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.

We study the cosmological dynamics of D-BIonic and DBI scalar field, which is
coupled to matter fluid. For the exponential potential and the exponential
couplings, we find a new analytic scaling solution yielding the accelerated
expansion of the Universe. Since it is shown to be an attractor for some range
of the coupling parameters, the density parameter of matter fluid can be the
observed value, as in the coupled quintessence with a canonical scalar field.
Contrary to the usual coupled quintessence, where the value of matter couple
giving observed density parameter is too large to satisfy observational
constraint from CMB, we show that the D-BIonic theory can give similar solution
with much smaller value of matter coupling. As a result, together with the fact
that the D-BIonic theory has a screening mechanism, the D-BIonic theory can
solve the so-called coincidence problem as well as the dark energy problem.

We study the modified gravity theory, names cuscuta-galileon gravity to explain the accelerated expansion of the universe by using a nondynamical scalar field. We find that the model can provide the sequence of the thermal history of the universe correctly, in the same time the gravitational coupling is modified from the Newtonian gravitational constant. Interestingly, the new gravitational coupling depends on time, then it leads to a changing on the Hubble parameter. As a result, the present value of the Hubble parameter in our model increases from the standard model of cosmology, the Lambda-Cold-Dark-Matter model, about 10%, then it might be a tendency to solve one of the recent problems in cosmology, so-called the Hubble tension problem. Observations in universe such as Big-Bang-Nucleosysthesis and Lunar-Laser-Ranging also provide constraints on parameters of the model, however the model can satisfy these observations. The above results have been done under some conditions, namely we assume an exponential form of potential for the scalar field. Thus other forms of potential may give other possibilities and results. We also improve the model by adding a small negative vacuum energy, it leads to a smaller of the present value of the Hubble parameter which satisfies the latest observations on Super Novae Type IA.

We study cosmology dynamics of the cuscuta-galileon gravity. This model has only two gravitational degrees of freedom as the General Relativity, thus it is satisfied observations in the solar system scale. We use dynamical system approach to find behavior under cosmological background. We found that the cuscuta-galileon model without a potential term cannot provide the thermal history of the universe correctly. After adding a potential term we successfully find radiation dominated, matter dominated, and dark energy dominated eras. By studying stabilities of the dynamical system we found that if we start from the radiation dominated era, it will follow by matter dominated and dark energy dominated eras respectively which corresponds to the thermal history of the universe. Therefore the model can be one of the candidates for the dark energy to explain the accelerated expansion of the universe. However, the amount of dark energy in the matter dominated and the radiation dominated eras is large comparing to the cosmological constant. Thus, some observations such as the Cosmic Microwave Background, the cuscuta-galileon gravity may not satisfy. Moreover, in the perturbations level we found that there is an instability in the model which arises from negative sound speed squared.