Is there really a Hubble tension?

The intrinsic magnitudes of nearby type Ia supernovae (SNe Ia) to which distances are known independently are characterised by a large scatter. However by exploiting the empirical (wavelength-dependent) correlation between the intrinsic supernova magnitude and the timescale of the luminosity decline [35], this scatter can be reduced to ∼0.1–0.2 mag, making them 'standardisable candles'. ³ Recently, the magnitude-redshift relation of SNe Ia in the nearby (z < 0.15) Universe has been leveraged with the local Cepheid-calibrated distance ladder to measure the Hubble constant to increasingly high precision [37] in what is claimed to be a model-independent manner. The single largest source of uncertainty in determining H₀ is now said to be the mean luminosity of the SNe Ia calibrators [38].

Such measurements have been used to argue [38] that there is a 4.4σ 'Hubble tension' between H₀ in the late Universe and its value inferred from the cosmic microwave background (CMB) assuming the standard flat ΛCDM cosmological model [1]. This is said to be robust with respect to choices of independent calibrators [39] so has stimulated an avalanche of explanations, many involving speculative new physics.

Today, publicly available SNe Ia data such as the SDSS-II/SNLS3 joint lightcurve analysis (JLA) [4] as well as the subsequent Pantheon [40] compilations come with observables already 'corrected' to account for the effects of peculiar velocities in the local Universe. However, as we describe in section 2, apart from modifying the magnitudes (redshifts) of low redshift SNe Ia by up to 0.2 mag (∼20%), these corrections appear to be arbitrary, error-prone and discontinuous within the data, while the datasets are discrepant with respect to each other in key observables. In section 3 we show that these discrepancies in heliocentric redshifts between JLA and Pantheon are on their own large enough that a significant 'Hubble tension' cannot be claimed between the late and early Universe determinations of H₀. We conclude (section 4) with a discussion of the model dependence of these corrections.

For 58 SNe Ia from SDSS-II [28] that are in common between the JLA and the Pantheon catalogues, the quoted heliocentric redshifts differ by between 5 to 137 times the quoted uncertainty in the redshift measurement. Their data is given in table 1, while their distribution on the sky is shown in figure 1. Many more SNe Ia in common between the catalogues have smaller shifts in their z_hel values. The uncertainty on the spectroscopic redshift measurement of the host galaxy is quoted as 0.0001–0.0002 in SDSS DR4, and 0.0005 for redshifts which were measured by the authors themselves [28]. Note that the ${\sigma }_{{z}_{\text{spec}}}$ arising from peculiar velocities mentioned in reference [28] is not a measurement uncertainty but rather the expected dispersion with respect to theoretical predictions. The quoted redshifts cannot have changed unless if they have been remeasured, a process that has not been documented in reference [40]. In fact the JLA z_hel values are in exact agreement with other public sources such as VizieR, whereas the Pantheon z_hel values are not independently verifiable. While some of these discrepancies may be due to the slight difference between the redshift of the supernova and of its host galaxy [45], no such distinction is made in the disseminated datasets and the covariances provided with the data are certainly not large enough to account for the shifts.

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It has been noted that the magnitudes of these SNe Ia are also inconsistent between the two catalogues ⁴ . These shifts seem to have been introduced on 27 November 2018 when new files were uploaded to purportedly rectify previously reported errors in the peculiar velocity corrections ⁵ . Such inconsistencies in publicly available SNe Ia data have also been noted earlier with respect to the choice of light curve fitter [2].

The supernova catalogues also provide z_cmb, the boosted redshift in the 'CMB rest frame' in which the CMB is supposed to look isotropic, assuming that its dipole asymmetry is entirely due to our motion with respect to this frame, and further corrected using a model of the local peculiar (non-Hubble) velocity field to account for the motion of the SNe Ia with respect to this frame ⁶ . The inconsistencies in the peculiar velocity corrections made in the JLA catalogue have been discussed elsewhere [11], in particular that while relying on the flow model [24], the corrections applied extend well beyond the extent of the survey (z ∼ 0.04) on which this model is based, and moreover abruptly fall to zero at z ∼ 0.06, even though the same model [24] reports a residual bulk velocity of 687 ± 203 km s⁻¹ beyond z ∼ 0.04 — which is over 4 times larger than the uncorrelated velocity dispersion of cσ_z = 150 km s⁻¹ included in the JLA error budget for cosmological fits. (There are other inconsistencies as well, e.g. SDSS2308 has the same z_cmb and z_hel despite being at a redshift of 0.14.)

Significantly more egregious errors are seen in the first version of the Pantheon compilation on Github [41] wherein peculiar velocity corrections were used to modify the redshifts of SNe Ia all the way up to z ≳ 0.2 although no survey has yet gone to such depths so the information required to make such corrections is simply not available. An illustrative example is SN2246, with z_cmb = 0.19422 which has been corrected by a peculiar velocity of 444.3 km s⁻¹. This object is 117° away from the CMB dipole, and 115° away from the direction of the external bulk flow reported by reference [7].

This issue is now said [41] to have been fixed by not making any peculiar velocity corrections for z > 0.08 in Pantheon. However the impact of this major change on the determination of H₀ or other cosmological parameters [40] has not been documented. In reference [37] the sample of SNe Ia is limited to 0.023 < z < 0.15 in order not to be affected by the coherent flow in the local volume, but these authors too account for such flows using the same model [7].

Peculiar velocities affect the distance modulus by (5/log 10)(v/cz) mag. Thus the peculiar velocity corrections shift the distance moduli of SNe Ia at redshifts smaller than the extent of the flow model by between 0.03 and 0.4 mag. Neither of the models used for correcting the supernova observables in JLA or Pantheon — respectively references [24] and [7] — report convergence between the correlated flow and the 'CMB rest frame'. Instead reference [7] provides definitive (>5σ) evidence for a residual bulk flow of V_ext = 159 ± 23 km s⁻¹ originating from sources beyond the extent (z = 0.067) of the model. This means that by choosing to make corrections up to the extent of the flow model and leaving the rest uncorrected, abrupt and arbitrary discontinuities of ∼0.07 (∼0.03) mag are introduced in the distance moduli of JLA (Pantheon) supernovae at these threshold redshifts. These discontinuities which introduce bias, are handled by adding just a variance term in the error budgets of the SNe Ia catalogues. We wish to draw attention to the following problematic issues in this regard:

• The modified data now encode a picture of the Universe in which a sphere of radius 170, 285 or 340 Mpc (depending upon whether the flow model is taken to extend to z = 0.04, 0.067 or 0.08) is smashing into the rest of the Universe — which is arbitrarily treated as at rest.
• Out of the 740 (1048) SNe Ia that make up JLA (Pantheon) catalogue, 632 (890) (including all the objects in table 1) are in the direction opposite to bulk flow reported in references [24] and [7].
• Corrections for peculiar velocities were introduced in supernova cosmology in 2011 [12] after low-to-intermediate redshift supernovae were first observed [28]. These are all in the hemisphere opposite to the CMB dipole (see figure 1). It is this set of SDSS-II SNe Ia, which crucially fill in the 'redshift desert' between low and high redshift objects, that are now alarmingly discrepant between JLA and Pantheon.

Errors in redshift measurements as small as Δz ∼ 0.0001 can have significant impact on the value of inferred cosmological parameters such as H₀ [14]. Since some of the shifts in z_hel reported in table 1 are as high as 0.1, we study the impact of these shifts on the measured value of the Hubble constant. Following reference [37], the distance modulus is

$\begin{equation}\mu =25+5\enspace {\mathrm{log}}_{10}(D/\mathrm{M}\mathrm{p}\mathrm{c}),\end{equation} \tag{ 1 }$

where D is the luminosity distance in Mpc, given in the cosmographic Taylor expansion (which is accurate to better than 7% for z < 1.3 [11] as in the JLA catalogue) by:

$\begin{equation}D(z)=\frac{cz}{{H}_{0}}\left[1+\frac{1}{2}(1-{q}_{0})z-\frac{1}{6}(1-{q}_{0}-3{q}_{0}^{2}+{j}_{0}-{{\Omega}}_{k}){z}^{2}+\cdots \enspace \right].\end{equation} \tag{ 2 }$

For each SNe Ia, the observational distance modulus is constructed as the difference in magnitudes of an apparent and absolute flux, μ^obs = m − M, where m is corrected for the specific light curve shape and colour by adding to it: αx₁ − βc. Here α and β are parameters assumed to be constants for all SNe Ia, while x₁, c and m are provided separately for each SNe Ia from the SALT2 template fit [4]. We then employ the standard 'χ² statistic'

$\begin{equation}{\chi }^{2}=\sum\limits _{i}\frac{{({\mu }_{i}-{\mu }_{i}^{\text{obs}})}^{2}}{{({\sigma }_{{\mu }_{i}})}^{2}+{({\sigma }_{\mu }^{\text{int}})}^{2}}.\end{equation} \tag{ 3 }$

Here σ_μi is the uncertainty for the ith SNe Ia provided with the catalogue, while ${\sigma }_{\mu }^{\text{int}}$ is the (unknown) intrinsic scatter.

Following reference [37], we set the deceleration parameter q₀ = −0.55, the jerk parameter j₀ = 1 and the curvature parameter Ω_k = 0 in the scans. While precise measurements of H₀ require the calibration of the absolute supernova luminosity M using a local distance ladder, we fix M to −19.10 [37] with the specific aim of studying the impact of redshift errors on H₀ since none of the 19 SNe Ia with Cepheid calibrations for the host galaxies reported in reference [37] are included in either the JLA or Pantheon compilations. This choice affects only the absolute value of H₀, and not the relative shift introduced by the choice of redshift. We use the Emcee code [19] to perform a Markov Chain Monte Carlo scan in likelihood over the parameter space of H₀, α, β and σ_int (the intrinsic dispersion), all with flat priors. The observed shift in H₀ is found to be robust with respect to alternative parametrisations of the likelihood which are provided in a Jupyter notebook [36]. Out of the 58 SNe Ia listed in table 1, 45 have z_Pantheon − z_JLA > 0.0025, while 13 have z_Pantheon − z_JLA < −0.0025. Figure 2 (top panel) shows that for supernovae with Δz > 0.0025, the Pantheon redshifts favour H₀ ∼ 72 km s⁻¹ Mpc⁻¹, while the JLA redshifts favour H₀ ∼ 68 km s⁻¹ Mpc⁻¹. If we change the criterion to an absolute difference |Δz| > 0.0025, figure 2 (bottom panel) shows that the Pantheon redshifts now favour H₀ ∼ 71 km s⁻¹ Mpc⁻¹, while the JLA redshifts favour H₀ ∼ 69 km s⁻¹ Mpc⁻¹. If all 117 SNe in JLA with 0.023 < z_CMB < 0.15, which are also in Pantheon, are used in the scans, as shown in figure 3 (left panel), the redshift discrepancies average out to give consistent values of H₀ as long as the JLA magnitudes are employed — nevertheless the magnitude discrepancies between the two catalogues are large enough to shift H₀ by 10% or more. Indeed figure 3 (right panel) shows that the 178 JLA SNe Ia in the relevant redshift range indicate a low H₀ compared to the 237 SNe in the Pantheon compilation in the same redshift range. The magnitudes of the SNe in both JLA [4] and Pantheon [40] were also corrected for bias assuming the ΛCDM model. The JLA catalogue provides sufficient detail to enable these corrections to be reversed. For the 117 SNe in JLA with 0.023 < z_CMB < 0.15, which are also in Pantheon, the impact of these corrections on the inferred value of H₀ is <2% as shown in figure 3 (left panel).

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Reference [38] finds H₀ = 74.03 ± 1.42 km s⁻¹ Mpc⁻¹ and emphasises the difference from H₀ ∼ 67.4 ± 0.5 km s⁻¹ Mpc⁻¹ inferred from Planck data on CMB anisotropies assuming flat ΛCDM [1]. However this ignores the various inconsistencies we have pointed out above, so it cannot be claimed [37, 38] that this 'Hubble tension' is significant.

In general relativity, metric expansion in the late-time Universe is an average effect, arising from the coarse-graining of physics at smaller scales [6]. This differs from the metric expansion in the FLRW solution to the field equations, wherein due to the isotropy and homogeneity imposed on the stress–energy tensor, all clocks remain synchronised and space expands isotropically and homogeneously, described by a single scale factor. However the observed inhomogenous Universe can only be approximately described by an FLRW metric. Fitting a Hubble diagram of observables not corrected for peculiar velocities, as was done in supernova cosmology analyses until 2011, and employing peculiar velocity corrections after transforming to the CMB frame, as was done subsequently [12], simply amounts to different choices of corresponding two-spheres within the 'null fitting' procedure described in reference [18]. Whether the dynamical evolution of the representative FLRW model can be precisely related to the average evolution of the inhomogeneous Universe is contingent on the resolution of a number of open questions [8].

Motivated by such considerations it has been argued that peculiar velocities should rather be thought of as variations in the expansion rate of the Universe [30] (see also references [46, 48]). Assuming the CMB dipole arises due to our motion with respect to a frame in which the Universe is isotropic, all matter in the local Universe seems to be flowing in a similar direction [7], however convergence of this flow to the hypothetical CMB rest frame has not been found [10]. This adds additional credence to the possibility that the CMB dipole is not purely kinematic [42, 43]. The Universe we observe appears to be anisotropic, and lacking an observationally consistent standard of rest.

A statistically significant variation of ∼9 km s⁻¹ Mpc⁻¹ across the sky was found in the Hubble constant measured in the Hubble key project [30]. Recently a similar anisotropy has been seen using ROSAT and Chandra data on x-ray clusters [31, 32]. Such a systematic variation of H₀ across the sky is as expected according to a cosmographic Taylor series expansion of the luminosity distance for a Universe without exact symmetries [23]. This study finds that the monopole of the generalised Hubble parameter Θ/3 is modified by a quadrupolar contribution (90° separation between poles) e^μ  e^ν   σ_μν from the shear tensor of the observer congruence σ_μν describing the anisotropic expansion of space [23]. It appears therefore that H₀ cannot be measured in a model-independent manner in the locally inhomogeneous and anisotropic Universe to a precision better than ∼10%.

To gain apparently better precision, reference [38] follows reference [37] who employed peculiar velocity corrections based on the flow model [7], presumably as was also done for Pantheon [40]. However reference [7] infers the velocity field from the density field using linear perturbation theory around an assumed FLRW background. These measurements cannot thus be said to be model-independent. Data thus 'corrected' have then been used to insist that the Hubble expansion is indeed isotropic [44] and also to argue that local structure has no impact on the measurement of the Hubble constant [27].

While other astronomical probes e.g. strong gravitational lensing are said to provide independent evidence for the 'Hubble tension', there appears to be a similar directional dependence to the ∼15% relative variation in the value of H₀ derived from the six lenses [47]. However systematic uncertainties in these measurements may have been underestimated [5, 16].

Within the concordance ΛCDM model, the effect of inhomogeneities is studied by linearising the field equations around a maximally symmetric solution (see reference [9]). Further making restricted 'gauge' choices [3] motivated by the idea that the Universe began with only scalar density perturbations left over from inflation, the usual perturbed FLRW framework is arrived at. However, as emphasised in reference [3] this eliminates known physical phenomena, and solutions to the linearized field equations can only be linearisations of the solutions to the fully nonlinear equations [15, 33]. From a general relativity perspective there is in fact locally inhomogeneous expansion beyond that expected in linear perturbation theory around a maximally symmetric background [21]. Studies of the impact of peculiar velocities [13, 25, 34] using Newtonian N-body simulations cannot capture such physics. A recent numerical relativity simulation [29] suggests significantly larger deviations from an isotropic Hubble law in the late Universe.

It has been argued that systematic calibration offsets within the distance ladder can account for the 'Hubble tension' [17, 22]. In fact the discrepancy we have established between the determinations of H₀ using the JLA and Pantheon catalogues is equivalent to the 'systematic bias of 0.1–0.15 mag in the intercept of the Cepheid period-luminosity relations of SH0ES galaxies' [17]. Indeed independent calibrations of SNe Ia [20] prefer lower values of H₀ which are consistent with the early Universe. We emphasise that any measurement with a claimed uncertainty smaller than that of reference [30] must be scrutinised for its understanding of, and correction for, peculiar velocities. It is clear that the corrections that have been applied so far are rather arbitrary. The fact that even the observed (uncorrected) heliocentric redshifts undergo unexplained changes from one catalogue (JLA) to another (Pantheon) thus inducing significant variations in the inferred value of H₀, undermines the 'precision cosmology' programme. This calls for the blinded testing of the isotropy of the Hubble diagram with forthcoming data from e.g. the Legacy Survey of Space and Time to be conducted at the Vera C. Rubin Observatory (https://lsst.org/).

We thank the anonymous referees for helpful comments and suggestions which have led to improvement of our paper.

The data that support the findings of this study are openly available at the following URL/DOI: https://github.com/rameez3333/SN1aDataandH0.