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

Abstract

We analyze the global behaviour of the growth index of cosmic inhomogeneities in an isotropic homogeneous universe filled by cold non-relativistic 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 $\gamma$ is finite from the asymptotic past to the asymptotic future is identified as the so-called heteroclinic orbit connecting the critical points $({\Omega }_m=0,{\gamma }_{\infty })$ in the future and $({\Omega }_m=1,{\gamma }_{-\infty })$ 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) $\epsilon {\Omega }_m^{\mathrm{tot}}$ remains unclustered, we find that the limit of the growth index in the past ${\gamma }_{-\infty }^{\epsilon }$ does not depend on the equation of state of DE, in sharp contrast with the case $\epsilon =0$ (for which ${\gamma }_{-\infty }$ is obtained). We show indeed that there is a mathematical discontinuity: one cannot obtain ${\gamma }_{-\infty }$ by taking ${\lim }_{\epsilon \to 0}{\gamma }_{-\infty }^{\epsilon }$ (i.e. the limits $\epsilon \to 0$ and ${\Omega }_m^{\mathrm{tot}}\to 1$ do not commute). We recover in our analysis that the value ${\gamma }_{-\infty }^{\epsilon }$ corresponds to tracking DE in the asymptotic past with constant $\gamma ={\gamma }_{-\infty }^{\epsilon }$ found earlier.

arXiv comment18 pages, 3 figures; v2: improved presentation, matches published version

References in corpus (49)

• Dynamics of dark energy
• The Cosmological Constant and Dark Energy
• Modified Gravity and Cosmology
• f(R) theories
• 4D Gravity on a Brane in 5D Minkowski Space
• Cosmological Constant - the Weight of the Vacuum
• The Case for a Positive Cosmological Lambda-term
• Observational Probes of Cosmic Acceleration
• Beyond the Cosmological Standard Model
• Cluster Abundance Constraints on Quintessence Models
• Cosmology and fundamental physics with the Euclid satellite
• Reconstruction of a scalar-tensor theory of gravity in an accelerating universe
• Dark Energy
• Reconstructing Dark Energy
• Dark Matter and Dark Energy Interactions: Theoretical Challenges, Cosmological Implications and Observational Signatures
• Beyond $\Lambda$CDM: Problems, solutions, and the road ahead
• Testing General Relativity in Cosmology
• Parameterized Beyond-Einstein Growth
• Weighing the Giants IV: Cosmology and Neutrino Mass
• Model independent evidence for dark energy evolution from Baryon Acoustic Oscillations
• Testing LCDM with the Growth Function δ(a): Current Constraints
• Viable f(T) models are practically indistinguishable from LCDM
• Current constraints on the cosmic growth history
• On the growth of linear perturbations
• The growth of matter perturbations in f(R) models
• Next Generation Redshift Surveys and the Origin of Cosmic Acceleration
• Growth factor parametrization and modified gravity
• Cosmic history of viable exponential gravity: Equation of state oscillations and growth index from inflation to dark energy era
• A parametrization of the growth index of matter perturbations in various Dark Energy models and observational prospects using a Euclid-like survey
• Phantom boundary crossing and anomalous growth index of fluctuations in viable f(R) models of cosmic acceleration
• Growth Index of DGP Model and Current Growth Rate Data
• Constraining $f(R)$ gravity by the Large Scale Structure
• Skewness as a test of the equivalence principle
• Bouncing Universes in Scalar-Tensor Gravity Models admitting Negative Potentials
• Constraints and tensions in testing general relativity from Planck and CFHTLenS including intrinsic alignment systematics
• A parametrization for the growth index of linear matter perturbations
• Agegraphic dark energy: growth index and cosmological implications
• Consistency of Modified Gravity with a decreasing $G_{\mathrm{eff}}(z)$ in a $\Lambda$CDM background
• Growth index with the exact analytic solution of sub-horizon scale linear perturbation for dark energy models with constant equation of state
• Growth index of matter perturbations in running vacuum models
• When is the growth index constant?
• Cosmological neutrino bounds for non-cosmologists
• Growth of spherical overdensities in scalar-tensor cosmologies
• Improving Fisher matrix forecasts for galaxy surveys: window function, bin cross-correlation, and bin redshift uncertainty
• The End of Cosmic Growth
• Global properties of the growth index of matter inhomogeneities in the universe
• Emulating $\Lambda$CDM-like expansion on the Phantom brane
• On the consistency of the expansion with the perturbations
• Testing consistency of general relativity with kinematic and dynamical probes

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