Global properties of the growth index: Mathematical aspects and physical relevance

Physical Review D - Particles, Fields, Gravitation and Cosmology. 2020. Vol. 101. No. 10. Article 103501.

Calderon R., Felbacq D., Gannouji R., Polarski D., Starobinsky A.

We analyze the global behavior of the growth index of cosmic inhomogeneities in an isotropic homogeneous universe filled by cold nonrelativistic matter and dark energy (DE) with an arbitrary equation of state. Using a dynamical system approach, we find the critical points of the system. That unique trajectory for which the growth index γ is finite from the asymptotic past to the asymptotic future is identified as the so-called heteroclinic orbit connecting the critical points (Ω_m=0,γ_{∞}) in the future and (Ω_m=1,γ_{−∞}) in the past. The first is an attractor while the second is a saddle point, confirming our earlier results. Further, in the case when a fraction of matter (or DE tracking matter) ϵΩ^{tot}_m remains unclustered, we find that the limit of the growth index in the past γ^{ϵ}_{−∞} does not depend on the equation of state of DE, in sharp contrast with the case ϵ=0 (for which γ_{−∞} is obtained). We show indeed that there is a mathematical discontinuity: one cannot obtain γ_{−∞} by taking lim_{ϵ→0} γ^{ϵ}_{−∞} (i.e., the limits ϵ→0 and Ω^{tot}_m→1 do not commute). We recover in our analysis that the value γ^{ϵ}_{−∞} corresponds to tracking DE in the asymptotic past with constant γ=γ^{ϵ}_{−∞} found earlier.

Keywords: growth index, cosmological models, inhomogeneous perturbations