# Which: scale or years? as parameter of change

## Which is most intuitive/convenient for tracking cosmic history?

26.7%

60.0%

13.3%
1. May 7, 2013

### marcus

The first stars formed when distances were 9% of their present size.
The solar system began forming when distances were 70% of what they are now.
Andromeda galaxy will make a first pass through Milkyway when they're 129% of present.

Which do you find more meaningful and intuitive for tracking the evolution of the universe, percentages like those or numbers of years?

Universe history (say from recombination onwards) is often laid out using years to parametrize event, but in some of Lineweaver's diagrams the scale factor (denoted "a") is used as well. If you have any thoughts about this: reasons to prefer one or the other, I'd be interested to learn what you think--and which parametrization is more intuitive for you.

If you have some other parameter that works better please post it, and discuss. We are not talking about the first few milliseconds of expansion, or the first three minutes, here, but about history after recombination and the emission of the ancient CMB light.

2. May 7, 2013

### marcus

Here's an example to aid comparison:
With help from Jorrie, I've made a table of history from FIRST PROTO-GALAXIES (when distances were 9% what they are today) to the first pass-thru in OUR COLLISION WITH ANDROMEDA (when distances will be 129% what they are today).

IOW I told the calculator to run from scalefactor 0.09 to 1.29, and chose to do it in 22 steps:

$${\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$ $${\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.09&0.466&0.700&31.525&2.837&4.321&2.19&4.05\\ \hline 0.10&0.558&0.838&30.560&3.102&4.776&2.12&3.70\\ \hline 0.11&0.668&1.004&29.536&3.382&5.270&2.05&3.37\\ \hline 0.13&0.801&1.202&28.448&3.674&5.804&1.98&3.06\\ \hline 0.15&0.959&1.438&27.294&3.976&6.378&1.90&2.76\\ \hline 0.16&1.149&1.721&26.069&4.284&6.993&1.81&2.49\\ \hline 0.19&1.376&2.057&24.771&4.591&7.647&1.72&2.23\\ \hline 0.21&1.647&2.457&23.396&4.891&8.338&1.62&1.99\\ \hline 0.24&1.971&2.931&21.940&5.173&9.061&1.52&1.77\\ \hline 0.27&2.356&3.489&20.402&5.426&9.811&1.42&1.56\\ \hline 0.30&2.815&4.144&18.781&5.634&10.580&1.30&1.36\\ \hline 0.34&3.358&4.903&17.077&5.779&11.357&1.19&1.18\\ \hline 0.38&4.000&5.774&15.293&5.837&12.130&1.06&1.01\\ \hline 0.43&4.753&6.755&13.437&5.785&12.883&0.93&0.86\\ \hline 0.49&5.630&7.837&11.520&5.594&13.600&0.80&0.71\\ \hline 0.55&6.643&8.995&9.558&5.235&14.265&0.66&0.58\\ \hline 0.62&7.798&10.193&7.573&4.679&14.865&0.53&0.46\\ \hline 0.70&9.097&11.382&5.594&3.898&15.387&0.39&0.34\\ \hline 0.79&10.536&12.508&3.651&2.869&15.828&0.25&0.23\\ \hline 0.89&12.105&13.525&1.772&1.571&16.188&0.12&0.12\\ \hline 1.00&13.787&14.400&0.000&0.000&16.472&0.00&0.00\\ \hline 1.13&15.566&15.121&1.690&1.906&16.691&0.12&0.13\\ \hline 1.21&16.592&15.458&2.570&3.100&16.788&0.18&0.20\\ \hline 1.29&17.640&15.750&3.410&4.398&16.870&0.24&0.28\\ \hline \end{array}}$$

You can see various events along the way. 9% size turns out to correspond to year 466 million.
The Milkyway disk structure formed around scale 43%, which you can see is year 4.8 billion
Solar system started forming at scale 70%, which the table shows is around year 9 billion., with the collapse of part of a giant molecular cloud.
The present day of course is at scale 100%, year 13.787 billion.
And the first pass of our collision with Andromeda galaxy will happen at scale 129% in year 17.6 billion.

3. May 7, 2013

### marcus

I'm still balancing the pros and cons, but I think on the whole I prefer the scale factor. It is directly READABLE from the light coming in from some object or event. If the wavelengths in the light are stretched out by a factor of two then the event occurred when distances were 50% of present. In that sense the process which is the universe can serve as its own "clock". It TELLS you what the scalefactor was when something happened.

Also there's a kind of rough order-of-magnitude proportionality with the year count over much of history, as you can see in the table:
43% and 4.8 billion
70% and 9 billion
100% and 13.8 billion
129% and 17.6 billion
If someone tells you the scale factor of an era, you can estimate the years (e.g loosely speaking, multiply by 10 and call it billions of years). That seems to work at least for any time in the past 10 billion years or, let's say, any era since scalefactor 30%.

4. May 7, 2013

### marcus

Another reason I find scalefactor "timing" convenient is because I can visualize it.
I can PICTURE when distances were 10%, or 30%, or 90% of what they are today.

You can see from the table that these scalefactors correspond to yearcounts
558 million
2.8 billion
12 billion
When I hear numbers of years like these I do not immediately get as clear a visual image as I do with the distance scales of 0.1, or 0.3, or 0.9. Time, as a quantity, is more nebulous. That could simply be a personal bias, so I'm interested to learn how the choice looks to other people. You may wish to set it out in different terms, using different examples.

BTW the collision trajectory with Andromeda was finally determined fairly precisely in a study published last year. Folks might like the link:
http://arxiv.org/abs/1205.6865
An estimate of when the solar system started forming is given here:
http://en.wikipedia.org/wiki/Formation_and_evolution_of_the_Solar_System
An article in Nature describing the formation of the thin disk spiral structure of Milkyway:
http://www.nature.com/news/galaxy-formation-the-new-milky-way-1.11517
An estimate of when the Milkyway thin disk formed:
http://en.wikipedia.org/wiki/Milky_Way#Age
http://arxiv.org/abs/astro-ph/0506458

Last edited: May 7, 2013
5. May 7, 2013

### Mordred

vote entered lol

6. May 7, 2013

### marcus

Thanks! I'm glad to get your opinion. Five voters already! which is reassuring since I realized afterwards that the thread-title was not worded all that well and people might not catch on to what it was about.

7. May 7, 2013

### Naty1

I voted for 'year' because of 'convenience' relative to other sources. Universe history seems to conventionally use years , for example a common style depiction is in the illustration here:

http://en.wikipedia.org/wiki/History_of_Universe

I suspect the 'year' parameter is popular because it's one in everyday use. Even I basically
know what they are!!

that's not quite as intuitive to me as years.....should I start worrying yet?? Probably not as its about 14B years in the future....
Will our sun last that long?? I think I will worry about that first!!

8. May 7, 2013

### MalcolmB

Vote duly entered!

I voted for scale. I'm fairly new to cosmology although I've been a visual and photographic astronomer for some time, and I find it easier to visualise percentages rather than age. I also like Marcus's take on the Universe serving as its own clock.

Just my 2P!

9. May 7, 2013

### Chronos

Pardon me for being eccentric, but, I like z. It's a directly measured quantity and you just add 1 to get the scale factor.

10. May 7, 2013

### Naty1

good point..if we were starting from a common unit among a variety of sources, nothing simpler than percentages....

As soon as one gets some cosmological knowledge, hard to ignore the appeal of z....
z was my first insight into what was meant by the 'scale factor' and because it's
'measured' [really calculated] its lends some 'reality' to otherwise difficult cosmological concepts.

11. May 7, 2013

### Mordred

This second part is one of the reasons why I like Jorrie's latest revisions. With the diversity of familiarity of units of measure. Flexibility is a key essential component in a calculator designed to provide educational concepts.

Some of the rows Jorrie added simply to increase that flexibility.

One notable example is stretch, z and scale factor.

At first glance I largely ignored the calculator. However upon further study and Marcus numerous and steady stream of examples. It didn't take long to appreciate its versatility.

Particularly since its true power lies in the ability to copy the resultant tables to excel. From there if your familiar with excel you can convert or graph the results.

12. May 7, 2013

### marcus

Thanks Malcolm! About percentages, I do too and there's a simple trick for gauging how far in future an event like meeting Andromeda is. The scale factor tag is 129% and that means essentially that how far in future it is is 29% of the present age of the universe. Or say 30% for round numbers.

Because currently and for a fair way into future the scale increase is proportional to time. 30% of 13.8 billion years is something you can do a quick approximation in your head---it's about 4 billion.
Naty was saying he liked years because he knew he didn't have to worry, it was so far in future. But you get about the same psychological sense by saying 30% of the age of the universe: in other words "a long time". It amounts using the age of universe as a unit of reference.

So I prefer scalefactor "time" as you do (universe its own clock ) but in many cases I guess if needed I could make a rough mental conversion to ordinary time using present age of universe as reference.

==============
Naty, your post may have a typo. You say we meet up with Andromeda "14 B years in the future".
I agree it's a long time from now but more like 4 B , not 14.
===============

Chronos, your name means time so you should be an authority here One problem with using redshift z to index events is what do you do about the FUTURE? What redshift do you associate with the Andromeda moment we talked about: scale factor 1.29?

Your z index decreases as time goes--goes to zero at the present and then presumably goes negative. Not so intuitive. By contrast the scale factor just keeps one increasing, like ordinary time does.

Last edited: May 7, 2013
13. May 7, 2013

### marcus

Be careful. When you add one to z you don't get the scale factor. You get one over the scale factor.
z+1 = 1/a
a, the scale factor in the Friedman metric, the thing used to define Hubble rate H = a'/a,
is a quantity that INCREASES over time.
z and z+1 are quantities that DECREASE.
It's so easy to mis-speak even if you are old hands and understand all this perfectly, but it can confuse or mislead newcomers.

Of course the scale factor a is also DIRECTLY MEASURED from an object's light, but it has the additional advantage that it is monotonically increasing (as we expect a "timely" measure to be)

If the received wavelengths in an object's light are TWICE what they were when emitted, then
the scalefactor a was 0.5 when the object emitted the light. Simple points but perhaps should be made.

14. May 7, 2013

### Chronos

Apologies, I agree 1/(z+1) is the definition of a. But, I think that should not confuse anyone with even a passing interest in cosmology. All our current distance measures are critically based on z, so even minor changes are highly significant. z=1 takes us back to when the universe was about 6 billion years old. Even Hubble has difficulty resolving objects that ancient.

15. May 8, 2013

### julcab12

Scale factor gives a better picture and perspective. Years better story. I guess i'm stuck with geometry. lol

16. May 8, 2013

### Naty1

oops. Right you are!!

17. May 8, 2013

### Chalnoth

I say "other" because it all depends upon what is being conveyed. Sometimes time is useful. For instance, I think time is useful for understanding the "coincidence problem" in that plotting cosmic history as a function of time makes it disappear. But because different cosmic events occurred at radically different rates at different times, attempting to show a plot with inflation, BBN, recombination, and reionization all on one plot with time as the horizontal axis would be hopeless. Using the scale factor or the logarithm of the scale factor would probably be better.

18. May 8, 2013

### Mordred

that's a good point I hadn't considered that when I voted. I'll let my vote stand though lol

19. May 8, 2013

### phinds

I find the % useful in some circumstances but I'm used to thinking in years and prefer that.

20. May 8, 2013

### marcus

Again thanks to everyone who answered! and especially those who gave reasons for different choices. It certainly makes sense to have a menu of several independent variables to use depending on the application, though that doesn't preclude having an overall "default" index that you normally imagine universe history in terms of.

Jorrie posted some charts that are very nice in the other thread earlier today. I'm going to try and duplicate one as an example. The red curve shows the shape of the past (plum-shaped) lightcone and the future (funnel-shaped) lightcone. The present (of course ) is where the stem of the plum joins the neck of the funnel.

The space between the Hubble radius (blue curve) and the event Horizon (gold curve) is the margin where the universe allows sources receding faster than light to send messages that will eventually reach us.

Since in this case scalefactor is serving as "time", the present comes where it equals 1.

The point where the the red curve past lightcone intersects the Hubble radius (blue) shows where a galaxy emitted light that initially "stood still" making no progress in our direction because its forward motion thru space was exactly canceled by the rate the distance to target was increasing. I guess everybody here is familiar with that observation but mention it just in case it might be a new reflection for someone. This happens right at scalefactor 1/2.6 whatever that is, probably just slightly less than 0.4.

Yeah, 38.5%, just a bit under 40%. That is the unique epoch in history when galaxies on our past lightcone were sending us light that "didn't get anywhere" at first. And it is where the past lightcone has its largest radius, namely properdistance 5.8 billion lightyears. So nice to see these familiar facts depicted on a chart!

Last edited: May 8, 2013