# Scale factor of the universe

Do we impose that the scale factor $a(t)$ of the Universe is a continuous function? Or there is a physical meaning?

Usually in physics we define functions to be continuous, such as the velocity of a point particle of mass m, but is this acceptable for all physical scales($10^{-10}$m to $10^{10}$m for example)?

• It isn't clear what you're asking. Of course the scale factor is continuous - the universe expands smoothly not in jumps. – John Rennie Feb 29 '16 at 17:01
• Why is smooth the expansion? – Saladino Feb 29 '16 at 18:33
• Saladino: may you precise what is "scale factor of the Universe", for you ? – Fabrice NEYRET Feb 29 '16 at 20:27
• The expansion is smooth because the model that is currently being used to describe it describes it as smooth and the model works. The question is no different from "Why does Newtonian mechanics use forces?". Same answer as always... because "it" works. I do understand that this may sound horribly trivial, but the basic assumptions of science happen to be horribly trivial: if something works to describe nature, then we can use it. – CuriousOne Feb 29 '16 at 20:49
• – Qmechanic Mar 1 '16 at 16:21

The Friedmann equations are differential equations: insofar they are a correct description of what happens expansion and contraction have to be continuous. Were $$a(t)$$ to discontinuously change value they would become ill-posed and not predict what the universe does next.

In fact, deep down General Relativity is based on the idea that space-time is a metric manifold $$(M,g)$$ where the metric is $$g$$ differentiable and the manifold $$M$$ locally always can be mapped to Minkowski space. Discontinuous jumps of $$a(t)$$ implies nondifferentiability of $$g$$, and were it to correspond to more than coordinate shenanigans then it would break the manifold topological structure.

Over what range this model is applicable remains to be seen, but at least large-scale it seems to be doing fine.

• Your answer sounded right until I realized it had me picturing an instant of existence in Minkowski space at the moment of local gravitational collapse, as on an object reaching the event horizon of an astrophysical black hole. I'd believed that the dilation of time might leave the passage into such a situation imperceptible, but wouldn't the locally sudden reversal of an expansion have to be described as a discontinuity at least temporal? Maybe you could say what I'm overlooking here....In spite of my criticism of oversimplifications by the OP, I see that I've seen it as a scale change. – Edouard Sep 9 '19 at 4:47
• Maybe the inseparability of space from time, and vice-versa, would maintain the continuity of spacetime: However, what I'd described in my previous comment seems more physical than an artifact (lengthened time, shortened space) of a coordinate system. – Edouard Sep 9 '19 at 5:28

The current form of the cosmological scale factor is continious function and can be precisely derived. See for example https://arxiv.org/abs/1109.2258

You may explore a connection to the Friedman equations in order to understand the significance of the scale factor as a function of time. Basically, it dictates the dynamic of the Universe, or (as some say) vice versa. The scale factor should be applicable to all distances for all scales while opinions may differ here, as the scale factor is not studied at the micro-scales.

• If you’re going to provide a formula, it should be correct. You left out the prefactor and another factor inside the sinh. Both are important. And it is worth mentioning that this is a (good) approximation which ignores the radiation in the universe, taking into account only dark energy and matter (both regular matter and dark matter). – G. Smith Sep 8 '19 at 23:12
• Thanks for your valuable comment, you are right it was a good approximation. I have removed the exact relation in order to do not extend the discussion on prefactor and another factor inside the $sinh$ though they are close to unity as you mentioned. – Eddward Sep 9 '19 at 7:59