Growth Index in the $γδ$CDM model

TL;DR

This work computes the growth index $\gamma$ within the recently proposed $\gamma\delta$CDM model—an $f(R)$-gravity-based framework solved on a Bianchi I background—and contrasts it with $\Lambda$CDM and $\omega$CDM. By combining growth-rate data $f(z)$, DESI2 BAO, DESY5 SNe Ia, and complementary $\sigma_8(z)$, $f\sigma_8(z)$ data, the authors derive both theoretical and numerical estimates of $\gamma$, finding it to be close to the GR-based values of standard models despite underlying modified gravity. The analysis shows the model yields suppressed growth at high redshift (via $f_{\infty}=1-\delta$) and growth-rate calibration within an extended framework, with $\gamma$-predictions aligning with observations within 1$\sigma$ and exhibiting competitive information criteria relative to $\Lambda$CDM and $\omega$CDM. Although the $S_8$ tension remains a separate issue, the $\gamma\delta$CDM model demonstrates potential to mitigate growth-related tensions, warranting further exploration with higher-redshift data and more extensive datasets.

Abstract

To better distinguish the nature of $H_0$ and $S_8$ tensions, it is necessary to separate the effects of expansion and the growth of structure. The growth index $γ$ was identified as the most important parameter that characterizes the growth of density fluctuations independently of the effects of cosmic expansion. In the $Λ$CDM model, analyses performed with various cosmological datasets indicate that the growth index has to be larger than its theoretically predicted value. Cosmological models based on $f(R)$ gravity theories have scale-dependent growth indices, whose values are even more at odds with the growth rate data. In this work, we evaluate the growth index in the $γδ$CDM model both theoretically and numerically. Although based on $f(R)$ gravity theory, we show through several analyses with different combinations of datasets that the growth index in the $γδ$CDM model is very close in value to the $Λ$CDM and the $ω$CDM models. The growth of structure is suppressed in the $γδ$CDM model, which is formulated with the extended gravitational growth framework. Upon analyzing cosmological data, we ascertain that the $γδ$CDM model is equally competitive as the $Λ$CDM and the $ω$CDM models.

Figures (5)

• Figure 1: Left panel: The density contrast, $\delta_m(a)$, is plotted versus scale factor, $a$. Equation (\ref{['dmgs']}) is used for the $\gamma\delta$CDM model and the growing solution given in Silveira:1994yqLee:2009gbNesseris:2017vorVelasquez-Toribio:2020xyf is used for the $\omega$CDM and the $\Lambda$CDM models. Right panel: The growth index undergoes a sudden increase near the present time. As in the $\Lambda$CDM and the $\omega$CDM models Linder:2018pth its value remains constant in the matter dominated era and then asymptotically approaches to new numerical value in the far future. In both panels, we use values of the cosmological parameters obtained from the analyses with $f(z)$+DESI2+DESY5 dataset.
• Figure 2: Left panels: The contour plots for 2D joint posterior distributions, with 1$\sigma$ and 2$\sigma$ confidence regions, for only free cosmological parameters (blue) and also together with the free growth indices (purple) of the $\gamma\delta$CDM, the $\Lambda$CDM and $\omega$CDM models, for the full $f(z)$+DESI2+DESY5 dataset, together with 1D marginalized posterior distributions. Right panels: Theoretical (dashed curves) and the numerically determined (solid curves) growth rate functions plotted together with the $f(z)$ growth rate data (see Table \ref{['fzdata']}) for the $\gamma\delta$CDM, the $\Lambda$CDM and $\omega$CDM models. Shaded bands indicate $1\sigma$ posterior regions.
• Figure 3: Comparison of theoretical growth rate $f(z)$ for various models. On the left panel the $\gamma\delta$CDM model is compared with the $\Lambda$CDM and the $\omega$CDM models. On the right panel the $\gamma\delta$CDM model is compared with the $f(R)$ gravity models. It is visually obvious that the growth rate in the $\gamma\delta$CDM model behaves similarly to the $\Lambda$CDM and the $\omega$CDM models, and distinctly to the $f(R)$ gravity models, even though $\gamma\delta$CDM model is based on $f(R)$ gravity.
• Figure 4: The $\sigma_8(z)$ predictions of three cosmological models plotted together with the $\sigma_8$ data points of Table \ref{['s8data']}. Here the red curve corresponds to the $\gamma\delta$CDM model, blue curve to the $\Lambda$CDM model with free $\Omega_{m0}$ parameter, and green curve to the $\Lambda$CDM model with $\Omega_{m0}=0.315$, which is the value obtained by the Planck collaboration Planck:2018vyg.
• Figure 5: The $f\sigma_8(z)$ predictions of three cosmological models plotted together with the $f\sigma_8$ data points of Table \ref{['fsigma8']}. Here the red curve corresponds to the $\gamma\delta$CDM model, blue curve to the $\Lambda$CDM model with free $\Omega_{m0}$ parameter, and green curve to the $\Lambda$CDM model with $\Omega_{m0}=0.315$, which is the value obtained by the Planck collaboration Planck:2018vyg.