History of speed of the expansion of the universe?

[SpaceBear]

I would like to know the history of the value of the speed of the expansion of the universe.
If today it's 74.3 ± 2.1 kilometres per second per megaparsec then how much it was a billion years ago, 5 billion years ago, 10 billion years ago, and how much it was in the first million years after the Big Bang?
Also when the universe was approximately 379,000 years old, when the photons started to travel freely through space.

 

[Simon Bridge]

That great! How were you intending to find out?

 

[GeorgeDishman]

In the absence of dark energy, the Hubble Constant would be inversely proportional to cosmic age. With dark energy, it initially falls that way but settles down to a constant value in the far future so the maths is more complex.

There is an applet called "CosmoCalc" in a number of varieties that tells you lots of interesting parameters. This one from Ned Wright tells you redshift from lookback time:

http://www.astro.ucla.edu/~wright/DlttCalc.html

This one takes redshift and tells you lots of stuff including the Hubble Constant

http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm

They need to use the same assumptions for current values so set H0 to 70.4 and Omega_M to 0.272 in Wright's. Put in say 5 for the light travel time in Gyr and press the "Flat" button. You get 0.492 for the redshift.

Now put that value in for redshift in the second applet and it will tell you everything you want to know (and more).

 

[marcus]

I would like to know the history of the value of the speed of the expansion of the universe.
...

The "speed" of the expansion of distances is proportional to the size of the distance. A distance twice as big increases at twice the "speed". Expansion rate is really not anyone particular "mph" or "km/s". It is a percentage growth rate of distance.

The current rate distances are growing is 1/140 percent per million years.

In the past the percentage growth rate was considerably larger. Here is a sample history:

$${\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&T_{Hub}(Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline10.000&0.100000&0.6&0.8&30.825&3.082&4.663&1.585\\ \hline9.000&0.111111&0.7&1.0&29.922&3.325&5.081&1.861\\ \hline8.000&0.125000&0.8&1.2&28.856&3.607&5.583&2.227\\ \hline7.000&0.142857&0.9&1.4&27.570&3.939&6.197&2.729\\ \hline6.000&0.166667&1.2&1.8&25.976&4.329&6.964&3.449\\ \hline5.000&0.200000&1.6&2.3&23.932&4.786&7.948&4.548\\ \hline4.000&0.250000&2.2&3.2&21.181&5.295&9.247&6.373\\ \hline3.000&0.333333&3.3&4.9&17.215&5.738&11.008&9.819\\ \hline2.000&0.500000&5.9&8.1&10.915&5.458&13.361&17.878\\ \hline1.000&1.000000&13.8&14.0&0.000&0.000&15.793&46.686\\ \hline\end{array}}$$

Past epochs are labeled by how much distances and wavelengths have been elongated since that time (the "stretch" factor). The table goes from S=10 to S=1 (the present moment).

You can read off the percentage growth rates from the "Hubble time" column (the fourth column).
In that column 14.0 Gy corresponds to the present rate of 1/140 percent per million years.
And 0.8 Gy corresponds to the rate of 1/8 percent per million years.
That was the rate back in year 600 million (i.e. in year 0.6 Gy) as you can see from the table.

The same calculator will easily tell you the distance growth rate at S=1090, the moment of clearing or transparency that you asked about. Around year 380,000 when the ancient CMB light originated. That light has been "stretched" by a factor of 1090 so you just have to put that number in the upper limit box, instead the number 10, which I put into make this table.

If instead of a percent growth rate, what you want is a km/s speed of some benchmark distance, I would suggest a million lightyears. Most people have some mental association with that distance---having heard the distance to a neighbor galaxy like Andromeda expressed in those terms. It is on the order of a million lightyears from us and its light takes on the order of a million years to get here.

When distances are growing at rate of 1/140% per million years, then a distance of 1 million lightyears is growing at a "speed" of 3000/140 km/s.
All you have to do, to convert, is multiply the percent rate 1/140 by 3000. That gives the km/s.

 

[marcus]

The same calculator will tell you that back at the time CMB originated distances were increasing by 157% per million years.

So if you want a "speed" of some sample benchmark distance, say 1 million lightyears. Just multiply that by 3000
3000 x 157 is some number of kilometers per second.

I would rather think of that in the equivalent form of 157% of the speed of light. Kilometers per second don't seem too meaningful in that range, an awkward unit to use.
But "speed" is an awkward way to visualize distance expansion in any case. What we are talking about is geometry change, not things traveling from one place to another. Nobody gets anywhere by expansion, all stationary observers just become farther apart.
And the expansion is proportional to the size of distance---you shouldn't have to pick a benchmark of a million this or that, you just make unnecessary work for yourself by doing that.

For convenience I keep the calculator link in my signature (TabCosmo7). Anyone who wants coaching in its use should ask. Several people here can help, and it's basically real easy to use.