# Where can I find source data for the graph of metric expansion over time, as shown in this ESA web page?

This web page from ESA/Hubble, shows the size of the universe over time. (I know that "size of the universe" is a gross simplification and open to multiple definitions and ambiguity, but this kind of graph is roughly what I'm after).

But no source data is cited. This is the closest I can find to what I'm after - where can I find source data to match this graph?

I was hoping a physicist might answer this, as I'm just an enthusiastic armchair scientist whose undergrad calculus was 30 years ago. But as no one else has had a go, let me try an answer.

The graph in the web page is a simplistic and somewhat exaggerated representation of the evolution of the cosmic scale factor a(t) over time. You can derive the "source data" yourself using the following metrics for the three distinct periods in the expansion of the Universe, since the only variable is t which in this case is the age of the Universe. The scale factor a(t) is dimensionless.

For the earliest period of the Universe up to about 47,000 years after the Big Bang, the Universe was dominated by radiation. The evolution of the scale factor in the FLRW metric for this period is given by: $$a(t) \propto \sqrt t$$

After that very brief time (in cosmological terms), matter took over as the dominant factor in the Universe for the next 9.8 billion years, and the evolution of the scale factor is given by: $$a(t) \propto t^{2/3}$$

Finally, the Universe in the current period is dominated by dark energy, and the evolution of the scale factor is given by: $$a(t) \propto exp (Ht)$$ where H is the Hubble constant.

In practical terms, if you'd like to generate a graph similar to that in the web page, you can set $t_1$ = 13.79 Gyr (i.e. the current age of the Universe). I found that the first era of radiation domination is so short that it's not worth including in the graph.

Here's my simplistic rendition:

• In this working, is the initial expansion at 10^-32 sec be to scale, and do the above factors include that initial expansion already? Or do those equations only govern expansion theory after say, a second or so, of cosmic time? That's a big concern for me, and I can't find a clear answer. And is a graph of the cosmic scale factor going to be the same (or proportionate to) a graph of the size of the universe over time (observable or entire, I'm not sure myself which it would be!), since that's what I'm actually after. – Stilez Jun 8 '18 at 16:08
• The hypothesised "cosmic inflation" is not included. The scale of that expansion is so extreme, and occurs in such a minuscule timeframe, that I think it's impossible to realistically represent cosmic inflation in the same chart as the scale factor. As I've noted in my answer, it's difficult to even represent the radiation-dominated era due to its relative shortness (47,000 years, or just 0.0003% of the age of the Universe), even using a logarithmic x axis. – Chappo Jun 9 '18 at 23:43
• Regarding the relationship between the cosmic scale factor and the "size" of the Universe, you note yourself that "size" is a simplification. I think (but stand to be corrected) that my graph of scale factor over time broadly represents "size", if by that you mean the distance between us and a given point (say, the CMB shell, i.e. the edge of the visible Universe). So, the distance to a given object at t=0.5 (i.e at half the Universe's current age, 6.9 Gyr) was about 0.63 of its current distance. – Chappo Jun 10 '18 at 1:10
• If you want a different visualisation of the expansion of the Universe, see this commonly used image, but note it is completely inaccurate regarding the inflationary epoch, which should not appear at all since it happened in a timescale too small to represent on the chart. You'll note that the slight curve to the top and bottom outlines of the Universe corresponds to the same curve in my graph and in the (more exaggerated) graph in the web page you cited. – Chappo Jun 10 '18 at 1:19
• Final comment: your question asked for the "source data" for the graph you cited, but as I've noted, such a simplistic graph only relies on one set of data, the age of the Universe. What you then need is the relevant metric to convert that data into the scale factor, the "scale" being the "size" of a unit measurement at that time. I've done my best to show this, but my chart isn't exact, as it needs a more sophisticated method of merging the two metrics. – Chappo Jun 10 '18 at 1:23