For example, in the 1930s, scientists measured groups of galaxies, and determined that there wasn’t enough visible stuff to exert enough gravity to hold them together. This was later discovered to be the same for individual galaxies. Thus, something has to be holding the galaxies together. They called this something dark energy.
Another example is that stars on the edges of a spinning spiral galaxy, according to standard physics, should be spinning more slowly than the stars in the center of the galaxy. However, it was found that they all spin at pretty much the same speed, which gives credence to the fact that dark matter exists, because something has to be exerting a force on the outer stars for them to be traveling at the same relative speed.
(This photo shows a spiral galaxy in the center, with the luminous halo around it with globular clusters throughout, and the dark matter halo surrounding it.)
Sources:
http://science.nationalgeographic.com/science/space/dark-matter/
https://www.youtube.com/watch?v=HneiEA1B8ks
http://scienceblogs.com/startswithabang/files/2012/07/I02-13-composition21.jpeg
darkmatter.jpeg
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