# How can we linearly extrapolate current expansion of universe to imply that it all started from a point?

We know that the universe is currently expanding. But how does that imply that it all started from an infinitesimally small point?

Its like given the stock price of Apple is increasing, it doesn't mean it was zero at IPO :)

• – Bruce Greetham Sep 21 '18 at 18:53
• Apple existed before its IPO. Whatever point one considers Apple to have come into existence, it is reasonable to treat its value as being zero immediately before that. – Acccumulation Sep 21 '18 at 19:11
• It's not true that the universe started from a point. – Ben Crowell Sep 21 '18 at 19:45

The glib answer is that we don't extrapolate back linearly. We extrapolate back using the Einstein field equations applied to a FRW metric.

The Einstein field equations are a bear to deal with in general. But in the case of the FRW metrics, they turn out to figuring out the behavior of a single function $$a(t)$$, which is called the scale factor of the Universe. This factor determines how the distances between distant galaxies change with time. For example, when the scale factor doubles, that means the distance between two distance objects (for example, the Milky Way and the quasar 3C 274) also doubles. If, at some time in the past, we had $$a = 0$$, then everything in the Universe would have been on top of each other; this time would correspond to the Big Bang.

The differential equation that the function $$a$$ must satisfy is $$\frac{\ddot{a}}{a} = - \frac{4 \pi G}{3} \left( \rho + \frac{3 p}{c^2} \right),$$ where $$\dot{a} = da/dt$$, $$\ddot{a} = d^2a/dt^2$$, $$\rho$$ is the average density of matter in the Universe, and $$p$$ is the average pressure. Since $$\dot{a}$$ tells us (roughly) about "how fast" the Universe is expanding, then $$\ddot{a}$$ tells us about the acceleration of the Universe.

If the Universe didn't actually have a Big Bang, then it can't have been expanding at its current rate arbitrarily far back in time. In other words, it must have been expanding less quickly at some point in the past. This means that at some point in the past, its expansion began accelerating, with $$\ddot{a} > 0$$. According to the equation above, this means that the matter in Universe at that point must have obeyed $$\rho + \frac{3p}{c^2} < 0.$$ But so far as we can tell, conventional matter and dark matter all obey $$\rho + 3p/c^2 \geq 0$$ instead. Dark energy (aka the cosmological constant) does have $$\rho + 3p/c^2 < 0$$; but if the dark energy had the same density back then as it does now, its density and pressure would have been swamped by the contributions from conventional matter (which was much denser and higher-pressure in the early Universe than it is now.)

This leads us to a contradiction: with the known sources of matter & energy in the Universe, and assuming that the dark energy density was the same throughout the history of the Universe, it's impossible for the expansion of the Universe to have accelerated in the early Universe. Since we see it expanding now, Universe must have had a singularity at some point in the past.

Now, maybe there are ways around this. Maybe the FRW metric is too simple to capture the details of the early Universe. Maybe the dark energy density was much higher (who knows why?) in the early Universe. Maybe the Einstein field equations need to be corrected when we get back to the early Universe. Serious people have looked at (and are looking at) each one of these options, and even wilder options to boot. But the problem is that there isn't much experimental evidence for any particular one of these fixes, and all of them go beyond well-established physical laws. So the debate goes on.

• Thanks for the thoughtful answer. Since we see it expanding now, Universe must have had a singularity at some point in the past. --> this is exactly the question btw. I contend that since its expanding now, it doesn't mean it started from size zero as proclaimed by Big Bang theory. Also if RHS in the eqn. is -ve then the scale factor will oscillate, a = exp(jwt). And if its +ve then a will be of the form a=exp(wt). In either case a is not zero at t = 0. – morpheus Sep 24 '18 at 5:00
• @morpheus: The factors of $\rho$ and $p$ are time-dependent in the above equation, so you can't necessarily say that the solutions to this equations are exponentials. Also, note that a solution of the form $a = \Re(e^{i \omega t})$ does go to zero at some time $t$, which is precisely the argument I was making: you need $\rho + 3 p/c^2 < 0$ at some point in the past to avoid a Big Band. – Michael Seifert Sep 24 '18 at 11:30

The big bang did not start from a point. At the moment of singularity, every particle's spacetime distance in the universe from everything else in the universe was 0.

This distance, and basically space itself is expanding in an accelerated way. Every point in space is getting further from another point, and this is true for every two points in space. That is why we cannot say that the big bang happened at some point in space. There is no such thing as the middle center of the universe where the big bang would have happened.

The big band happened at every point in space. That is why space is expanding everywhere in the observable universe. It is true, that the expansion is observable in the inter galactic voids of space, because gravity is not dominant there, but dark energy is.

The reason that we cannot directly see the expansion inside our galaxies or the solar system is because gravity is dominant in these regions and it is stronger then the expansion (the correct statement is that the effects of gravity are so that we cannot observe the expansion inside the solar system or the milky way).

• Not answer question? – Bruce Greetham Sep 21 '18 at 18:49
• Your response, besides not really addressing the question, argues against the idea that there is a point in the current universe at which the big bang occurred, which isn't really the same thing as whether, at the singularity, the universe was a single point. – Acccumulation Sep 21 '18 at 19:08

The trick that lets us deduce that the universe is expanding and that it had a hot dense beginning is the observation that the farther from us a light source is located the faster it is receding from us, and that this observation holds no matter in which direction we look into the sky.

This is a special sort of expansion and geometrically speaking, the easiest way to accomodate it into our world view is to conclude that the universe has been expanding for some time. This means the universe had a past in which it was smaller, and there you go.

A simple backwards extrapolation leads to the simple conclusion that there was a time when the universe began, and at that time all the matter we see in it today was piled together on top of itself into a point at infinitely high temperature and density.

But we know that the picture is not this simple because as our models of the process bring us back in time to that pointlike origin, those models become mathematically invalid, and presently we have no testable models that take over from there and lead us all the way back to that ultimate point.

However the model we do have (the hot big bang) not only takes almost all the way back but it furnishes a list of testable consequences which we can compare with that we see, and the match between the model and the observations is impressive.

• You are simply recapitulating the argument that the OP expressed a problem with: that the universe used to be smaller does not necessarily entail that its size was once zero. – Acccumulation Sep 21 '18 at 19:10
• In my answer I stated that our models fail before the universe got to zero volume. – niels nielsen Sep 21 '18 at 19:52
• That doesn't really address my objection. You state that the models conclude that the size was infinitely small from the fact that the size is increasing. Noting that this conclusion is problematic does not negate the fact that the reasoning, as presented, should have been rejected to begin with. Furthermore, the fallacy of "the size is currently increasing, therefore the size was once zero" is not cured by changing it to "the size is currently increasing, therefore the size was once really close to zero". – Acccumulation Sep 21 '18 at 20:00
• why not write up your own answer to the OP's question? – niels nielsen Sep 21 '18 at 20:13