Growth-Index Parametrizations Beyond ΛCDM: A Review of Interacting Dark Energy, Modified Gravity, and Observational Tests

The growth relation f = d ln D/d ln a ≃ Ωγm provides a compact description of linear structure formation whose interpretation depends on the background, perturbation equations, amplitude convention, and scale range. This critical review compares constant, redshift-dependent, density-based, and bounded-transition growth indices in general relativity (GR), interacting dark energy (IDE), and modified gravity (MG). A homogeneous energy-transfer rate does not determine IDE perturbations, while scale-dependent MG generally requires more than a redshift-only index. We distinguish density growth from the velocity response entering redshift-space distortions. A reproducible radiation-free flat-ΛCDM illustration gives m aximum f σ8 approximation errors of 1.123% for constant 6/11, 0.108% for an endpoint-matched Chevallier–Polarski–Linder (CPL)-like form, and 0.159% for an unoptimized bounded transition on 0 ≤ z ≤ 2. A separate phenomenological calculation illustrates scale dependence without asserting a viable MG theory. These examples provide neither observational constraints nor a physical IDE/MG model comparison. We identify weakly constrained transition limits and set out the perturbative and likelihood requirements for testing additional parametrization freedom and reassessing the reported high-growth-index indication.