Evidence for Suppression of Structure Growth in the Concordance Cosmological Model

We present evidence for a suppressed growth rate of large-scale structure during the dark-energy-dominated era. Modeling the growth rate of perturbations with the “growth index” $\gamma$, we find that current cosmological data strongly prefer a higher growth index than the value $\gamma =0.55$ predicted by general relativity in a flat Lambda cold dark matter cosmology. Both the cosmic microwave background data from Planck and the large-scale structure data from weak lensing, galaxy clustering, and cosmic velocities separately favor growth suppression. When combined, they yield $\gamma =0.633_{-0.024}^{+0.025}$, excluding $\gamma =0.55$ at a statistical significance of $3.7\sigma$. The combination of $f{\sigma }_8$ and Planck measurements prefers an even higher growth index of $\gamma =0.639_{-0.025}^{+0.024}$, corresponding to a $4.2\sigma$ tension with the concordance model. In Planck data, the suppressed growth rate offsets the preference for nonzero curvature and fits the data equally well as the latter model. A higher $\gamma$ leads to a higher matter fluctuation amplitude $S_8$ inferred from galaxy clustering and weak lensing measurements, and a lower $S_8$ from Planck data, effectively resolving the $S_8$ tension.

Figure 1

Marginalized constraints on the growth index $\gamma$ from CMB (PL18) and LSS data. The latter includes $f{\sigma }_8$, DESY1, and BAO measurements. The legend indicates different combinations of the datasets. The vertical dashed line marks the concordance model prediction of $\gamma =0.55$.

Figure 2

Marginalized posterior on the theoretical $f(z){\sigma }_8(z)$ assuming the growth-index parametrization in Eq. (2). Shaded bands show the 68% and 95% posteriors from our baseline analysis that includes $f{\sigma }_8$ and PL18 data (orange), and the corresponding constraints in the concordance model with $\gamma =0.55$ (black). The data points indicate actual $f{\sigma }_8$ measurements.

Figure 3

68% and 95% marginalized constraints on parameters in the concordance model allowing for a free growth index $\gamma$, from $f{\sigma }_8+\mathrm{DES}\mathrm{Y}1+\mathrm{BAO}$ (blue), PL18 alone (red), and $f{\sigma }_8+\mathrm{DES}\mathrm{Y}1+\mathrm{BAO}+\mathrm{PL}18$ (violet). Contours contain 68% and 95% of the corresponding projected 2D constraints. The horizontal black dashed lines mark the concordance model prediction of $\gamma =0.55$. The horizontal bars in the $\gamma -S_8$ panel indicate the 68% limits on $S_8$ for a fixed $\gamma =0.55$ (see text); they are vertically offset from $\gamma =0.55$ for visibility.

Figure 4

Degeneracy between $\gamma$ and ${\mathrm{\Omega }}_K$ in the PL18 $\mathrm{temp}+\mathrm{pol}$ analysis when both parameters are allowed to vary. Contours show the 68% and 95% credible intervals. The dashed lines mark the point $[{\mathrm{\Omega }}_K=0,\gamma =0.55]$ corresponding to the concordance flat $\mathrm{\Lambda }\mathrm{CDM}$ model.

Figure 5

Residuals in the CMB TT power spectrum $D_{\ell }\equiv \ell (\ell +1)C_{\ell }/(2\pi )$ between the best-fit model with free $\gamma$ (orange), best-fit model with curvature (blue), and best-fit model with free CMB lensing amplitude $A_{\mathrm{lens}}$ (green). The data points and error bars represent the Planck 2018 (binned) TT power spectrum residuals and the 68% uncertainties. All residuals are computed with respect to the best-fit concordance model.