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Late time approaches to the Hubble tension deforming $H(z)$, worsen the growth tension

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G. Alestas, L. Perivolaropoulos

Cosmology and Nongalactic Astrophysics · Submitted 6 Mar 2021 · 74 references

Pith's verdict: A clean, deterministic result — CPL deformations tuned to fix H0 grow perturbations faster than Planck18/ΛCDM — undercut by a Δσ tension metric that compares contours from two different models.

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arXiv ID2103.04045 (version 3)
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CitationG. Alestas, L. Perivolaropoulos. "Late time approaches to the Hubble tension deforming $H(z)$, worsen the growth tension." arXiv:2103.04045 (2021). https://pith.science/paper/CJLJPXA4
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The paper

The paper claims

Smooth late-time deformations of the expansion rate that raise $H_0$ to the locally measured value also raise the predicted growth of cosmic structure, deepening the separate disagreement between the cosmic microwave background and galaxy-s

The reading

The Hubble tension is the disagreement between the expansion rate inferred from the cosmic microwave background ($H_0 = 67.4$ km/s/Mpc) and the value measured from local distance ladders ($H_0 = 74.0$ km/s/Mpc). One widely tried family of fixes leaves the early universe untouched and instead bends the expansion history $H(z)$ at low redshift, holding the physical matter density $Ω_{0m}h^2$ and the comoving distance to the last scattering surface fixed so the microwave background spectrum is preserved, while pushing $H_0$ up to the local value. This paper asks whether that same bending also relieves a second, milder disagreement: the one between how much clustering the microwave background predicts and how much is actually seen in galaxy surveys. The authors' answer is no. Working through the standard two-parameter family of evolving dark energy equations of state, they find that the deformations which succeed at raising $H_0$ predict more growth of structure than the standard model does, so the mismatch with measured growth rates grows rather than shrinks.

What carries the argument

The argument is carried by the growth index $γ$, defined through $f=dlnΔ/dlna = Ω_m(a)^γ$ with $γ = 3(1-w)/(5-6w)$, which equals $6/11$ for a cosmological constant. Because the growth factor can be written as an integral of $Ω_m^{γ}$ over the expansion history, and because $γ$ stays in the narrow band $0.45$–$0.65$, that integral closely resembles the comoving-distance integral. The models under study are constructed to hold the comoving distance to recombination fixed, so their growth integrals are also nearly fixed, and the growth factor at $z=0$ ends up tracking $ω_m^{γ(w_∞)}$. Since $γ$ rises as $w_∞$ moves above $-1$, the deformations that lift $H_0$ also lift the growth factor. This analytic sketch is checked against the full numerical solution of the growth equation, and the two agree to within about four percent over $w_∞$ between $-2$ and $-0.5$.

Extended reading notes

Core claim

For the CPL family $w(z)=w_0+w_1 z/(1+z)$, the authors select the pairs $(w_0,w_1)$ that satisfy three conditions simultaneously: $Ω_{0m}h^2$ fixed to the Planck value $0.143$, the comoving distance to recombination fixed to $4.62$ Gpc, and $H_0$ fixed to the local value $74$ km/s/Mpc. This is the condition set that lets a late-time deformation mimic the cosmic microwave background while matching the local expansion rate. Solving the linear growth equation numerically for these pairs, they find the growth factor at $z=0$ exceeds the Planck18/$Λ$CDM value whenever the early-time asymptotic equation of state $w_∞ = w_0+w_1$ is greater than about $-0.5$. Fitting the same models to a compilation of redshift-space-distortion $fσ_8$ data and comparing the resulting $Ω_{0m}$–$σ_8$ contours with the Planck contours, the offset between the two best-fit $σ_8$ values comes out at $Δσ = 2.9$ to $3.4$, against $Δσ = 2$ for $Λ$CDM. So the models that are built to close the $H_0$ gap open the growth gap wider. The authors add two further costs: the same models fit supernova, BAO and CMB distance data worse than $Λ$CDM by $Δχ^2$ of $54$ to $279$, and their best-fit supernova absolute magnitude sits below the Cepheid-calibrated value.

Load-bearing premise

The claim rests on treating the CPL family, with three conditions imposed by hand, as a fair stand-in for every smooth late-time deformation of the expansion rate, and on reading the offset between the two best-fit $σ_8$ values as a valid measure of how bad the tension is.

Editorial extensions

If this is right

  • A late-time fix for the Hubble tension cannot be counted as free: any smooth deformation that raises $H_0$ while preserving the microwave background distance scale inherits a stronger $Ω_{0m}$–$σ_8$ mismatch with redshift-space-distortion data.
  • The class of models is already disfavoured by low-redshift geometry alone, since the best fit to supernova, BAO and CMB distance data lies at $w_∞ ≈ -1.1$ while the Hubble-solving pairs run to $w_∞ > -0.5$, more than $3σ$ away.
  • The same models push the best-fit supernova absolute magnitude below the Cepheid calibration, adding a third independent point of friction beyond distance fits and growth.
  • Approaches built on a rapid transition of supernova intrinsic luminosity near $z_t ≈ 0.01$ escape this particular problem, which the authors present as a reason to prefer them over smooth deformations.
  • If the growth prediction is generic, then future growth measurements tightening $σ_8$ will sharpen the exclusion of smooth late-time Hubble fixes rather than rescue them.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The mechanism suggests a trade-off rather than a coincidence: holding the distance to last scattering fixed while raising $H_0$ forces the dark energy to behave differently at low and high redshift, and the growth integral is sensitive to exactly that difference. Any deformation with the same three constraints should show a similar sign of effect, even if the magnitude varies.
  • Because the growth offset is computed from the difference of two best-fit $σ_8$ values rather than from a calibrated tension statistic, a reanalysis using a concordance-based measure could reorder the models even if the underlying growth factors are unchanged. The qualitative direction, however, follows from the growth-index argument and would survive a change of metric.
  • A model that raises $H_0$ by altering the expansion rate only over a narrow redshift window, rather than smoothly across all low redshift, might evade the growth penalty while keeping the distance to recombination fixed. That is a testable structural prediction: sharpness in redshift, not amplitude, would be what buys relief.
  • The authors' preference for a supernova luminosity transition near $z_t ≈ 0.01$ carries its own exposure: it predicts a step in the Hubble diagram at that redshift and a corresponding change in the effective gravitational strength, both of which upcoming standard-siren and peculiar-velocity data can probe directly.
Original abstract

Many late time approaches for the solution of the Hubble tension use late time smooth deformations of the Hubble expansion rate $H(z)$ of the Planck18/$\Lambda$CDM best fit to match the locally measured value of $H_0$ while effectively keeping the comoving distance to the last scattering surface and $\Omega_{0m} h^2$ fixed to maintain consistency with Planck CMB measurements. A well known problem of these approaches is that they worsen the fit to low $z$ distance probes. Here we show that another problem of these approaches is that they worsen the level of the $\Omega_{0m}-\sigma_8$ growth tension. We use the generic class of CPL parametrizations corresponding to evolving dark energy equation of state parameter $w(z)=w_0+w_1\frac{z}{1+z}$ with local measurements $H_0$ prior and identify the pairs $(w_0, w_1)$ that satisfy this condition. This is a generic class of smooth deformations of $H(z)$ that are designed to address the Hubble tension. We show that for these models the growth tension between dynamical probe data and CMB constraints is worse than the corresponding tension of the standard Planck18/$\Lambda$CDM model. We justify this feature using a full numerical solution of the growth equation and fit to the data, as well as by using an approximate analytic approach. The problem does not affect recent proposed solutions of the Hubble crisis involving a SnIa intrinsic luminosity transition at $z_t\simeq 0.01$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,501 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.34 · human participants 0.02 · controlled comparison 0.87 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

Many late time approaches for the solution of the Hubble tension use late time smooth deformations of the Hubble expansion rate $H(z)$ of the Planck18/$\Lambda$CDM best fit to match the locally measured value of $H_0$ while effectively keeping the comoving distance to the last scattering surface and $\Omega_{0m} h^2$ fixed to maintain consistency with Planck CMB measurements. A well known problem of these approaches is that they worsen the fit to low $z$ distance probes. Here we show that another problem of these approaches is that they worsen the level of the $\Omega_{0m}-\sigma_8$ growth tension. We use the generic class of CPL parametrizations corresponding to evolving dark energy equation of state parameter $w(z)=w_0+w_1\frac{z}{1+z}$ with local measurements $H_0$ prior and identify the pairs $(w_0, w_1)$ that satisfy this condition. This is a generic class of smooth deformations of $H(z)$ that are designed to address the Hubble tension. We show that for these models the growth tension between dynamical probe data and CMB constraints is worse than the corresponding tension of the standard Planck18/$\Lambda$CDM model. We justify this feature using a full numerical solution of the growth equation and fit to the data, as well as by using an approximate analytic approach. The problem does not affect recent proposed solutions of the Hubble crisis involving a SnIa intrinsic luminosity transition at $z_t\simeq 0.01$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Many late time approaches for the solution of the Hubble tension use late time smooth deformations of the Hubble expansion rate $H(z)$ of the Planck18/$\Lambda$CDM best fit to match the locally measured value of $H_0$ while effectively keeping the comoving distance to the last scattering surface and $\Omega_{0m} h^2$ fixed to maintain consistency with Planck CMB measurements. background 1.00 0.05 0.04 asserted 1.00
s1 A well known problem of these approaches is that they worsen the fit to low $z$ distance probes. background 0.99 0.06 0.05 asserted 1.00
s2 Here we show that another problem of these approaches is that they worsen the level of the $\Omega_{0m}-\sigma_8$ growth tension. result 0.76 0.82 0.77 asserted 1.00
s3 We use the generic class of CPL parametrizations corresponding to evolving dark energy equation of state parameter $w(z)=w_0+w_1\frac{z}{1+z}$ with local measurements $H_0$ prior and identify the pairs $(w_0, w_1)$ that satisfy this condition. method 0.93 0.94 0.04 asserted 0.98
s4 This is a generic class of smooth deformations of $H(z)$ that are designed to address the Hubble tension. background 0.54 0.11 0.03 asserted 1.00
s5 We show that for these models the growth tension between dynamical probe data and CMB constraints is worse than the corresponding tension of the standard Planck18/$\Lambda$CDM model. result 1.00 0.95 0.81 asserted 1.00
s6 We justify this feature using a full numerical solution of the growth equation and fit to the data, as well as by using an approximate analytic approach. method 0.99 0.94 0.04 asserted 0.97
s7 The problem does not affect recent proposed solutions of the Hubble crisis involving a SnIa intrinsic luminosity transition at $z_t\simeq 0.01$. result 0.76 0.59 0.04 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.89 · evidence from simulation or computation 0.53.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Left panel:The relative growth factor 𝛿(1) 𝐶𝑃𝐿/𝛿(1) Λ𝐶𝐷𝑀 (red line), the best fit ratio 𝜎𝐶𝑃𝐿 8 /𝜎Λ𝐶𝐷𝑀 8 (green line, fixed 𝜔𝑚 = 0.143) and the ratio 𝜔 𝛾 𝑚/𝜔 6 11 𝑚 (orange dot-dashed line) all have similar dependence on 𝑤∞. The five thick dots correspond to the CPL parameter values (𝑤0, 𝑤1) pairs, (−1, −0.93), (−1.1, −0.497), (−1.22, 0), (−1.5, 1.05) and (−1.73, 1.72) of the five panels of
Figure 2. The Planck18/ΛCDM form of 𝐻 (𝑧)/(1+ 𝑧) (blue dot-dashed line) is compared with the same function obtained with various pairs of CPL parameters that address the Hubble tension. Some BAO data are also shown. tions. Notice that for a given measured value of 𝑓 𝜎8 (𝑎), weaker growth (smaller 𝛿(𝑎 = 1)) implies a lower value of 𝜎8 (assuming that 𝛿 0 (𝑎) does not change significantly for a given value of 𝑎). This is demonstrated in
Figure 3. The cyan and the red contours correspond to the Growth and the Plank 18 CMB data respectively, for the ΛCDM and various (𝑤0, 𝑤1) pairs of the CPL model. The Δ𝜎 differences between the best fit values produced by the two contours in each case, are shown in Tab. 1. It is clear that the tension does not ease in the case of the CPL model despite the fact that it appears to solve the 𝐻0 tension for the same 𝑤0 and 𝑤1 values used. behavior as 𝜔 𝛾(𝑤∞) 𝑚 . This is demonstrated in
Figure 4. The best fit values of the absolute magnitude 𝑀, for the (𝑤0, 𝑤1) pairs displayed in Tab. 1. These values are consistently lower than the cor￾responding value of 𝑀 implied by local Cepheid calibrators (upper dashed line) even though this tension is not as large as for the best fit value of 𝑀 obtained in the context of the standard Planck18/ΛCDM model (lower dashed line). for parameter values where the growth factor is less than that of Planck18/ΛCDM (lower right panel corresponding to 𝑤∞ < −0.5). In constructing
Figure 5. The values of the absolute magnitude corresponding to the binned SnIa data in the context of the CPL model (𝑤0 = −1.22, 𝑤1 = 0) that addresses the 𝐻0 tension via an 𝐻0 prior, shown to be in tension with the Cepheid calibrated 𝑀 range. lower best fit values of the SnIa absolute magnitude than the Cepheid calibrator absolute magnitude. One could argue that there are several physical cases not covered by the CPL model (Scherrer 2015;Busti et al. 2015), thus casting a doubt on the generality of our results. However these issues arise due to CPL being accurate only around the present day…

Figures reproduced from arXiv:2103.04045 by the authors.

References
How the literature uses it

43 arXiv papers cite this in 46 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 27 sentences without a confident label

Cites it as support 4 papers
Cites it as background 15 papers

What it cites · 14 sentences without a confident label

Uses data or values from 2 papers
  • However, the scale $H(z=0)=H_0$ (the Hubble constant) obtained by the sound horizon approach ($H_0^P18=67.4 ±0.5 km sec^-1 Mpc^-1$ [ref] ) is lower by $9%$ compared to the Hubble constant obtained by the SnIa distance ladder approach ($H_0^R20=74.03 ±1.42 km sec^-1 Mpc^-1$ [ref] ) assuming that the absolute luminosity of SnIa remains unchanged before and after $z=0.01$[This is a crucial assumption that has been put under intense scrutiny recently [ref] ].

    A $w-M$ phantom transition at $z_t<0.1$ as a resolution of the Hubble tension · Introduction · 0.92

  • However, the scale $H(z=0)=H_0$ (the Hubble constant) obtained by the sound horizon approach ($H_0^P18=67.4 ±0.5 km sec^-1 Mpc^-1$ [ref] ) is lower by $9%$ compared to the Hubble constant obtained by the SnIa distance ladder approach ($H_0^R20=74.03 ±1.42 km sec^-1 Mpc^-1$ [ref] ) assuming that the absolute luminosity of SnIa remains unchanged before and after $z=0.01$[This is a crucial assumption that has been put under intense scrutiny recently [ref] ].

    A rapid transition of $G_{\rm eff}$ at $z_t \simeq 0.01$ as a solution of the Hubble and growth tensions · Introduction · 0.92

Cites as background 2 papers
  • A significant number of theoretical approaches attempting to address this issue have been proposed [ref] , and can be divided in three broad classes: * 'Early time' models that attempt to recalibrate the scale of the standard ruler (the sound horizon at recombination) by introducing new physics during the prerecombination epoch that deform $H(z)$ at prerecombination redshifts $z>1100$ (early dark energy [ref] , new types of neutrinos [ref] etc).

    Is gravity getting weaker at low z? Observational evidence and theoretical implications · Introduction · 0.99

  • A significant number of theoretical approaches attempting to address this issue have been proposed [ref] , and can be divided in three broad classes: * 'Early time' models that attempt to recalibrate the scale of the standard ruler (the sound horizon at recombination) by introducing new physics during the prerecombination epoch that deform $H(z)$ at prerecombination redshifts $z>1100$ (early dark energy [ref] , new types of neutrinos [ref] etc).

    In the Realm of the Hubble tension $-$ a Review of Solutions · Introduction · 0.99

Discussion

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