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An introduction (containing the info I gathered from several Wikipedia articles).

A galaxy cluster, or cluster of galaxies, is a structure that consists of anywhere from hundreds to thousands of galaxies that are bound together by gravity with typical masses ranging from 1014–1015 solar masses. They are the largest known gravitationally bound structures in the universe. Notable galaxy clusters in the relatively nearby Universe include the Virgo Cluster, Fornax Cluster, Hercules Cluster, and the Coma Cluster. A very large aggregation of galaxies known as the Great Attractor, dominated by the Norma Cluster, is massive enough to affect the local expansion of the Universe. Notable galaxy clusters in the distant, high-redshift Universe include SPT-CL J0546-5345 and SPT-CL J2106-5844, the most massive galaxy clusters found in the early Universe.

The Virgo Supercluster (Virgo SC) or the Local Supercluster (LSC or LS) is a mass concentration of galaxies containing the Virgo Cluster and Local Group, which in turn contains the Milky Way and Andromeda galaxies.

Some of the galaxies within the galaxy cluster (aka cluster of galaxies) have already collided into each other and merged, while some others will eventually do so. Galaxy mergers can occur when two (or more) galaxies collide. They are the most violent type of galaxy interaction. The gravitational interactions between galaxies and the friction between the gas and dust have major effects on the galaxies involved. The exact effects of such mergers depend on a wide variety of parameters such as collision angles, speeds, and relative size/composition, and are currently an extremely active area of research.

Galaxy mergers are important because the merger rate is a fundamental measurement of galaxy evolution. The merger rate also provides astronomers with clues about how galaxies bulked up over time. For example, the Milky Way and Andromeda galaxies in the Local Group of the Virgo Supercluster are destined to collide and merge in about 4.5 billion years from now.

The Andromeda Galaxy was formed roughly 10 billion years ago from the collision and subsequent merger of smaller protogalaxies, while the Milky Way in it's current form got created via a merger/collision, which happened 11.5 billion years ago when a small galaxy called Gaia- Enceladus slammed into what then originally existed of the Milky Way, which itself wass about 13.5 billion years old.

In physical cosmology, the age of the universe is the time elapsed since the Big Bang. A measurement based on direct observations of an early state of the universe, indicates this age to be 13.772±0.040 billion years.

The questions

Since Andromeda and Milky Way galaxies are getting closer to each other and eventually will merge, I am assuming that predecessors of Andromeda and Milky Way galaxies, as well as other galaxies in the Virgo Supercluster, were always "neighbors" in the Universe and traveled in space in relative proximity to each other since they were formed after the Big Bang event - is this assumption correct?

What more could be said about the cause and the time of formation of such galaxy clusters (or clusters of galaxies) and galaxies mergers within them?

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