In the present paper we study the geodesic structure of the Janis-Newman-Winicour(JNW) space-time which contains a strong curvature naked singularity. This metric is an extension of the Schwarzschild geometry when a massless scalar field is included. We find that the strength parameter µ of the scalar field effects on the geodesic structure of the JNW space-time. By solving the geodesic equation and analyzing the behavior of effective potential, we investigate all geodesic types of the test particle and the photon in the JNW space-time. At the same time we simulate all the geodesic orbits corresponding to the energy levels of the effective potential in the JNW space-time.

We investigate the strong gravitational lensing for black hole with scalar charge in massive gravity.We find that the scalar charge and the type of the black hole significantly affect the radius of the photon sphere, deflection angle, angular image position, angular image separation, relative magnifications and time delay in strong gravitational lensing. Our results can be reduced to that of the Schwarzschild and Reissner-Nordström black holes in some special cases.

We study strong gravitational lensing for photons coupled to Weyl tensor in a regular phantom black hole spacetime. It is generally accepted that photons with different polarizations have different trajectories which yields a phenomenon of birefringence. As a result, there are two sets of relativistic images on each side of the object, this is quite different from the uncoupled case in which there is only one set of images. Nevertheless, we focus our attention on the relativistic images on one side of the object and investigate the difference between them by discussing how the coupling constant and phantom hair affect the difference of photon sphere radius, minimum impact parameter and deflection angle. After that, we find that the closer the light gets to the black hole, the larger the deflection angle will be. Then, we investigate the difference in angular image position and relative magnitudes of the first relativistic image between the two types of polarized photons, and find that the two images for different polarizations will separate further and be distinguished more easily in the cases that the phantom hair decreases or the absolute value of the coupling constant increases. Furthermore, the image is brighter when it seats closer to the optical axis.

In the present paper we analyze the geodesic structure of black hole spacetime in massive gravity with the scalar charge Q representing the modification to Einstein's general relativity. By solving the geodesic equation and analyzing the behavior of effective potential, we investigate the time-like geodesic types of the test particle around a black hole in massive gravity. At the same time, all kinds of orbits, which are allowed according to the energy levels of the effective potential, are numerically simulated in detail.

In this paper, we analyze the universe evolution and phase space behavior of the Umami Chaplygin model, where the Umami Chaplygin fluid replaces both a dark energy and a dark and baryonic matter. We find the Umami Chaplygin model can be stable against perturbations under some conditions and can be used to explain the late-time cosmic acceleration. The results of phase space analysis show that there exists a late-time accelerated expansion attractor with [Formula: see text], which indicates the Umami Chaplygin fluid can behave as a cosmological constant. Moreover, the Umami Chaplygin model can describe the expansion history of the universe. The evolutionary trajectories of the statefinder diagnostic pairs and the finite time future singularities are also discussed.

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The emergent scenario provides a possible way to avoid the big bang singularity by assuming that the Universe originates from an Einstein static state. Therefore, an Einstein static Universe stable under perturbations is crucial to a successful implementation of the emergent mechanism. In this paper, we analyze the stability of the Einstein static Universe against the scalar perturbations in the mimetic theory and find that stable Einstein static solutions exist under certain conditions in this theory. In the original mimetic gravity, the Einstein static Universe is unstable. Then, we find that the Universe can naturally exit from the initial static state, evolve into an inflationary era and then exit from the inflationary era. Thus, the emergent scenario can be used to resolve the big bang singularity in the mimetic theory.

We analyze the Schrödinger-type scalar wave equation of an extended black hole in 𝑓 (𝑅) gravity, and numerically investigate its absorption/scattering cross sections using the partial wave method. It is found that the dimension of length 𝛼 makes the peak value of the effective scattering potential fall down, and the absorption cross section oscillates around the geometric optical value in the high frequency regime. We can also see that the scattering flux becomes stronger and its angle width becomes narrower in the forward direction, the glory peak becomes lower and the glory width becomes narrower along the backward direction when the coupling parameter 𝛼 increases.