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If predictions are correct, the Milky Way and Andromeda are set to collide in around 4 billion years. If, when this occurs, a Quasar is formed by matter being accreted to the common galactic center (perhaps with the two supermassive black holes forming a binary), what would be the outcome for life in the new galaxy?

It's known that galactic collisions were much more common in the early universe; most Quasars are observed at high redshifts. It isn't clear if there are other conditions necessary to produce a quasar other than just the galactic collision itself, so obviously this is just speculation, but it is clear that Quasars are some of the most energetic systems in the Universe; if one were ignited so close to home, would every trace of life in the galaxy go out in a blaze of glory?

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Quasars have very pointed jets perpendicular to the rotational axis of the accretion disc around the central black hole. As long as this jet doesn't happen to point into the galactic plane, the cosmic radiation of these jets won't be a too serious problem.

But a collision of two galaxies can become an eventual problem in a different way. A merger of galaxies triggers star formation and sweeps most of the galaxy free of interstellar dust and gas. So, after a short (hundreds of millions of years) period of time, we'll get lots of new stars. But they'll eventually exceed their main sequence lifetime. Due to lack of interstellar matter only very few new stars will form afterwards. That's a difficult settings for formation of planets and life.

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