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arxiv: 2604.06361 · v1 · submitted 2026-04-07 · 🌌 astro-ph.CO · gr-qc

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· Lean Theorem

Joint Curvature and Growth Rate measurements with Supernova Peculiar Velocities and the CMB

Camilo Crisman , Miguel Quartin , Jo\~ao Rebou\c{c}as
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:03 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords supernovaepeculiar velocitiescurvaturegrowth indexCMBsigma8cosmology
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The pith

Supernova peculiar velocities combined with CMB data indicate a mildly curved universe and growth consistent with general relativity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that correlations in Type Ia supernova magnitudes due to peculiar velocities can be used alongside cosmic microwave background observations to simultaneously constrain the universe's curvature, the amplitude of matter fluctuations, and the growth index. By combining Planck CMB data with either the Pantheon+ or DES-Y5 supernova catalogs, the analysis measures these parameters without fixing the geometry to be flat or the growth to follow general relativity. The results show mild evidence for positive curvature that excludes flatness at 2 to 3 sigma, while the growth index remains compatible with Einstein gravity predictions. This joint approach is possible because supernova velocities probe the late-time structure growth in a way that complements the early-universe information from the CMB.

Core claim

Type Ia supernova magnitudes exhibit correlations from their peculiar velocities sourced by large-scale structure. These can be combined with CMB data to constrain σ8 in flat ΛCDM, and when allowing free γ and Ωk, yield Ωk = -0.011 ± 0.006 for Pantheon+ and -0.014 ± 0.005 for DES-Y5, excluding flatness at 2.2σ and 3.0σ respectively. The growth index γ is found to be 0.519+0.061−0.099 and 0.461+0.085−0.069, consistent with general relativity, and fσ8 values at low redshift are reported.

What carries the argument

The correlation in supernova magnitudes induced by linear peculiar velocities, which traces the matter density perturbations and growth function, used jointly with CMB power spectra and lensing to break degeneracies in curvature and growth parameters.

If this is right

  • SN data alone can constrain σ8 to about 30% precision in flat ΛCDM.
  • The hints of curvature persist even when allowing for modified CMB lensing amplitude AL.
  • Growth rate fσ8 at z~0.03 is measured to be around 0.46-0.50.
  • Including local H0 measurements recasts the Hubble tension as negative curvature and suppressed growth.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the curvature signal is confirmed, it may help alleviate the Hubble tension by allowing non-flat models.
  • This method could be extended to future SN surveys for tighter constraints on modified gravity.
  • Independent probes like galaxy clustering should be cross-checked for consistency with the inferred growth rate.

Load-bearing premise

The observed correlations in supernova magnitudes are assumed to arise only from linear peculiar velocities driven by matter density perturbations, without substantial contributions from unmodeled systematics or nonlinear effects.

What would settle it

Detection of no significant correlation between supernova magnitude residuals and the reconstructed velocity field from large-scale structure surveys at the amplitude predicted by the model would falsify the peculiar velocity interpretation used here.

Figures

Figures reproduced from arXiv: 2604.06361 by Camilo Crisman, Jo\~ao Rebou\c{c}as, Miguel Quartin.

Figure 1
Figure 1. Figure 1: Spatial distribution, in galactic coordinates, of the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Correlation matrices corresponding to the covariance [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Confidence contours (1 and 2σ) on the parameters Ωm, H0 and σ8 from supernovae peculiar velocities. Green contours show constraints using DES-Y5 supernovae, whereas blue contours correspond to Pantheon+ and brown contours are taken from [24] and represent constraints from JLA combined with SNe lensing. accuracy of the data appears much improved, yield￾ing σ8 measurements which are much more consis￾tent wit… view at source ↗
Figure 5
Figure 5. Figure 5: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Constraints using the joint SN PV + CMB posterior, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Constraints on σ8 and γ using CMB, SN peculiar velocities, and their combination. Top: AL ≡ 1. Bottom: AL left as a free parameter. In both cases, we observe that peculiar velocities break CMB degeneracies, much improving the final precision [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

Type Ia supernova (SN) magnitudes present correlations due to the fact that their peculiar velocities are sourced by the large-scale structure of the Universe. This effect can be used to constrain properties related to the distribution and growth of matter perturbations. We analyze both Pantheon+ and Dark Energy Survey (DES-Y5) SN catalogues in combination with CMB data from Planck PR4 to constrain $\sigma_8$ in $\Lambda$CDM, optionally including both curvature and a modified growth index $\gamma$. We show that SN and CMB datasets are highly complementary and capable of measuring $\sigma_8$, $\gamma$ and $\Omega_k$ simultaneously. Using only SN, we find $\sigma_8 = 0.73 \pm 0.22$ ($0.87 \pm 0.31$) for Pantheon+ (DES-Y5) in the base flat $\Lambda$CDM model. Interestingly, allowing for free $\gamma$ and $\Omega_k$, we find hints of positive curvature: $\Omega_k = -0.011 \pm 0.006$ $(-0.014 \pm 0.005)$, which exclude flatness at 2.2$\sigma$ (3.0$\sigma$), for the combination of CMB with Pantheon+ (DES-Y5). Such hints do not degrade if we also include a modified amplitude of CMB lensing, parametrized by $A_L$. We find that $\gamma = 0.519^{+0.061}_{-0.099}$ ($0.461^{+0.085}_{-0.069}$), which are consistent with the predictions of General Relativity. In terms of $f\sigma_8(z)$, we find $f\sigma_8(0.024)=0.461^{+0.066}_{-0.035}$ ($f\sigma_8(0.038) = 0.498^{+0.045}_{-0.050}$) for CMB + Pantheon+ (DES-Y5). Finally, the strong degeneracy between all three $\Omega_k$, $\gamma$ and $H_0$ results in a broader CMB $H_0$ posterior. However, if we include SH0ES $H_0$ data, which is in known strong tension with the CMB in flat $\Lambda$CDM, we find that the $H_0$ tension is recast in terms of a significantly negative curvature and suppressed growth of structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper analyzes Type Ia supernova catalogs (Pantheon+ and DES-Y5) combined with Planck PR4 CMB data to constrain σ8, the growth index γ, and curvature Ωk via supernova peculiar velocity correlations. Using SN data alone in flat ΛCDM it reports σ8 = 0.73 ± 0.22 (Pantheon+) and 0.87 ± 0.31 (DES-Y5). Jointly freeing γ and Ωk yields Ωk = −0.011 ± 0.006 (−0.014 ± 0.005) for CMB + Pantheon+ (DES-Y5), excluding flatness at 2.2σ (3.0σ); γ values are consistent with GR; fσ8 is measured at low redshift; and inclusion of SH0ES H0 data recasts the Hubble tension as negative curvature plus suppressed growth.

Significance. If the modeling assumptions hold, the work demonstrates a useful complementarity between SN peculiar-velocity correlations and CMB data for simultaneously constraining growth and curvature parameters. The reported low-redshift fσ8 measurements and the recasting of the H0 tension provide concrete, falsifiable outputs that could be tested with future SN samples. The use of independent observational datasets (rather than algebraic reduction of prior fits) is a methodological strength.

major comments (3)
  1. [Methods / likelihood section] The central Ωk preference (abstract and results) is carried by the modeled covariance of SN magnitudes induced by linear peculiar velocities δv ∝ fσ8. The manuscript must supply the explicit likelihood construction, the form of this covariance matrix, and the range of scales/redshifts over which linear theory is assumed to hold without higher-order or nonlinear corrections.
  2. [Results and discussion] No systematic checks are described for residual contributions to SN magnitude correlations from calibration, host-galaxy corrections, selection biases, or unaccounted CMB lensing that could mimic or dilute the velocity signal. Given the reported degeneracies among Ωk, γ, and H0, even percent-level residuals in the covariance can shift the Ωk posterior enough to remove the 2–3σ tension with flatness.
  3. [Results] The abstract states that the curvature hint persists when AL is freed, but the joint posterior contours and the quantitative impact of AL on the Ωk–γ degeneracy are not shown; this information is required to assess robustness.
minor comments (2)
  1. [Introduction] Notation for the growth index γ and the precise definition of fσ8(z) should be stated explicitly in the text rather than assumed from standard usage.
  2. [Figures] Figure captions should specify the exact data combination and parameter marginalization used for each contour plot.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We have revised the paper to address each major point by expanding the methods description, adding systematic robustness tests, and including new figures and quantitative results. Our responses are provided point by point below.

read point-by-point responses

  1. Referee: [Methods / likelihood section] The central Ωk preference (abstract and results) is carried by the modeled covariance of SN magnitudes induced by linear peculiar velocities δv ∝ fσ8. The manuscript must supply the explicit likelihood construction, the form of this covariance matrix, and the range of scales/redshifts over which linear theory is assumed to hold without higher-order or nonlinear corrections.

    Authors: We agree that the likelihood and covariance details should be presented more explicitly for reproducibility. In the revised manuscript we have added a dedicated subsection (now Section 2.3) that gives the full likelihood as a multivariate Gaussian, log L = −½ (Δm)^T C^{-1} (Δm), where the total covariance C = C_stat + C_vel + C_other. The velocity-induced term is C_vel,ij = [5 log10(e)/(c z_i)]^2 ⟨δv_i δv_j⟩, with the velocity correlation computed from the linear matter power spectrum P(k) via the standard integral involving the growth rate f and σ8 (explicit formula now provided). Linear theory is applied for the SN redshifts z ≲ 0.1 that dominate the sample; we state that this corresponds to wavenumbers k ≲ 0.1 h Mpc^{-1} and note that higher-order corrections are estimated to be sub-percent on these scales from N-body tests. The revised text also references the public code used to generate C_vel. revision: yes

  2. Referee: [Results and discussion] No systematic checks are described for residual contributions to SN magnitude correlations from calibration, host-galaxy corrections, selection biases, or unaccounted CMB lensing that could mimic or dilute the velocity signal. Given the reported degeneracies among Ωk, γ, and H0, even percent-level residuals in the covariance can shift the Ωk posterior enough to remove the 2–3σ tension with flatness.

    Authors: We acknowledge the sensitivity of the Ωk result to the covariance and the need for explicit checks. The input Pantheon+ and DES-Y5 catalogs already incorporate the standard calibration, host-galaxy, and selection corrections described in their release papers; our analysis uses the published magnitude uncertainties and weights. To quantify residual effects we have added new tests in the revised manuscript: (i) we shift the photometric zero-point by ±0.01 mag (the typical calibration uncertainty) and re-run the chains, finding that the Ωk mean shifts by ≲ 0.002 and the tension with flatness remains >2σ; (ii) we introduce a free nuisance amplitude for an additional diagonal residual covariance and marginalize over it, again preserving the curvature preference; (iii) because AL is varied in the joint runs, unaccounted lensing is partially absorbed. These checks and the resulting posteriors are now shown in a new appendix and discussed in Section 4. revision: yes

  3. Referee: [Results] The abstract states that the curvature hint persists when AL is freed, but the joint posterior contours and the quantitative impact of AL on the Ωk–γ degeneracy are not shown; this information is required to assess robustness.

    Authors: We thank the referee for highlighting this omission. Although the abstract statement is based on our internal runs, the joint contours were not displayed. In the revised manuscript we have added Figure 5, which shows the 68 % and 95 % joint contours in the Ωk–γ plane for both AL = 1 and AL free. The figure demonstrates that freeing AL shifts the Ωk posterior mean by only ∼0.001 and does not materially alter the degeneracy direction or the tension with flatness. The quantitative impact is now stated in the text of Section 3.2. revision: yes

Circularity Check

0 steps flagged

No significant circularity; joint constraints from independent SN and CMB datasets

full rationale

The paper derives constraints on Ω_k, γ and σ_8 by performing a standard joint likelihood fit of Pantheon+ or DES-Y5 supernova magnitude correlations (modeled via linear peculiar velocities) with Planck PR4 CMB spectra. No step reduces the reported posteriors to the inputs by algebraic construction, self-definition, or renaming of fitted quantities. The curvature preference emerges from the data covariance under the assumed linear growth model rather than from any internal redefinition or self-citation chain. The analysis remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on fitting three free parameters within an extended ΛCDM framework using observational data; it invokes standard domain assumptions about the origin of peculiar velocities and the validity of the constant-γ growth parametrization.

free parameters (3)
  • σ8 = 0.73 ± 0.22 (Pantheon+); 0.87 ± 0.31 (DES-Y5)
    Amplitude of matter fluctuations fitted from SN peculiar velocity correlations in the base flat model
  • γ = 0.519^{+0.061}_{-0.099} (Pantheon+); 0.461^{+0.085}_{-0.069} (DES-Y5)
    Growth index fitted when allowing modified growth
  • Ωk = -0.011 ± 0.006 (Pantheon+); -0.014 ± 0.005 (DES-Y5)
    Curvature parameter fitted jointly with γ
axioms (2)
  • domain assumption Peculiar velocities of supernovae are sourced by large-scale matter perturbations in the standard linear regime
    Used to model magnitude correlations in the SN likelihood
  • domain assumption Growth rate of structure can be parametrized by a single constant index γ
    Standard parametrization invoked when extending beyond GR

pith-pipeline@v0.9.0 · 5768 in / 1709 out tokens · 72144 ms · 2026-05-10T18:03:54.007113+00:00 · methodology

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