On the growth of linear perturbations

TL;DR

This work addresses how dark energy and modified gravity affect the linear growth of matter perturbations by using the growth index $\gamma$ and its redshift dependence $\gamma(z)=\gamma_0+\gamma'_0 z$ at low $z$. It derives a general constraint at $z=0$ linking $\gamma_0$, $\gamma'_0$, $\Omega_{m,0}$ and $w_{DE,0}$ (assuming $\frac{G_{\rm eff,0}}{G_{N,0}}=1$), showing $\gamma'_0$ is determined by $\gamma_0$, $\Omega_{m,0}$ and $w_{DE,0}$ and is well approximated by a linear relation with slope $\sim3$. Across ΛCDM and constant-$w_{DE}$ models, $|\gamma'_0|$ is typically $\lesssim 0.02$, with $\gamma'_0$ around $-0.02$ for constant $w_{DE}$; smoothly varying EOS scenarios do not exceed this bound. The key implication is that precise low-$z$ measurements of $\gamma(z)$ can help distinguish GR from modified gravity and other DE models, and ignoring $\gamma'_0$ can bias inferences of $\Omega_{m,0}$ and $w_{DE,0}$.

Abstract

We consider the linear growth of matter perturbations in various dark energy (DE) models. We show the existence of a constraint valid at $z=0$ between the background and dark energy parameters and the matter perturbations growth parameters. For $Λ$CDM $γ'_0\equiv \frac{dγ}{dz}_0$ lies in a very narrow interval $-0.0195 \le γ'_0 \le -0.0157$ for $0.2 \le Ω_{m,0}\le 0.35$. Models with a constant equation of state inside General Relativity (GR) are characterized by a quasi-constant $γ'_0$, for $Ω_{m,0}=0.3$ for example we have $γ'_0\approx -0.02$ while $γ_0$ can have a nonnegligible variation. A smoothly varying equation of state inside GR does not produce either $|γ'_0|>0.02$. A measurement of $γ(z)$ on small redshifts could help discriminate between various DE models even if their $γ_0$ is close, a possibility interesting for DE models outside GR for which a significant $γ'_0$ can be obtained.

Figures (4)

• Figure 1: a) The left panel shows the constraint (\ref{['dgamma0b']}) for $\Omega_{m,0}=0.3$ and various values of $w_{DE,0}$. We have from top to bottom: $w_{DE,0}=-1.4,~-1.3,~-1.2,~-1,~-0.8$. For given $\Omega_{m,0}$ and $w_{DE,0}$, the couple $\gamma_0,~\gamma'_0$ is on the corresponding line for any model while $\gamma'_0$ will depend on the value $\gamma_0$ realized in a particular model. b) On the right panel the constraint (\ref{['dgamma0b']}) is shown in function of $\Omega_{m,0}$. From top to bottom we have $w_{DE,0}=-1.2,~-1,~-0.8$. We see that the coefficient $b$ defined in (\ref{['dgamma0f']}) increases for increasing $\Omega_{m,0}$ and decreasing $w_{DE,0}$.
• Figure 2: a) On the left panel, the blue line shows the degeneracy in the $\Omega_{m,0},~w_{DE,0}$ plane for $\gamma_0=0.555$assuming $\gamma'_0=0$. The red, resp. green, dashed lines correspond to $\gamma'_0=-0.02$ (top) and $\gamma'_0=0.02$ (bottom), resp. $\gamma'_0=-0.05$ (top) and $\gamma'_0=0.05$ (bottom). Ignoring the true non vanishing value of $\gamma'_0$ increases significantly the uncertainty on the couples $\Omega_{m,0},~w_{DE,0}$. b) On the right panel it is seen that models with very close $\gamma_0$ can be discriminated if $\gamma$ is measured for $0\le z\le 0.5$ assuming $\gamma$ is linear on small $z$, as often is the case. The lower the values of $\gamma_0$, the easier it is to discriminate these models through the difference in their slope $\gamma'_0$. For illustration, we have assumed here an error of $1\%$.
• Figure 3: a) The lines in colour on the left panel are the same as in Figure 1. The black line gives the true value of $\gamma_0$ realised in models with $w_{DE}=w_{DE,0}$ = constant and $\Omega_{m,0}=0.3$. It is seen that all models with $w_{DE}$ = constant shown here have practically the same non vanishing $\gamma'_0$, $\gamma'_0\approx -0.02$. Note that $\gamma_0$ increases when $w_{DE}$ increases. b) On the right, $\gamma_0$ is displayed in function of $\Omega_{m,0}$ for the $\Lambda$CDM model.
• Figure 4: The parameters $\gamma_0$ (left) and $\gamma'_0$ (right) are shown in function of $\beta\equiv w_1$ and $\Omega_{m,0}$ for a model with variable equation of state parameter $w_{DE}= -1.2 + \beta~\frac{z}{1+z}$. Hence all the points on the two surfaces have $w_{DE,0}=-1.2$. The results for $w_{DE}=-1.2$ are recovered for $\beta=0$. We note for the left figure that $\gamma'_0=0$ is obtained for some particular combinations $\beta,~\Omega_{m,0}$.