Testing LCDM with the Growth Function δ(a): Current Constraints

TL;DR

The paper tests ΛCDM by constraining the growth index $\gamma$ from the growth function $δ(z)$ using data on the growth rate via redshift distortions and Ly-α–derived $σ_8(z)$, employing the Wang–Steinhardt relation $f = Ω_m(a)^γ$. A joint χ^2 fit with $Ω_m=0.3$ yields $γ = 0.674^{+0.195}_{-0.169}$, which is consistent at 1σ with the ΛCDM value $γ = 6/11$. A derivative-free null test using only $H(z)$ and $δ(z)$ also supports ΛCDM, indicating current linear growth data are well described by the standard model. The results, while limited by data accuracy, demonstrate ΛCDM as a robust description of growth and highlight the value of dynamical growth tests as a complement to geometric probes; future large-scale surveys like DUNE could significantly sharpen these constraints.

Abstract

We have compiled a dataset consisting of 22 datapoints at a redshift range (0.15,3.8) which can be used to constrain the linear perturbation growth rate f=\frac{d\lnδ}{d\ln a}. Five of these data-points constrain directly the growth rate f through either redshift distortions or change of the power spectrum with redshift. The rest of the datapoints constrain f indirectly through the rms mass fluctuation σ_8(z) inferred from Ly-αat various redshifts. Our analysis tests the consistency of the LCDM model and leads to a constraint of the Wang-Steinhardt growth index γ(defined from f=Ω_m^γ) as γ=0.67^{+0.20}_{-0.17}. This result is clearly consistent at $1σ$ with the value γ={6/11}=0.55 predicted by LCDM. A first order expansion of the index γin redshift space leads to similar results.We also apply our analysis on a new null test of LCDM which is similar to the one recently proposed by Chiba and Nakamura (arXiv:0708.3877) but does not involve derivatives of the expansion rate $H(z)$. This also leads to the fact that LCDM provides an excellent fit to the current linear growth data.

Figures (4)

• Figure 1: The numerically obtained solution of eq. (\ref{['lnda']}) for the normalized growth of eq. (\ref{['gzdef']}) in the case of $\Lambda$CDM ($\Omega_{0 {\rm m}}=0.3$) (black dashed line) along with the corresponding approximate result with $\gamma=\frac{6}{11}$ obtained from eq. (\ref{['fomgam1']}) (blue continuous line). The agreement between the two approaches is excellent.
• Figure 2: The cosmological data for the growth rate $f(z)$ along with the best theoretical fit $f=\Omega_m(z)^\gamma$ with $\Omega_{0 {\rm m}}=0.3$ (black continuous line) and the corresponding $1\sigma$ errors (shaded region). The errorboxes on f are obtained using the ratios at the specific redshifts. Clearly, the best fit shows a minor difference from $\Lambda$CDM (blue dashed line) only at low redshifts.
• Figure 3: The $1\sigma$ range (shaded region) of the lhs of eq. (\ref{['nulltest']}) (black continuous line). Notice that by construction it is independent of the redshift $z$ since we have assumed that the geometric part of eq. (\ref{['nulltest']}) ($H(z)$) behaves like $\Lambda$CDM ($\Omega_{0 {\rm m}}=0.3$). The value $0$ corresponding to $\Lambda$CDM for both $H(z)$ and $\delta(z)$ (dashed line) is clearly well within $1\sigma$ from the best fit (continuous line). This is to be expected since the range of $\gamma$ in eq. (\ref{['gambf']}) includes the value $\gamma=\frac{6}{11}$.
• Figure 4: Same as Fig. 2 with a generalized parametrization (\ref{['gamexp']}) where either both parameters $\gamma_0$, $\gamma_0'$ are allowed to vary (a) or only $\gamma_0'$ is allowed to vary while $\gamma_0$ is fixed to its $\Lambda$CDM value (b).