basically the same results: Supercal loc = . ± . −1 −1, H0 74 85 1 57 km s Mpc (B3) loc =− . ± . . q0 0 98 0 24 (B4)

APPENDIX C: ANALYSIS RELATIVE TO THAT OF RIESS et al. Here we report the results relative to the measurement by 0 Riess et al. in [6]: R19 = . ± . −1 −1. −0.5 H0 74 03 1 42 km s Mpc (C1) The calibration prior is 0 −1 q Mdm =−19.2178 ± 0.0407 mag, (C2) −1.5 B which gives the determinations −2 loc = . ± . −1 −1, H0 75 89 1 70 km s Mpc (C3) 69 72 75 78 81 −2.0 −1.5 −1.0 −0.5 0.0 loc =− . ± . , H0 q0 q0 1 08 0 29 (C4) which are in tension with the corresponding CMB-only con- FIG. 4. Comparison between the analysis that uses the Supercal . σ . σ supernova sample and the one of Fig. 3 that uses the Pantheon straints from the Planck Collaboration at the 4 8 and 1 9 sample. The agreement is good. levels, respectively. The tension in the H0-q0 plane is at the 4.8σ level.

[1]P.A.R.Adeet al. (Planck Collaboration), Planck 2013 results. [11] D. Camarena and V. Marra, Impact of the cosmic variance on

XVI. Cosmological parameters, Astron. Astrophys. 571, A16 H0 on cosmological analyses, Phys.Rev.D98, 023537 (2018). (2014). [12] W. Cardona, M. Kunz, and V. Pettorino, Determining H0 with [2] A. G. Riess, L. Macri, S. Casertano, H. Lampeitl, H. C. Bayesian hyper-parameters, J. Cosmol. Astropart. Phys. 1703, Ferguson et al., A 3 space telescope and wide ﬁeld camera 3, 056 (2017). Astrophys. J. 730, 119 (2011). [13] B. R. Zhang, M. J. Childress, T. M. Davis, N. V. Karpenka, [3] D. N. Spergel, R. Flauger, and R. Hložek, Planck data reconsid- C. Lidman, B. P. Schmidt, and M. Smith, A blinded determi-

ered, Phys. Rev. D 91, 023518 (2015). nation of H0 from low-redshift type Ia supernovae, calibrated [4] N. Aghanim et al. (Planck Collaboration), Planck intermediate by Cepheid variables, Mon. Not. R. Astron. Soc. 471, 2254 results. XLVI. Reduction of large-scale systematic effects in (2017). HFI polarization maps and estimation of the reionization optical [14] S. M. Feeney, D. J. Mortlock, and N. Dalmasso, Clarifying the depth, Astron. Astrophys. 596, A107 (2016). Hubble constant tension with a Bayesian hierarchical model of [5] G. Pietrzynski,´ D. Graczyk, A. Gallenne, W. Gieren, I. B. the local distance ladder, Mon. Not. R. Astron. Soc. 476, 3861 Thompson, B. Pilecki, P. Karczmarek, M. Górski, K. (2018). Suchomska, M. Taormina et al., A distance to the Large Mag- [15] K. Vattis, S. M. Koushiappas, and A. Loeb, Dark matter decay-

ellanic Cloud that is precise to one per cent, Nature (London) ing in the late Universe can relieve the H0 tension, Phys.Rev.D 567, 200 (2019). 99, 121302 (2019). [6] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. [16] V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Early Scolnic, Large Magellanic Cloud Cepheid standards provide a Dark Energy Can Resolve the Hubble Tension, Phys. Rev. Lett. 1% foundation for the determination of the Hubble constant and 122, 221301 (2019).

stronger evidence for physics beyond LambdaCDM, Astrophys. [17] R.-Y. Guo, J.-F. Zhang, and X. Zhang, Can the H0 tension J. 876, 85 (2019). be resolved in extensions to CDM cosmology? J. Cosmol. [7] M. J. Reid, D. W. Pesce, and A. G. Riess, An improved distance Astropart. Phys. 1902, 054 (2019). to NGC 4258 and its implications for the hubble constant, [18] E. Ó. Colgáin, M. H. P. M. van Putten, and H. Yavartanoo, de

Astrophys. J. Lett. 886, L27 (2019). Sitter Swampland, H0 tension & observation, Phys. Lett. B 793, [8] N. Aghanim et al. (Planck Collaboration), Planck 2018 results. 126 (2019). VI. Cosmological parameters, arXiv:1807.06209. [19] E. Di Valentino, A. Melchiorri, and J. Silk, Reconciling Planck

[9] W. L. Freedman, Cosmology at a crossroads, Nat. Astron. 1, with the local value of H0 in extended parameter space, Phys. 0121 (2017). Lett. B 761, 242 (2016). [10] L. Verde, T. Treu, and A. G. Riess, Tensions between the early [20] L. Knox and M. Millea, The Hubble hunter’s guide, and late Universe, Nat. Astron. 3, 891 (2019). arXiv:1908.03663.

013028-7 DAVID CAMARENA AND VALERIO MARRA PHYSICAL REVIEW RESEARCH 2, 013028 (2020)

[21] V. Marra, L. Amendola, I. Sawicki, and W. Valkenburg, Cosmic [31] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, Variance and the Measurement of the Local Hubble Parameter, emcee: The MCMC hammer, Publ. Astron. Soc. Pac. 125, 306 Phys. Rev. Lett. 110, 241305 (2013). (2013). [22] C. Cattoen and M. Visser, The Hubble series: Convergence [32] A. Lewis, GetDist: A Python package for analysing Monte properties and redshift variables, Class. Quantum Grav. 24, Carlo samples, arXiv:1910.13970, available at https://getdist. 5985 (2007). readthedocs.io. [23] M.-J. Zhang, H. Li, and J.-Q. Xia, What do we know about [33] esa.int/Planck. cosmography, Eur. Phys. J. C 77, 434 (2017). [34] W. Lin and M. Ishak, Cosmological discordances: A new [24] A. G. Riess et al., A 2.4% determination of the local value of measure, marginalization effects, and application to geometry the Hubble constant, Astrophys. J. 826, 56 (2016). versus growth current data sets, Phys. Rev. D 96, 023532 [25] D. M. Scolnic et al., The complete light-curve sample of (2017). spectroscopically conﬁrmed SNe Ia from Pan-STARRS1 and [35] W. Valkenburg, V. Marra, and C. Clarkson, Testing the Coper- cosmological constraints from the combined Pantheon sample, nican principle by constraining spatial homogeneity, Mon. Not. Astrophys. J. 859, 101 (2018). R. Astron. Soc. 438, L6 (2014). [26] D. Scolnic et al., Supercal: Cross-calibration of multiple pho- [36] M. Scrimgeour et al., The WiggleZ Dark Energy Survey: The tometric systems to improve cosmological measurements with transition to large-scale cosmic homogeneity, Mon. Not. R. type Ia supernovae, Astrophys. J. 815, 117 (2015). Astron. Soc. 425, 116 (2012). [27] M. Betoule et al. (SDSS Collaboration), Improved cosmo- [37] V. Marra and D. Sapone, Null tests of the standard model logical constraints from a joint analysis of the SDSS-II using the linear model formalism, Phys. Rev. D 97, 083510 and SNLS supernova samples, Astron. Astrophys. 568, A22 (2018). (2014). [38] R. S. Gonçalves, G. C. Carvalho, C. A. P. Bengaly, J. C. [28] M. Schmelling, Averaging correlated data, Phys. Scr. 51, 6769 Carvalho, and J. S. Alcaniz, Measuring the scale of cosmic ho- (1995). mogeneity with SDSS-IV DR14 quasars, Mon. Not. R. Astron. [29] Y.-L. Kim, M. Smith, M. Sullivan, and Y.-W. Lee, Environ- Soc. 481, 5270 (2018). mental dependence of type Ia supernova luminosities from a [39] E. Macaulay et al. (DES Collaboration), First cosmological sample without a local-global difference in host star formation, results using type Ia supernovae from the Dark Energy Survey: Astrophys. J. 854, 24 (2018). Measurement of the Hubble constant, Mon. Not. R. Astron. Soc. [30] M. Rigault et al. (Nearby Supernova Factory Collaboration), 486, 2184 (2019). Strong dependence of type Ia supernova standardization on the [40] D. Camarena and V. Marra, A new method to build the (inverse) local speciﬁc star formation rate, arXiv:1806.03849. distance ladder, arXiv:1910.14125.

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