How do astronomers conclude that vast quantities of dark matter surround our Galaxy? How is this dark matter distributed in space?

One of the most compelling pieces of evidence for the existence of dark matter comes from its gravitational effects. Even though dark matter itself remains unseen, we can observe its influence on visible matter in the cosmos. It acts as an invisible scaffold that structures the universe, affecting the motion of galaxies and clusters of galaxies. For instance, the stars at the outer edges of galaxies move at speeds that only make sense if there's a considerable amount of unseen mass influencing their motion. Without this additional mass, provided by dark matter, the outer stars would fly away into space due to the insufficiency of gravitational pull from the visible matter alone.
In cosmology, we also encounter the concept of galactic rotation curves, which are another indicator of the presence of dark matter. These curves are plots of the rotational speeds of stars against their distance from the galactic center. According to Newton's laws, we'd expect the rotation speeds to decrease with distance from the center; however, the curves remain flat or even rise, suggesting there's more mass than we can see—mass that must be dark matter. The study of these curves has led to groundbreaking revelations about the composition of the universe.

Galaxy rotation curves offer a vital clue in unraveling the dark matter mystery. They essentially graph the velocity of stars against their distance from a galaxy’s center. In a galaxy, astronomers observe that the inner stars rotate at different speeds compared to those on the outside. According to the laws of physics, as we move away from the galactic center, the rotational speed should drop presuming most of a galaxy's mass is concentrated at its core. This expectation arises from the distribution of visible matter, such as stars and gas clouds.
However, strangely, the rotation curves stay flat as we move outward, implying that the outer stars are moving much faster than expected. This anomaly can be understood if there's additional mass that doesn't emit light—dark matter. Envisage this invisible matter as a sprawling halo enveloping the galaxy, providing the necessary gravitation to keep the outer stars in speedy, stable orbits, a discovery that reshaped our understanding of the universe’s structure and contents.

Gravitational lensing is a fascinating phenomenon that acts as a cosmic magnifying glass. As predicted by Einstein's General Theory of Relativity, massive objects like planets, stars, and galaxies can bend the fabric of space-time, causing the path of light traveling nearby to curve. This bending results in an effect similar to a lens, where the light coming from a more distant object, like a quasar or a background galaxy, is magnified and distorted.
Astronomers use this effect to probe the invisible dark matter. When light from distant galaxies passes near a galaxy or cluster rich in dark matter, it’s lensed, producing arcs or rings in telescope images—marked as gravitational arcs. These phenomena not only allow scientists to detect the presence of dark matter but also to map its distribution. By analyzing how light bends, they can infer the mass of the dark matter causing the lensing, offering a kind of indirect 'photograph' of dark matter's gravitational footprint in the universe. Gravitational lensing has become an essential tool for astrophysicists studying the enigmatic substance that cloaks our universe's visible matter.