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OK, I was thinking of the big bang and how it relates to an infinite universe.

If the universe is infinite is mass and there was a big bang, then at the big bang there was an infinite amount of matter in an infinitely small point.

If space expanded at a finite speed then it would still be a finite size, but an infinite amount of mass in a finite space would have infinite density.

as this is not the case, it means that space must have expanded infinitely fast, the universe would have gone from a point to infinite in size instantly.

This doesn't seem to make much sense to me!

For example, what about the CMB? Would it have to have infinite wavelength?

Any thoughts?

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    $\begingroup$ You are assuming the big bang occurred at some point in the universe, and everything is expanding from that. The model is instead that there was a singularity, and this singularity was 'everywhere' (there is no 'center' of the universe). Imagine the real number line. Now multiply it by a scaling constant. The number line doesn't become any less infinite when scaling it, but we approach a singularity as the scaling approaches zero. Regardless of the scaling factor, the number line looks equally 'dense' everywhere. It's not a perfect conceptual model, but gets some of the main points across. $\endgroup$ – John Sep 23 '14 at 5:09
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    $\begingroup$ The Big Bang didn't happen at a point; it happened everywhere. However you're correct that the FLRW metric predicts an infinite density at $t = 0$. I think the expansion rate is also infinite ($\dot{a}/a$ is certainly infinite) but I would have to check to be sure. Few physicists think this makes sense either, and we expect quantum effects to enforce a finite maximum density. The CMB didn't exist until 380,000 years after the Big Bang. $\endgroup$ – John Rennie Sep 23 '14 at 5:09
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/1915/2451 and links therein. $\endgroup$ – Qmechanic Dec 5 '14 at 18:50
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It is true that FRW metric, if redirected backwards in time, predicts a singularity. However, when the universe's size is comparable to the Planck scale, no one really knows what truly happens.

To date there is no successful and consistent theory of quantum gravity, although a lot of partially successful ones exist.

String theory, for instance has become a leading competitor in the race for QG.

As for the expansion rate, that is the subject of Inflationary Cosmology. The FRW metric, along with Einstein's Field Equations, yield equations of motion for the cosmos. $$ H^2=\frac{8\pi G \rho(t)}{3} $$

These yield different solutions depending on different mixtures of energy density constituents, if the universe is matter-dominated, an expansion rate is given, and if the universe is radiation-dominated a different one is yielded.

A truly exponential growth factor is yielded if the universe is "Dark Energy" dominated, meaning an energy density which is constant in time, regardless of the expansion of the universe. $$ H^2=\frac{8\pi G \rho_0}{3}\Rightarrow \frac{\dot{a}}{a}=\sqrt{\frac{8\pi G \rho_0}{3}} $$ Where $\rho_0$ is a constant. One solution for DE was vacuum energy, but that description has it's own set of problems. (say about 120 orders of magnitude of discrepancy...)

By the way, evidence suggests that the universe is right now expanding exponentially albeit very gradually.

Having said that, the universe might or might not be infinite per-say, it is enough that it spans a space that is larger than our event horizon.

There is also a finite amount of energy at every instant of time in the universe, even though, the existence of dark energy suggests that energy is added to the universe all the time.

A really nice book to "set you straight" is S. Dodelson's Introduction to Cosmology. You only need to read the first chapter, and MAYBE chapter 8 (I think) that deals with inflation.

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