# How would you teach quantitative cosmology? What examples to use?

1. May 17, 2013

### marcus

By quantitative cosmology I mean with real times, distances, expansion rates, horizons, CMB stationary observers etc , derived from the standard model fitted to data. I don't mean ideas about conditions shortly before or after the start of expansion, although that is very interesting too.

The question has been on my mind: how would you approach teaching that? Say you were tutoring some interested person. I'd like to hear other people's ideas. Probably the scale factor is the most important thing to get one's mind around.

There would be two main levels to choose from: with and without the Friedman equation. I want to focus on the WITHOUT case. To deal briefly with the other case: if the person you are tutoring were good with simple differential equations and you went WITH the Friedman then it seems fairly straightforward. The conceptual structure might be like this:
the stretch in light we can observe: S = 1+z = anow/athen.
the distance growth rate H = a'/a, obviously a reciprocal time or percentage growth rate.
the (energy equivalent) matter density ρ comprising ordinary matter&radiation plus dark matter
the spatial flat case of the Friedman: H2 - Λ/3 = [const] ρ
where the LHS is reciprocal time and the [const] converts energy density into reciprocal time.

The without-Friedman case seems like it's a lot more challenging: how do you wean learners away from purely verbal thinking and accustom them to quantities? You have to use quantitative EXAMPLES: get the learner used to seeing numbers and imagining the growth process in real terms rather than merely verbally. What sorts of examples would it be good to work through?

Last edited: May 17, 2013
2. May 17, 2013

### marcus

One example I like, and I don't know if it would fit in--perhaps later rather than sooner--is 'catching up to a galaxy' https://www.physicsforums.com/showthread.php?t=689610
You take a two-row table (Supper=1, Slower=0.01)

$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&1.13&94.38\\ \hline \end{array}}$$

You point out that the event horizon (a kind of communication limit) is NOW around 16.47
so any nearer galaxy can be reached with a flash of light we send today. But when will it arrive?
Let's pick a target that is now 16.33 from us (that is less than 16.47 so it should be reachable).
Already today the target is receding faster than light, but our message can still get there---though not until year 92.3 billion

Last edited: May 17, 2013
3. May 17, 2013

### Staff: Mentor

4. May 17, 2013

### Spourk

This is awesome. I also love at 109 meters they show the size of a Minecraft world, which is roughly the size of Neptune.

Now, if you can do the same thing with time/future/past/scale from the BB to Heat Death...

5. May 17, 2013

### marcus

Thanks! The universe is a process that occurs in TIME and that's the main thing one wants to be able to visualize, for cosmology, I think. The powers-of-ten movie is wonderful and Ive seen some other great scale things (that did not have temporal change, treated the world as frozen). A lot of the distances that we're concerned with are increasing faster than light. You can see this from the brief tables, sometimes important ones grow 100s of times faster than light, and this has a power of tens angle, I guess. Any ideas on this?

Here's another example that a tutor might work thru with learners---it's very simple and basic:
$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.00092&1090.000&0.000373&45.33&0.042&0.057&0.001&3.15&66.18\\ \hline 1.00000&1.000&13.787206&0.00&0.000&16.472&46.279&0.00&0.00\\ \hline \end{array}}$$

I'm trying to imagine what a tutor might say: Notice that today our particle horizon is 46.28 which is the farthest matter is NOW which could have sent us light we'd be receiving today. It is the maximum distance from home a flash of light could have reached in the whole time since start of expansion (through its own efforts aided by expansion).
But notice that the matter that emitted CMB (the most ancient light we actually do see today) is only at 45.33. That's because light emitted earlier by more distant matter DID NOT GET THRU because the partially ionized gas was effectively opaque. That opacity caused the difference between 45.33 and 46.28.

Get familiar with the fact that the wavelengths of the ancient light are by now expanded 1090-fold, and the moment the gas became transparent was year 373,000, when distances were 1/1090 of their present-day size.

Also from the table you see that in year 373,000 our cosmic event horizon was 57 Mly, anything farther could not reach us with its light. But the matter that emitted the CMB we're seeing now was even closer: only 42 million lightyears. It was receding at 66 times the speed of light back then, but the light it emitted did, in fact, eventually reach us. We are receiving it today.

Last edited: May 17, 2013
6. May 17, 2013

### marcus

Pear-shaped past lightcone

Another thing if you were tutoring cosmology would be to work thru the fact of the past lightcone being pear-shaped. The past lightcone is the location in space and time of every source we are getting light from right now. The gold curve is the lightcone's radius. You could think of it as the history of a flash of light aimed towards us that is at first swept back by expansion (its distance from us increasing) and then is briefly at a constant distance when its progress is exactly canceled by expansion (the widest part of the pear), and then begins to make progress towards us.

The gold curve is also an outline of the FUTURE or "forward" light cone. The times and places we could contact if we sent a flash of light today. The gold curve, radius of the past and future lightcones, is a plot of the Dthen column of the table. To repeat, the pear-shaped past part gives the distance of the source when it emitted the light we're getting now, and also the distance of a flash of light that started towards us long ago in the past and only just got here today (having being swept back at first).

The blue curve of the Hubble radius is to show it intersects with the past lightcone at the latter's maximum. Hubble radius plots the size of distances which, at the given time, are expanding at speed c. The widest part of the pear is where the light's forward progress is exactly canceled by expansion and it hangs suspended at a constant distance from its target. This occurred around the year 4 billion.

Here's the corresponding table.

$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly) \\ \hline 0.067&0.10&38.87&0.97&1.38\\ \hline 0.112&0.17&37.38&1.31&1.88\\ \hline 0.186&0.28&35.61&1.74&2.55\\ \hline 0.308&0.46&33.52&2.29&3.42\\ \hline 0.510&0.77&31.05&2.97&4.54\\ \hline 0.844&1.27&28.12&3.76&5.96\\ \hline 1.394&2.08&24.67&4.61&7.70\\ \hline 2.297&3.40&20.62&5.39&9.70\\ \hline 3.759&5.45&15.94&5.83&11.86\\ \hline 6.048&8.33&10.68&5.46&13.89\\ \hline 9.395&11.63&5.17&3.70&15.49\\ \hline 13.787&14.40&0.00&0.00&16.47\\ \hline 18.924&16.05&4.37&6.10&16.95\\ \hline 22.085&16.57&6.42&10.90&17.09\\ \hline 25.326&16.88&8.16&16.80&17.16\\ \hline 28.613&17.06&9.61&24.01&17.20\\ \hline \end{array}}$$

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7. May 17, 2013

### cepheid

Staff Emeritus
Addressing a small portion of the problem you've posed:

I like to explain the scale factor a(t) like this. Consider any two points in space at cosmological distances. You can compare their separation at time t to their separation now. The ratio between these two separations is the scale factor at time t. The key point to make here is that it applies to any two points you choose. So if galaxyA and galaxyB are presently separated by 1 Gpc, when the scale factor was 0.5, they would have been separated by 0.5 Gpc. The same thing applies to another pair of galaxies, C and D, at present separation 4.5 Gpc. They would have been separated by 2.25 Gpc of proper distance at time t. So, the specific distances involved are different for the two pairs, as is the *amount* by which their distance has increased between the two epochs. However, the *factor* by which their separation has increased between these two epochs is the same. This factor is the same for any two points when comparing between a given epoch and now.

Here, I've implicitly assumed that we can set a0 = 1 at the present time, which I believe is a normalization that you are free to choose only if there is no spatial curvature. Otherwise it's a/a0, as you stated.

Last edited: May 17, 2013
8. May 17, 2013

### marcus

Thanks, Cepheid. I'm glad to get suggestions or actually worked-out contributions (like this one of yours) to a tutorial repertory. I'm thinking now of another worked example called 'Getting an Answer'. We can use the CEH (cosmic event horizon labeled Dhor in the table) to help in choosing whom to "message".

Or here's a slightly different type of example: We get a message and measure the stretch. It's S=2, the carrier wavelength has doubled while the light was in transit. The message says "Immediate answer requested" Should we respond? Will our reply, even if we send it immediately, ever get back to them?

The first row of the table shows that because S=2 their present distance (Dnow) from us is 11.
The second row shows the present communication range for messages sent today (Dhor) is 16.
They are within range: 11 < 16.
$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly) \\ \hline 0.500&2.000&5.86&8.11&11.05&5.52&13.77\\ \hline 1.000&1.000&13.79&14.40&0.00&0.00&16.47\\ \hline 3.175&0.315&32.70&17.18&11.06&35.12&17.22\\ \hline \end{array}}$$
The table range was Supper=2, Slower=0.315, and Steps=2. Incoming stretch being 2 implies that Dnow ≈ 11. And this is included in our forward lightcone. Our reply will reach them in year 32.7 billion of the expansion.

Last edited: May 18, 2013
9. May 17, 2013

### Staff: Mentor

Perhaps someone could do a balloon expansion with a light beam traveling from point A to point B along a spherical geodesic while the balloon expands at some rate. You could even make the expansion rate changeable by the student so they could the effects better.

10. May 18, 2013

### marcus

That's a really good idea! I think you mean a simplified more DIAGRAMMATIC version of Ned Wright's (link in my signature). You can see the effect if you watch his version closely: the photons always travel at the same speed and the percentage rate of expansion is declining.

What I picture for a clearer more diagram-like version is you just have TWO galaxies (he has them scattered all over the balloon which is distracting).

And at a certain moment galaxy A emits a photon aimed at galaxy B.
But at first the photon does not make any progress because of expansion.
And then after a while, because the percentage expansion rate declines (i.e. the Hubble radius increases), the photon is just keeping at the same distance from its target.
And then with further decline of rate, you see it begin to gain ground and approach its target.

Meanwhile at the side you have a couple of numbers, that keep changing: maybe the scalefactor a(t) and the percentage per second growth rate of distances: essentially the model's H(t).
And maybe a third number which is the distance from the photon to galaxy B--which initially increases and then stays the same for a little while and then goes down to zero.

Maybe the Hubble radius could be illustrated by a dotted line circle drawn around galaxy B, which keeps expanding until it finally takes the photon in, which is the moment when the photon stops being dragged back and begins closing in on target.

So the universe (as in Ned Wright's balloon movie) is an expanding blue disk,
but it has just two white dots, the galaxies, which stay at the same latitude longitude throughout the show.
Plus there is the little wiggly thing that travels between them.
And plus there is an expanding circle around the destination galaxy, to show the Hubble radius.

That is how I picture what you are saying. I do not like letting user control speed of light because it is not necessary and makes it more of an interactive computer game. I like a diagrammatic focus on the central realization: light can travel between galaxies which are receding from each other faster than the speed of light, and here's how it does it.
Once you get that idea clearly in mind it is time to move on to the next idea and the next diagram (if there is one), and not get delayed and engrossed with interactive "features". It could be just a difference in sensibility/style--I like your basic idea, as a clean diagram, very much.

11. May 18, 2013

### marcus

Here is another worked example tutorial idea. It could be called "Get to know your matter-radiation equality"

In the early universe radiation dominated: the energy density (joules per cubic meter) of radiation was much higher than the equivalent energy density represented by ordinary and dark matter. Expansion reduces the energy density of radiation more rapidly than that of matter, because it lengthens each photon's wavelength, weakening it, as well as thinning the photons out.
ρmat falls off as the cube of the scale factor (same matter in larger volume) but ρrad falls off as the fourth power (same number of photons in larger volume and each one with a longer wavelength).

At present the density of matter (as an E=mc2 energy equivalent is 0.23 nanojoules per cubic meter.) ρmat(now) = 0.23 nJ/m3.
As a rough estimate the radiation density, mostly in the form of CMB photons, can be put at 1/3400 of that.

More exactly, it's estimated that the two densities were equal at the epoch S = 3400
Back then, when distances and wavelengths were 1/3400 present size, matter density would have been
ρmat(then) = 34003ρmat(now)

and radiation density would have been

So the two densities matched and that stage in expansion history is called "matter-radiation equality"--in Jorrie's calculator it's labeled Seq = 3400.

Just out of curiosity, what was the radiation density in joules per cubic meter at that epoch?

Answer: the same as the matter density, take the "now" matter density 0.23 nJ/m3 and multiply by 34003.

NOW LET'S GO BACK FARTHER IN TIME TO WHEN S IS TWENTY THOUSAND.

$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline S&T (Gy)&D_{now} (Gly)&D_{then}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 20000.000&0.000001873&46.18&0.002309&3.21&659.18\\ \hline 3400.000&0.000051270&45.88&0.013494&3.19&145.58\\ \hline \end{array}}$$

It is year 1873 of the expansion. You may want to calculate the radiation density at the epoch S=20000. Then radiation density vastly outweighs matter, by a factor of 20000/3400. If you are curious you can find the matter density in year 1873. Just multiply the density now by 200003

Last edited: May 18, 2013
12. May 18, 2013

### Mordred

Marcus I was trying to print a history for this thread but the copy paste latex kept messing up from my phone.

this is the thread I wanted to print out to. Can you do me a fav and test the small latex printout on that thread for a full expansion history?
Make sure its just my phone as opposed to a latex printout error?

13. May 18, 2013

### marcus

Sure, what limits and number of steps do you want? for now I will just do the default history in 10 steps from S = 1090 to 0.01. Say if you want something different.

14. May 18, 2013

### Mordred

I was trying a 20 step default full option row display default decimal

edit just saw your post in the other thread thanks that works its just my phone fighting me.