  The origin and nature of dark energy is one of the most significant
challenges in modern science. This research aims to investigate dark energy on
astrophysical scales and provide a cosmology-independent method to measure its
equation-of-state parameter $w$. To accomplish this, we introduce the concept
of a perfect fluid in any static, curved spacetime, and express the
energy-momentum tensor of the perfect fluid in a general isotropic form, namely
Weinberg's isotropic form. This enables us to define an equation-of-state
parameter in a physical and global manner. Within this theoretical framework,
we demonstrate that the energy-momentum tensor of dark energy on different
scales can take the general isotropic form. Furthermore, we explore the
SdS$_{w}$ spacetime and establish its connection with dark energy in cosmology
through the equation-of-state parameter $w$. In the SdS$_{w}$ spacetime, a
repulsive dark force can be induced by dark energy locally. We then apply the
concept of the dark force to realistic astrophysical systems using the Poisson
equation. Finally, we find that an anomaly in the Milky Way rotation curve can
be quantitatively interpreted by the dark force. By fitting the galactic curve,
we are able to obtain the value of the equation-of-state parameter of dark
energy, independently of specific dark energy models.

General Relativity and Quantum Cosmology