A sound horizon independent measurement of $H_0$ from BOSS, DESI and DES Y3

Th\'eo Simon; Vivian Poulin; Yifu Cai; Zhiyu Lu

REVIEW 3 major objections 6 minor 126 references

A sound horizon independent measurement of $H_0$ from BOSS, DESI and DES Y3

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A sound-horizon-free multiprobe large-scale structure analysis measures H0 to 3–4% precision, yielding h = 0.702 ± 0.02.

desk verdict A careful, transparent multiprobe analysis with a plausible headline H0, but the sound-horizon-free label for the BOSS bispectrum is not yet fully secured. read the letter →

arxiv 2602.21733 v2 pith:HZE5KXS7 submitted 2026-02-25 astro-ph.CO

Zhiyu Lu , Théo Simon , Vivian Poulin , Yifu Cai This is my paper · ORCID

classification astro-ph.CO

keywords Hubbleconstantsoundhorizonlarge-scalestructurebaryonacousticoscillationswiggle-no-wigglesplitH0tensiondarkenergycosmologicalparameterestimation

topics Dark Energy The Hubble Tension

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that the expansion rate of the universe can be estimated from the clustering of galaxies and other large-scale structure data without calibrating the cosmic sound horizon. It does this by splitting the matter power spectrum into a smooth part and a baryon acoustic oscillation part, then marginalizing over the BAO scale parameter. Combining the BOSS galaxy power spectrum and bispectrum with galaxy lensing, imaging clustering, and CMB lensing yields h = 0.702 with 3–4% precision. This matters because the Hubble tension depends on whether early-universe estimates share a common sound-horizon calibration, and the method provides an independent route. The paper also shows how the same analysis can diagnose new physics, finding consistency with hints of evolving dark energy.

What carries the argument

The wiggle-no-wiggle split: the linear matter power spectrum is decomposed into a smooth broadband component and an oscillatory BAO component, P_lin(k) = P_nw(k) + P_w(alpha_rs k). Marginalizing over alpha_rs removes the acoustic peaks that would otherwise calibrate the sound horizon. The method relies on the claim, from the literature, that sound-horizon information entering through the baryon suppression scale is negligible for current survey precision. This split is applied to the BOSS power spectrum and bispectrum, and the other probes are effectively sound-horizon-independent because their BAO signal is weak or projection-smeared.

What would settle it

A concrete test: vary the BBN prior on the baryon density omega_b (which sets the sound horizon) and see if the inferred H0 shifts by more than the quoted error. If shifting omega_b within its BBN uncertainty moves h by more than about 0.02, residual sound-horizon information is leaking through the broadband shape. Additionally, the alpha_rs deviation grows with the bispectrum scale cut; a measurement at higher k with larger survey volume would either confirm the deviation as real physics or reveal it as a modeling artifact.

Watch

Extended reading notes

Core claim

The paper claims that a sound-horizon-free multiprobe analysis of large-scale structure can determine H0, Omega_m, and sigma_8 at 3–4% precision. The key step is the wiggle-no-wiggle split of the linear matter power spectrum, P_lin(k) = P_nw(k) + P_w(alpha_rs k), with alpha_rs marginalized over so the BAO oscillation carries no weight. Combining the BOSS power spectrum and bispectrum (for the first time in this context) with the DESI angular power spectra, DES Y3 3x2pt, and Planck PR3 CMB lensing gives h = 0.702^{+0.022}_{-0.024}, Omega_m = 0.310 ± 0.013, sigma_8 = 0.799 ± 0.020. The paper also reports a 1.8σ deviation of alpha_rs from unity generated by bispectrum modes above k = 0.18 h/Mpc

Load-bearing premise

The analysis assumes that after marginalizing over alpha_rs, the remaining broadband shape of the power spectrum—in particular the baryon suppression scale proportional to the sound horizon—carries negligible sound-horizon information for current survey precision; if that residual is non-negligible, the 'sound-horizon-free' label on the H0 constraint fails.

Editorial extensions

If this is right

A sound-horizon-free H0 from LSS alone at ~3–4% precision, h = 0.702, lies between Planck and SH0ES, with 1.2σ and 2.1σ tensions respectively.
Adding Pantheon+ supernovae gives h = 0.686 ± 0.018 (2.6% precision), shifting the baseline toward lower H0 and higher Omega_m.
The BOSS bispectrum improves the h–Omega_m figure of merit by a factor of ~2 and induces a 1.8σ deviation of alpha_rs from unity; this deviation disappears when the bispectrum is restricted to k_B^max = 0.18 h/Mpc.
The sound-horizon-free analysis acts as a new-physics diagnostic: mock early-dark-energy cosmologies show no alpha_rs shift, while evolving-dark-energy mocks reproduce a ~1.9σ shift similar to the real data.
The methodology is directly applicable to upcoming surveys such as Euclid and LSST, which should yield tighter sound-horizon-independent constraints.

Reading between the lines

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

If the wiggle-no-wiggle split fully removes sound-horizon information, the agreement between this H0 and Planck's sound-horizon-calibrated value is a nontrivial internal consistency check of Lambda CDM: the Hubble tension cannot be ascribed solely to sound-horizon systematics.
The sensitivity of the alpha_rs deviation to the bispectrum scale cut suggests that the effect could be a modeling artifact; future surveys with higher signal-to-noise will determine whether it is real physics or a consequence of pushing one-loop effective field theory too far.
The same principle could be used with the matter-radiation equality turnover as an independent standard ruler, providing an orthogonal check on the wiggle-no-wiggle results.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is the first sound-horizon-free H0 analysis to include the BOSS power spectrum and bispectrum, and the combination with DESI Cℓ, DES Y3, and Planck lensing gives a 3–4% h constraint that sits between Planck and SH0ES — a reasonable, useful result. Second, the paper is admirably transparent about its own soft spots: the bispectrum scale cut matters, and the authors show it explicitly.

What is genuinely new: the joint multiprobe pipeline, the wiggle-no-wiggle procedure extended to the bispectrum, and the mock-based EDE/DDE diagnostic. The robustness checks are real: PyBird vs CLASS-PT in the alpha_rs-free context, prior sensitivity, and a tension metric between datasets. I believe the central h=0.702±0.023 is not a numerical artifact.

The main concern is the sound-horizon-free claim for the bispectrum. In Sec II B they dismiss residual r_s information through the baryon suppression scale by citing Ref [19], but that demonstration was for power spectra. Their own Fig 3 shows that adding bispectrum scales above k=0.18 shifts h and Ωm by ~1.4σ and makes alpha_rs deviate 1.8σ from unity; the deviation goes away at k=0.18. That is precisely the channel where r_s sensitivity could re-enter after alpha_rs marginalization. So I would not call the EFTBOSS bispectrum likelihood fully sound-horizon-free. The authors instead interpret this as a possible DDE signal, and their DDE mock does reproduce a similar shift — suggestive, but not yet a demonstration that the r_s channel is closed. This affects the label more than the headline number, since the full baseline returns alpha_rs=0.997±0.022, but the systematic sits in the core likelihood.

Other soft spots, in proportion: the internal 2.6–2.7σ tension between EFTBOSS and EFTDES is reported but not modeled; the EFTDES contribution is a Gaussian prior approximation; and the DESI Cℓ likelihood comes from the authors' own preprint. None of these is disqualifying, and all are stated.

Bottom line: worth a serious referee. The paper is careful, reproducible in spirit (public likelihood), and the question it addresses matters. The referee should push on the bispectrum wiggle-no-wiggle validation, ideally with mocks that have known r_s-dependent features, and ask whether the combined posterior is robust under a tension-aware combination. I would accept after moderate revision.

Referee Report

3 major / 6 minor

Summary. Using the wiggle/no-wiggle decomposition of the linear matter power spectrum (Eq. 1) and marginalizing over the BAO rescaling parameter α_rs, the authors construct a 'sound-horizon-free' EFTofLSS likelihood from BOSS power spectrum and bispectrum at one loop, and combine it with DESI Legacy Survey DR9 angular C_ℓ's, DES Y3 3×2pt constraints (entering through Gaussian priors on {h, Ω_m, σ8}), and Planck PR3 CMB lensing, plus BBN and Planck priors on ω_b and n_s. The baseline gives h = 0.702^{+0.022}_{-0.024}, Ω_m = 0.310 ± 0.013, σ8 = 0.799 ± 0.020. Inclusion of Pantheon+ lowers h to 0.686 ± 0.018; adding a DESI DR2 BAO-based Ω_m prior yields h = 0.708^{+0.015}_{-0.017}. The EFTBOSS bispectrum analysis yields α_rs = 1.063 ± 0.035 at k_Bmax = 0.23 h/Mpc (1.8σ from unity), which becomes consistent with unity at k_Bmax = 0.18. The authors also run ΛCDM fits on EDE and DDE mocks and interpret the real-data α_rs-free/α_rs-fixed difference as consistent with evolving dark energy.

Significance. If valid, the result is one of the tightest sound-horizon-free H0 determinations from LSS alone (~3–4% precision), and the first to incorporate the BOSS bispectrum in the α_rs-marginalized framework. The paper's strengths include the multiprobe combination, the explicit internal-consistency quantification, the scale-cut study, the PyBird/CLASS-PT cross-check for the 2pt case, and the EDE/DDE mock tests. These are useful and go beyond many previous analyses. However, the sound-horizon-free label and the quoted precision are not yet supported by the analysis as presented: the bispectrum at the baseline scale cut introduces a scale-cut-dependent shift, and the baseline combination includes two likelihoods with ~2.6σ internal tension. These issues require revision before the central claims can be accepted.

major comments (3)

[Sec. II B, Eq. (1); Sec. III B, Fig. 3] The assertion that marginalizing α_rs fully removes sound-horizon information is borrowed from Ref. [19], which is a power-spectrum-only demonstration. The present paper extends this to the one-loop bispectrum with k_Bmax = 0.23 h/Mpc. Sec. III B and Fig. 3 show that this extension is not stable: at k_Bmax = 0.23, α_rs = 1.063 ± 0.035 (1.8σ from unity), and the α_rs-free vs α_rs=1 analyses differ by 1.8σ in h and 1.4σ in Ω_m; lowering k_Bmax to 0.18 reduces the discrepancy to 0.6σ and restores α_rs = 1. Since the small-scale bispectrum is exactly where broadband shape features (including the baryon-suppression scale β r_s) can re-enter P_nw after α_rs marginalization, the baseline EFTBOSS likelihood is not demonstrated to be sound-horizon-free. The quoted h error bars therefore miss a systematic that the scale-cut dependence reveals. Please make k_Bmax = 0.18 the baseline or propagate th
[Sec. III A, Tab. I; Sec. III C] The baseline combination of EFTBOSS, DESICℓ and EFTDES contains internal tension: the paper reports a 2.6–2.7σ disagreement between EFTBOSS and EFTDES in the {h, Ω_m, σ8} space, using Eq. (2). These likelihoods are then multiplied as if statistically compatible. Combining two incompatible posteriors with different preferred regions (EFTBOSS h ≈ 0.63; EFTDES h ≈ 0.77) can shift the joint mode and artificially shrink uncertainties. The headline h = 0.702 ± 0.022 therefore depends on an unmodeled tension. The authors should either introduce a tension/shift parameter, report pairwise removals (e.g., baseline without EFTDES), or use a conservative hyperparameter combination. A robustness table showing that the central h and its error are stable under these choices is needed to support the claimed 3–4% precision.
[Sec. II A (EFTDES entry)] The DES Y3 3×2pt contribution is not a full likelihood but Gaussian priors on {h, Ω_m, σ8} taken from Ref. [93]. This compression assumes a Gaussian posterior and discards correlations with other parameters such as ω_cdm and ln(10^10 A_s) that are free in the EFTBOSS analysis. Since EFTDES dominates the Ω_m constraint in the combination, a non-Gaussian or correlated tail could bias the final Ω_m and h. Please validate the Gaussian compression by comparing with the full likelihood in a common parameter space, or explicitly quantify the information loss; alternatively rerun the baseline with the full EFTDES likelihood.

minor comments (6)

[Sec. II A] The redshift ranges for CMASS and LOWZ appear swapped: CMASS should be 0.43 < z < 0.7 (z_eff = 0.57) and LOWZ should be 0.2 < z < 0.43 (z_eff = 0.32). Please correct.
[Sec. II A (DESICℓ)] The link for the public likelihood is incomplete ('here /github'). Provide the full URL.
[Sec. III B / Fig. 3] The y-axis labels for k_Bmax are easy to misread because the tick order is non-monotonic; consider reordering and adding explicit units.
[Sec. III C] The phrase '2.4% measurement on h free from sound horizon' should be 'free of'; also 'EFBOSS' appears as a typo in the bottom-left panel description.
[Table II] Define the abbreviations 'scf' and 'log10 ac' in the EDE row; currently only experts will recognize these parameters.
[Sec. III D] The DDE mock test shows a 1.9σ difference in one realization. The statement 'consistent with recent hints of DDE' is stronger than the evidence; please soften or provide multiple mock realizations / a p-value.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: H0 is obtained by standard MCMC parameter estimation with alpha_rs marginalized; the sound-horizon-free assumption rests on an external result and is explicitly stress-tested, not derived from the paper's own outputs.

full rationale

The central h measurement is a conventional parameter fit to LSS angular, 3x2pt, and lensing likelihoods; alpha_rs is a free nuisance parameter that is marginalized over (Eq. 1), not a fitted quantity that is then relabeled as a prediction. The claim that residual r_s information via the baryon-suppression scale beta r_s is negligible is imported from Ref. [19] (Farren, Philcox & Sherwin), which is external to the present author set, and the paper explicitly flags this residual channel and tests it with k_Bmax variations (Sec. III B) and a PyBird/CLASS-PT cross-check (App. A). Self-citations, chiefly Ref. [84] for the DESIC_l likelihood and Ref. [21] for the wiggle-no-wiggle split, are not load-bearing circular supports: Ref. [21] is accompanied by external Refs. [19,26,27] for the same method, and Ref. [84] provides a standalone pre-existing likelihood used as an input dataset, not an output of the current fit. No parameter from the final posterior is fed back into any likelihood by definition, no uniqueness theorem from the authors' prior work is invoked to force the sound-horizon-free construction, and no known empirical result is merely renamed. The observed alpha_rs deviation and scale-cut sensitivity are honest systematic diagnostics rather than a reduction of the H0 result to its inputs. Consequently, there is no exhibitable circular step.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper fits standard cosmological parameters plus the BAO scale parameter to existing public data; it introduces no new particles, forces, or entities. The method rests on several domain assumptions about the sound-horizon-free procedure and data independence.

free parameters (5)

h = 0.702^{+0.022}_{-0.024} (baseline); 0.686±0.018 with Pantheon+
Hubble parameter; free parameter in the MCMC, central result.
ω_cdm = not reported directly
Physical cold dark matter density; free parameter in the MCMC.
ln(10^10 A_s) = not reported directly
Primordial amplitude; free parameter in the MCMC.
α_rs = 0.997±0.022 (full baseline); 1.063^{+0.034}_{-0.038} (EFTBOSS alone)
BAO scale parameter in wiggle-no-wiggle split; fitted and marginalized to remove sound horizon.
EFT nuisance parameters (BOSS bias, counterterms, stochastic; DESI Cℓ bias) = marginalized analytically or sampled
Necessary for EFTofLSS modeling; not the focus but fit to data.

assumptions (5)

domain assumption The wiggle-no-wiggle split with marginalized α_rs removes all sound-horizon information; residual baryon suppression scale β r_s information is negligible
Sec II.B, Eq. 1; relies on Ref [19].
domain assumption EFTofLSS at one loop accurately describes BOSS power spectrum and bispectrum up to k_max = 0.23/0.20 h/Mpc
Sec II.A; standard but unverified within this paper.
domain assumption The four probes (BOSS, DESI Cℓ, DES Y3, Planck lensing) are statistically independent after partial overlap removal; residual correlations are subdominant
Sec II.A; some overlap handled by remeasuring C_gg, but not all cross-correlations.
domain assumption DES Y3 constraints are fully captured by Gaussian priors on {h,Ωm,σ8} from Ref [93]
Sec II.A; compression of full 3x2pt likelihood.
domain assumption CMB lensing is approximately sound-horizon independent
Sec II.B; from Ref [24].

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Cite this review

Pith. "Pith review of A sound horizon independent measurement of $H_0$ from BOSS, DESI and DES Y3." pith.science (2026). https://pith.science/paper/HZE5KXS7

@misc{pith2026260221733,
  author       = {Pith},
  title        = {Pith review of: A sound horizon independent measurement of $H_0$ from BOSS, DESI and DES Y3},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZE5KXS7}},
  note         = {Machine review of arXiv:2602.21733}
}

abstract

We present a sound horizon independent measurement of the Hubble parameter using a multiprobe large-scale structure analysis. Removing the dependency on the sound horizon with a rescaling procedure at the matter power spectrum level, we analyse the BOSS full-shape power spectrum and bispectrum (for the first time) using the effective field theory of large-scale structure up to one loop. We combine this analysis with the auto- and cross-angular power spectra from the DESI Legacy Imaging Survey DR9, the $3 \times 2$pt analysis from DES Y3, and the CMB gravitational lensing power spectrum from Planck PR3. Our baseline analysis, that does not rely on supernovae data, yields $h = 0.702^{+0.022}_{-0.024}$, $\Omega_m = 0.310 \pm 0.013$, and $\sigma_8 = 0.799 \pm 0.020$, corresponding to $3-4 \%$ precision measurements. When adding supernovae data from Pantheon+, we obtain a $2.6 \%$ measurement of $h$, with $h = 0.686 \pm 0.018$. We further note that our EFTBOSS analysis indicates a slight deviation of the BAO scale parameter (at $1.8 \sigma$) from its $\Lambda$CDM value, caused by the small scales of the bispectrum. We finally use the sound horizon-free EFTBOSS analysis as a diagnosis for the presence of new physics, finding that our results are consistent with the recent hints of evolving dark energy.

Figures

Figures reproduced from arXiv: 2602.21733 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: 1D posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: 2D posterior distributions from the full dataset, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5: 2D posterior distributions from the ΛCDM fit to the EFTBOSS 2+3pt mock data generated with the early [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

126 extracted references · 112 linked inside Pith

[19]

G. S. Farren, O. H. E. Philcox, and B. D. Sherwin, Phys. Rev. D105, 063503 (2022), arXiv:2112.10749 [astro-ph.CO]

arXiv 2022
[93]

D’Amico, A

G. D’Amico, A. Refregier, L. Senatore, and P. Zhang, (2025), arXiv:2510.24878 [astro-ph.CO]

arXiv 2025
[1]

Aghanimet al.(Planck), Astron

N. Aghanimet al.(Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

arXiv 2020
[2]

A. G. Riesset al., Astrophys. J. Lett.934, L7 (2022), arXiv:2112.04510 [astro-ph.CO]

arXiv 2022
[3]

Casertanoet al.(H0DN), (2025), arXiv:2510.23823 [astro-ph.CO]

S. Casertanoet al.(H0DN), (2025), arXiv:2510.23823 [astro-ph.CO]

arXiv 2025
[4]

Louiset al.(Atacama Cosmology Telescope), JCAP 11, 062 (2025), arXiv:2503.14452 [astro-ph.CO]

T. Louiset al.(Atacama Cosmology Telescope), JCAP 11, 062 (2025), arXiv:2503.14452 [astro-ph.CO]

arXiv 2025
[5]

Camphuiset al.(SPT-3G), (2025), arXiv:2506.20707 [astro-ph.CO]

E. Camphuiset al.(SPT-3G), (2025), arXiv:2506.20707 [astro-ph.CO]

arXiv 2025
[6]

Abdul Karimet al.(DESI), Phys

M. Abdul Karimet al.(DESI), Phys. Rev. D112, 11 0.65 0.75 h 0.95 1.00 1.05 αrs 0.74 0.80 0.86 σ8 0.30 0.35 Ω m 0.3 Ω m 0.75 0.85 σ8 1 αrs Lensing + EFTBOSS + DESI Cℓ + EFTDES Lensing + EFTBOSS + DESI Cℓ + EFTDES (wide prior) 0.62 0.71 h 1.00 1.08 αrs 0.75 0.80 0.85 σ8 0.30 0.35 Ω m 0.31 0.37 Ω m 0.76 0.84 σ8 0.98 1.07 αrs Lensing + PanPlus + EFTBOSS 2pt (...

arXiv 2025

Show all 126 references

[7]

T. M. C. Abbottet al.(DES), (2026), arXiv:2601.14559 [astro-ph.CO]

2026
[8]

T. M. C. Abbottet al.(Kilo-Degree Survey, DES), Open J. Astrophys.6, 2305.17173 (2023), arXiv:2305.17173 [astro-ph.CO]

2023 arXiv
[9]

Asgariet al.(KiDS), Astron

M. Asgariet al.(KiDS), Astron. Astrophys.645, A104 (2021), arXiv:2007.15633 [astro-ph.CO]

2021 arXiv
[10]

Pantos and L

I. Pantos and L. Perivolaropoulos, (2026), arXiv:2602.12238 [astro-ph.CO]

2026
[11]

A. H. Wrightet al., Astron. Astrophys.703, A158 (2025), arXiv:2503.19441 [astro-ph.CO]

2025 arXiv
[12]

Simon, P

T. Simon, P. Zhang, and V. Poulin, JCAP07, 041 (2023), arXiv:2210.14931 [astro-ph.CO]

2023 arXiv
[13]

Neveuxet al.(eBOSS), Mon

R. Neveuxet al.(eBOSS), Mon. Not. Roy. Astron. Soc. 499, 210 (2020), arXiv:2007.08999 [astro-ph.CO]

2020
[14]

A. G. Adameet al.(DESI), JCAP07, 028 (2025), arXiv:2411.12022 [astro-ph.CO]

2025 arXiv
[15]

Broutet al., Astrophys

D. Broutet al., Astrophys. J.938, 110 (2022), arXiv:2202.04077 [astro-ph.CO]

2022 arXiv
[16]

Rubinet al., Astrophys

D. Rubinet al., Astrophys. J.986, 231 (2025), arXiv:2311.12098 [astro-ph.CO]

2025 arXiv
[17]

T. M. C. Abbottet al.(DES), Astrophys. J. Lett.973, L14 (2024), arXiv:2401.02929 [astro-ph.CO]

2024 arXiv
[18]

Popovicet al.(DES), (2025), arXiv:2511.07517 [astro-ph.CO]

B. Popovicet al.(DES), (2025), arXiv:2511.07517 [astro-ph.CO]

2025 arXiv
[20]

O. H. E. Philcox, G. S. Farren, B. D. Sherwin, E. J. Baxter, and D. J. Brout, Phys. Rev. D106, 063530 (2022), arXiv:2204.02984 [astro-ph.CO]

2022 arXiv
[21]

T. L. Smith, V. Poulin, and T. Simon, Phys. Rev. D 108, 103525 (2023), arXiv:2208.12992 [astro-ph.CO]

2023 arXiv
[22]

Alamet al.(BOSS), Mon

S. Alamet al.(BOSS), Mon. Not. Roy. Astron. Soc. 470, 2617 (2017), arXiv:1607.03155 [astro-ph.CO]

2017 arXiv
[23]

Poulin, T

V. Poulin, T. L. Smith, R. Calder´ on, and T. Simon, Phys. Rev. D111, 083552 (2025), arXiv:2407.18292 [astro-ph.CO]

2025 arXiv
[24]

E. J. Baxter and B. D. Sherwin, Mon. Not. Roy. Astron. Soc.501, 1823 (2021), arXiv:2007.04007 [astro-ph.CO]

2021 arXiv
[25]

O. H. E. Philcox, B. D. Sherwin, G. S. Farren, and E. J. Baxter, Phys. Rev. D103, 023538 (2021), arXiv:2008.08084 [astro-ph.CO]

2021 arXiv
[26]

Ghaemi, N

K. Ghaemi, N. Sch¨ oneberg, and L. Verde, JCAP11, 029 (2025), arXiv:2504.10578 [astro-ph.CO]

2025
[27]

E. A. Zaborowskiet al., JCAP06, 020 (2025), arXiv:2411.16677 [astro-ph.CO]

2025 arXiv
[28]

Brieden, H

S. Brieden, H. Gil-Mar ´ ın, and L. Verde, JCAP12, 054 (2021), arXiv:2106.07641 [astro-ph.CO]

2021 arXiv
[29]

Brieden, H

S. Brieden, H. Gil-Mar ´ ın, and L. Verde, JCAP04, 023 (2023), arXiv:2212.04522 [astro-ph.CO]

2023 arXiv
[30]

Pogosian, G.-B

L. Pogosian, G.-B. Zhao, and K. Jedamzik, Astro- phys. J. Lett.904, L17 (2020), arXiv:2009.08455 [astro- ph.CO]

2020 arXiv
[31]

Garc ´ ıa Escudero, S

H. Garc ´ ıa Escudero, S. H. Mirpoorian, and L. Pogosian, Astrophys. J. Lett.998, L41 (2026), arXiv:2509.16202 [astro-ph.CO]

2026
[32]

R. K. Sharma and J. Lesgourgues, (2025), arXiv:2510.15835 [astro-ph.CO]

2025
[33]

E. A. Zaborowskiet al., (2025), arXiv:2510.19149 [astro-ph.CO]

2025 arXiv
[34]

Cunnington, Mon

S. Cunnington, Mon. Not. Roy. Astron. Soc.512, 2408 (2022), arXiv:2202.13828 [astro-ph.CO]

2022 arXiv
[35]

Y. Lai, C. Howlett, and T. Davis, JCAP10, 071 (2025), arXiv:2507.11823 [astro-ph.CO]. 12

2025 arXiv
[36]

Bahr-Kalus, D

B. Bahr-Kalus, D. Parkinson, and E.-M. Mueller, Mon. Not. Roy. Astron. Soc.524, 2463 (2023), [Erra- tum: Mon.Not.Roy.Astron.Soc. 526, 3248–3249 (2023)], arXiv:2302.07484 [astro-ph.CO]

2023 arXiv
[37]

Krolewski, W

A. Krolewski, W. J. Percival, and A. Woodfinden, (2024), arXiv:2403.19227 [astro-ph.CO]

2024 arXiv
[38]

Krolewski and W

A. Krolewski and W. J. Percival, (2024), arXiv:2403.19236 [astro-ph.CO]

2024 arXiv
[39]

Pantos and L

I. Pantos and L. Perivolaropoulos, (2026), arXiv:2601.00650 [astro-ph.CO]

2026
[40]

Audren, J

B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, JCAP1302, 001 (2013), arXiv:1210.7183 [astro-ph.CO]

2013 arXiv
[41]

MontePython 3: boosted MCMC sampler and other features,

T. Brinckmann and J. Lesgourgues, “MontePython 3: boosted MCMC sampler and other features,” (2018), arXiv:1804.07261 [astro-ph.CO]

2018 arXiv
[42]

D. Blas, J. Lesgourgues, and T. Tram, JCAP07, 034 (2011), arXiv:1104.2933 [astro-ph.CO]

2011 arXiv
[43]

Aghanimet al.(Planck), Astron

N. Aghanimet al.(Planck), Astron. Astrophys.641, A8 (2020), arXiv:1807.06210 [astro-ph.CO]

2020 arXiv
[44]

J. J. M. Carrasco, M. P. Hertzberg, and L. Senatore, JHEP09, 082 (2012), arXiv:1206.2926 [astro-ph.CO]

2012 arXiv
[45]

Baumann, A

D. Baumann, A. Nicolis, L. Senatore, and M. Zal- darriaga, JCAP07, 051 (2012), arXiv:1004.2488 [astro- ph.CO]

2012 arXiv
[46]

R. A. Porto, L. Senatore, and M. Zaldarriaga, JCAP 05, 022 (2014), arXiv:1311.2168 [astro-ph.CO]

2014 arXiv
[47]

Pajer and M

E. Pajer and M. Zaldarriaga, JCAP08, 037 (2013), arXiv:1301.7182 [astro-ph.CO]

2013 arXiv
[48]

A. A. Abolhasani, M. Mirbabayi, and E. Pajer, JCAP 05, 063 (2016), arXiv:1509.07886 [hep-th]

2016 arXiv
[49]

Senatore and M

L. Senatore and M. Zaldarriaga, (2014), arXiv:1409.1225 [astro-ph.CO]

2014 arXiv
[50]

Baldauf, M

T. Baldauf, M. Mirbabayi, M. Simonovi´ c, and M. Zaldarriaga, Phys. Rev. D92, 043514 (2015), arXiv:1504.04366 [astro-ph.CO]

2015 arXiv
[51]

Senatore and M

L. Senatore and M. Zaldarriaga, JCAP02, 013 (2015), arXiv:1404.5954 [astro-ph.CO]

2015 arXiv
[52]

Senatore and G

L. Senatore and G. Trevisan, JCAP05, 019 (2018), arXiv:1710.02178 [astro-ph.CO]

2018 arXiv
[53]

Lewandowski and L

M. Lewandowski and L. Senatore, JCAP03, 018 (2020), arXiv:1810.11855 [astro-ph.CO]

2020 arXiv
[54]

D. Blas, M. Garny, M. M. Ivanov, and S. Sibiryakov, JCAP07, 028 (2016), arXiv:1605.02149 [astro-ph.CO]

2016 arXiv
[55]

J. J. M. Carrasco, S. Foreman, D. Green, and L. Sen- atore, JCAP07, 056 (2014), arXiv:1304.4946 [astro- ph.CO]

2014 arXiv
[56]

J. J. M. Carrasco, S. Foreman, D. Green, and L. Sen- atore, JCAP07, 057 (2014), arXiv:1310.0464 [astro- ph.CO]

2014 arXiv
[57]

McDonald and A

P. McDonald and A. Roy, JCAP08, 020 (2009), arXiv:0902.0991 [astro-ph.CO]

2009 arXiv
[58]

Senatore, JCAP11, 007 (2015), arXiv:1406.7843 [astro-ph.CO]

L. Senatore, JCAP11, 007 (2015), arXiv:1406.7843 [astro-ph.CO]

2015 arXiv
[59]

Mirbabayi, F

M. Mirbabayi, F. Schmidt, and M. Zaldarriaga, JCAP 07, 030 (2015), arXiv:1412.5169 [astro-ph.CO]

2015 arXiv
[60]

Angulo, M

R. Angulo, M. Fasiello, L. Senatore, and Z. Vlah, JCAP 09, 029 (2015), arXiv:1503.08826 [astro-ph.CO]

2015 arXiv
[61]

Fujita, V

T. Fujita, V. Mauerhofer, L. Senatore, Z. Vlah, and R. Angulo, JCAP01, 009 (2020), arXiv:1609.00717 [astro-ph.CO]

2020 arXiv
[62]

Perko, L

A. Perko, L. Senatore, E. Jennings, and R. H. Wechsler, (2016), arXiv:1610.09321 [astro-ph.CO]

2016 arXiv
[63]

E. O. Nadler, A. Perko, and L. Senatore, JCAP02, 058 (2018), arXiv:1710.10308 [astro-ph.CO]

2018 arXiv
[64]

D’Amico, Y

G. D’Amico, Y. Donath, M. Lewandowski, L. Senatore, and P. Zhang, JCAP07, 041 (2024), arXiv:2211.17130 [astro-ph.CO]

2024 arXiv
[65]

O. H. E. Philcox, M. M. Ivanov, G. Cabass, M. Si- monovi´ c, M. Zaldarriaga, and T. Nishimichi, Phys. Rev. D106, 043530 (2022), arXiv:2206.02800 [astro-ph.CO]

2022 arXiv
[66]

D’Amico, Y

G. D’Amico, Y. Donath, M. Lewandowski, L. Senatore, and P. Zhang, JCAP05, 059 (2024), arXiv:2206.08327 [astro-ph.CO]

2024 arXiv
[67]

H. A. Feldman, N. Kaiser, and J. A. Peacock, Astro- phys. J.426, 23 (1994), arXiv:astro-ph/9304022

1994 arXiv
[68]

Yamamoto, M

K. Yamamoto, M. Nakamichi, A. Kamino, B. A. Bas- sett, and H. Nishioka, Publ. Astron. Soc. Jap.58, 93 (2006), arXiv:astro-ph/0505115

2006 arXiv
[69]

N. Hand, Y. Li, Z. Slepian, and U. Seljak, JCAP07, 002 (2017), arXiv:1704.02357 [astro-ph.CO]

2017 arXiv
[70]

Bianchi, H

D. Bianchi, H. Gil-Mar ´ ın, R. Ruggeri, and W. J. Per- cival, Mon. Not. Roy. Astron. Soc.453, L11 (2015), arXiv:1505.05341 [astro-ph.CO]

2015 arXiv
[71]

Beutleret al.(BOSS), Mon

F. Beutleret al.(BOSS), Mon. Not. Roy. Astron. Soc. 443, 1065 (2014), arXiv:1312.4611 [astro-ph.CO]

2014 arXiv
[72]

Scoccimarro, Phys

R. Scoccimarro, Phys. Rev. D92, 083532 (2015), arXiv:1506.02729 [astro-ph.CO]

2015 arXiv
[73]

Gil-Mar ´ ınet al.(BOSS), Mon

H. Gil-Mar ´ ınet al.(BOSS), Mon. Not. Roy. Astron. Soc.460, 4188 (2016), arXiv:1509.06386 [astro-ph.CO]

2016 arXiv
[74]

Gil-Mar ´ ın, W

H. Gil-Mar ´ ın, W. J. Percival, L. Verde, J. R. Brown- stein, C.-H. Chuang, F.-S. Kitaura, S. A. Rodr ´ ıguez- Torres, and M. D. Olmstead, Mon. Not. Roy. Astron. Soc.465, 1757 (2017), arXiv:1606.00439 [astro-ph.CO]

2017 arXiv
[75]

Kitauraet al., Mon

F.-S. Kitauraet al., Mon. Not. Roy. Astron. Soc.456, 4156 (2016), arXiv:1509.06400 [astro-ph.CO]

2016 arXiv
[76]

Beutler, E

F. Beutler, E. Castorina, and P. Zhang, JCAP03, 040 (2019), arXiv:1810.05051 [astro-ph.CO]

2019 arXiv
[77]

Reidet al.(BOSS), Mon

B. Reidet al.(BOSS), Mon. Not. Roy. Astron. Soc.455, 1553 (2016), arXiv:1509.06529 [astro-ph.CO]

2016 arXiv
[78]

D’Amico, J

G. D’Amico, J. Gleyzes, N. Kokron, K. Markovic, L. Senatore, P. Zhang, F. Beutler, and H. Gil-Mar ´ ın, JCAP05, 005 (2020), arXiv:1909.05271 [astro-ph.CO]

2020 arXiv
[79]

D’Amico, L

G. D’Amico, L. Senatore, and P. Zhang, JCAP01, 006 (2021), arXiv:2003.07956 [astro-ph.CO]

2021 arXiv
[80]

Reeves, P

A. Reeves, P. Zhang, and H. Zheng, (2025), arXiv:2507.20991 [astro-ph.CO]

2025 arXiv
[81]

Hadzhiyska, K

B. Hadzhiyska, K. Wolz, S. Azzoni, D. Alonso, C. Garc ´ ıa-Garc ´ ıa, J. Ruiz-Zapatero, and A. Slosar, (2023), 10.21105/astro.2301.11895, arXiv:2301.11895 [astro-ph.CO]

2023 arXiv
[82]

Chudaykin, M

A. Chudaykin, M. M. Ivanov, O. H. E. Philcox, and M. Simonovi´ c, Phys. Rev. D102, 063533 (2020), arXiv:2004.10607 [astro-ph.CO]

2020 arXiv
[83]

O. H. E. Philcox and M. M. Ivanov, Phys. Rev. D105, 043517 (2022), arXiv:2112.04515 [astro-ph.CO]

2022 arXiv
[84]

Lu and T

Z. Lu and T. Simon, (2025), arXiv:2511.10616 [astro- ph.CO]

2025
[85]

Zhouet al., JCAP11, 097 (2023), arXiv:2309.06443 [astro-ph.CO]

R. Zhouet al., JCAP11, 097 (2023), arXiv:2309.06443 [astro-ph.CO]

2023 arXiv
[86]

Zhouet al.(DESI), Astron

R. Zhouet al.(DESI), Astron. J.165, 58 (2023), arXiv:2208.08515 [astro-ph.CO]

2023 arXiv
[87]

Carron, M

J. Carron, M. Mirmelstein, and A. Lewis, JCAP09, 039 (2022), arXiv:2206.07773 [astro-ph.CO]

2022 arXiv
[88]

Akramiet al.(Planck), Astron

Y. Akramiet al.(Planck), Astron. Astrophys.643, A42 (2020), arXiv:2007.04997 [astro-ph.CO]

2020 arXiv
[89]

Saileret al., JCAP06, 008 (2025), arXiv:2407.04607 [astro-ph.CO]

N. Saileret al., JCAP06, 008 (2025), arXiv:2407.04607 [astro-ph.CO]. 13

2025 arXiv
[90]

Kimet al., JCAP12, 022 (2024), arXiv:2407.04606 [astro-ph.CO]

J. Kimet al., JCAP12, 022 (2024), arXiv:2407.04606 [astro-ph.CO]

2024 arXiv
[91]

Alonso, J

D. Alonso, J. Sanchez, and A. Slosar (LSST Dark En- ergy Science), Mon. Not. Roy. Astron. Soc.484, 4127 (2019), arXiv:1809.09603 [astro-ph.CO]

2019 arXiv
[92]

Reymond, A

L. Reymond, A. Reeves, P. Zhang, and A. Refregier, (2025), arXiv:2505.22718 [astro-ph.CO]

2025
[94]

Abbottet al.(DES), Mon

T. Abbottet al.(DES), Mon. Not. Roy. Astron. Soc. 460, 1270 (2016), arXiv:1601.00329 [astro-ph.CO]

2016 arXiv
[95]

Sch¨ oneberg, J

N. Sch¨ oneberg, J. Lesgourgues, and D. C. Hooper, JCAP10, 029 (2019), arXiv:1907.11594 [astro-ph.CO]

2019 arXiv
[96]

Consiglio, P

R. Consiglio, P. F. de Salas, G. Mangano, G. Miele, S. Pastor, and O. Pisanti, Comput. Phys. Commun. 233, 237 (2018), arXiv:1712.04378 [astro-ph.CO]

2018 arXiv
[97]

R. J. Cooke, M. Pettini, and C. C. Steidel, Astrophys. J.855, 102 (2018), arXiv:1710.11129 [astro-ph.CO]

2018 arXiv
[98]

E. Aver, K. A. Olive, and E. D. Skillman, JCAP07, 011 (2015), arXiv:1503.08146 [astro-ph.CO]

2015 arXiv
[99]

P. L. Taylor and K. Markoviˇ c, Phys. Rev. D106, 063536 (2022), arXiv:2205.14167 [astro-ph.CO]

2022 arXiv
[100]

Mauset al., JCAP11, 077 (2025), arXiv:2505.20656 [astro-ph.CO]

M. Mauset al., JCAP11, 077 (2025), arXiv:2505.20656 [astro-ph.CO]

2025
[101]

M. M. Ivanov, J. M. Sullivan, S.-F. Chen, A. Chu- daykin, M. Maus, and O. H. E. Philcox, (2026), arXiv:2601.16165 [astro-ph.CO]

2026 arXiv
[102]

Deyet al.(DESI), Astron

A. Deyet al.(DESI), Astron. J.157, 168 (2019), arXiv:1804.08657 [astro-ph.IM]

2019 arXiv
[103]

T. M. C. Abbottet al.(DES), Phys. Rev. D105, 023520 (2022), arXiv:2105.13549 [astro-ph.CO]

2022 arXiv
[104]

Hamann, S

J. Hamann, S. Hannestad, J. Lesgourgues, C. Rampf, and Y. Y. Y. Wong, JCAP07, 022 (2010), arXiv:1003.3999 [astro-ph.CO]

2010 arXiv
[105]

Reeves, A

A. Reeves, A. Nicola, and A. Refregier, JCAP06, 042 (2025), arXiv:2502.01722 [astro-ph.CO]

2025 arXiv
[106]

Reeves, S

A. Reeves, S. Ferraro, A. Nicola, and A. Refregier, (2025), arXiv:2510.06114 [astro-ph.CO]

2025
[107]

Albrechtet al., (2006), arXiv:astro-ph/0609591

A. Albrechtet al., (2006), arXiv:astro-ph/0609591

2006 arXiv
[108]

Poulin, T

V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Phys. Rev. Lett.122, 221301 (2019), arXiv:1811.04083 [astro-ph.CO]

2019 arXiv
[109]

Y. Cai, X. Ren, T. Qiu, M. Li, and X. Zhang, (2025), arXiv:2505.24732 [astro-ph.CO]

2025 arXiv
[110]

Y.-F. Cai, E. N. Saridakis, M. R. Setare, and J.-Q. Xia, Phys. Rept.493, 1 (2010), arXiv:0909.2776 [hep-th]

2010 arXiv
[111]

Feng, X.-L

B. Feng, X.-L. Wang, and X.-M. Zhang, Phys. Lett. B 607, 35 (2005), arXiv:astro-ph/0404224

2005 arXiv
[112]

Xia, Y.-F

J.-Q. Xia, Y.-F. Cai, T.-T. Qiu, G.-B. Zhao, and X. Zhang, Int. J. Mod. Phys. D17, 1229 (2008), arXiv:astro-ph/0703202

2008 arXiv
[113]

G.-B. Zhao, R. G. Crittenden, L. Pogosian, and X. Zhang, Phys. Rev. Lett.109, 171301 (2012), arXiv:1207.3804 [astro-ph.CO]

2012 arXiv
[114]

Guo, Y.-S

Z.-K. Guo, Y.-S. Piao, X.-M. Zhang, and Y.-Z. Zhang, Phys. Lett. B608, 177 (2005), arXiv:astro-ph/0410654

2005 arXiv
[115]

Zhao, J.-Q

G.-B. Zhao, J.-Q. Xia, M. Li, B. Feng, and X. Zhang, Phys. Rev. D72, 123515 (2005), arXiv:astro- ph/0507482

2005
[116]

Xia, G.-B

J.-Q. Xia, G.-B. Zhao, B. Feng, H. Li, and X. Zhang, Phys. Rev. D73, 063521 (2006), arXiv:astro- ph/0511625

2006
[117]

T. L. Smith, V. Poulin, and M. A. Amin, Phys. Rev. D101, 063523 (2020), arXiv:1908.06995 [astro-ph.CO]

2020 arXiv
[118]

Laureijset al.(EUCLID), (2011), arXiv:1110.3193 [astro-ph.CO]

R. Laureijset al.(EUCLID), (2011), arXiv:1110.3193 [astro-ph.CO]

2011 arXiv
[119]

Ivezi´ cet al.(LSST), Astrophys

ˇZ. Ivezi´ cet al.(LSST), Astrophys. J.873, 111 (2019), arXiv:0805.2366 [astro-ph]

2019 arXiv
[120]

D’Amico, Y

G. D’Amico, Y. Donath, L. Senatore, and P. Zhang, JCAP03, 032 (2024), arXiv:2012.07554 [astro-ph.CO]

2024 arXiv
[121]

Fujita and Z

T. Fujita and Z. Vlah, JCAP10, 059 (2020), arXiv:2003.10114 [astro-ph.CO]

2020 arXiv
[122]

Nishimichi, G

T. Nishimichi, G. D’Amico, M. M. Ivanov, L. Sen- atore, M. Simonovi´ c, M. Takada, M. Zaldarriaga, and P. Zhang, Phys. Rev. D102, 123541 (2020), arXiv:2003.08277 [astro-ph.CO]

2020 arXiv
[123]

Simon, P

T. Simon, P. Zhang, V. Poulin, and T. L. Smith, Phys. Rev. D107, 123530 (2023), arXiv:2208.05929 [astro- ph.CO]

2023 arXiv
[124]

E. B. Holm, L. Herold, T. Simon, E. G. M. Ferreira, S. Hannestad, V. Poulin, and T. Tram, Phys. Rev. D 108, 123514 (2023), arXiv:2309.04468 [astro-ph.CO]

2023 arXiv
[125]

Carrilho, C

P. Carrilho, C. Moretti, and A. Pourtsidou, JCAP01, 028 (2023), arXiv:2207.14784 [astro-ph.CO]

2023 arXiv
[126]

Gsponer, R

R. Gsponer, R. Zhao, J. Donald-McCann, D. Bacon, K. Koyama, R. Crittenden, T. Simon, and E.-M. Mueller, Mon. Not. Roy. Astron. Soc.530, 3075 (2024), arXiv:2312.01977 [astro-ph.CO]

2024 arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [19]

G. S. Farren, O. H. E. Philcox, and B. D. Sherwin, Phys. Rev. D105, 063503 (2022), arXiv:2112.10749 [astro-ph.CO]

arXiv 2022

[2] [93]

D’Amico, A

G. D’Amico, A. Refregier, L. Senatore, and P. Zhang, (2025), arXiv:2510.24878 [astro-ph.CO]

arXiv 2025

[3] [1]

Aghanimet al.(Planck), Astron

N. Aghanimet al.(Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

arXiv 2020

[4] [2]

A. G. Riesset al., Astrophys. J. Lett.934, L7 (2022), arXiv:2112.04510 [astro-ph.CO]

arXiv 2022

[5] [3]

Casertanoet al.(H0DN), (2025), arXiv:2510.23823 [astro-ph.CO]

S. Casertanoet al.(H0DN), (2025), arXiv:2510.23823 [astro-ph.CO]

arXiv 2025

[6] [4]

Louiset al.(Atacama Cosmology Telescope), JCAP 11, 062 (2025), arXiv:2503.14452 [astro-ph.CO]

T. Louiset al.(Atacama Cosmology Telescope), JCAP 11, 062 (2025), arXiv:2503.14452 [astro-ph.CO]

arXiv 2025

[7] [5]

Camphuiset al.(SPT-3G), (2025), arXiv:2506.20707 [astro-ph.CO]

E. Camphuiset al.(SPT-3G), (2025), arXiv:2506.20707 [astro-ph.CO]

arXiv 2025

[8] [6]

Abdul Karimet al.(DESI), Phys

M. Abdul Karimet al.(DESI), Phys. Rev. D112, 11 0.65 0.75 h 0.95 1.00 1.05 αrs 0.74 0.80 0.86 σ8 0.30 0.35 Ω m 0.3 Ω m 0.75 0.85 σ8 1 αrs Lensing + EFTBOSS + DESI Cℓ + EFTDES Lensing + EFTBOSS + DESI Cℓ + EFTDES (wide prior) 0.62 0.71 h 1.00 1.08 αrs 0.75 0.80 0.85 σ8 0.30 0.35 Ω m 0.31 0.37 Ω m 0.76 0.84 σ8 0.98 1.07 αrs Lensing + PanPlus + EFTBOSS 2pt (...

arXiv 2025