Linear Perturbations and Multi-Probe Diagnostics in Dark-Sector Selective $f(R,T_χ)$ 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, $T_χ$, defined from a canonical dark-matter field $χ$. 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.

Figures (6)

• Figure 1: Translated late-time constraint in the $(\hat{\alpha},\tilde{\xi})$ plane for the polynomial model $f(R,T_\chi)=R+\alpha R^2+\xi R T_\chi+\beta T_\chi^2$, using the benchmark cap $\log_{10}|f_{R0}|<-4.18$ and the dust-level approximation $T_{\chi0}\simeq-\rho_{\chi0}$. Here $\hat{\alpha}\equiv \alpha H_0^2$ and $\tilde{\xi}\equiv \xi\rho_{c0}$. The shaded strip shows the allowed region satisfying $|f_{R0}|=|2\alpha R_0+\xi T_{\chi0}|<10^{-4.18}$, illustrating the parameter-combination degeneracy beyond the pure-$\alpha$ and pure-$\tilde{\xi}$ limits (dashed/dotted lines).
• Figure 2: Redshift evolution of the quasi-static transition scale $k_{\rm trans}(z)\simeq a\,M_{\rm eff}(z)$ for viable benchmark points in the $(\hat{\alpha},\tilde{\xi})$ plane, using the polynomial model $f(R,T_\chi)=R+\alpha R^2+\xi R T_\chi+\beta T_\chi^2$ and the effective mass expression in Eq. \ref{['eq:Meff_Bcoef_model_safe']}. The plotted curves use the dust-regime benchmark closure $\langle V\rangle\simeq \rho_\chi/2$ (oscillating quadratic scalar), yielding $M_{\rm eff}^2/H_0^2=[1-2\tilde{\xi}\Omega_{\chi0}(1+z)^3]/(6\hat{\alpha})$. This figure identifies the scale at which scalaron-mediated modifications are expected to transition in the quasi-static regime DeFeliceTsujikawa2010fRReviewTurner1983CoherentScalarOsc.
• Figure 3: Diagnostic quasi-static scalaron response kernel, $\mathcal{R}_s(k,z)=k^2/[k^2+k_{\rm trans}^2(z)]$, for viable benchmark points of the selective dark-sector $f(R,T_\chi)$ model, where $k_{\rm trans}(z)\simeq a\,M_{\rm eff}(z)$ is determined by the effective mass derived in Sec. \ref{['subsec:model_QS_trace']}. Solid (dashed) curves denote $z=0$ ($z=2$). This plot isolates the scale at which scalaron-mediated modifications become active and provides a theory-level bridge between the effective-mass sector and the scale-dependent growth/lensing observables reconstructed in the quasi-static regime.
• Figure 4: Diagnostic QS growth-modification map and transition scale. Two-dimensional map of the diagnostic growth modification $\mu_{\mathrm{proxy}}(k,z)-1$ in the quasi-static sub-horizon regime, displayed over mode scale $k/H_0$ and redshift $z$. The proxy $\mu_{\mathrm{proxy}}(k,z)$ is constructed from the QS algebraic closure for the scalaron perturbation in Eq. \ref{['eq:QS_deltafR_model_safe']} with the background-defined coefficients $M_{\rm eff}(a)$ and $\mathcal{B}(a)$ given in Eq. \ref{['eq:Meff_Bcoef_model_safe']}. The solid curve shows the evolving transition locus $k_{\rm trans}(z)/H_0$, illustrating that the onset of scale-dependent growth departures tracks the effective-mass scale $k_{\rm trans}(z)\simeq aM_{\rm eff}(z)$. This figure therefore provides a compact pre-closure diagnostic of where, in the $(k,z)$ plane, the model predicts the strongest QS growth response.
• Figure 5: Transition-sensitivity ridge of the quasi-static (QS) growth diagnostic. Heat map of the logarithmic response measure $\mathcal{T}_{\mu}(k,z)\equiv \partial \mu_{\mathrm{proxy}}(k,z)/\partial\ln k$ across the $(k,z)$ plane (horizontal axis: $\log_{10}(k/H_0)$; vertical axis: redshift $z$), for the same Benchmark C setup adopted in the preceding analysis. The bright ridge delineates the scale band where the reconstructed QS growth response becomes maximally sensitive to mode scale, marking the effective transition between GR-like behavior and the modified-gravity scalar sector in the QS limit. The solid curve overlays the predicted transition locus $k=k_{\mathrm{trans}}(z)$, highlighting that the strongest scale sensitivity tracks the evolving QS transition scale inferred from the effective-mass sector. The color bar reports the computed range of $\mathcal{T}_{\mu}$ for this benchmark.