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Scientists used to use following reasoning:

most galaxies are red-shifted $\implies$ there was a 'Big Bang'

Why this is being considered valid since not every galaxy has such property?

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  • $\begingroup$ Actually almost every galaxy does indeed have this property, out of billions of galaxies. You're just thinking of the exceptional cases of a handful of galaxies near us. (Those are just "bouncing around" with local motion.) $\endgroup$ – Fattie Oct 27 '16 at 13:39
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The evidence for expansion is that the redshift is proportional to distance.

The redshift of a galaxy can be divided into two components: that due to the cosmological expansion, which stretches the wavelength of light whilst it travels towards us; and a peculiar motion with respect to the cosmological expansion, which causes a straightforward doppler shift.

The former term increases with distance - this is known as Hubble's law. The redshift here is always positive (ie always a stretching of wavelength). It is this that tells us the universe is expanding. The latter term is caused by the gravitational effects of other nearby galaxies and clusters of galaxies on the galaxy in question. It is typically of order a few hundred km/s and can be positive or negative (ie it can cause a redshift or blueshift).

Redshift due to cosmological expansion completely dominates at distances greater than a few hundred million light years. Up till then (and Andromeda is only 2 million light years distant), peculiar motions can result in blueshifts for some galaxies.

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    $\begingroup$ With Hubble constant at 68km/sec Mpc, Andromeda is only receding via space expansion at 41.7 km/sec That leaves plenty of room for peculiar velocities to dominate. $\endgroup$ – Wayfaring Stranger Mar 26 '16 at 17:06
  • $\begingroup$ There is a wonderful video on YouTube about the expansion of space. It also mentions why Andromeda and our Galaxy are moving towards each other. youtube.com/watch?v=gzLM6ltw3l0 $\endgroup$ – RichS Apr 3 '16 at 18:20
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Scientists used to use following reasoning:
most galaxies are red-shifted $\implies$ there was a 'Big Bang'

That is an oversimplification, and a rather large one at that. There are multiple lines of evidence that all lead to the concept of the Big Bang, one of which is that galaxies tend to have a redshift that is strongly proportional to distance.

While the correlation between distance and redshift is very strong, it is not perfect. Nearby galaxies do not exhibit this behavior. The reason is simple. That redshift is proportional to distance means that nearby galaxies would have a very small redshift due to recession. (Ordinary) gravitation can easily overwhelm this tiny recessional velocity, and that is exactly what happens.

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