Testing $\Lambda \mathrm{CDM}$ with the growth function $\delta (a)$: Current constraints

We have compiled a data set consisting of 22 data points at a redshift range (0.15, 3.8) which can be used to constrain the linear perturbation growth rate $f=\frac{d\ln \delta }{d\ln a}$. 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 data points constrain $f$ indirectly through the rms mass fluctuation ${\sigma }_8(z)$ inferred from $L_y\mathrm{\text{−}}\alpha$ at various redshifts. Our analysis tests the consistency of the $\Lambda \mathrm{CDM}$ model and leads to a constraint of the Wang-Steinhardt growth index $\gamma$ [defined from $f={\Omega }_m(a{)}^{\gamma }$] as $\gamma ={0.67}_{-0.17}^{+0.20}$. This result is clearly consistent at $1\sigma$ with the value $\gamma =\frac{6}{11}=0.545$ predicted by $\Lambda \mathrm{CDM}$. We also apply our analysis on a new null test of $\Lambda \mathrm{CDM}$ 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 $\Lambda \mathrm{CDM}$ provides an excellent fit to the current linear growth data.

Figure 1

The numerically obtained solution of Eq. (2.4) for the normalized growth of Eq. (2.11) in the case of $\Lambda \mathrm{CDM}$ (${\Omega }_{0\mathrm{m}}=0.3$) (black dashed line) along with the corresponding approximate result with $\gamma =\frac{6}{11}$ obtained from Eq. (2.13) (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 ${\Omega }_{0\mathrm{m}}=0.3$ (black continuous line) and the corresponding $1\sigma$ errors (shaded region). The error boxes on $f$ are obtained using the ratios at the specific redshifts. Clearly, the best fit shows a minor difference from $\Lambda \mathrm{CDM}$ (blue dashed line) only at low redshifts.

Figure 3

The $1\sigma$ range (shaded region) of the lhs of Eq. (3.6) (black continuous line). Notice that by construction it is independent of the redshift $z$ since we have assumed that the geometric part of Eq. (3.6) [$H(z)$] behaves like $\Lambda \mathrm{CDM}$ (${\Omega }_{0\mathrm{m}}=0.3$). The value 0 corresponding to $\Lambda \mathrm{CDM}$ 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. (2.25) includes the value $\gamma =\frac{6}{11}$.

Figure 4

The same as Fig. 2 with a generalized parametrization (2.14) where either both parameters ${\gamma }_0$, ${\gamma }_0^{\prime }$ are allowed to vary (a) or only ${\gamma }_0^{\prime }$ is allowed to vary while ${\gamma }_0$ is fixed to its $\Lambda \mathrm{CDM}$ value (b).