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REVIEW 4 major objections 4 minor 49 references

Testing Quasi-Linear Coasting Cosmologies with Late-Time Large-Scale Structure Growth

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Coasting cosmologies fit late-time structure-growth data, and in spherical geometry the S8 tension drops to 0.62σ.

desk verdict A clean, honest growth-rate consistency test for coasting cosmologies; the S8 tension-reduction claim is overframed and should be treated as a re-parameterized check, not a resolution. read the letter →

arxiv 2506.11826 v5 pith:K6JSGG2O submitted 2025-06-13 astro-ph.CO

Dávid A. Ködmön , Péter Raffai This is my paper
classification astro-ph.CO
keywords coastingcosmologyquasi-linearexpansiongrowthfactordensity-weightedratefσ8(z)redshift-spacedistortionsS8tensionmodifiedBesselfunctionsBayesianmodelselection
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

The paper asks whether quasi-linear coasting cosmologies—universes whose scale factor grows as $a(t)\propto t$ at late times—can describe the growth of large-scale structure, and whether they can relieve the late-time $S_8$ tension. It derives closed-form expressions for the growth factor $D(z)$ and the density-weighted growth rate $f\sigma_8(z)$ that depend only on the linear expansion law, not on any specific coasting model. Fitting these expressions to 35 largely uncorrelated redshift-space-distortion measurements at $z<2$ yields parameter estimates for three curvature geometries ($k=-1,0,+1$) plus a flat $\Lambda$CDM reference. The paper finds that all four models are statistically consistent with the data, that $\Lambda$CDM is preferred by Bayes factors, and that the $S_8$ tension with the Planck value shrinks from about $2\sigma$ to $1.21\sigma$ ($k=0$) and $0.62\sigma$ ($k=+1$). If correct, this makes late-time linear expansion a viable, testable alternative for structure growth and identifies the spherical coasting geometry as the variant most capable of easing the growth tension.

What carries the argument

The load-bearing object is the pair of closed-form equations (2.6) and (2.8). Equation (2.6) is the growing-mode solution of $\ddot\delta+2(\dot a/a)\dot\delta=4\pi G\bar\rho\,\delta$ under the coasting assumption $a(t)=H_0 t$; after the change of variables $y=\sqrt{6\Omega_{\mathrm{m},0}/a}$ it becomes a Bessel equation, and the decaying $I_1$ mode is discarded, leaving $D(z)\propto y\,K_1(y)$. Equation (2.8) converts this into the growth-rate observable via $\sigma_8(z)=\sigma_{8,0}D(z)$ and $f(a)=d\ln D/d\ln a$. On the data side, the Alcock–Paczyński factor $q(z,\Omega_{\mathrm{m},0},\Omega^{\mathrm{fid}}_{\mathrm{m},0})=H(z)D_A(z)/(H^{\mathrm{fid}}(z)D_A^{\mathrm{fid}}(z))$, with $H(z)=H_0(1+z)$ and curvature-dependent angular diameter distances, maps each published $f\sigma_8$ value from its fiducial flat-$\Lambda$CDM cosmology onto the tested model. These two ingredients together allow all four models to be compared on a single, recalibrated dataset.

What would settle it

Numerically integrate the perturbation equation (2.3) with $a(t)=H_0 t$ while retaining the decaying $I_1$ mode, or run a high-resolution cosmological $N$-body simulation with a quasi-linear coasting expansion history; if the resulting $D(z)$ departs from Eq. (2.6) by more than the reported uncertainties across $z<2$, the analytic growth-factor claim fails. A complementary check is to obtain a new $f\sigma_8$ measurement at $z<2$ with a full covariance matrix that shifts the fitted $S_8$ by more than about $1\sigma$.

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Extended reading notes

Core claim

The central claim is that the linearized growth of matter perturbations in any universe with $a(t)=H_0 t$ is governed by a closed-form solution: substituting the matter density $\bar\rho=(3H_0^2/8\pi G)\Omega_{\mathrm{m},0}a^{-3}$ into the fluid perturbation equation gives a Bessel equation whose growing mode yields $D(z)\propto K_1(\sqrt{6\Omega_{\mathrm{m},0}(1+z)})$, normalized to unity today; the density-weighted growth rate follows from $f\sigma_8(z)=\sigma_{8,0}D(z)\,d\ln D/d\ln a$. The paper asserts that fitting this prediction to the 35-point $f\sigma_8$ dataset after Alcock–Paczyński recalibration gives $\Omega_{\mathrm{m},0}=\{0.206^{+0.073}_{-0.061},\,0.297^{+0.085}_{-0.073},\,0.412^{+0.097}_{-0.086}\}$ and $\sigma_{8,0}=\{1.071^{+0.213}_{-0.151},\,0.867^{+0.128}_{-0.097},\,0.725^{+0.080}_{-0.065}\}$ for $k=\{-1,0,+1\}$, and that all models pass a normality test on the uncertainty-normalized residuals. It also claims the corresponding $S_8$ values $\{0.890,\,0.865,\,0.850\}$ leave the Planck 2018 value at $2.12\sigma$, $1.21\sigma$, and $0.62\sigma$ respectively, while flat $\Lambda$CDM gives $S_8=0.746$ with a $2.00\sigma$ discrepancy.

Load-bearing premise

The load-bearing premise is stated in Section 3: the 35 'largely uncorrelated' data points can be treated with a diagonal covariance matrix, and the Alcock–Paczyński recalibration factor of Eq. (3.1) fully removes each point's fiducial-cosmology dependence, with residual differences among published forms of $q$ neglected.

Editorial extensions

If this is right

  • Any cosmological model whose late-time expansion is linear in cosmic time now has an analytic prediction for $D(z)$ and $f\sigma_8(z)$ that involves no microphysical assumptions about how the coasting phase is realized.
  • The 35-point $z<2$ redshift-space-distortion dataset does not rule out any of the three coasting geometries; each passes the Anderson–Darling residual test with $p>0.6$.
  • Flat $\Lambda$CDM is preferred over the coasting models with $\log_{10}\mathcal{B}=\{1.79,1.55,1.42\}$, a preference that weakens under Student-$t$ likelihoods but does not disappear.
  • The inferred $S_8$ for the $k=0$ and $k=+1$ coasting models ($0.865$ and $0.850$) sits only $1.21\sigma$ and $0.62\sigma$ from the Planck 2018 value, so the growth tension is effectively absent for these geometries.
  • $\Lambda$CDM, despite winning the model comparison, shows the largest overfitting signal in the posterior predictive check ($p_b=0.926$), so the paper's own conclusion is that the preference is qualified.

Reading between the lines

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

  • Editorial extension: because Eqs. (2.6) and (2.8) use only the functional form $a\propto t$, the same Bessel-function growth law could be tested against any other expansion history that is locally linear in the data window, such as a broken power law $a(t)\propto(t-t_0)$, to see how sensitive the $S_8$ conclusions are to the exact coasting assumption.
  • Editorial extension: the near-vanishing $S_8$ tension for $k=+1$ suggests a concrete next step the paper leaves open—computing the CMB-derived $S_8$ inside a coasting framework; if that value lands near $0.85$, the growth tension would be resolved by spatial geometry rather than by new dark-sector physics.
  • Editorial extension: the residual-based consistency checks treat the recalibrated dataset as if correlations were absent; re-running the fit with a covariance matrix estimated from overlapping survey footprints, or removing points whose fiducial $\Omega_{\mathrm{m},0}$ differs most from the tested model, would test whether the coasting consistency result survives a more conservative data treatment
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper derives closed-form expressions for the linear growth factor D(z) and the density-weighted growth rate fσ8(z) in late-time coasting cosmologies, defined by a(t)∝t but otherwise not tied to a specific dark-energy or K-matter implementation. The authors recalibrate 35 fσ8 measurements from the Skara--Perivolaropoulos compilation using an Alcock--Paczyński correction, fit three coasting models (k=-1,0,+1) and a flat ΛCDM model with nested sampling, and compare the models with Anderson--Darling tests, Bayes factors, leave-one-out cross-validation, and posterior predictive checks. They report that all four models are consistent with the data, that ΛCDM is strongly preferred by Bayes factors but has the largest posterior-predictive p-value, and that the k=0 and k=+1 coasting models reduce the nominal S8 discrepancy with the Planck 2018 value to 1.21σ and 0.62σ, respectively.

Significance. If the derivation in Section 2 is correct, the paper provides a useful analytic growth-factor solution for an entire family of coasting models and demonstrates that these models are not trivially excluded by late-time RSD data. The statistical analysis is thorough: nested sampling with 10,000 live points, prior-robustness checks, heavy-tailed likelihood tests, LOO-CV, posterior predictive checks, and a public code repository are all genuine strengths. The S8-tension interpretation, however, is the weakest part of the paper and needs to be reframed before the claims in the abstract and Table 1 can be accepted.

major comments (4)
  1. [Section 4, Table 1, Abstract] The claim that the coasting models 'reduce the S8 tension' to 1.21σ and 0.62σ compares model-dependent S8 values, obtained by fitting fσ8(z) within a coasting cosmology, to the Planck 2018 value S8=0.832±0.013, which was derived under flat ΛCDM. As the paper itself notes in Section 4 and the Conclusion, all S8 determinations are model-dependent and a genuine resolution test would require a CMB analysis within the coasting framework. The abstract and Table 1 nevertheless present the reduced discrepancy as a central result. This is an over-interpretation: the numbers are a re-parameterized consistency check, not a measurement of tension. The authors should either remove the 'resolve the S8 tension' language or explicitly label these as conditional discrepancies that do not constitute a resolution test.
  2. [Section 3, Eq. (3.1)] For the ΛCDM fit, the text states that the dataset is homogenized with respect to the Planck 2018 baseline value Ωm,0=0.3153 before fitting. But the Alcock--Paczyński factor q(z,Ωm,Ω_fid) in Eq. (3.1) depends on Ωm through H(z) and D_A(z) for a ΛCDM target. If the data are multiplied once by q evaluated at the reference Ωm and then fitted with Ωm free, the likelihood for Ωm≠0.3153 compares the theory to data corrected with the wrong fiducial mapping. The fit should instead evaluate q at each sampled Ωm, or equivalently compare the theoretical fσ8 to the data divided by q. As written, this can bias the reported ΛCDM parameters, S8, and the Bayes factors in Table 1. This issue does not affect the coasting fits, because q happens to be independent of the fitted parameters there.
  3. [Equations (2.6) and (2.8)] As typeset, the normalization of D(z) in Eq. (2.6) appears to give D(0)=6Ωm,0 rather than the stated normalization D(0)=1, since y(0)=√(6Ωm,0) and the numerator contains y(z)√(6Ωm,0)K1(√(6Ωm,0)) while the denominator contains only K1(y(z)). If this is not an OCR artifact, the missing factor propagates directly into Eq. (2.8) and changes all fitted σ8,0 and S8 values. If it is a rendering artifact, the equations should be rewritten unambiguously; the appearance of 'b²' in Eq. (2.8), which should presumably be 'y²', adds to the concern that the printed formulas are not faithful to the implemented ones.
  4. [Section 4, Table 1] The discrepancy values ΔS8 are quoted in units of σ with no explicit formula. The authors should state how the asymmetric posterior uncertainties on S8 are combined with the Planck uncertainty to compute the significance; otherwise the 2.12σ, 1.21σ, 0.62σ numbers are not reproducible from the information in the paper.
minor comments (4)
  1. [Section 3] The diagonal-covariance approximation for the 35-point subset is mentioned, but its impact on the quoted parameter uncertainties and Bayes factors is never quantified; a brief test or a stated limit would make the approximation's role clearer.
  2. [Section 4, Eq. (4.2)--(4.4)] The LOO-CV section uses notation y−i and p(yi|y−i) without explicitly stating that the same recalibrated data values are used in every leave-one-out step; this should be stated for reproducibility.
  3. [Section 4, Figure 3] The text says that more than 95% of p-values exceeded 0.05 for all models, but the main text also quotes single best-fit AD p-values; it would be helpful to state explicitly that the histogram p-values are from posterior draws while the quoted p-values in Table 1 are for the best-fit residuals.
  4. [Throughout] Some equation renderings in the posted text are ambiguous, e.g., the 'p' preceding '6Ωm,0' in Eqs. (2.6) and (2.8) and the 'b2/4' term; the authors should ensure the final typeset version uses explicit fraction bars and consistent symbols for y and b.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the growth-factor derivation is analytic and the S8 comparison is a transparent post-fit parameter transformation, not a fitted input masquerading as an independent prediction.

full rationale

The paper's derivation chain is self-contained. Equation (2.6) for D(z) is obtained by solving the linear growth equation (2.3) with the coasting scale factor a=H0 t, reducing to a Bessel equation (2.4); Equation (2.8) for f sigma8(z) follows from the standard relation (2.7). There is no way in which these equations assume the fitted values of Omega_m0 or sigma8,0; rather, the model is fitted to the data after the derivation is complete. The Alcock-Paczynski recalibration q in Equation (3.1) uses only H(z) and D_A(z) for the coasting background, Equations (3.4)-(3.5), which are fixed functions of z and k and are independent of the fitted Omega_m0; the paper explicitly notes that 'no prior knowledge of Omega_m,0 was required to perform this step'. Thus the data recalibration is not a hidden fit loop. The consistency claims and Bayes factors are computed from the likelihood of the 35 f sigma8 points, with external Planck values used only as an outside benchmark, not as part of the fit. The S8 values in Table 1 are deterministic combinations of the fitted Omega_m0 and sigma8,0 via Equation (1.1), and the paper labels them as obtained by curve fitting rather than as independent predictions; this is standard parameter inference, not a fitted input renamed as a prediction. The paper itself flags the only real limitation of the S8-tension comparison in Section 4: 'as noted in [49], however, all determinations of S8 are inherently model-dependent', and the Conclusion states that resolving the tension 'would require obtaining a value for S8 from CMB measurements within a coasting cosmology framework'. That admission prevents the S8 comparison from being circular, while also weakening the headline tension-reduction interpretation as a scientific claim. The only self-citations in the introduction (e.g., [14] for expansion-probe preference) are motivational and are not load-bearing for the growth derivation or the fits. No circular step can be exhibited from the paper's equations, so the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or exotic components are introduced. The central claim rests on the coasting background assumption, the standard linear perturbation equation, and the statistical treatment of recalibrated fσ8 data. The two fitted parameters Omega_m0 and sigma8,0 carry the entire empirical content, and the S8 values are derived from them.

free parameters (2)
  • Omega_m0 for each model = 0.206, 0.297, 0.412, 0.286 (k=-1,0,+1, LambdaCDM)
    Present-day matter density parameter, fitted to fσ8 data via Eq. (2.8) for coasting models or the Lambda-CDM growth model. Uncertainties are quoted in Table 1.
  • sigma8,0 for each model = 1.071, 0.867, 0.725, 0.764 (k=-1,0,+1, LambdaCDM)
    Present-day rms matter fluctuation amplitude in 8 h^-1 Mpc spheres, fitted jointly with Omega_m0. The quoted S8 values are derived from these two fitted parameters through Eq. (1.1).
assumptions (6)
  • domain assumption The background expansion is exactly a(t)=H0 t (coasting) over the redshift range probed (z<2).
    Central to the derivation of D(z); invoked in Section 2 after Eq. (2.3).
  • domain assumption Growth of perturbations follows the pressureless, isentropic, sub-Jeans linear fluid equation (Eq. 2.3) with matter-only background density.
    Used to obtain Eq. (2.4); assumes c_s≈0, S_k=0, and rho=3H0^2 Omega_m0/(8πG) a^-3.
  • domain assumption The decaying mode of the Bessel solution (the I1 term) is negligible and D(z) is normalized to unity at present.
    Neglect of C1 in Eq. (2.5) and normalization at a=1; standard for late-time growth after matter domination.
  • domain assumption Redshift-space distortion fσ8 measurements are unbiased tracers of matter fluctuations independent of galaxy bias (f_g sigma8,g = f sigma8).
    Justifies direct use of fσ8 data from galaxy surveys without bias correction.
  • domain assumption The 35 selected data points are largely uncorrelated, so their covariance matrix is diagonal.
    Section 3; no full covariance matrix is available for the overlapping surveys in the [20] compilation.
  • domain assumption The Alcock-Paczyński recalibration factor q in Eq. (3.1) correctly converts data from each source's fiducial flat Lambda-CDM to the model under study.
    Section 3; approximate correction, and the paper notes that different forms of q exist in the literature.

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Cite this review

Pith. "Pith review of Testing Quasi-Linear Coasting Cosmologies with Late-Time Large-Scale Structure Growth." pith.science (2026). https://pith.science/paper/K6JSGG2O

@misc{pith2026250611826,
  author       = {Pith},
  title        = {Pith review of: Testing Quasi-Linear Coasting Cosmologies with Late-Time Large-Scale Structure Growth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K6JSGG2O}},
  note         = {Machine review of arXiv:2506.11826}
}
abstract

We derive analytical expressions for the growth factor, $D(z)$, and density-weighted growth rate, $f\sigma_8(z)$, for cosmologies in which $a\propto t$ at late times. We fit the resulting $f\sigma_8(z)$ predictions to redshift-space-distortion measurements in the range $z<2$ using the `dynesty` implementation of nested sampling. Three coasting models, with curvature parameters ${k=\{-1,0,+1\}}$ in $H_{0}^{2}c^{-2}$ units, and a flat $\Lambda$CDM model are tested. We evaluate each model's consistency with the data using the Anderson--Darling test for normality applied to the uncertainty-normalised residuals, supplemented by posterior predictive checks. For the coasting models, we obtain ${\Omega_\mathrm{m,0}=\{0.206^{+0.073}_{-0.061},\,0.297^{+0.085}_{-0.073},\,0.412^{+0.097}_{-0.086}\}}$ and ${\sigma_{8}(z=0)=\{1.071^{+0.213}_{-0.151},\,0.867^{+0.128}_{-0.097},\,0.725^{+0.080}_{-0.065}\}}$, respectively. For $\Lambda$CDM, we obtain $\Omega_\mathrm{m,0}=0.286^{+0.053}_{-0.047}$ and $\sigma_{8}(z=0)=0.764^{+0.039}_{-0.035}$. All models are consistent with the data, although $\Lambda$CDM is favoured over the coasting models, with log Bayes factors ${\log_{10}{\mathcal{B}}=\{1.79,\,1.55,\,1.42\}}$. This preference is robust against alternative priors and prior parametrisations, but weakens when heavier-tailed likelihoods are adopted. Predictive performance is assessed using the expected log predictive density computed by leave-one-out cross-validation. The $\Lambda$CDM model has the largest predictive performance, but its advantage is statistically significant only relative to the ${k=\{-1,0\}}$ coasting models.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.