Direct Probe of Dark Energy through Gravitational Lensing Effect
  We show that gravitational lensing can provide a direct method to probe the
nature of dark energy at astrophysical scales. For lensing system as an
isolated astrophysical object, we derive the dark energy contribution to
gravitational potential as a repulsive power-law term, containing a generic
equation of state parameter $w$. We find that it generates $w$-dependent and
position-dependent modification to the conventional light orbital equation of
$w=-1$. With post-Newtonian approximation, we compute its direct effect for an
isolated lensing system at astrophysical scales and find that the dark energy
force can deflect the path of incident light rays. We demonstrate that the
dark-energy-induced deflection angle $\Delta\alpha_{DE}^{}\propto
M^{(1+\frac{1}{3w})}$ (with $1+\frac{1}{3w}>0$), which increases with the
lensing mass $M$ and consistently approaches zero in the limit $M \to 0$. This
effect is distinctive because dark energy tends to diffuse the rays and
generates concave lensing effect. This is in contrast to the conventional
convex lensing effect caused by both visible and dark matter. Measuring such
concave lensing effect can directly probe the existence and nature of dark
energy. We estimate this effect and show that the current gravitational lensing
experiments are sensitive to the direct probe of dark energy at astrophysical
scales. For the special case $w=-1$, our independent study favors the previous
works that the cosmological constant can affect light bending, although our
prediction qualitatively and quantitatively differ from the literature,
including our consistent realization of $\Delta\alpha_{DE}\to 0$ (under $M \to
0$) at the leading order.
