Approximate LCDM Expansion in Simplified Math (Part 2) - Comments

[Jorrie]

Jorrie submitted a new PF Insights post

Approximate LCDM Expansion in Simplified Math (Part 2)

Continue reading the Original PF Insights Post.

 

[RyanH42]

Great work :)

This time I don't have any questions,Thanks Marcus

 

[marcus]

I have to say, Ryan, that I actually learned quite a bit by trying to answer all your questions occasioned by Part 1 of Jorrie's Insights article. (We took those questions over to a Cosmology forum thread, A2z2Z, instead of the regular Insights comments section.)
I really must say that struggling with your persistent questions brought me some new understanding of the approach to basic cosmology that Jorrie and I have been working on. It's good to have someone around with that much curiosity and drive.

I'm looking forward to seeing Jorrie's Part 3, and hope it will be posted soon.

 

[marcus]

This could be seen as slightly off topic. I'm trying to think of ways to supplement Jorrie's treatment of the basic math of standard cosmic model. The treatment is concise and clear. How could we ADD stuff (like examples, exercises, graphics...) to make it more concrete.
For example in the A2z2Z thread Wabbit posted this:

==quote==
...because the redshift+1 of the CMB is estimated at 1090, the temperature of the hot gas at last scattering, that emitted the ancient background light, was 2.725*1090 ≈ 3000 kelvin
==endquote==
A pale orange hue then :

Unfortunately the current color is off-the chart, in the radio spectrum, so we cannot compare : (

https://www.physicsforums.com/threa...ndard-cosmic-model.811718/page-4#post-5121570
It makes it possible for a newcomer to imagine the original CMB light, on the day it stopped getting scattered and got loose to run freely---and it impresses the idea that expansion cools heat-glow light in proportion as it lengthens the wavelengths. this may or may not be worth adding as a footnote somewhere.
Or maybe not even adding, just having a link to it handy as a teaching resource. Wabbit may have other ideas as well.

If it is not too much clutter---or too off-topic---I will try to think up some supplemental material. I like examples that make cosmology more concrete.

 

[marcus]

Imagine you are an astronomer observing some patch of sky last night and some light comes in with stretch S=2.6
Imagine that light when it was just emitted by its home galaxy. Obviously the light is aimed in our direction because it eventually got here.
Try to picture the situation.
When it was first emitted what kind of progress towards us was it making?
Was it approaching us or was it being swept back by expansion?
At what speed was the distance of the home galaxy from us increasing?

 

[marcus]

Later that night you measured the spectrum of light from a different galaxy and found the stretch of that light was S = 1.65.
Think about the situation back when that light was emitted---the distance from that galaxy to the matter that eventually became us, the Earth as we know it, the stuff around us etc.---that distance was expanding, obviously, but was its speed of expansion slowing or speeding up?

Is this exercise too hard? Can someone suggest some more congenial, perhaps easier examples.

The thing is, if you know S then we have formulas for H(S) and D(S) in Jorrie's treatment (if not in part 2 in part 3) and v=HD
so we have a handle on the speed of that distance's expansion. "was its speed of expansion slowing or speeding up?" maybe the answer is simply "no".
: ^)

Still not sure if this kind of supplemental material is sufficiently on-topic. I collected some such "cosmology enrichment" stuff in the "hypersine" thread starting around post #39
https://www.physicsforums.com/threads/the-hypersine-cosmic-model.819954/page-2#post-5155318

 

[marcus]

Anybody, please tell me if you think these exercise problems are too hard, or don't belong here. We can always make examples and exercises easier, or eliminate them altogether. But we won't know if they belong here and are the right compatible level if we don't try. Here's one

Explain why this gives, as of today, the distance to the farthest galaxy we could reach with a signal sent today
$$ \int_0^1 ((s/1.3)^3 + 1)^{-1/2}ds$$
and explain why this other integral gives, again in terms of today's distance, the radius of the currently observable universe
$$ \int_1^\infty ((s/1.3)^3 + 1)^{-1/2}ds$$
that is the current distance to the farthest matter that can have emitted a signal which is arriving now.

What represents the present era, the "now", in these integrals? How would you change them so as to give the same quantities but for people in the past or future---the radius of observable for some people in the past---the farthest now reachable (cosmic event horizon) for some people in the future, etc.

these integrals are really easy to do at numberempire.com, it even remembers the integrand ((s/1.3)^3 + 1)^{-1/2} so you don't have to type it in each time. all you do is put in different limits and press calculate.
==============proceed at your own risk================
"For extra credit" what does this signify?$$ \int_0^\infty ((s/1.3)^3 + 1)^{-1/2}ds$$spoiler: the distance now to the farthest matter that will EVER be part of our observable. IOW the ultimate max of the "particle horizon" in what is called "comoving distance" ( jargon terms, professional technical vocabulary are often unhelpful, this might be a case of unhelpful jargon, but the idea of permanently assigning to each patch of matter a number which is its distance from us as of today turns out to be very handy)

 

[marcus]

Is there any merit to getting "transdisciplinary" about this and mixing some biohistory in with our cosmology?
Does it do anything for you to be told that the galaxy disk formed at t=0.29 and Earth formed at expansion age t=0.54 zeit
and that the first evidence of microbial life on Earth dates from t=0.59?
Or that our ancestors the fish did not appear until t=0.77, and even then they did not have jaws---had not even figured out how to have jaws!
As far as I'm concerned it's helpful because it helps me get an idea of the passage of time in cosmology to picture the passage of time in other developments like evolution.

 

[Jorrie]

Explain why this gives, as of today, the distance to the farthest galaxy we could reach with a signal sent today

\int_0^1 ((s/1.3)^3 + 1)^{-1/2}ds

You are running ahead of Insights Part 3 now.

It may be good to make readers think about the horizons in advance, but they are quite tricky to comprehend and I am still working on the best way to present them in concise form.

 

[marcus]

Good. I'll wait on horizons. Let me know if I should erase any spoilers or edit any of my posts here (intended to explore possible examples and supplemental stuff).

 

[marcus]

There's something slightly awkward about the Insights format.

Now that part 3 is posted at Insights, I would like to comment about two equations in part 2, namely equations 2.3 and 2.4. But Insights does not seem to have LaTex.

So I cannot comment in the regular Insights comment section. And in this thread we cannot see the original part 2, so I cannot use the REPLY function to obtain copies of equations 2.3 and 2.4. this means my comment is likely to be obscure because it won't be clear what I am talking about.

 

[marcus]

Basically equation 2.3 introduces the scale factor as a function of the "time then", t, namely S(t) = 1.3/sinh^2/3(1.5t).

And equation 2.4 gives you the distance integral with S as the variable and H(S) in the denominator.

As a comment or addendum it is also possible (though certainly not essential) at that point to give an alternative form of the distance integral with t as the variable and S(t) as the integrand. IOW cdt is the little step the light takes at time t, and S(t) is the factor by which that step is blown up or shrunk down between time t and the present. So the distance integral is the sum of little steps, each of which has been adjusted for expansion.

The two horizons become$$\int_.00001^.8 S(t) dt$$ $$\int_.8^\infty S(t) dt$$and the transition to the equation 2.4 form of the distance integral is a somewhat messy (but at least not mysterious) change of variable. The actual change-of-variable process is either footnote material or IMHO something hairy to be politely ignored. : ^)

 

[Jorrie]

So I cannot comment in the regular Insights comment section. And in this thread we cannot see the original part 2, so I cannot use the REPLY function to obtain copies of equations 2.3 and 2.4. this means my comment is likely to be obscure because it won't be clear what I am talking about.

Yea, there is a bit of 'disconnect' (incompatibility) between the Insights system and the normal PF editor. It is also difficult to copy and paste from PF into Insights, but there are workarounds.

You can copy and paste text and equations, but the equations are converted to plain text. I use the windows right-click function on the original equation to show the equation as Tex commands, which can then be copied either way. One need to put the 'tex' or 'itex' tags in by hand. Cumbersome, but workable.

 

[Jorrie]

As a comment or addendum it is also possible (though certainly not essential) at that point to give an alternative form of the distance integral with t as the variable and S(t) as the integrand.

Yes, time is a very intuitive variable for us humans, but unfortunately it is not one of the observables in cosmology. The time values we use are very model and parameter dependent. Time against S can also be obtained by a simple integration, similar to the ones discussed here. I am trying to give all derived quantities in terms of the observable (S, z, a and H₀) in as simple as possible a fashion.

I think the idea of showing some of the other relationships in end-notes is good (or perhaps in an extra appendix).

 

[marcus]

...
I think the idea of showing some of the other relationships in end-notes is good (or perhaps in an extra appendix).

Good. I could contribute a postscript of some sort. I don't want to intrude on your main text article. It wouldn't need to be attached.
I see the point of basing the main development on observable quantities, like S. Time is intuitive but elusive. But that would be an easy afterword-type thing for me to write---if you want---the two horizon integrals dt instead of ds.