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

Dávid A. Ködmön, Péter Raffai

TL;DR

The paper investigates quasi-linear coasting cosmologies with a late-time $a(t)\propto t$ expansion and tests them against late-time large-scale-structure growth data. It derives a closed-form growth solution and an analytic expression for $f\sigma_8(z)$, then compares three curved coasting geometries ($k=−1,0,+1$) and flat $\Lambda$CDM to redshift-space distortion measurements using dynesty nested sampling, AD normality tests, Bayes factors, and predictive checks. The analysis finds that all models are consistent with the data, but $\Lambda$CDM is strongly preferred by Bayes factors and predictive performance, while quasi-linear coasting models can reduce the $S_8$ tension in some cases. The results emphasize that a CMB-based $S_8$ evaluation within a coasting framework is needed to fully gauge these models’ viability, pointing to future theoretical and observational work to explore late-time linear expansion as a potential alternative to standard cosmology.

Abstract

We derive analytical expressions for the growth factor, $D(z)$, and density-weighted growth rate, $fσ_8(z)$, for cosmologies in which $a\propto t$ at late times. We fit $fσ_8(z)$ to data from redshift-space distortion measurements in the redshift range $z<2$ using the `dynesty` implementation of nested sampling. Three coasting models, with curvature parameters $k=\{-1, 0, +1\}$ in $H^2_0c^{-2}$ units, and a flat $Λ$CDM model are tested. We evaluate each model's consistency with the data by applying the Anderson--Darling test for normality on the normalized residuals. We obtained $Ω_\mathrm{m,0}=\{ 0.206^{+0.073}_{-0.061},\, 0.297^{+0.085}_{-0.073},\, 0.412^{+0.097}_{-0.086}\}$} and $σ_{8}(z=0)=\{1.071^{+0.213}_{-0.151},\,0.867^{+0.128}_{-0.097},\,0.725^{+0.080}_{-0.065}\}$ for the coasting models, while for $Λ$CDM $Ω_\mathrm{m,0}=0.286_{-0.047}^{+0.053}$ and $σ_{8}(z=0) = 0.764_{-0.035}^{+0.039}$. All models are consistent with the data, though the $Λ$CDM model is strongly favored over the coasting models, with $\log$ Bayes factors of $\log_{10}{\mathcal{B}} = \{1.79,\, 1.55,\,1.42\}$. A predictive performance metric and posterior predictive check confirmed that while $Λ$CDM achieves the highest predictive accuracy, it also shows the strongest indication of overfitting. We also examined whether the $S_8$ tension can be resolved by linear expansion for $z<2$. Curve fitting yielded $S_8 = \{0.890^{+0.024}_{-0.024},\,0.865^{+0.024}_{-0.024},\,0.850^{+0.026}_{-0.026}\}$ for the coasting models, resulting in $ΔS^\mathrm{Coasting}_8=\{2.12σ,\,1.21σ,\,0.62σ\}$ discrepancies with the standard \textit{Planck} 2018 value. A value of $S_8=0.746^{+0.041}_{-0.039}$ was obtained for the $Λ$CDM model, indicating a tension level of ${ΔS_8^{Λ\mathrm{CDM}}=2.00σ}$.

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

TL;DR

The paper investigates quasi-linear coasting cosmologies with a late-time expansion and tests them against late-time large-scale-structure growth data. It derives a closed-form growth solution and an analytic expression for , then compares three curved coasting geometries () and flat CDM to redshift-space distortion measurements using dynesty nested sampling, AD normality tests, Bayes factors, and predictive checks. The analysis finds that all models are consistent with the data, but CDM is strongly preferred by Bayes factors and predictive performance, while quasi-linear coasting models can reduce the tension in some cases. The results emphasize that a CMB-based evaluation within a coasting framework is needed to fully gauge these models’ viability, pointing to future theoretical and observational work to explore late-time linear expansion as a potential alternative to standard cosmology.

Abstract

We derive analytical expressions for the growth factor, , and density-weighted growth rate, , for cosmologies in which at late times. We fit to data from redshift-space distortion measurements in the redshift range using the `dynesty` implementation of nested sampling. Three coasting models, with curvature parameters in units, and a flat CDM model are tested. We evaluate each model's consistency with the data by applying the Anderson--Darling test for normality on the normalized residuals. We obtained } and for the coasting models, while for CDM and . All models are consistent with the data, though the CDM model is strongly favored over the coasting models, with Bayes factors of . A predictive performance metric and posterior predictive check confirmed that while CDM achieves the highest predictive accuracy, it also shows the strongest indication of overfitting. We also examined whether the tension can be resolved by linear expansion for . Curve fitting yielded for the coasting models, resulting in discrepancies with the standard \textit{Planck} 2018 value. A value of was obtained for the CDM model, indicating a tension level of .

Paper Structure

This paper contains 5 sections, 22 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Growth of Large Scale Structure in Coasting Cosmologies
  3. Data and Analysis
  4. Results and Discussion
  5. Conclusion

Figures (3)

  • Figure 1: Posterior distributions of $\Omega_\mathrm{m,0}$ and $\sigma_{8,0}$ for coasting models with different geometries (upper panels, lower left panel) and a flat $\Lambda$CDM cosmology (lower right panel), obtained from fitting thirty-five $f\sigma_8$ data points from cit:Perivolaropoulos-Skara. The posteriors lie within the $(0, 2)$ interval of the prior distribution for both parameters. The programs used to obtain the posterior distributions are available in our public code repository cit:Zenodo.
  • Figure 2: Comparison of the density-weighted growth rate, $f\sigma_8$, for coasting models with different geometries (upper panels, lower left panel) and a $\Lambda$CDM cosmology (lower right panel). The fits are shown for redshift-space distortion data collected by cit:Perivolaropoulos-Skara, recalibrated for each model using a correction factor based on the Alcock--Paczyński effect cit:APeffect, as detailed in cit:Kazantzidis-Perivolaropoulos and cit:Perivolaropoulos-Skara. The dark blue lines represent the best-fit curves. Dashed lines indicate the 68% confidence intervals. The dataset consists of thirty-five largely uncorrelated data points. The programs used to perform curve fitting are available in our public code repository cit:Zenodo.
  • Figure 3: Histograms of AD-test $\log_{10}p$ values for one hundred and thirty-seven thousand realizations of the coasting and $\Lambda$CDM models, sampled from the joint posterior distribution of $\Omega_{\mathrm{m},0}$ and $\sigma_{8,0}$ obtained from fitting to thirty-five $f\sigma_8$ data points from cit:Perivolaropoulos-Skara. More than 95% of $p$-values exceeded 0.05 for all four models.