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I have a question that I was not able to find (and think of) an intuitive answer.

It is well known that density perturbations in the early universe grows with time such that some eventually form the non-linear objects that are stars and galaxies. One can follow a spherical region with enclosed mass M and determine the time and size at which it eventually decouples from the Hubble flow. Solving the equation in parametric form allows us to determine that the average density inside that sphere which now starts to collapse is 5.5 times bigger than the background density (given by the critical density of the universe).

Now this ends up being true with no dependence on the original mass M enclosed in the sphere. What I don't understand is: if I choose an initial M to be such that the average density in the sphere is bigger than 5.5 times the critical density, by the time its expanding outermost shell turnaround the sphere would enclose an average density bigger than what is predicted. Where is the flaw in my reasoning? And why is the turnaround over-density contrast independent of the initial enclosed mass and density?

Thank you

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# Density contrast inside a region at turnaround.

Can you offer guidance or do you also need help?

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