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Is the reason the universe doesn't collapse into itself due to gravity because there is an infinite amount of bodies in infinite space, therefore there is an infinite amount of gravitational pull on an infinite amount of objects so it all balances out?

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No, the reason the universe doesn't collapse is because it's not dense enough. This can be seen from the Friedmann Equations, the main equations of cosmology. If you work through the derivation you'll find that there's a so-called critical density,

$\rho_c = \frac{3H^2}{8\pi G}$

If the universe's average density is above this, then it will collapse under its own gravity into a big crunch. Adding up all the stars, galaxies, etc that we can see gives a density about ~5% of this. Adding dark matter still gives only about ~23%. There isn't enough matter to cause the universe to collapse.

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    $\begingroup$ No mention of dark energy? Are you assuming it's zero? $\endgroup$ – D. Halsey Apr 6 '18 at 21:05
  • $\begingroup$ @D.Halsey yes - to quote the Wikipedia article, "An expression for the critical density is found by assuming Λ [aka. dark energy] to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, k, equal to zero." $\endgroup$ – Allure Apr 6 '18 at 21:06

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