IndisputableMonolith.ILG.Reciprocity
The ILG.Reciprocity module defines the dimensionless variable X equal to k tau0 over a and records the associated reciprocity identities. Cosmologists working on first-order corrections to structure growth in modified gravity cite this module to nondimensionalize the ILG kernel. The module consists of a short sequence of definitions followed by direct algebraic identities that follow from the definition of X.
claimThe module centers on the dimensionless variable $X = k τ_0 / a$, where $k$ is the wave number and $τ_0$ a reference time scale, together with reciprocity identities relating quantities evaluated at $X$ and at $1/X$.
background
The Infra-Luminous Gravity framework modifies the growth of density perturbations through a kernel that depends on wave number and scale factor. The upstream Kernel module supplies the explicit form $w(k,a) = 1 + C · (a / (k τ_0))^α$. The Reciprocity module introduces the combination $X = k τ_0 / a$ so that the kernel becomes a function of $X$ alone.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the variable $X$ and reciprocity identities required by the GrowthODE module. That module derives the prefactor for the first-order ILG growth correction in an EdS background by substituting the ansatz $D = a(1 + B a^α)$ into the growth ODE.
scope and limits
- Does not assign numerical values to the constants C or alpha.
- Does not derive the functional form of the kernel w(k,a).
- Does not treat backgrounds other than Einstein-de Sitter.
- Does not include higher-order terms in the growth expansion.