From Aeon to Zeon to Zeit, simplifying the standard cosmic model

[marcus]

But on the other hand most of it is accessible to someone with no such prior knowledge - provided a brief intro is added.

That's a good idea, something that Jorrie may choose to do. It's part of what interests me about this concise layout as an example if how a section or chapter of a tutorial could go. As for the intended audience, I think Jorrie's blog readers have been exposed to Friedmann equation and LambdaCDM. So they've had an introduction of some sort and this is more giving them a glimpse of a variant version of the Friedmann equation (which has some nice featues) when they've already got some idea of the standard cosmic model. I think. don't know for sure.

I recall he started off this way:
==quote==
Quite a lot has been written on this ... Blog about the standard Lambda-Cold-Dark Matter (LCDM) cosmological model and its equations...
==endquote==

 

[marcus]

BTW Jorrie and Wabbit and whoever else might be reading,

since we decided for good reasons (I thought) to go with MASS density of matter instead of the energy density equivalent, I wanted to use our basic equation and basic parameters 14.4 and 17.3 to calculate the present-day density in familiar imaginable terms, like micrograms and a cubic volume that is 1000 km on a side. Everybody at the BBQ should be able to picture 1000 km, because we fly distances like that to visit relatives.

The basic equation, e.g. in Jorrie's blog, is
$$H^2 - H_\infty^2 = \frac{8 \pi G}{3} \rho$$

So I typed this into google
((14.4 billion years)^-2 - (17.3 billion years)^-2)*3/(8 pi G)
which is
$$\rho = (H^2 - H_\infty^2) \frac{3}{8 \pi G} $$
and google gave back
((((14.4 billion years)^(-2)) - ((17.3 billion years)^(-2))) * 3) / ((8 * pi) * G) =
2.66045729 × 10^-27 kg / m³
which looks to me like 2.66 micrograms in a cube which is 1000 km on a side.

So I tried to force google to tell me the answer in micrograms per cubic megameter and I typed in
((14.4 billion years)^-2 - (17.3 billion years)^-2)*3/(8 pi G) in micrograms/(1000 km)^3
and the google calculator bought it and came back with
2.66045729 micrograms / ((1000 km)^3)

 

[marcus]

Google calculator. It loves to show off by putting in all the parentheses.
There should be a google calculator at every engineer's barbecue! It turns out the calculator knows "megameter"! I typed in:
((14.4 billion years)^-2 - (17.3 billion years)^-2)*3/(8 pi G) in micrograms/megameter^3
and it came back with
((((14.4 billion years)^(-2)) - ((17.3 billion years)^(-2))) * 3) / ((8 * pi) * G) =
2.66045729 micrograms / (megameter^3)

 

[wabbit]

Interesting : )
Following your lead I tried ((14.4 billion years)^-2 - (17.3 billion years)^-2)*3/(8 pi G) in u/m^3
Google tells me that's 1.602 atomic mass per cubic meter - I guess we can call that 1.6 hydrogen atom per cubic meter.

I also note that humans are frighteningly dense at ~10^30 times the average density of the universe. And that ratio is getting higher everyday

 

[marcus]

Interesting : )
Following your lead I tried ((14.4 billion years)^-2 - (17.3 billion years)^-2)*3/(8 pi G) in u/m^3
Google tells me that's 1.602 atomic mass per cubic meter - I guess we can call that 1.6 hydrogen atom per cubic meter.

I also note that humans are frighteningly dense at ~10^30 times the average density of the universe. And that ratio is getting higher everyday

That might be even better as a way to communicate the average density. I'm glad to learn that google calculator understands the atomic mass unit "u"

As I recall if one calculates it in terms of energy density equivalent (which we've decided not to use) it comes out around 0.24 nanojoules per cubic meter.
I always liked that because it seemed communicable. Everybody knows cubic meter and you can demonstrate a joule of energy by raising a book 10 centimeters off the table and letting fall back with a little thump. 10 Newtons to raise a 1 kilogram book, by a tenth of a meter. A one joule thump.
So the universe has about 0.24 joules worth of matter in a cubic kilometer.

 

[marcus]

Justwanttoobserve that a joule of energy is something we can experience and feel and a cubic kilometer is something we can visualize so 0.24 joule per cubic km is something definite and comprehensible. But we're giving up on the energy density measure of matter concentration, for good and sufficient reasons!

How about mass density measures? a lot of people only have a foggy idea of a gram (the weight of a paperclip?), and nobody ever felt the weight of an hydrogen atom or of a microgram.
Those are more verbal ideas, rather than experiential. Still, we want to re-ify the universe's average mass density.

WE'RE IN LUCK. In a sense we are confronted by the distance to the Sun all the time. 150 million km. If you can't already imagine a cube which is a million km on a side then maybe give it a try? Learn how? A million km is a gigameter. We're in luck because google calculator knows gigameter.

What Wabbit tried is rather beautiful (the calculator knows u, the atomic mass unit) but instead of this:

Interesting : )
... I tried ((14.4 billion years)^-2 - (17.3 billion years)^-2)*3/(8 pi G) in u/m^3
...

let's try this:
((14.4 billion years)^-2 - (17.3 billion years)^-2)*3/(8 pi G) in kg per gigameter^3

I pasted that in and google liked it and came back with:
2.66045729 kg per (gigameter^3)

That has the advantages that
We can calculate it from our two basic numbers 14.4 and 17.3
Everybody has felt the weight of a kilogram and can probably picture something with around that 2.66 kg mass (a gallon of milk from the supermarket isn't TOO far off especially if it has been in the fridge a day or so and is only 3/4 full.)

Admittedly a cubic gigameter is something of a stretch to visualize. But maybe the million km distance itself should be part of our intuitive repertory given that an AU is 150 of them.

 

[marcus]

Interesting : )
Following your lead I tried ((14.4 billion years)^-2 - (17.3 billion years)^-2)*3/(8 pi G) in u/m^3
Google tells me that's 1.602 atomic mass per cubic meter - I guess we can call that 1.6 hydrogen atom per cubic meter.

I keep coming back to this. one atomic mass unit per cubic meter is a nice density. It has a classic sound.
Visualized as one hydrogen atom per cubic meter.

If I temporarily think of that as standard density, then the current density of the universe is 1.6, as you point out. or 1.602.

the word "gigameter" is clunky. But one could use it to explicate this classic standard density.
One u per cubic meter is the same as 1.66 kg per cubic gigameter.

To a trained physicist that's completely trivial because he knows that an atomic mass unit is 1.66 x 10^-27 kg. But as an amateur bystander, I regularly forget such things.

Can we re-ify or "thing-ify" a million kilometers? Should we even want to?

Distance to Sun 150
Distance to our moon 0.38
Distance from Jupiter to its two largest moons: 1.07 and 1.88
Distance from Jupiter to the other two large ones: 0.42 and 0.67
Distance from Saturn to its four largest moons: 0.38, 0.53, 1.22, 3.56

Is this too Baroque? I look in the fridge and see a 3/4 full gallon of milk. I think of the distance from Saturn to its largest moon Titan. I imagine a CUBE that size. I explode the 3 quarts of milk to fill that cube and that is the average density of the universe, at present.

Or I think of the distance from Jupiter to ITS largest moon, Ganymede. That is even closer to right distance: 1.07. And base the cube on that.

I suppose it is ridiculous and doesn't quite make it. I'm using 3 quarts of milk to represent 2.66 kilograms. Taking the 1.66 of "u" and the 1.6 that Wabbit calculated, multiplying 1.6 x 1.66 to get 2.66.
2.66 x 10^-27 kg per m³ = 2.66 kg per cubic gigameter

How else can I imagine a million kilometers? 2.6 times the distance to our moon?
1/150 of the distance to the sun?

 

[wabbit]

How about using the Earth moon distance itself as a unit
1.6 u/m^3 in kg/(4 pi/3 (384400km)^3)
630g per sphere of earth-moon radius?

(Anything without gigameters really, I have to mentally convert them - twice to make sure - each time I see that unit)

 

[marcus]

Great suggestion!
I put this in google
4 pi/3 *2.6^-3*2.66 kg in pounds
and it gives me 1.4 pounds. that much butter would have to be dispersed in a sphere of earth-moon radius!

 

[marcus]

I ventured to our fridge and found approximately 1.4 pounds of butter on the shelf in the door.
Huge lasers are trained on this butter and in an instantaneous flash it is vaporized, becoming a glowing cloud
which expands to fill a sphere of earth-moon radius.
This is the current density of the universe.

 

[wabbit]

Somwhere along this thread we went from natural, universal units based on the cosmological constant and the Planck length, to pounds of butter per moon-sphere... next week we'll show you how to cook a universe in your microwave (oven, not background)

 

[marcus]

There's a real tension, even if we both joke about it. I think H_∞ is a great unit and when you use it the equations do simplify nicely.
From that unit rate of fractional growth we get an e-fold time unit of 17.3 billion years.
That's an awful mental stretch.

Or maybe it's not. Maybe one just adopts the right perspective, the present Age is 0.8 zeon, the present distance growth rate is 1.201 per zeon, and everything is nice.

I've gotten side-tracked, in a way, worrying about how to make that huge (lovely) time unit palatable.

The business about matter density is merely a minor distraction. Rho is just a footnote.

How much fuss do you think needs to be made about assimilating the zeon time unit?

Do we break it down into billionths and practice imagining and remembering passages of "zeits" of 17.3 years? Talk about H_∞ as fractional growth of one billionth per zeit?

Or just present the main unit with a blank stare and count on some readers being sophisticated enough to find, if they care to, their own private ways to assimilate it.

 

[marcus]

Wabbit, do you have some ideas about how to communicate the fact that an instantaneous fractional growth rate of 1/zeon does not correspond to a doubling time of 1 zeon?
Because of compounding the doubling time is actually ln2 zeon. It is shorter than the e-fold time by a factor of ln 2 = 0.693.

Maybe one just states this? Or offers a brief explanation like:

$$a(t) = e^{\frac{1}{zeon}t}$$
$$a(zeon) = e^{\frac{1}{zeon}zeon} = e^1 = 2.718...$$

If that turns out to be sufficient, it removes one of my concerns that led to proposing a billionth unit.
I was motivated to suggest a nanozeon or zeit unit because it is short enough that one can neglect the effect of compounding and say straight out that, with H_∞ = a billionth per zeit, a typical distance grows by one billionth of its size in one zeit.

 

[wabbit]

About the zeon or zeit unit, I think.since most people have some familiarity about the age of the universe already, and about the timescale of earth, geology, life evolution, these would be good references - Earth is about 0.3 zt old etc. One issue is that I think.its easier to grasp units when things are measured as a few to.a few thousand units, rather than a fraction of a unit - we could introduce millizeits but that would be one more unit again.

$H_\infty=1$ is a bit hard to get as a fractional growth rate, not sure how to get around this. A growth rate of 1 is just not directly apprehensible.

As to e-folds, it is natural because of the exponential function - this reminds me of the questions asked in another thread, and somehow the key seemed to be $(1+x/n)^n\rightarrow e^x$.

So $H_\infty=1$ means that in the distant future distances will grow by 1 percent for each percent of a zeit.

But this doesn't help much wrt to doubling time... is it really necessary to introduce doubling time?

 

[marcus]

Probably we don't have to introduce doubling time. It sounds like you agree with Jorrie about "zeit" He indicated, if I remember, that he liked zeit as a name for the main unit, instead of zeon.

I had come up with zeit as a provisional name for a billionth of the main time unit. Jorrie said he didn't like changing units and would go with whatever we had rather than start renaming but if we had to do it over he would favor zeit and if you need a billionth of that just call it "nanozeit".

I'm of a mind to follow your and Jorrie's leanings on this. You say make main unit "zeit" and suggest abbreviation "zt" and the present age is 0.8 zt.
Jorrie suggested abbreviation nz for nanozeit.

To paraphrase your above post, H_∞ = 1 means that in the distant future distances will grow by 1 ppb per nz.
That is by one part per billion every nanozeit.

Also noting what you said about people finding it easier to grasp fractions and multiples on the order of 0.001 and 1000 or even closer to one than that. I'll try out a thousandth.
To paraphrase again, H_∞ = 1 means that in the distant future a distance will grow by a thousandth of its size each millizeit.
That is 0.001 per mz.

In Earth orbit terms, one mz is 17.3 million years. OK, I can kind of picture that. they had wooly mammoths back then, didn't they? Warm-blood hairy things faintly resembling ourselves in many respects.
Heck, it was just a millizeit ago!

Maybe this is going to be all right. From Aeon to Zeon to zeit.
zeit should not be capitalized, that way it's less apt to be confused with the capitalized German noun.

 

[wabbit]

Actually it wasn't conscious choice I was following what I thought you had more or less settled on - but I do agree, either zeit or zeon is fine but I think its better if that one unit designates directly a natural value such as $H_\infty=1$.

Nothing wrong with nanozeit but isn't that a bit small ? millizeit would give nice results for geological ages with ~230 millizeits for the age of the earth. Or maybe microzeit ? Andromeda is about 0.1 microzeit away and it is not part of the Hubble flow, I guess that might starts at microzeit scale (or a bit higher ?) so maybe ~1 microzeit is about the smallest "cosmological" distance.

 

[marcus]

I see the point you are making about geological ages and millizeits. It's actually a convenient scale
http://en.wikipedia.org/wiki/Age_of_the_Earth
4.54/17.3 Basically the age of the earth, given there, is 260 mz.
Let's see what some geological ages look like in millizeits.

This is from UC Berkeley Museum of Paleontology, I think it is part of their public outreach educational website. http://www.ucmp.berkeley.edu/education/explorations/tours/geotime/guide/geologictimescale.html
The Cambrian would be how many millizeits ago?
543/17.3 = 31
OK, 31 millizeits ago there were a lot of trilobites and fish got started.
and then 29 millizeits ago the first land plants
and then 25 mz ago there were insects and fish developed jaws
and 24 mz ago *amphibians*, like frogs! and a whole lot of fish!
and 21 mz ago insects developed wings and there were forests
and 19 mz ago, the first reptiles, and really big forests
and 17 mz ago, amphibians were dominant, but later there was a major extinction over land and sea.
Then 14 mz ago, first dinosaurs, also (according to one definition) first mammals.
12 mz ago, dinosaurs dominant, first birds appear
8.4 mz ago, marsupials, bees, butterflies, flowering plants, then a mass extinction esp. of large animals.
3.8 mz ago, *placental mammals*, modern birds, first primates (things with thumbs)
3.1 rodents, primitive whales, grasses
2.2 pigs, cats, rhinos
1.3 dogs and bears---insects and flowering plants coevolve
0.3 millizeit ago, first hominids

 

[marcus]

The Wikipedia article on Mammals allows for several definitions including one that puts the first mammals around 167 million years ago or roughly 10 mz. (the most recent common ancestor of all extant mammals--a "clade"-type definition.)
http://en.wikipedia.org/wiki/Mammal#Varying_definitions.2C_varying_dates
"Clade" (phylogenetic) classification has gained adherents over recent decades so maybe one wants to go with that and say the mammal clade started around 10 millizeit ago. "Crown group" is a technical term in cladistics, not a term of approbation. A set of species for which one is going to find the most recent common ancestor.
http://en.wikipedia.org/wiki/Crown_group
==quote Mammal article==
If Mammalia is considered as the crown group, its origin can be roughly dated as the first known appearance of animals more closely related to some extant mammals than to others. Ambondro is more closely related to monotremes than to therian mammals while Amphilestes and Amphitherium are more closely related to the therians; as fossils of all three genera are dated about http://tools.wmflabs.org/timescale/?Ma=167 million years ago in the Middle Jurassic, this is a reasonable estimate for the appearance of the crown group.[6]
==endquote==

That said, I don't think there is any need for us to get off topic and onto geology/paleontology. I wanted to do two things (1) pick up on your comment suggesting millizeit could be a good scale for geological ages (thus giving people a way to get used to and assimilate that scale: something to practice on.) and
(2) try it out, myself, to see how well the scale worked.

It seems to work OK.

 

[marcus]

I think this is the ZEIT thread now. Jorrie first suggested doing away with "zeon" and it has taken me a while to come around to that. I'm not sure why but zeit just seems to sound better and it's one syllable: easier to say.

'Zeit' is actually a very cute choice, because it is German for 'time'; pity we did not originally hit upon it instead of 'zeon'...

My conservative choice would be to do away with zeon and replace it with 'zeit' and then use 'nanozeit' (or simply nz) for the 17.3 years period (only one new name)...

Your mentioning blogs led me to find the 12 May blog entry where you gave a presentation of the cosmic model using H_∞ units. I hadn't seen it earlier because I was looking only for discussion board posts. It's clear and succinct. I'd like to copy it in the AeonZeon thread. It would fit into one medium sized post, I think. Let me know if that would be undesirable for any reason.

...
BTW, if you want to copy my humble Engineering Blog post here, you are welcome. Edit as you think necessary.

I added it to the AeonZeon thread (tinkered with the wording some, couldn't resist : ^)
Let me know if unsatisfactory for any reason. I can change or delete as you wish.
https://www.physicsforums.com/threa...ndard-cosmic-model.811718/page-4#post-5134159

I want to recopy Jorrie's brief blog summary putting in zeit and see how it looks.
===tinkered with copy, putting in zeit for zeon===
With Jorrie's permission, I'll copy here a piece he posted on his Blog, with some suggested edits. It sums up the simplified approach to cosmology we've been working out and exploring in this thread. So it contributes to this thread, but also I want us to be able to suggest edits, and comment. This is supposed to be introductory, is anything potentially confusing, can the wording be improved? I'll make any changes Jorrie approves, or delete this draft version if that seems better.
==draft version==
How Aeons turned into Zeits
Posted May 12, 2015 12:00 PM by Jorrie

Quite a lot has been written on this ... Blog about the standard Lambda-Cold-Dark Matter (LCDM) cosmological model and its equations. Arguably the most important equation of the model is the evolution of the expansion rate over cosmological time. In other words, how the Hubble constant H has changed over time. If one knows this function, most of the other LCDM equations can be derived from it, because it fixes the expansion dynamics.

The changing H is most simply expressed in this variant of the Friedman equation, an exact solution of Einstein's field equations for a spatially flat and perfectly homogeneous universe.

(1) H²−Λc²/3 = 8πG/3 ρ

Here H is the fractional expansion rate at time t, Λ is Einstein's cosmological constant, G is Newton's gravitational constant and ρ is the changing concentration of matter and radiation (at time t) expressed as a mass density. This density includes dark matter, but no 'dark energy', because Λ appears as a spatial curvature on the left side of the equation.

As you can check, the right-hand side gives SI units of 1/s², also the units of H² since it is the square of an instantaneous fractional growth rate. Since Λ is a constant curvature, its SI units would be reciprocal area 1/m² and multiplying by c² again gives a 1/s² quantity. Hence both sides' units agree. it is convenient to replace Λc²/3 with the square of a constant growth rate H_∞² representing the square of the Hubble constant of the 'infinite future', when cosmic expansion will effectively have reduced matter density to zero.

(2) H²−H_∞² = 8πG/3 ρ

Since we can measure the present value of H, labeled H₀ (H-naught) and also how it has changed over time, it allows us to use Einstein's GR and his cosmological constant to determine the value of H_∞. If we assume that radiation energy is negligible compared to other forms (as is supported by observational evidence), then we can express eq. (2) as:

(3) H²−H_∞² = (H₀²−H_∞²)S³

H₀ is the present observed rate of expansion per unit distance, which tells us that all large scale distances are presently increasing by 1/144 % per million years. This gives us a Hubble radius of 14.4 billion light years (Gly). S is the 'stretch factor' by which wavelengths of all radiation from galaxies have increased since they were emitted.[1]

Clearly the distance growth rate H is changing, it is declining and leveling out at the constant value H_∞. The point of this equation is to understand how it is changing over time and how this effects the expansion history. But let's imagine that H remains constant. Then, as you can check, the size a(t) of a generic distance would increase as e^Ht. The time interval 1/H would then be a natural time-scale of the expansion process. In that length of time distances would increase by a factor of e = 2.718. For constant H, the time 1/H can be called the "e-fold time", by analogy with "doubling time" . An e-fold is like a doubling except by a factor of 2.718 instead of 2.

In the long run the universe's expansion process will be exponential at nearly the constant rate H_∞, so eventually all large scale distances will undergo an e-fold expansion every 17.3 Gy. Or stated differently, all distances will eventually grow at about H_∞ = 1/173 % per million years.

The 17.3 Gy 'e-fold time' is a natural time scale set by Einstein's cosmological constant. An informal study by a group of Physics-Forums contributors suggested that the 17.3 Gy time-span could be a natural timescale for the universe.[2] For lack of an 'official name' for it, the group called it a 'zeit'. The longterm expansion rate, the reciprocal of the e-fold time, is therefore H_∞ = 1 per zeit.

One light-zeit is 17.3 Gly in conventional terms. If the current rate H₀ were to continue unchanged, distances would expand by e every 14.4/17.3 = 0.832 zeit. H₀ = 17.3/14.4 =1.201 per zeit.[3] Our present time is 13.8/17.3 ~ 0.8 zeit.

We can easily normalize equation (3) to the new (zeit) scale by dividing through by H_∞ (which then obviously equals 1).

(4) H²−1 = (1.201²−1)S³ = 0.443 S³

or

(5) H² = 1 + 0.443 S³ !NB!

This remarkably simple equation forms the basis of a surprisingly large number of modern cosmological calculations, as will be discussed in a follow-on Blog entry.

Here is a graph of the normalized H changing over time, expressed in zeits.

The blue dot represents our present time, 0.8 zeit and a Hubble constant of 1.2 zeit^-1. The long term value of H approaches 1.
...
...

[1] 'Stretch factor' S = 1/a, where a is the scale factor, as used in the LightCone calculator. S is also simply related to cosmological redshift z by S=z+1.

[2] [A] group of PhysicsForums members [fleshed] out of this "universal scale", based on the cosmological constant.

[3] The traditional unit of the Hubble constant as used by Edwin Hubble is kilometers per second per Megaparsec. From an educational p.o.v. it was an unfortunate choice, because it seems to indicate a recession speed, while it is really a fractional rate of increase of distance. It is a distance divided by a distance, all divided by time. So its natural unit is 1/time, or simply time^-1.

 

[Jorrie]

With Jorrie's permission, I'll copy here a piece he posted on his Blog, with some suggested edits. It sums up the simplified approach to cosmology we've been working out and exploring in this thread. So it contributes to this thread, but also I want us to be able to suggest edits, and comment. This is supposed to be introductory; is anything potentially confusing, can the wording be improved? I'll make any changes Jorrie approves, or delete this draft version if that seems better.

Marcus, I'm happy with using this as a 'development platform' for a simplified introduction to cosmology. It needs some development, because I'm not happy with my own writing on it. It feels like 'full of holes' in terms of clarity.

It also received a completely underwhelming response on my Engineering Blog. Generally, engineers don't like reading equations that do not fit into their daily-use regime. They then want to read text in their own lingo that tells them exactly what an equation is 'trying to tell' them, without having to puzzle it out themselves. Almost (but not quite) 'barbeque ("braai") level'.

I'm not sure to what degree this holds for physicists as well (?)

However, I think we may be able to find common ground and so serve both communities...

 

[marcus]

I'd say let's not worry just yet about making this palatable to the Engineering community right away, let's make a coherent package:
you've suggested some good moves, some of which I'm repeating here
get rid of zeon, use zeit consistently
don't name a small piece, use millizeit (mz) if needed
keep the nomenclature simple
get rid of playful terms like "eepling" and "eebling"
steer clear of "doubling time" and make consistent use of "e-fold time"
keep drumming that in, eventually it will be acceptable. (already is to some people, I expect)

 

[wabbit]

The only thing missing I think (perhaps not on Jorrie's blog since that is covered elsewhere - though that blog entry doesn't link to it) is that short intro/motivation/pointer to other source explaining where that Friedman equation comes from : )

I realize the text says its a solution of GR for a flat universe - but why? How come the equation says that increasing matter density increases the speed of expansion, isn't gravity supposed to be atractive?

 

[marcus]

Wabbit, do you have (or does Jorrie in another of his blog entries have) a short intro/motivation like that, or a pointer to something you think would be satisfactory, that you could share with us?

Jorrie, you suggested replacing zeon by zeit. The main work involved AFAICS would fall on your shoulders because Lightcone7z would need the replacement done in some 4 or so column headings, and also in the graphing section where charts are labeled in the upper right corner.
I hesitate to urge this because I don't know how much of a bother it would be. If you are willing, then I've come around to that take on things and would be happy to make the switch final.

I also like the simple phrase "e-fold time" for the time it takes something to expand by factor of e. It's short, I like terms with few syllables. I think it will either go over with various communities or that it already has and I just haven't heard it used.
====================

Another thing. I think a lot of us (including some engineers I've known) like to be able to calculate stuff. Knowledge has a practical purpose in action, to build, to control, to answer a question on one's own reckoning instead of having to look it up .

To the extent that a cosmic model like this can get people involved with it (not just admiring from a distance) it might help to present of a string of questions an ordinary person can answer using the model. Here are some ideas that have come up. Can you think of others?

1. you wake up some time in the future and the CMB is a different (lower) temperature, what time is it?
2. your friend is studying a galaxy and tells you the redshift, what time was the light emitted?
3. say the Earth formed 0.26 zeit ago, what was the expansion rate back then? That would have been at age 0.54.
4. or maybe that's too recent. We are told our galaxy's disk formed at age 0.29 zeit. (that is 0.51 ago.) What was the expansion rate back then? What redshift does that correspond to?
5. somebody tells you the first stars were around 13.3 billion years ago, what was the matter density then compared with now?

I'm having a difficult time thinking of question challenges like this, at the moment. Way to get readers to imagine putting the model to work.
Maybe they never actually solve the problems, but merely look briefly at the questions, but it opens up the interactive aspect. The model is something you can do with. With nothing more than a scientific calculator or some other hand-held device (log, square root...)

(the Milky way halo stars are older, the galaxy's thin disk formed more recently, the age of disc stars is put at 8.8 billion years, i.e. disk formed around year 5 billion. maybe too esoteric...)

 

[Jorrie]

Jorrie, you suggested replacing zeon by zeit. The main work involved AFAICS would fall on your shoulders because Lightcone7z would need the replacement done in some 4 or so column headings, and also in the graphing section where charts are labeled in the upper right corner.
I hesitate to urge this because I don't know how much of a bother it would be. If you are willing, then I've come around to that take on things and would be happy to make the switch final.

This would be a simple search and replace operation in the source code, so no problem.

A more acute problem would be to change the two main inputs to 'zeit' as well - they are currently in Gly, so it's a bit of a mixed bag. The simplest would be to only change the input names to Hubble time in place of Hubble radius and of course change the symbols and units. Then all the input validity checks remain untouched in the software.

If we do venture into the bigger change, I would suggest that we switch to more conventional inputs anyway - at least to ones closer to the published data sets. This means inputting the Hubble constant and Omega_lambda in place of Hubble times; then the latter pair can appear as conversions on the top-right (essentially swapping the top two rows around). It sounds simple, but the programming effort might be significant.

What do you think?

 

[wabbit]

Wabbit, do you have (or does Jorrie in another of his blog entries have) a short intro/motivation like that, or a pointer to something you think would be satisfactory, that you could share with us?

Hah, not so easy as I thought : ) I'll have a look around. It's easy to give a simple Newtonian argument for the Friedman equation with curvature but without cosmological constant using just $\ddot a=-C/a^2$ but I'm not sure what is the natural way of introducing the CC... Where does $\ddot a=-C/a^2+Da$ come from ?

 

[marcus]

This would be a simple search and replace operation in the source code, so no problem.

A more acute problem would be to change ...

Great, if it's really that simple to change the output units from zeon to zeit, let's see how it looks!

I like very much having the two main inputs be in traditional units because it serves as a conceptual bridge. I could be wrong. Sometimes you see farther than I do. But to me it seems like an advantage that a newcomer to the Lightcone7z sees something that he/she recognizes or can connect with past exposure to cosmology. "Oh that's the Hubble radius, sure, 14.4 billion light years."

I'd say be gradual/incremental about changing. It is a beautiful gadget as it stands. I'd suggest just changing zeon to zeit and letting us play around with it a bit more.

I could see maybe eventually changing the two main inputs to be Hubble times, with the default values stated in years: 14.4 Gy and 17.3 Gy.

And then (parenthetically or over on the righthand side where you list some equivalents) you could indicate that these correspond to 0.797 zeit and 1 zeit. The latter is by definition. So the input list already shows the user the definition of the primary unit.

Anyway, I'd argue for keeping the two main inputs in Gly or Gy, as the most direct interface between our simple "zeit" model and the conventional cosmology world at large.

 

[Jorrie]

OK, here is first trial of LightCone7zeit by means of search and replace.
Do not create a signature link to it yet, because it may contain errors/omissions...
I think the Intro and possibly some tooltips should be changed to make the term 'zeit' more understandable.

 

[marcus]

Thanks Jorrie! I'll check it out (probably several of us will).

I'm interested in the "look and feel" and seeing if we can make up easy concrete "quasi-real world" exercises to go along with it.

 

[marcus]

Maybe we can exploit this equation for some exercise challenges:
$$a = \frac{sinh^{2/3}(\frac{3}{2}t)}{1.3}$$ The scale factor is essentially the same information as the stretch, or the stretch-1 = redshift. So it relates the two most intuitive things without introducing H as intermediate.
Here is the inverse function for getting from stretch-type information back to time.
$$t = \frac{2}{3} \ln \big((1.3a)^{3/2} + \sqrt{(1.3a)^3 + 1}\big)$$

EDIT: had to run an errand in the middle of this. Got back and corrected some errors.
This number "1.3" is 1.31146.. but for working back-of-envelope and sample exercises we can just say 1.3. It's nice that the two significant figures version is so close.

In terms of our two main parameters the number is ( (17.3/14.4)² - 1)^-1/3 = 1.31146...

 

[marcus]

Yay!
I like the look of the http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7zeit/LightConeZeit1.html
It even has the two main input parameters be the Hubble times.
I'll check how the graphs look when you select the chart feature.

Neat! Look at this. It is a graphic reminder of the many nice relations among the quantities that make up cosmic history. Each curve has a story:

 

[marcus]

Jorrie, Wabbit (Wabbit glad you like the looks of the zeit version!) what we seem to be moving towards is a presentation that might at least put some ideas in the heads of Jorrie's constituency even if they don't take hold and work with it right away, and could be used in other contexts as well, that might go like this:
1. an introductory essay along the lines of what Jorrie already wrote. (but possibly even more explicit about "e-fold", for instance "e-fold is like three-fold except with e=2.718 instead of 3, the e-fold time is the time it takes something to expand by a factor of e. If f(t) is any positive increasing function with instantaneous fractional growth rate f'(t)/f(t) constant then it's a fact of calculus that the reciprocal of that rate, f(t)/f'(t), is the function's e-fold time. And the function must be of the form f(t) = e^rtwhere r equals the constant growth rate f'(t)/f(t).")

2. graphic introduction to the zeit version of Lightcone, particularly with a bunch of curves that illustrate the model.

3. the equations in part 1 have hand-calculable solutions that closely approximate the Lightcone tables and curves. In particular the expansion of the universe follows a hyperbolic sine curve: $$a = \frac{\sinh^{2/3}(\frac{3}{2}t)}{1.3}$$ and that relation of distance size to time can be inverted to tell us time as a function of the scale factor a. $$t = \frac{2}{3} \ln \big((1.3a)^{3/2} + \sqrt{(1.3a)^3 + 1}\big)$$

So here's a sample exercise. The present age (as it says in Jorrie's part 1.) is 0.8 zeit. You fall asleep and when you wake up you discover the cosmic microwave background is much colder. It must be some time in the distant future! The CMB is only one tenth its present temperature. what time is it?

Solution. The scale factor a has increased from 1 (present) to 10. So just put that 10 into the previous equation and calculate: 1.3a = 13
$$t = \frac{2}{3} \ln \big((13)^{3/2} + \sqrt{(13)^3 + 1}\big)$$
t will be the age, in zeits, at that future time.

 

[marcus]

As a check, I put in 2/3*ln(13^(3/2)+(13^3 + 1)^(1/2))
and got an age 3.027123...zeits

Again, I put in (sinh(3/2*3.027123)^(2/3)/1.3 and got 10 (correct to six significant figures : ^) so it seems ok.
As a further check, I went to Lightcone Zeit
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7zeit/LightConeZeit1.html
and put in S_lower = 0.1 and got that the corresponding age was 3.0356...zeit.
Close enough! The calculator uses numerical integration and implements the full LambdaCDM standard cosmic model. With simple formulas we should be happy to get within 1 percent and the agreement here is better than 1%.

then main thing is not to do these exercises (except as a check) but to invent some new exercises to challenge readers with. Can you do this? Or do you have to fall back on Ned Wright cosmocalculator, or Jorries Lightcone? Can you hand-calculate the U expansion history?

the Earth formed 0.26 zeit ago, i.e at expansion age 0.54 zeit. Suppose right now somebody with a telescope is studying our galaxy as it was when the Earth formed. What redshift do they measure?

Solution. $$a = \frac{\sinh^{2/3}(\frac{3}{2}0.54)}{1.3}$$

and z = s-1 = 1/a - 1

I put in (sinh(3/2*0.54)^(2/3)/1.3 and got a = 0.71786...
the stretch s = 1/a = 1.39302.. so redshift .39
I'm experimenting to see how it might be to just give novices one formula, initially, and make the first batch of exercise challenges all be be applications of that one formula (and its inverse).

 

[marcus]

Another sample exercise.
the Earth formed 0.26 zeit ago, i.e at expansion age 0.54 zeit. Suppose right now somebody with a telescope is studying our galaxy as it was when the Earth formed. How far away are they?Solution. use the same hyperbolic sinh formula for the scale factor. c=1 so distance comes out in lightzeits.
$$D_{now} = \int_{.54}^{.8} \frac{1.3dt}{\sinh^{2/3}(\frac{3}{2}t)}$$
Each tiny step cdt along there way gets expanded (between then and now) by a factor of 1/a. And they all get added up.
I googled "definite integral calculator" which got me to numberempire.com and put in
1.3*(sinh(1.5*t))^(-2/3)
for the function to integrate, with t running from 0.54 to 0.8
and it gave 0.30460...
That basically means the astronomer is currently 0.3 lightzeit from us.

 

[marcus]

Another sample problem. You are studying a galaxy at redshift z = s-1 = 2. that is to say wavestretch factor s = 3. Suddenly you notice an exploding supernova! What time in history did it happen? What was the expansion age when that star exploded?

Solution. a = 1/3. Use the inverse of our one-formula-with-many-uses.
$$t = \frac{2}{3} \ln \big((1.3a)^{3/2} + \sqrt{(1.3a)^3 + 1}\big)$$
1.3a = 1.3/3 = 0.4333
putting in 2/3*ln(.4333^(3/2)+(.4333^3 + 1)^(1/2)) gives 0.18765... zeit
Better check that, seems like a long time ago. 0.188 zeit. Over 0.61 zeit ago!
I put S_lower = 3 in LightconeZeit and it said 0.18989...
Not too good but we have to make allowances for the fact that we're using a rough 1.3 instead of a more accurate larger 1.31146... That would make the 0.188 larger and improve the approximation.
Still pretty good considering.

( (17.3/14.4)² - 1)^-1/3 = 1.31146...

 

[marcus]

I'd welcome any comments including critical, at this point. What I see coming out of this discussion is a rough 3-part outline of a "simple introduction to (H_∞) cosmology" for a fairly wide audience, anybody who's interested basically. Part A is something like Jorrie's brief account. Part B is graphic. You show the curves that come out of equations like those in Part A, and give the reader a chance to get hands-on experience with the model via Lightcone.
and part C might involve hand-calculation exercise to convey a sense of empowerment---"look at what you can do on your own, you can hand-calculate facts of expansion history..."

Part C (some calculator challenges) might or might not appeal. I'm not sure at this point.

I'm thinking now that Part A needs one more thing tacked on at the end.
1. We already get that H(t) ≈ coth(1.5 t) and we get Jorrie's equation #4, I think it is: H² - 1 ~ s³

2. From this we could go a step further and prove that a(t) ~ sinh^2/3(1.5 t)

That is because of the identity coth² - 1 = 1/sinh²
which gives you 1/sinh²(1.5 t) ~ s(t)³
You just flip this over and you have the result. The constants in these proportionalities are things like .4433 and 1.3115 which are simply related to each other.
The identity comes from coth² - 1 = (cosh² - sinh²)/sinh² = 1/sinh²
because of that thing about cosh² - sinh² ≡ 1

So from Part A we can get the multipurpose formula a(t) = sinh^2/3(1.5 t)/1.311..., or for rough work simply use 1.3
that is the thing that graphs like a pair of antelope horns and shows a slight inflection between deceleration and acceleration. Nice curve.
So I think it might be nice to base the calculator challenges mostly on that formula. (but that might be wrong, there are simpler formulas one might use that don't involve a 2/3 power.)