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Am I right in understanding that current theory states that Big Bang originated from a single point of singularity?

If so, would this mean that this was a uniform point?

If so, as the universe expanded, what factors contributed to transforming its expansion in a non-uniform manner, to the point of what we can now observe? E.g., non uniform densities of matter and non-uniform cosmic microwave background noise, the latter we are told being the imprint of the very early universe.

I'm making assumptions here but I would be interested in understanding how, if it started uniformly, what made the universe become diverse.

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This is just an expansion of user1929959's answer, which is correct.

Firstly, the Big Bang didn't happen at point. This is a common misconception. I won't pursue this point any further because there are loads of answers about it on the site. See my answer to What's the evidence supporting 1 singular Big Bang? or search the site for "FLRW metric". Alternatively I gave a more "popular science" answer on the SciFi Stack Exchange.

Anyhow, re the uniformity: we currently believe that the early universe went through a period of rapid expansion called inflation. The rapid expansion hid any features of the universe that predated it, so we can only see what happened from the moment inflation ended. With any quantum system the uncertainty principle means the energy of the system will be slightly uncertain, and in the case of inflation it means inflation ended at very slightly different times in different places. This caused differences in the energy density of the universe immediately after inflation ended, and these are the primordial quantum fluctuations that user1929959 mentioned.

The fluctuations meant some regions of the universe had a slightly greater gravitational attraction that others, and the regions of greater gravitational attration increased in density while the other regions decreased in density. This caused the differences in temperature that we see in the cosmic microwave background. It's possible to use the theory of inflation to calculate what the variations in the CMB should be, and experimental data matches the calculations very well.

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There are various theories about how the Big Bang occurred and also there are many theories about what happened before the Big Bang. Primordial structure in universe are due to primordial quantum fluctuations and inflation primordial quantum fluctuations and inflation.

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