Scale Dependent Memory in Cosmological Backreactions, From Local Viscosity to Global Information Drag

This work extends $\Lambda$CDM to a non--Markovian cosmological backreaction
model by introducing a causal memory kernel that links the present expansion to
a weighted history of nonlinear structure. In the Infinite Transformation
Principle (ITP) framework, a negative, long--horizon kernel can fit late--time
expansion and growth data while remaining compatible with early--time
inferences, and shifts the CMB--inferred Hubble rate by a few per cent. Separately, direct measurements in IllustrisTNG show that virialisation and bulk flows generate short--horizon, viscosity--like kernels on nonlinear coarse--graining scales. This paper unifies these results in a scale--dependent memory picture.

A multiscale Volterra closure is formulated in which the backreaction term
$\delta H^{2}(t;L)$ at coarse--graining scale $L$ is sourced by a structural
variable $\Sigma(t;L)$ through a kernel $K_{L}(\Delta t)$. Under a mild
separability assumption for the structural source across scales,
$\Sigma(t;L)\approx S(L)\Sigma_{\rm bg}(t)$, the background backreaction can be
written as a convolution with an effective kernel
$\Keff(\Delta t)=\int \dd\ln L\,W(L)S(L)K_{L}(\Delta t)$ that remains a kernel
in the usual sense. Even if each $K_L$ is a short--horizon exponential, their
mixture need not be.

Using TNG300--1 and TNG50--1, kernels are measured at multiple domain sizes to
build a ladder of memory parameters $(\tau(L),A(L)\tau(L))$ with bootstrap
uncertainties. The results show negative, viscosity--like kernels at all
nonlinear scales, with $|A\tau|$ increasing towards smaller $L$ and
$\tau(L)\lesssim 0.05$--$0.1\,{\rm Gyr}$. An SDSS DR8 domain analysis reproduces
the same sign and scale trend in the real Universe on
$L\sim 60$--$240\,{\rm Mpc}/h$ scales. At the horizon, a Planck 2018 ITP fit
fixes a long--memory kernel whose integrated drag
$|A_{\rm hor}\tau_{\rm hor}|\simeq 0.03$ accounts for the few--percent reduction
in $H_{0}$ relative to flat $\Lambda$CDM.

Taken together, these measurements define a ``cosmic memory ladder'' from
micro-- to horizon scales. Simple power--law fits show that the effective memory
time grows roughly as $\tau(L)\propto L^{2/3}$ once the horizon point is
included, while the integrated drag $|A\tau|$ becomes a slowly varying,
scale--dependent coupling. A controlled kernel--mixture demonstration illustrates
how fitting a broad mixture of short--horizon kernels with a single exponential
can inflate the inferred effective horizon. In this sense, the long--range
``information drag'' in non--Markovian cosmology can be read as the infrared
limit of the same virial friction that couples structure and expansion inside
the cosmic web.