Non-exotic traversable wormholes with strong deflection angle in King and Dekel-Zhao dark matter halos under f(R,Lm) gravity

In this article, we investigate asymptotically flat non-exotic traversable wormhole geometries within the King and Dekel-Zhao dark matter halos in the framework of $f(R, L_m)$ gravity. Two functional forms of the theory are considered: Model-I: $f(R, L_m)=(R/2) + L_m^α$ and Model-II: $f(R, L_m)=(R/2) + (1 + λR)L_m$. For both models, wormhole solutions are obtained and analyzed using the King and Dekel-Zhao dark matter density profiles, allowing us to explore how the underlying matter distribution influences the wormhole structures. The energy conditions are examined to verify the feasibility of sustaining the wormhole geometries with non-exotic matter, while embedding surfaces, proper radial distance, and total gravitational energy are studied to illustrate the wormhole’s physical viability and traversability. Moreover, we test the strong deflection angle and its implications for gravitational lensing and show possible observational signatures of such wormhole configurations. Our results indicate that within $f(R, L_m)$ gravity, and for appropriate parameter choices, dark matter environments can sustain physically consistent non-exotic traversable wormhole geometries with distinct gravitational lensing signatures, providing new insights into the interplay between modified gravity, dark matter, and astrophysical observations.

💡 Research Summary

The paper investigates asymptotically flat, traversable wormhole geometries that can be supported by non‑exotic matter within the framework of f(R, L_m) gravity. Two specific functional forms of the theory are examined: Model I, f(R, L_m)=R/2+L_m^α, and Model II, f(R, L_m)=R/2+(1+λR)L_m. For each model the authors construct static, spherically symmetric wormhole solutions by embedding the metric into two widely used dark‑matter halo density profiles – the King model and the Dekel‑Zhao (DZ) model. The King profile represents a truncated isothermal sphere with a flat core and a r⁻³ outer decline, while the DZ profile is a double‑power‑law distribution that smoothly interpolates between inner and outer slopes via adjustable parameters.

The field equations derived from the f(R, L_m) action are presented, together with the modified conservation law that links the matter Lagrangian L_m, the Ricci scalar R, and the energy‑momentum tensor T_{μν}. By assuming the matter Lagrangian equals the energy density (L_m=ρ) and choosing a shape function b(r) expressed through an auxiliary function ξ(r), the authors enforce the usual wormhole throat conditions b(r₀)=r₀ and b′(r₀)<1. Numerical integration of the shape function for representative parameter sets (e.g., β=0.65, γ=1, η=−0.5 for the King halo) yields well‑behaved b(r) that satisfy the flare‑out condition.

A central part of the analysis is the verification of the energy conditions. For Model I, the exponent α>½ ensures that the null, weak, strong, and dominant energy conditions are satisfied throughout the spacetime. In Model II, a positive coupling λ (typically 0.1–0.3) plays a similar role, with the curvature‑dependent term (1+λR)L_m providing additional effective pressure that prevents NEC violation. Consequently, the wormholes can be sustained by ordinary (non‑exotic) matter, with the dark‑matter halo itself furnishing the required stress‑energy.

Geometrical diagnostics include embedding diagrams obtained from z(r)=±∫