History of a Particle Bounded to the Cosmological LTB Black Hole Surrounded by the Quintom Field

Abstract: In this paper, we derived the complete set of time-dependent geodesic equations for an LTB black hole surrounded by a Quintom field and investigated the evolution of effective potential and photon orbits across cosmic epochs. Our findings demonstrate that in an accelerated universe, the peak of an effective potential decreases in height and shifts toward smaller radii. Additionally, the probability of stable orbit formation decreases as cosmic expansion progresses. We classified the possible trajectories into four types: terminating bound orbits, stable orbits, scattering flyby orbits, and terminating escape orbits. The results indicate that stable bound orbits are more prevalent in the early universe, whereas at late time epochs, flyby orbits become dominant due to the expansion-driven weakening of gravitational potential. We further analyzed the impact of angular momentum on the evolution of orbits, showing that as it increases, the ISCO radius decreases while the peak of the effective potential shifts outward. This suggests that particles with higher angular momentum follow extended bound orbits, and more energetic photons are more likely to be captured by the black hole. Conversely, an increase in angular momentum reduces the probability of flyby orbits while increasing the likelihood of direct fall into the black hole. Our study provides new insights into how cosmic acceleration influences black hole geodesics, revealing that the progressive shrinking of ISCO and stable orbits eventually disappear as the universe approaches the Big Rip singularity. These findings contribute to a deeper understanding of the dynamical nature of cosmological black holes and may offer new perspectives for observational tests through gravitational lensing, accretion disk evolution, and quasi-periodic oscillations (QPOs) in evolving black hole spacetimes.