# Spatial Expansion -- Coincidence?

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• stuart100
In summary, the Hubble constant is a measure of the expansion rate of the universe and is currently estimated to be around 70 km/sec/Mpc. This means that it would take approximately 14 billion years to cover a distance of 1 megaparsec at this rate. However, this is only an approximation and is not a coincidence. The actual age of the universe would be different if the recession velocities of objects in the past were different. The Hubble time, which is similar to the age of the universe, is also not a coincidence and is only approximately equal to the age of the universe at this moment in time. In the future, these two values will diverge and will no longer be considered a coincidence. f

#### stuart100

I was trying to follow how Hubble constant supposedly works. If it is about 70 km/sec/Mpc, then
the the expansion at 13.8 Bly distance is about the speed of light. That, if I got it right, seems too
convenient don't you think. I know it's not the same start to finish. Thanks if you can correct my thought.

It's not a coincidence, but also it is.

Here's why it is not. The meaning of the Hubble constant is that just as you need approx. 14 billion years to cover one megaparsec at approx. 70 km/s, you also need that much time to cover a proportionally larger distance at a proportionally larger speed. If you set the speed to c, you end up with c*14 By = 14 Bly.
That is to say, if everything were always receding at the velocities given today by the Hubble's law, it'd take the reciprocal of today's Hubble constant for all distances in the universe to shrink to zero.

But the recession velocities were not constant throughout the history of the universe. If, for example, we'd have the recession velocities be higher in the past, then the actual age of the universe would be lower than what we get from the Hubble's law today (i.e. you don't need 14 By to cover 1 Mpc if for half the journey you traveled at e.g. 140 km/s and slowed down to 70 km/s only later). If, on the other hand, the recession velocities were lower in the past, then the age of the universe thus calculated would be higher.

Here's where the coincidence comes in: it's a coincidence that we live at a time in the history of the universe when the periods of deceleration and acceleration took about the same amount of time, with about the same but opposite nett effect, and the resultant expansion curve happens to be - to an o.k. ballpark - approximated by a line.
Yet, it is only a ballpark figure. Today, what you get from the detailed model differs by a good few hundred million years from this approximation. Some aliens living some couple billion years in the future will find the approximation exactly right (and they'll think it a massive coincidence, I'm sure).
But all the aliens living afterwards will find this approximation ever more inaccurate, as the Hubble parameter asymptotically approaches a constant value - and so, does too its reciprocal - while the time on the universe's clocks keeps ticking for all eternity. The two values will keep diverging.
An alien 10 billion years from now will find the reciprocal of their Hubble constant show approx 17 billion years. Another one living 100 billion years later will calculate almost the same value. Nobody will ponder a coincidence then.

stuart100, vanhees71, PeroK and 2 others
That, if I got it right, seems too
convenient don't you think.
This turns out to be equivalent to saying that the Hubble time is the same as the age of the universe. It's a coincidence. Due to the way the Hubble constant evolves over time it wasn't true in the past and it won't be true in the future, and it's only approximately true now (it is exactly true at only one instant).

Edit: @Bandersnatch beat me to it, I see.

stuart100 and vanhees71
Just to pile on...

The Hubble time isn't the age of the universe. They are maybe 5% different. Dimensional analysis suggests the two times be "close", but close might mean "the same", "within a factor of two" or, as it happens today "within 5 or 6 percent".

stuart100 and vanhees71
It's not a coincidence, but also it is.

Here's why it is not. The meaning of the Hubble constant is that just as you need approx. 14 billion years to cover one megaparsec at approx. 70 km/s, you also need that much time to cover a proportionally larger distance at a proportionally larger speed. If you set the speed to c, you end up with c*14 By = 14 Bly.
That is to say, if everything were always receding at the velocities given today by the Hubble's law, it'd take the reciprocal of today's Hubble constant for all distances in the universe to shrink to zero.

But the recession velocities were not constant throughout the history of the universe. If, for example, we'd have the recession velocities be higher in the past, then the actual age of the universe would be lower than what we get from the Hubble's law today (i.e. you don't need 14 By to cover 1 Mpc if for half the journey you traveled at e.g. 140 km/s and slowed down to 70 km/s only later). If, on the other hand, the recession velocities were lower in the past, then the age of the universe thus calculated would be higher.

Here's where the coincidence comes in: it's a coincidence that we live at a time in the history of the universe when the periods of deceleration and acceleration took about the same amount of time, with about the same but opposite nett effect, and the resultant expansion curve happens to be - to an o.k. ballpark - approximated by a line.
Yet, it is only a ballpark figure. Today, what you get from the detailed model differs by a good few hundred million years from this approximation. Some aliens living some couple billion years in the future will find the approximation exactly right (and they'll think it a massive coincidence, I'm sure).
But all the aliens living afterwards will find this approximation ever more inaccurate, as the Hubble parameter asymptotically approaches a constant value - and so, does too its reciprocal - while the time on the universe's clocks keeps ticking for all eternity. The two values will keep diverging.
An alien 10 billion years from now will find the reciprocal of their Hubble constant show approx 17 billion years. Another one living 100 billion years later will calculate almost the same value. Nobody will ponder a coincidence then.
Thanks for clarification. My main aim was to see if I could work thru the arithmetic.

Just to pile on...

The Hubble time isn't the age of the universe. They are maybe 5% different. Dimensional analysis suggests the two times be "close", but close might mean "the same", "within a factor of two" or, as it happens today "within 5 or 6 percent".
Yes, some numbers do line up occassionally, like slot machines( I avoid them). I am suspicious of too much good luck also.

I am suspicious of too much good luck also.
Lots of unrelated numbers are close to each other. The sun and the moon's apparent diamater. What causes that? Why is $\pi \approx \sqrt{10}$?

stuart100 and vanhees71