Recent low-redshift observations give value of the present-time Hubble
parameter $H_{0}\simeq 74~\rm{km s}^{-1}\rm{Mpc}^{-1}$, roughly 10\% higher
than the predicted value $H_{0}=67.4~\rm{km s}^{-1}\rm{Mpc}^{-1}$ from Planck's
observations of the Cosmic Microwave Background radiation~(CMB) and the
$\Lambda$CDM model. Phenomenologically, we show that by adding an extra
component X with negative density in the Friedmann equation, it can relieve the
Hubble tension without changing the Planck's constraint on the matter and dark
energy densities. For the extra negative density to be sufficiently small, its
equation-of-state parameter must satisfy $1/3\leq w_{X}\leq1$. We propose a
quintom model of two scalar fields that realizes this condition and potentially
alleviate the Hubble tension. One scalar field acts as a quintessence while
another "phantom" scalar conformally couples to matter in such a way that
viable cosmological scenario can be achieved. The model depends only on two
parameters, $\lambda_{\phi}$ and $\delta$ which represent rolling tendency of
the self-interacting potential of the quintessence and the strength of
conformal phantom-matter coupling respectively. The toy quintom model with
$H_{0}=73.4~\rm{km s}^{-1}\rm{Mpc}^{-1}$~(Quintom I) gives good Supernovae-Ia
luminosity fits, decent $r_{\rm BAO}$ fit, but slightly small acoustic
multipole $\ell_{A}=285.54$. Full parameter scan reveals that quintom model
provide better model than the $\Lambda$CDM model in certain region of the
parameter space, $0.02<\delta<0.10, \Omega_{m}^{(0)}<0.31$, while significantly
relieving Hubble tension even though not completely resolving it. A benchmark
quintom model, Quintom II, is presented as an example.

We investigate gravitational lensing and particle motions around
non-asymptotically flat black hole spacetime in non-linear, ghost-free massive
gravity theory, called de Rham-Gabadadze-Tolley (dRGT) massive gravity.
Deflection angle formulae are derived in terms of perihelion parameter. The
deflection angle can be positive, zero or even negative with various perihelion
distance. The negative angle reveals repulsive behaviour of gravity from a
linear term $\gamma$ in the dRGT black hole solution. We also find an
analytically approximated formula of deflection angle in two regimes: large and
small $\gamma$ term regimes which are shown to be consistent with direct
numerical integration. Null and timelike geodesic motions on equatorial plane
are explored. Particle trajectories around the dRGT black hole are plotted and
discussed in details.

We investigate cosmological dynamics and screening mechanism of the
Dirac-Born-Infeld (DBI) Galileon model. The model has been divided into two
regimes, one has positive signs in front of scalar field kinetic terms
so-called the DBI galileon, another one has negative signs and it is dubbed as
the DBIonic galileon. We find de Sitter solution and evolution of the Universe
starting from radiation dominated era to late-time accelerated expansion in the
DBI galileon model without the presence of potential term. In one of the
attractors, the ghost and Laplacian instabilities vanishes for the whole
evolution. We find mixing of screening mechanisms between the Vainshtein
mechanism and the DBIonic screening mechanism in the DBIonic galileon model, in
which a scale changing between these two mechanisms depends on a mass of a
source.

We investigate effects of massive graviton on the rotation curves of the
Milky Way, spiral galaxies and Low Surface Brightness~(LSB) galaxies. Using a
simple de Rham, Gabadadze, and Tolley (dRGT) massive gravity model, we find
static spherically symmetric metric and a modified Tolman-Oppenheimer-Volkoff
(TOV) equation. The dRGT nonlinear graviton interactions generate density and
pressures which behave like a dark energy that can mimic the gravitational
effects of a dark matter halo. We found that rotation curves of most galaxies
can be fitted well by a single constant-gravity parameter $\gamma \sim
m_{g}^{2}C \sim 10^{-28}~{\rm m^{-1 } }$ corresponding to the graviton mass in
the range $m_g \sim 10^{-21}-10^{-30} {\rm eV}$ depending on the choice of the
fiducial metric parameter $C\sim 1-10^{18}~\text{m}$. Fitting rotation curve of
the Milky Way puts strong constraint on the Yukawa-type coupling of the massive
graviton exchange as a result of the shell effects.