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

[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.)

 

[Jorrie]

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.)

Yea, for simplicity and given that S = z+1 = 1/a, I'm a little biased towards basing everything on

$H = 1 + 0.443 S^3$ and $t = \ln(\frac{H+1}{H-1})/3$

probably because that's the way I started my Blog entry. I actually shied away from the hyperbolic functions as far as possible because they do not come naturally to my niche readership. We discussed this briefly in https://www.physicsforums.com/threads/from-aeon-to-zeon-simplifying-the-standard-cosmic-model.811718/page-3#post-5121465.

Values like proper distances might require integration, but that (I think) is unavoidable. Quite a few of the exercises that you mentioned requires only the two equations above.

 

[marcus]

Yea, for simplicity and given that S = z+1 = 1/a, I'm a little biased towards basing everything on

$H^2 = 1 + 0.443 S^3$ and $t = \ln(\frac{H+1}{H-1})/3$

probably because that's the way I started my Blog entry. I actually shied away from the hyperbolic functions as far as possible because they do not come naturally to my niche readership. We discussed this briefly in https://www.physicsforums.com/threads/from-aeon-to-zeon-simplifying-the-standard-cosmic-model.811718/page-3#post-5121465.

Values like proper distances might require integration, but that (I think) is unavoidable. Quite a few of the exercises that you mentioned requires only the two equations above.

Hi Jorrie, I put the exponent on the H² in the quote. I'm often forgetting to square the H and things like that but fortunately we can go back and edit. Your judgement about style and notation has been a guide to me recently, about using zeit and e-fold and keeping terminology simple.
Also your understanding of what communicates to that particular audience.
Maybe we should develop two versions, one avoiding the hyperbolic sine (and the 2/3 power) and one employing it.

The two equations you like are remarkably versatile, and the one expressing H as a function of S² is just what we need to do the integral for distance D_now based on an observable quantity S. That simplifies the exposition. We don't observe time, we observe the stretch S.

On the other hand the curve of expansion history a(t) is intuitively appealing to some newcomers (not the engineers necessarily, maybe another niche). Because they have been wondering about it. What is this expansion I've been hearing about? What is this "acceleration" they talk about?
and the a(t) curve shows it. So it has that plus, to balance the off-putting strangeness of the hyperbolic function. Maybe there should be both.

How if you push ahead with a "4engineers" version? Maybe replace "doubling" by "e-fold time" in part A, and show them some graphs with LightconeZeit, as per part B? Add one or two sample problems if you feel like it? You could make a development thread here at PF which would allow Wabbit and me to comment and help, if you want. And transfer to your blog later. Or build it there, in situ, if you prefer.

 

[Jorrie]

On the other hand the curve of expansion history a(t) is intuitively appealing to some newcomers (not the engineers necessarily, maybe another niche). Because they have been wondering about it. What is this expansion I've been hearing about? What is this "acceleration" they talk about?
and the a(t) curve shows it. So it has that plus, to balance the off-putting strangeness of the hyperbolic function. Maybe there should be both.

Hi Marcus, thanks for correcting the H-typo.
One can obviously get a(t) without reverting to the hyperbolic functions, but I agree that it becomes a more intimidating equation, as per this attachment that you have posted before.

Mostly, I will just use the two equations $H^2 = 1 + 0.443 S^3$ and $t= \ln(\frac{H+1}{H-1})/3$
without solving them together. I would rather explain how they can be used in an algorithm to calculate specific values of a(t) =1/S and plot expansion curves. My feeling is that for beginners, it takes too much explaining on how the hyperbolic solution is arrived at, but it's just a personal preference.

PS (edit): I changed the Intro text box and the Time tooltip somewhat and then uploaded LightCone7zeit into the place of the original LightCone7z, so you do not have to change your signature link. The 7zeit link will still work for compatibility with older posts.

 

[marcus]

...PS (edit): I changed the Intro text box and the Time tooltip somewhat and then uploaded LightCone7zeit into the place of the original LightCone7z, so you do not have to change your signature link. The 7zeit link will still work for compatibility with older posts.

Jorrie thanks! Lightcone is at the core of this modest little project. I think you were right about a one-syllable name for the unit and am glad you modified Lightcone7z accordingly. In your blog, is it possible that the niche readership could be introduced to the term "e-fold time" to replace doubling (i.e. two-fold) time?

Mostly, I will just use the two equations $H^2 = 1 + 0.443 S^3$ and $t= \ln(\frac{H+1}{H-1})/3$
without solving them together. I would rather explain how they can be used in an algorithm to calculate specific values of a(t) =1/S and plot expansion curves. My feeling is that for beginners, it takes too much explaining on how the hyperbolic solution is arrived at, but it's just a personal preference..

In fact I'm eager to see the next chapter. I think your judgment (which you say is just a personal preference : ^) may well be right. Those two equations are simple and not alienating, I think, and they suffice for natural sorts of exercise problems. I hope you pursue development of that approach and I'll try to help with suggestions if I have any, or comment.
It looks to me as if we should have a parallel development. I'm trying to think of ways to make sinh^2/3(1.5t) palatable and not scary.

 

[RyanH42]

You are preparing a PF insight post I guess.I am in high school which it makes hard to understand the things.I see great effort here Marcus and of course Jorrie and the others.

I saw your part 1 LCMD basic math calculations.And it has been deleted.(I don't know why).Or you are working on it.

The problem is the who you are writing to it.You are making just the theory part I guess.

My suggestion is If you get a number or equation please explain how it evolve(If you are doing that then there's no problem)

I just want to say for an amateur I can't understand the main idea.

 

[RyanH42]

Your math is understandble but teaching part is not enough(For insight post).My first language is not english(Maybe that's the reason).
I asked some questions which shows I didnt understand maybe I should wait part 2 and part 3.

 

[Jorrie]

I saw your part 1 LCMD basic math calculations. And it has been deleted.(I don't know why). Or you are working on it.

The Insights post has not been deleted, AFAIK. https://www.physicsforums.com/insights/approximate-lcdm-expansion-simplified-math/

My suggestion is If you get a number or equation please explain how it evolve(If you are doing that then there's no problem)

Ryan, my Insights post is an article, not a tutorial, so it is does not include mathematical derivations. You are welcome to ask questions as you have done in the comments section: https://www.physicsforums.com/threads/approximate-lcdm-expansion-in-simplified-math-comments.823929/

I am contemplating writing an appendix with the reasoning behind the specific equations and the derivations, but that does not belong in an article and will come later. Marcus and Wabbit have provided many of the reasoning in the various threads of PF if you want to search for them, but they are quite scattered. Part of the motive for my Insights series is to gather things together for easier reading and reference.

 

[RyanH42]

I am so so sorry.I will not ask any question anymore.

 

[Jorrie]

I am so so sorry. I will not ask any question anymore.

No, No! You are entitled to ask questions - this is the main purpose of forums like this one. For the Insights posts, stick to questions about the specifics of the thread, like you have done up to now. That's a good way to learn.

You may sometimes be directed to go and read certain answers that have already been covered somewhere else, but if physics questions are to the point and not obvious, they are mostly answered here.

 

[marcus]

This thread might be a good place to ask basic questions. I can try to answer.

If you don't dislike my asking, what is your first language? You said it is not English. Sometimes it helps me to know what the other person's main language is.

I noticed in another thread (where I did not want to jump in, because it was not my thread) that you did not seem to understand the exponential function e^x.

that is how you calculate with an instantaneous growth rate like H.

If you have a distance like 1 km. And H is the growth rate of 10% per year. Then it is not true that after one year the distance is 1.1 km.

The way you calculate the distance after one year is you paste this into google:
e^(.1*1)
that is e^(rate*time) = e^(.1 per year x 1 year)

Do you understand what I am saying?

 

[marcus]

If you use a different calculator we might experience confusion. If you use the google calculator then we are both using the same one and it is easier to talk. But with something simple like e^(.1*1) we should both get 1.10517...

Try it and see.

And try it for longer periods of time, like 5 years and 10 years. What do you get?

If you have a distance of 1 km, and it is growing 0.1 per year (instantaneous growth rate) then how big is it after 5 years?

You should get 1.6487...

If you don't get 1.6487... then we are in big trouble! Big confusion!

 

[RyanH42]

I get that number Marcus.My english level is good actually.I don't know.The problem is I am just cannot understand the idea.Which it seems it my problem.I will wait next articles to understand the iasue better.Today maybe 1 hour later I will going to come and I will going to ask my questions.
Thank ypu

 

[RyanH42]

H is know 1/144% per million year so it mean now distance R grows e^(1/144.t) t in million years.So 100,000 thousand years R will be increase e^(1/144.0.1)=1,00069.So I will multiply R*1,00069=A distance in 100.000 thousand years later.

Time in zeit means t/17.3(unit billion years)
Normalized H means H/17.3(I guess).Which current value is 1.2 zeit^-1.


I understand the other parts.

H=1.2 zeit^-1=1/144% per million year or 1/14.4 billion year.
Is that mean H=1/14.4 billion years ?
Can you check my idea

Thank you.

 

[Jorrie]

If you have a distance like 1 km. And H is the growth rate of 10% per year. Then it is not true that after one year the distance is 1.1 km.

The way you calculate the distance after one year is you paste this into google:
e^(.1*1)
that is e^(rate*time) = e^(.1 per year x 1 year)

Marcus, my feeling is that the e-folding is complicating the understanding part a bit, especially for beginners. I would rather relate it to things that high school students know about, e.g. interest calculation and monetary growth. Granted, it uses an approximation, with discrete periods like months or years, which is not quite accurate for an instantaneous rate. But then, this whole approach is an approximation of the LCDM model, which is again an approximation of reality.

Because a million years is such a small time interval in the cosmological times we are working with here, I see no harm in using the calculation I showed to Ryan in comment: https://www.physicsforums.com/threa...simplified-math-comments.823929/#post-5173700. I agree that we should gradually introduce the e-fold idea of expansion.

 

[marcus]

...
Because a million years is such a small time interval in the cosmological times we are working with here, I see no harm in using the calculation I showed to Ryan in comment: https://www.physicsforums.com/threa...simplified-math-comments.823929/#post-5173700. I agree that we should gradually introduce the e-fold idea of expansion.

OK you are the best judge of the timing, I think. Eventually we have to get to the understanding that H is not tied to a particular interval of time like a year, or a million years, or something else, because it is a continuous growth rate. So the only way to compute it really precisely is e^Ht.
But the key word is "gradually introduce". So I will keep quiet and not mess things up by intruding.

One thing I'm curious about though. Who is the painter of the lovely avatar picture? I have the feeling that it is Spanish of maybe 18th or 19th century. But I don't really know, it could be painted anywhere in the world, in any century---I just have a feeling that it is a portrait of a Spanish lady by a contemporary of Goya or Ingres. Would Ryan be willing to tell us who the painter is, and where he found the image? Or even just the painter's name and I could look it up.