TY - JOUR

T1 - Global properties of the growth index

T2 - Mathematical aspects and physical relevance

AU - Calderon, R.

AU - Felbacq, D.

AU - GANNOUJI, RADOUANE

AU - Polarski, D.

AU - Starobinsky, A. A.

N1 - Publisher Copyright:
© 2020 American Physical Society.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/5/15

Y1 - 2020/5/15

N2 - 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 (ωm=0,γ∞) in the future and (ωm=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) µωmtot remains unclustered, we find that the limit of the growth index in the past γ-∞ µ does not depend on the equation of state of DE, in sharp contrast with the case µ=0 (for which γ-∞ is obtained). We show indeed that there is a mathematical discontinuity: one cannot obtain γ-∞ by taking limµ→0γ-∞ µ (i.e., the limits µ→0 and ωmtot→1 do not commute). We recover in our analysis that the value γ-∞ µ corresponds to tracking DE in the asymptotic past with constant γ=γ-∞ µ found earlier.

AB - 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 (ωm=0,γ∞) in the future and (ωm=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) µωmtot remains unclustered, we find that the limit of the growth index in the past γ-∞ µ does not depend on the equation of state of DE, in sharp contrast with the case µ=0 (for which γ-∞ is obtained). We show indeed that there is a mathematical discontinuity: one cannot obtain γ-∞ by taking limµ→0γ-∞ µ (i.e., the limits µ→0 and ωmtot→1 do not commute). We recover in our analysis that the value γ-∞ µ corresponds to tracking DE in the asymptotic past with constant γ=γ-∞ µ found earlier.

UR - http://www.scopus.com/inward/record.url?scp=85085981583&partnerID=8YFLogxK

U2 - 10.1103/PhysRevD.101.103501

DO - 10.1103/PhysRevD.101.103501

M3 - Article

AN - SCOPUS:85085981583

VL - 101

JO - Physical Review D

JF - Physical Review D

SN - 2470-0010

IS - 10

M1 - 103501

ER -