Linear Perturbations and Multi-Probe Diagnostics in Dark-Sector Selective $f(R,T_{\chi})$ Gravity

TL;DR

This paper introduces a novel dark-sector selective $f(R,T_{\chi})$ gravity model that depends solely on dark matter trace, deriving its equations, perturbations, and multi-probe observational signatures to test deviations from General Relativity.

Contribution

It develops a new gravity extension with dark-sector trace coupling, providing a clear theoretical foundation and a multi-probe framework for observational constraints.

Findings

The model produces scale- and time-dependent deviations from GR in structure growth and lensing.

Derived exact field equations and perturbation systems in gauge-ready form.

Established a framework for applying linear-regime constraints to dark-sector modified gravity.

Abstract

We develop a dark-sector selective trace-coupled extension of gravity in which the matter--curvature coupling depends exclusively on the trace of the dark-matter energy--momentum tensor, $T_{\chi}$, defined from a canonical dark-matter field $\chi$. This construction provides a microphysically specified trace sector, removes the usual matter-Lagrangian ambiguity of $f(R,T)$-type models, and preserves minimal coupling of visible matter by design. We derive the full field equations, the exact dark-sector exchange structure, and the linear scalar-perturbation system in gauge-ready form. In the sub-horizon regime, we derive effective modified-gravity functions governing structure growth and light deflection, and show that the model generically produces correlated, scale- and time-dependent departures from General Relativity in growth and lensing observables. Building on this structure, we formulate a perturbation-focused multi-probe framework based on redshift-space distortions, weak lensing, and CMB lensing, explicitly targeting degeneracy breaking beyond background-expansion tests. The analysis establishes the action-level and perturbation-level foundations of the model and provides a conservative, reproducible framework for translated linear-regime constraints within a dark-sector selective modified-gravity setting.