Cosmo calculators with tabular output

[marcus]

Experience and conventional wisdom say that whenever there is a really good idea ready to makes its entrance it often occurs to several individuals or groups. That might be happening, or going to happen, with online cosmic model calculators. So I'm hoping to hear comments about this including if you have seen this new kind of tabular output calculator at other websites I don't know about.

It's very interesting and has a lot of teaching/learning potential. It goes beyond the one-shot format you get with Ned Wright or with Morgan's calculator.
http://www.astro.ucla.edu/~wright/CosmoCalc.html

The idea is that what you are really trying to do when you play around with a model of the expansion history is get an idea of the SHAPE OF COSMIC EVOLUTION. You want to grasp the overall shape of the expansion process.

So instead of just putting in one scalefactor or one z and getting just one row of the table that describes only one slice of the whole spacetime shebang, why not let the user put in a range (start to end) and a STEPSIZE and get out a TABLE giving the key dimensional quantities for a whole bunch of slices?

there are some interesting features, like the pear-shape or tear-drop shape (in proper or contemporary distance terms) of the light cone, that only stand out clearly when you see a tabulation with, say, ten or more rows.

And the fact that the Hubble expansion rate has been decreasing so rapidly for much of the time---which translates into the Hubble TIME (the reciprocal rate) increasing---but less rapidly now and as time goes on. You also see that when you look at a table.

So anyway, do others have some thoughts about this? Are there online tabulating cosmo calculators that you have used or know something about? Do they work for you and do something more for you than the one-shots?

 

[marcus]

I found one online!
http://dotastronomy.com/blog/2012/08/cosmology-calculator-os-x-widget/

It was posted 16 August 2012, just a couple of weeks ago, by an astrophysicist at Oxford named Brooke Simmons.
I don't especially like how Brooke and her friends implemented the idea, but it does have a kind of tabular output.

Some of the output columns are technical in nature as would be interesting to a specific line of astrophysics research, distances in parsecs, luminosity distance, comoving volume...
That's fine, it is what Brooke wanted. the general idea of tabular output is the main thing.
I think it might become popular (I sure as heck like it better than oneshot!)

 

[Jorrie]

I have been using various spreadsheets for cosmology simulations for many years. They are tabular by definition and very useful for painlessly plotting graphs. Spreadsheets do have some limitations, though. To mention a few:

1. They become cumbersome for huge ranges of input values at small intervals, as is usually required for accurate numerical integration.
2. They are not generally portable between systems; the user needs a compatible spreadsheet program.
3. They can become large in file size.

Web calculators solve the above problems, being directly runnable on any web browser. However, they require much more difficult programming in order to achieve nicely formatted tabular output - as I have discovered in an attempt to convert one of my spreadsheets into a HTML/Javascript program. I've got it to work, but there is always the danger of some data exception crashing the program and hanging up the user's computer. I'm not a real programmer and Javascript is surely not my cup of tea.

Nevertheless, after a few more rounds of testing between Marcus and myself, I think it may be time to 'publish' it here for broader testing. Maybe within a day or two...

Attached is a screenshot of the test model's inputs and outputs. It implements some of the concepts that Marcus recently discussed in his Balloon analogy sticky. The info (i) buttons present explanations of the in- and outputs. They are the result of a collaborative forum project.

 

[marcus]

... I think it may be time to 'publish' it here for broader testing. Maybe within a day or two...

YAY!
AFAICS tt's turning out to be a beauty. It'll be fun to share.

 

[marcus]

We never see anything when it was more than 5.8 billion lightyears away from us. Most of the stuff we see was quite a bit nearer (to our matter) when it emitted the light we are getting from it. It's an interesting fact. Just now, to "test drive" a draft version of the tabular output calculator Jorrie's working on, I set the stretch limits and step so that S would run from 2 to 3 in steps of 0.5, to illustrate the fact by generating this:

stretch S  scale a         age t[SUB]S[/SUB]    Hubbletime Y[SUB]S[/SUB]         D[SUB]S,now[/SUB]       D[SUB]S,then[/SUB]
                            (Gy)            (Gy)           (Gly)          (Gly)
   3	   0.33333	  3.36025	 4.88601	 17.2221	5.7406
   2.95	   0.33898        3.44332	 4.99916	 16.975	        5.7542
   2.9	   0.34483	  3.52978	 5.11636	 16.7221	5.7663
   2.85	   0.35088	  3.61982	 5.2378	         16.4633	5.7766
   2.8	   0.35714	  3.71363	 5.36364	 16.1983        5.7851
   2.75	   0.36364	  3.81147	 5.4941	         15.9268	5.7916
   2.7	   0.37037	  3.9135	 5.62934	 15.6488        5.7958
   2.65	   0.37736	  4.02002	 5.7696	         15.3639	5.7977
   2.6     0.38462	  4.13131	 5.91509	 15.0717	5.7969
   2.55	   0.39216	  4.24765	 6.06601	 14.7722	5.7931
   2.5	   0.4    	  4.36931	 6.22258	 14.465	        5.786
   2.45	   0.40816	  4.49665	 6.38503	 14.1499	5.7754
   2.4	   0.41667	  4.63006	 6.5536	         13.8264	5.761
   2.35	   0.42553	  4.76987	 6.7285	         13.4944	5.7423
   2.3     0.43478	  4.91651	 6.90998	 13.1535	5.7189
   2.25	   0.44444	  5.07046	 7.09825	 12.8033	5.6903
   2.2	   0.45455	  5.23215	 7.29353	 12.4436	5.6562
   2.15	   0.46512	  5.40217	 7.49605	 12.0738	5.6158
   2.1	   0.47619	  5.58099	 7.70598	 11.6938	5.5685
   2.05	   0.4878	  5.76931	 7.92351	 11.3031	5.5137
   2	   0.5    	  5.96774	 8.1488	         10.9013	5.4507

You can see the maximum distance of around 5.8 Gly appear rather clearly as 5.7977. The corresponding time is about 4 billion years into the the expansion process (so a bit less than 10 billion years ago.) Objects we see that are earlier or later in expansion history were all closer to us than that, when they emitted the light.

The 5.8 is so to speak the bulging waistline of the pear-shaped lightcone. The "maximum girth" radius. It's interesting to try to understand why that should correspond to where D_S,then is just equal to the Hubble radius cY_S

A sample exercise one might give students using this calculator could be something like this: "Light from a galactic cluster has stretch factor 2.65 (incoming wavelengths expanded by that factor) and appears to be about 1 degree wide. Given that angular width in the sky, how many lightyears wide is the cluster?"

 

[marcus]

A couple of useful graphs to go with the table output:
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg

Another thing that can be read off the table is which of the galaxies that we can see are currently receding >c.
The speed a distance is growing is found simply by dividing the distance by the contemporaneous Hubble time, which at present is 13.9 Gy. For example the current distance to a S=3 galaxy is 17.222 Gly. Dividing that by 13.9 Gy you get 17.222/13.9 c. The units work out: Gly/Gy = c.
So you can basically read recession speeds off the table, as D_now/Y_now.
The present is S=1, so Y_now and Y₁ mean the same thing: 13.9 Gy in this case.
The upshot is that current distances are growing faster than c for all S > 2.4.

You can also read off the table that D_S,then > cY_S for all S > 2.65.
In all those cases the distance back then (when the light now arriving was emitted) was growing faster than c.
To find the speed that such distances were increasing, just divide D_S,then > Y_S

 

[marcus]

I had a look at the first few hundred million years of expansion, with the draft calculator.
S=10 corresponds to the appearance of the first galaxies. So this is the period from 6 million years to 560 million years leading up to that.

stretch S  scale a  age t[SUB]S[/SUB]   Hubbletime Y[SUB]S[/SUB]  D[SUB]S,now[/SUB]     D[SUB]S,then[/SUB]
                     (Gy)       (Gy)        (Gly)     (Gly)
 
  200     0.005     0.00622   0.00938      43.9703   0.2199
  190     0.00526   0.00672   0.01013      43.8728   0.2308
  180     0.00556   0.00729   0.01099      43.7672   0.2433
  170     0.00588   0.00795   0.01198      43.6525   0.2567
  160     0.00625   0.00871   0.01312      43.5272   0.272
  150     0.00667   0.0096    0.01446      43.3894   0.2894
  140     0.00714   0.01065   0.01603      43.2373   0.3087
  130	  0.00769   0.01191   0.01792      43.0679   0.3312
  120	  0.00833   0.01343   0.02021      42.8776   0.3572
  110     0.00909   0.01531   0.02303      42.6621   0.3878
  100     0.01      0.01767   0.02657      42.4147   0.4241
   90     0.01111   0.0207    0.03113      42.1271   0.468
   80     0.0125    0.02471   0.03715      41.7872   0.5223
   70     0.01429   0.0302    0.0454       41.3768   0.5913
   60     0.01667   0.03807   0.05721      40.8676   0.6813
   50     0.02      0.05007   0.07522      40.2123   0.8042
   40     0.025     0.07001   0.10514      39.3244   0.9831
   30     0.03333   0.10783   0.16189      38.0232   1.2673
   20     0.05      0.19818   0.29741      35.8402   1.792
   10     0.1       0.56056   0.84035      30.9144   3.0914

Jorrie, I see from your next post that you have an efficient way of posting tables output from the calculator. In the above I had to align columns by hand, but eventually I will learn some technique for doing it more efficiently.
In any case, let's see how fast some of these distances are increasing, say for S=200.
D_now/Y_now = 43.97 Gly/13.9 Gy = 3.16 c
D_then/Y_then = 0.2199 Gly/0.00938 Gy = 23.44 c

I see that I may want to learn to use spreadsheets, have to think about that tomorrow. For now it's bedsheets for me.

 

[Jorrie]

I had a look at the first few hundred million years of expansion, with the draft calculator.
I haven't finished transferring the table so that it prints right. Have to do something else for an hour or so, but then will get back to this.
...

BTW, the data block is easier to copy via a spreadsheet, which can do some formatting for you. With MS Excel, I could copy the table straight out of the calculator, paste into an empty Excel sheet and set the format to a number of decimal places. I then copied into the CODE block of the Forum editor.
I do not know how to get the headers right, though - the CODE function has a mind of its own...

200	0.00500	0.00622	0.00938	43.97030	0.21990
190	0.00526	0.00672	0.01013	43.87280	0.23080
180	0.00556	0.00729	0.01099	43.76720	0.24330
170	0.00588	0.00795	0.01198	43.65250	0.25670
160	0.00625	0.00871	0.01312	43.52720	0.27200
150	0.00667	0.00960	0.01446	43.38940	0.28940
140	0.00714	0.01065	0.01603	43.23730	0.30870
130	0.00769	0.01191	0.01792	43.06790	0.33120
120	0.00833	0.01343	0.02021	42.87760	0.35720
110	0.00909	0.01531	0.02303	42.66210	0.38780
100	0.01000	0.01767	0.02657	42.41470	0.42410
90	0.01111	0.02070	0.03113	42.12710	0.46800
80	0.01250	0.02471	0.03715	41.78720	0.52230
70	0.01429	0.03020	0.04540	41.37680	0.59130
60	0.01667	0.03807	0.05721	40.86760	0.68130
50	0.02000	0.05007	0.07522	40.21230	0.80420
40	0.02500	0.07001	0.10514	39.32440	0.98310
30	0.03333	0.10783	0.16189	38.02320	1.26730
20	0.05000	0.19818	0.29741	35.84020	1.79200
10	0.10000	0.56056	0.84035	30.91440	3.09140

Edit: will see if calculator outputs can be formatted so as to make it easier to copy over.
Copying blocks of output into a spreadsheet is also handy for producing graphs...

 

[Jorrie]

...
Nevertheless, after a few more rounds of testing between Marcus and myself, I think it may be time to 'publish' it here for broader testing. Maybe within a day or two...

Attached is a screenshot of the test model's inputs and outputs. It implements some of the concepts that Marcus recently discussed in his Balloon analogy sticky. The info (i) buttons present explanations of the in- and outputs. They are the result of a collaborative forum project.

OK, here is the link to an 'alpha-test' version of the CosmoLean calculator. The 'lean' refers to minimal output parameters in tabular form, based on Marcus's ideas in the Balloon analogy sticky thread. His main emphasis was ease of use and educational utility.

The interface is mainly self-explanatory through its info popups, but here are a few introductory remarks.

• It works strictly for the spatially flat LCDM case, self-adjusting the values of the density Omegas from the three inputs: present Hubble time, long-term (constant) Hubble time and the redshift for matter-radiation density equality.
• 'Stretch' is a factor coined collaboratively for the inverse of the scale factor, i.e. S = 1/a. It is the factor by which distances 'stretched' while light was on its way to us. It is obviously also just z+1, but this has a decidedly Doppler shift connotation, which is not quite what cosmological redshift is about. Also, z+1 appears multiple times in many cosmology equations, making them slightly more awkward to write than is strictly necessary.
• The inevitable round-off errors of numerical integration are present, but they are well within 1% for the intended range of S =1 to 3500, well past matter-radiation equality. It could work for higher S, but it becomes quite slow then. It goes through two 40,000 step loops to achieve this accuracy.
• You will get a good idea of the workings if you click 'Calculate' with the default values. Then play around, but do not make S_Step too small when asking for a large range - you may have to wait quite some time...

Marcus and I would appreciate comments for improving the info texts and other features, provided we keep things fairly simple. Some updates, like exception checking/reporting, changeable number of decimals per column and perhaps optional units are considered at present.

 

[marcus]

Hey congratulations!

I just saw your post. This is a really fine achievement. And I can say that because I actually had very little to do with it.

Not realizing it was out, I just used the preproduction version to get a table of times, expansion rates, and distances from the era when the first galaxies formed, up to the present, and posted it in another thread. It worked really neatly. I didnt have to align columns by hand when I copy-pasted the table.

I like very much the simplicity and intuitiveness. Only three parameters to set:
two Hubbletimes (reciprocal expansion rates)---the now and the future limit ones
and the scale at which radiation and matter densities reach parity.

And I really really like the idea of tabular output, for a cosmology calculator. You can see things when you look at a whole progression that you just don't get with a one-shot.

Good work.

 

[marcus]

We should post some hints on ways to use the CosmoLean calculator.

For example, how to use it as a one-shot. Suppose you want data on just one stage in past history---say when scale was 1/1090, or when reciprocal scale (stretch) was 1090.

You can put 1090 in for the upper limit, and 1000 in for the step.


It is quick and just gives a table with two entries:
one for S=1090
and one for S=90

It's a very good feature that it builds the table working down in steps of the size specified. The stepsize must always be smaller than the overall range (between upper and lower limits) but if there is room for only one step, then so be it. You just get a table with two rows.

So in this case, leave the "start" box alone, where it gives the lower limit of the range. Let that stay S=1
which represents the present era.
Just change the "end" box to 1090
and make the step large enough so that there is room for only one more step in the whole range. Something like 1000 will do.

====================

But in this case I would actually INVITE you to get MORE information about your one-shot value: let the calculator tell you how much or how little difference it makes whether you say S=1090 or 1089
In effect, put in a confidence interval for the scale
EDITED AT JORRIE'S SUGGESTION:
say lower limit = 1080
upper limit = 1090
step = 5

Then it teaches you that things like times, Hubble expansion rates, distance now, distance then actually change very little if you make a small change around 1090. And in fact it tells you just how much these things vary in that neighborhood.

That's part of understanding a number. So this calculatory can be used as a oneshot calculator and it is actually a BETTER oneshot. It is better e.g. than Ned Wright oneshot because it tells us something which his does not tell. Namely how much the outputs wiggle if you wiggle the input. Which we ought to know as well as the outputs themselves.

=======

So far I know of only two online tabular output cosmo calculators. I would like to hear of more if anyone knows. I don't mean do-it-yourself Java or spreadsheets, I mean ready-mades. Friendly to the clueless. Please let us know if you find others!

Besides the one here there is the one posted by Brooke Simmons' at Oxford. Somebody should write her or post a comment to her blog.
Jorrie's http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A3.html
Brooke's http://dotastronomy.com/blog/2012/08/cosmology-calculator-os-x-widget/

 

[marcus]

This CosmoLean machine is a real pleasure to use. I put the link in my signature. So far my favorite output table is where you put
start S = 1 (i.e. present)
end S = 10 (i.e. first galaxies forming, distances 1/10 of today size)
step = 0.33333 (five digits is enough to get effective step of 1/3)
then what you get is this:

S=1/a    scalefactor a    time(Gy)   Hubbletime(Gy)   D[SUB]now[/SUB](Gly)      D[SUB]then[/SUB](Gly)

10.00	0.100000	0.558619	0.839348	30.904551	3.090455
9.67	0.103448	0.587799	0.883047	30.617708	3.167349
9.33	0.107143	0.619654	0.930686	30.315192	3.248056
9.00	0.111111	0.654446	0.982733	29.996387	3.332932
8.67	0.115385	0.692615	1.039801	29.659359	3.422234
8.33	0.120000	0.734549	1.102548	29.303064	3.516368
8.00	0.125000	0.780996	1.171897	28.923900	3.615487
7.67	0.130435	0.832503	1.248777	28.520615	3.720080
7.33	0.136364	0.889918	1.334397	28.090224	3.830485
7.00	0.142857	0.954152	1.430165	27.630118	3.947160
6.67	0.150000	1.026561	1.537915	27.135608	4.070341
6.33	0.157895	1.108514	1.659755	26.603238	4.200511
6.00	0.166667	1.201987	1.798433	26.027216	4.337869
5.67	0.176471	1.309229	1.957280	25.402101	4.482723
5.33	0.187500	1.433317	2.140615	24.720163	4.635030
5.00	0.200000	1.578263	2.353993	23.971943	4.794388
4.67	0.214286	1.749255	2.604580	23.146333	4.959928
4.33	0.230769	1.953045	2.901717	22.230355	5.130081
4.00	0.250000	2.199343	3.258071	21.205492	5.301372
3.67	0.272727	2.501266	3.690535	20.049940	5.468165
3.33	0.300000	2.877818	4.222240	18.734447	5.620333
3.00	0.333333	3.356917	4.884836	17.220673	5.740223
2.67	0.375000	3.980585	5.721191	15.458441	5.796914
2.33	0.428571	4.814342	6.787256	13.381146	5.734775
2.00	0.500000	5.964059	8.147995	10.900901	5.450448
1.67	0.600000	7.604379	9.852421	7.910657	4.746392
1.33	0.750000	10.030831	11.858689	4.298519	3.223887
1.00	0.999999	13.754769	13.899959	0.000026	0.000026

Now here's a neat thing: we can read off the COMOVING HUBBLE RADIUS at various past epochs from this. YOU JUST HAVE TO MULTIPLY S TIMES THE HUBBLETIME that corresponds to that stretch S!
I like this feature. the output is lean but also rich in possibilities.
For example for S=10 the Hubbletime is 0.84 Gy, so you get 8.4 Gly
and for S = 1.67 the Hubbletime is 9.85 Gy, so by multiplying you get 16.4 Gly.
Now to check that we can go to Lineweaver's figure 1 because he plots curves for things like the lightcone and the Hubble radius in comoving distance.
And it checks! You see that 1/1.67 = 0.6 and look at the bottom strip of the figure
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
at the level marked scale = 0.6, and behold, the comoving distance of the Hubble radius is about 16 Gly.
And at the level marked scale = 0.1, the Hubble radius should be about 8.4 according to the calculator's table, and so it is.

 

[Jorrie]

...
But in this case I would actually INVITE you to get MORE information about your one-shot value: let the calculator tell you how much or how little difference it makes whether you say S=1090 or 1089
In effect, put in a confidence interval for the scale

say lower limit = 1088
upper limit = 1091
step = 1

Then it teaches you that things like times, Hubble expansion rates, distance now, distance then actually change very little if you make a small change around 1090. And in fact it tells you just how much these things vary in that neighborhood.

This is a cool idea, but unfortunately not one guaranteed to work...
On my Firefox browser, it goes into a loop that times out and gives no outputs. The reason can be traced to internal round-off errors that are different for different browsers. The S-values are converted to a=1/S internally and then the differences are very small for large S and small spans.

To be on the safe side, when S is large, make the difference between S_end and S_start at least 5; it takes about the same time and will work every time with a step of 1.

Bugs/limitations like this one is bound to crop up for some time to come. This is what alpha-testing is all about...

 

[marcus]

I edited my previous post in conformance. Is this OK?

EDITED AT JORRIE'S SUGGESTION:
say lower limit = 1080
upper limit = 1090
step = 5

That should also give an intuitive feel for how much the numbers are changing when you wiggle the input.

But in fact, Jorrie, with my browser (on Mac notebook) I don't need to take that precaution. I just put in this input:
start 1080
end 1090
step 1
and it did not hang.
It said this table:

1090.00 0.000917 0.000381 0.000642 45.904946 0.042115
1089.00 0.000918 0.000382 0.000643 45.904303 0.042153
1088.00 0.000919 0.000382 0.000644 45.903659 0.042191
1087.00 0.000920 0.000383 0.000645 45.903014 0.042229
1086.00 0.000921 0.000384 0.000646 45.902368 0.042267
1085.00 0.000922 0.000384 0.000647 45.901721 0.042306
1084.00 0.000923 0.000385 0.000648 45.901074 0.042344
1083.00 0.000923 0.000385 0.000649 45.900425 0.042383
1082.00 0.000924 0.000386 0.000650 45.899775 0.042421
1081.00 0.000925 0.000387 0.000651 45.899124 0.042460
1080.00 0.000926 0.000387 0.000652 45.898473 0.042499

The response was instantaneous.
=================

It is a nice feature that the table in the previous post
that you get for
start 1
end 10
step 0.33333
is both simple and rich in what you can read off it.
I was talking about this in the other thread and showed how you can get the lightcone in proper distance
and the lightcone in comoving distance.
You can also get the Hubble radius for past epochs, in comoving distance
and all these things agree with the curves that Lineweaver has plotted in his Figure 1.

another thing we can get off the table is the ANGLE that something makes in the sky, if it is some given real size, like say a cluster that is 100 million lightyears across. What angle it makes in the sky will depend on the scale of the era in which it was living and emitting the light that we are getting from it. And we can tell that scale from the stretch of the light itself.

So it's simple and rich in possibilities, and I like how it meshes with Lineweaver figure 1, which itself is a really enlightening graphic about expansion history at a quantitative level.

 

[Jorrie]

I edited my previous post in conformance. Is this OK?

Yes, that will work fine.

I will attempt to build some warning into the code when such 'infinite loops' happen. Next update should be out in a week or so.

 

[Jorrie]

But in fact, Jorrie, with my browser (on Mac notebook) I don't need to take that precaution. I just put in this input:
start 1080
end 1090
step 1
and it did not hang.

Neither does Firefox with those inputs. It is when end - start = step (with large S) that hangups may occur.

 

[marcus]

Neither does Firefox with those inputs. It is when end - start = step (with large S) that hangups may occur.

So the rule should be to make the range at least 5, and the step no larger than 1, to be safe. Correct me if I'm wrong Have to go out but will check back in a couple of hours.

 

[Jorrie]

So the rule should be to have at least 3 steps? Or 5 steps?
When venturing into high S territory please make the step size less than some fraction of the range, like 1/3 or 1/5?

It also depends on how high is high S. It looks like problems start to occur when a single step represents some 0.1% of S. The problem may however be solved in the next update, e.g. by making it small enough or by catching and preventing it from causing trouble.

I am working on flexible rounding of output column data; it's working, but a few issues still prevent it from being released.

 

[marcus]

Since we just turned a page, I will bring forward the sample table output from post#12 earlier and also copy some relevant comment. At this point we are mostly talking about how to use the new tabular-output calculator. It can be used basically as a one-shot if that is all you want but there are things you can see from a table. Also it's nice having the input be a range of (reciprocal) scale.

===quote Jorrie; 4059998===
... I also prefer the stretch or scale factor over time for more than one reason. Firstly, it is relatively easy to visualize the matter-radiation equality epoch at some 1/3350 th of the present scale, but how easy is it to visualize 50 thousand years on a scale of 14 billion years?

Secondly, cosmic models run more efficiently with scale factor as independent variable; we know the limits in advance, being 'a' from near 0 to 1, with the upper limit model independent. Time runs from near zero to some unknown time today, which is model dependent.
===endquote===

===marcus;4060517===
This is a clear statement of motivation and could be included in an online "user's booklet" for the CosmoLean if there were one. Another thing that would be nice in such a booklet would be this figure:
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
Because when you make a table like the one I just posted some of the columns correspond to curves in the figure.

For example look at the middle strip of the figure. the D_now column corresponds to the LIGHTCONE curve. That spreads farther and farther out as you go back further to smaller scalefactor. that is because D_now is the same as comoving distance, and you can see that scalefactor is the measure plotted along the righthand side of the strip.

As a check, looking back at post #12 you see from the table that by the time you get down to scale 0.1 the comove distance of the lightcone should be around 31 Gly. So let's look at Lineweaver's figure and see.

Yes. It checks. Lineweaver's 2003 parameters are not exactly the same as the 2010 ones so the plot does not exactly agree but it's pretty close. You can see the agreement even better on the lower strip, which also has comoving distance. but has the scalefactor marks more spread out. It is easier to find scale=0.1 on the righthand edge of the strip (i.e. S=10)

Also the D_then column of the table in post #12 should correspond to the lightcone in the TOP strip because in that one the distance coordinate is PROPER distance.

The lightcone should bulge out to 5.8 Gly at around scale 0.375 (S=2.666) and then should be back to 3 Gly by the time it gets to scale 0.1 (S=10). So let's check. Well the figure is a bit cramped and smudgy but it looks about right. there is only a tick-mark at proper distance 10 Gly, so you have to judge by eye where 5.8 is.
===endquote===

I will go fetch a copy of that sample output, so readers can see what we're talking about.

 

[marcus]

I'll bring forward post#12 of previous page
===quote marcus;4060562===
So far my favorite output table is where you put
start S = 1 (i.e. present)
end S = 10 (i.e. first galaxies forming, distances 1/10 of today size)
step = 0.33333 (five digits is enough to get effective step of 1/3)
then what you get is this:

S=1/a    scalefactor a    time(Gy)   Hubbletime(Gy)   D[SUB]now[/SUB](Gly)      D[SUB]then[/SUB](Gly)

10.00	0.100000	0.558619	0.839348	30.904551	3.090455
9.67	0.103448	0.587799	0.883047	30.617708	3.167349
9.33	0.107143	0.619654	0.930686	30.315192	3.248056
9.00	0.111111	0.654446	0.982733	29.996387	3.332932
8.67	0.115385	0.692615	1.039801	29.659359	3.422234
8.33	0.120000	0.734549	1.102548	29.303064	3.516368
8.00	0.125000	0.780996	1.171897	28.923900	3.615487
7.67	0.130435	0.832503	1.248777	28.520615	3.720080
7.33	0.136364	0.889918	1.334397	28.090224	3.830485
7.00	0.142857	0.954152	1.430165	27.630118	3.947160
6.67	0.150000	1.026561	1.537915	27.135608	4.070341
6.33	0.157895	1.108514	1.659755	26.603238	4.200511
6.00	0.166667	1.201987	1.798433	26.027216	4.337869
5.67	0.176471	1.309229	1.957280	25.402101	4.482723
5.33	0.187500	1.433317	2.140615	24.720163	4.635030
5.00	0.200000	1.578263	2.353993	23.971943	4.794388
4.67	0.214286	1.749255	2.604580	23.146333	4.959928
4.33	0.230769	1.953045	2.901717	22.230355	5.130081
4.00	0.250000	2.199343	3.258071	21.205492	5.301372
3.67	0.272727	2.501266	3.690535	20.049940	5.468165
3.33	0.300000	2.877818	4.222240	18.734447	5.620333
3.00	0.333333	3.356917	4.884836	17.220673	5.740223
2.67	0.375000	3.980585	5.721191	15.458441	5.796914
2.33	0.428571	4.814342	6.787256	13.381146	5.734775
2.00	0.500000	5.964059	8.147995	10.900901	5.450448
1.67	0.600000	7.604379	9.852421	7.910657	4.746392
1.33	0.750000	10.030831	11.858689	4.298519	3.223887
1.00	0.999999	13.754769	13.899959	0.000026	0.000026

Now here's a neat thing: we can read off the COMOVING HUBBLE RADIUS at various past epochs from this. YOU JUST HAVE TO MULTIPLY S TIMES THE HUBBLETIME that corresponds to that stretch S!
I like this feature. the output is lean but also rich in possibilities.
For example for S=10 the Hubbletime is 0.84 Gy, so you get 8.4 Gly
and for S = 1.67 the Hubbletime is 9.85 Gy, so by multiplying you get 16.4 Gly.
Now to check that we can go to Lineweaver's figure 1 because he plots curves for things like the lightcone and the Hubble radius in comoving distance.
And it checks! You see that 1/1.67 = 0.6 and look at the bottom strip of the figure
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
at the level marked scale = 0.6, and behold, the comoving distance of the Hubble radius is about 16 Gly.
And at the level marked scale = 0.1, the Hubble radius should be about 8.4 according to the calculator's table, and so it is.
===endquote===

So to summarize, what we're seeing is that you can read stuff off the table that corresponds to the curves in Lineweaver's Figure 1
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
namely
the lightcone in proper distance
the lightcone in comoving (now) distance
the Hubble radius in proper distance (just interpret the Hubbletime Gy as Gly)
the Hubble radius in comoving distance (just multiply by S)

These things are a cinch to read directly off the table. Other things you can get from the table are "recession speeds" (now or then)as multiples of the speed of light. Just divide the then distance by the then Hubbletime, or divide the now distance by the now Hubbletime. They should not be thought of as speeds of anything traveling in the usual sense, but as the speeds distances are growing.

Further things you can get from the table are the angles which something of a given size makes in the sky (which will be found using the table, from its wavelength stretch).

 

[Jorrie]

... It looks like problems start to occur when a single step represents some 0.1% of S. The problem may however be solved in the next update, e.g. by making it small enough or by catching and preventing it from causing trouble.

I am working on flexible rounding of output column data; it's working, but a few issues still prevent it from being released.

The 'single-step problem' has been solved in CosmoLean_A17 and the 'few issues' with the flexible rounding of column data are gone as well, or so I hope. Please try it out and report any anomalies.

The most important differences are:

1. The info-popups have been mostly reworded and include comments as received in PMs.
2. The stretch range inputs arranged to be more consistent with the output table, from highest to lowest stretch.
3. Some 'logic' built into the input processor so that 'one-shot' outputs are intuitively achieved by either making s_step zero, which gives output for s_upper only; or by making s_upper and s_lower equal to each other, irrespective of s_step.
4. The number of decimals (rounding) of column data are adjustable individually. Becomes active on clicking Calculate and will remain so until changed again, reset clicked, or the page is refreshed.
5. Overall accuracy has been improved by resolving some coding issues. It now seems to work accurately up to s =10 000.
6. Some input validation and protection against crash of program are included. More to be considered.
7. On the drawing board: "into the future" (s < 1).

 

[Lino]

Marcus / Jorrie, Would you mind posting the link again please? I'm afraid that the mobile version of PF does not include your signatures so I'm having difficulty finding it.

Regards,

Noel.

 

[Jorrie]

Marcus / Jorrie, Would you mind posting the link again please? I'm afraid that the mobile version of PF does not include your signatures so I'm having difficulty finding it.

Regards,

Noel.

Did you mean the link to the calculator? If so, it is in my prior post, labelled: CosmoLean_A17, but I copied it here. The one in Marcus's sig may still be the old release.

 

[Lino]

Thanks Jorrie. Much appreciated.

Regards,

Noel.

 

[marcus]

The new version is a pleasure to use.
We should accumulate a bit of "user manual" type information like that in your post#21, three or four posts back.

Putting step=0 makes it very simple to use as a one-shot.
It's mostly self-explanatory how to use it, so not very much by way of "user manual" seems necessary. but at least the hint about setting step to zero.

The feature of deciding on how many decimal places to show is quite nice. It rounds off for you. I like seeing 3-place precision but knowing I'm riding on 6-place (like a new set of tires on the car, you just feel better.)

Visually clean, sufficient but just what's essential.

 

[Jorrie]

The new version is a pleasure to use.
We should accumulate a bit of "user manual" type information like that in your post#21, three or four posts back.

Good idea. Will add a on-page popup in next update with some general tips.

BTW, work-shopping the 'back to the future' options on a spreadsheet, it struck me that things like D_now and D_then are then ill defined. For the past is is easy; we think in terms of a source observed at (say) stretch 2 and we ask the questions: how far was it at time of emission and how far is it now?

The equivalents for the future are more obscure. Since we cannot observe anything from the future, we may have to think in terms of emitting some signal now and answer the question: How far must an observer be from us to receive that signal at a stretch of (say) 2, both now and then?

Or is there a better way?

PS: Lean gives for s=2, D_now=11.1 Gly and D_then half of this. For a future s=1/2, I get via spreadsheet: D_now=7.4 and D_then twice this.

 

[marcus]

I think the present Lean format is perfect WITHOUT a future extension because it communicates easily and directly. Beginners can get the concepts without trouble.

I would not try to add on to the existing Lean A17 version. It might diminish its value as a beginner's cosmology tool.

But the future is really interesting and a new format like "Lean⁺" with future would be new in cosmology calculators AFAIK. I will to help think it out and perhaps you can bounce things off me and use my reactions.
=====================

The way I think about it, there are two ways it could be. One way is about what galaxies could we send a signal to in future? How far away is target galaxy now? How far away will it be when the signal gets there---say, in S=1/2, when distances are twice what they are today?

So the Distance labels are more general they are D_emit and D_receive.

Or maybe call it D_send and D_receive.

If it is the PAST segment of the table then these are just the same as Dthen and Dnow. The distance from us back then when it sent us the light and the distance now on the day we get it.

But in the FUTURE segment of the table Dsend is the distance from us to the target on the day we send the signal
and Dreceive is the distance from us to the target on the day they get the signal.

Dsend (the distance of the target today when we send the signal) cannot be more than around 16 billion ly (proper) or it will never get there. That's what event horizon means.

However Dreceive (the proper distance of the target when the message finally gets there) can be very very far.
I don't know the present event horizon---somewhere around 16 and eventually will converge to 16.3. But whatever it is, say it exactly 16. then the closer the galaxy is to 16 when we send the message the longer it is going to take for the message to reach target, and the farther away target will be when it arrives. I don't think there is any theoretical limit on how big Dreceive can be.
===============

The other way I don't see how to implement and can't tell on short notice whether it's good or not. The other way is what I think LINEWEAVER does in bottom panel of his Figure 1. that is he PUTS THE EARTH INFINITELY FAR IN THE FUTURE and looks back in a somewhat similar way as before
So then every finite S, not only S>1 but also S=1 and S<1 corresponds to some era which is in the past of the Earth at this imaginary infinite future. I think now that it would be unwise to attempt at this point because it appears to involve "conformal time" and only comoving distance (not proper) seems well-defined. the infinite future is not a definite time so proper time might break down as a concept. It might be a big headache to try this second way. But it is what Lineweaver seems to do in one of the panels in his figure 1. Sorry about the unprepared response. Need more time to think.

 

[Jorrie]

The 'single-step problem' has been solved in CosmoLean_A17 and the 'few issues' with the flexible rounding of column data are gone as well, or so I hope. Please try it out and report any anomalies.
...

The latest version (as in Marcus's signature) is CosmoLean_A20, which adds an 'Introduction' button with some hints for usage. It is supposed to be fairly stable now and it is perhaps time to give an idea of the underlying formulas and conventions. It follows the development of the 13.9/16.3 simplified model proposed by Marcus, but with inclusion of the early stage radiation energy density.

The basic input parameters are:
present Hubble time Y_{now}, long term Hubble radius Y_{inf} and the redshift for radiation/matter equality z_{eq}. Since the factor z + 1 occurs so often, an extra parameter S = z + 1 is defined. From these, the Friedman equation terms for the cosmological constant, radiation and matter can respectively be found for a perfectly flat LCDM model.

\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, <br /> \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \,<br /> \Omega_m = S_{eq}\Omega_r

The 'heart' of any simple cosmological calculator is the time variable Hubble constant H, which comes from the Friedman equation as:

H = H_0 \sqrt{\Omega_\Lambda + \Omega_r S^4 + \Omega_m S^3}

For perfect flatness, it can be expressed as

H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}

It can be interpreted in terms of the "13.9/16.3 factors" as follows: \Omega_\Lambda = 0.7272 and \Omega_m (1+S/S_{eq})= 0.2728, which of course sum to 1 (required for perfect flatness). It also shows at a glance how the influence of the various energy densities changes with S. Since S_eq ~ 3350, radiation dominated when S > 3350 and matter dominated for S < 3350, until such time as \Omega_m (1+S/S_{eq}) &lt; 0.7272, when the cosmological constant started to dominate the equation.

From H, the following calculator outputs are readily available:

Hubble time Y(a) = 1/H

Cosmic time T(S) = \int_{S}^{\infty}{\frac{dS}{S H}}

Proper distances to a source at stretch S, "now" and "then" respectively,

D_{now} = \int_{1}^{S}{\frac{dS}{H}},\ \ \ \ D_{then} = \frac{D_{now}}{S}

The integration for T(S) to S_infinity is problematic, but is usually stopped at a suitably high S (effectively close enough to time zero).

In principle, the equations can be used for projecting into the future as well. This has been "secretly" sneaked into version A20. If you want to try it out, enter 1 into S_upper and 0.1 into both S_lower and S_Step. Note the time going to some 50 Gy, T_Hubble to around 16.3 Gy and the distances to negative values.

As Marcus has pointed out before, D_now for this scenario is the present distance to a target that will receive our signals with a wavelength stretch S at future time T(a). D_then means the proper distance of the target when they eventually receive our signal, obviously 1/S times farther.

This 'trial feature' can go down to S = 0.01 in steps of 0.01, but not lower at this time.

 

[Jorrie]

Tabular Cosmo calculator with Event Horizon

...
In principle, the equations can be used for projecting into the future as well. This has been "secretly" sneaked into version A20. If you want to try it out, enter 1 into S_upper and 0.1 into both S_lower and S_Step. Note the time going to some 50 Gy, T_Hubble to around 16.3 Gy and the distances to negative values.

As Marcus has pointed out before, D_now for this scenario is the present distance to a target that will receive our signals with a wavelength stretch S at future time T(a). D_then means the proper distance of the target when they eventually receive our signal, obviously 1/S times farther.

This 'trial feature' can go down to S = 0.01 in steps of 0.01, but not lower at this time.

Hand-in-hand with the 'future option' goes the cosmic event horizon. It has been included in CosmoLean_A22.

For completeness, I'll repeat the prior post's equations together with D_CEH.

Given present Hubble time Y_{now}, long term Hubble time Y_{inf} and the redshift for radiation/matter equality z_{eq}. Since the factor z + 1 occurs so often, an extra parameter S = z + 1 = 1/a is defined, making the equations neater.

\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, <br /> \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \,<br /> \Omega_m = S_{eq}\Omega_r

Hubble parameter
H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}

Hubble time, Cosmic time
Y = 1/H, \ \ \, <br /> T = \int_{S}^{\infty}{\frac{dS}{S H}}

Proper distance 'now', 'then' and cosmic event horizon
D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \, <br /> D_{then} = \frac{D_{now}}{S}, \ \ \, <br /> D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}

This essentially means integration for S from zero to infinity, but practically it has been limited to 10^{-7} &lt; S &lt; 10^{7} with quasi-logarithmic step sizes, e.g. a small % increase between integration steps.

 

[Jorrie]

Hand-in-hand with the 'future option' goes the cosmic event horizon. It has been included in CosmoLean_A22.

Following some tests and hints from others, some changes have been made to the user interface and it is now at version Cosmolean_A25.

The troublesome "range check pop-ups" of the input boxes have been replaced by a feature that simply changes the color of the "range text" to red when a problem is detected. Out of range values and non-numerical inputs are actually accepted and you can calculated with them if you wish. AFAIK, the calculator do not crash, but may throw out funny values.

At the same time the conventional values of Ho, Omega_Lambda and Omega_matter is calculated and displayed so that the influence of your (changed) input becomes visible to you. Also, 'down-step' values S_step <= 0 now have the special meaning of setting the number of steps between S_upper and S_lower to the rounded-abs of the value given. A 'one-shot' (single row) is hence still produced by S_step = zero.

Here is what it the input section now looks like.

The outputs with 100 steps:

S	a	T	T_Hub	D_now	D_then	D_hor
10.00	0.100	0.559	0.839	30.890	3.089	4.653
9.90	0.101	0.567	0.852	30.805	3.112	4.691
9.80	0.102	0.576	0.865	30.720	3.135	4.730
9.70	0.103	0.585	0.878	30.633	3.158	4.770
9.60	0.104	0.594	0.892	30.544	3.182	4.810
9.50	0.105	0.603	0.906	30.454	3.206	4.851
9.40	0.106	0.613	0.921	30.363	3.230	4.893
9.30	0.108	0.623	0.936	30.270	3.255	4.936
9.20	0.109	0.633	0.951	30.176	3.280	4.979
9.10	0.110	0.643	0.966	30.080	3.305	5.023
9.00	0.111	0.654	0.983	29.983	3.331	5.068
8.90	0.112	0.665	0.999	29.884	3.357	5.114
8.80	0.114	0.677	1.016	29.783	3.384	5.161
8.70	0.115	0.688	1.034	29.681	3.411	5.208
8.60	0.116	0.700	1.052	29.577	3.439	5.257
8.50	0.118	0.713	1.070	29.471	3.467	5.306
8.40	0.119	0.726	1.089	29.363	3.495	5.356
8.30	0.120	0.739	1.109	29.253	3.524	5.407
8.20	0.122	0.752	1.129	29.142	3.553	5.460
8.10	0.123	0.766	1.150	29.028	3.583	5.513
8.00	0.125	0.781	1.171	28.912	3.613	5.567
7.90	0.127	0.795	1.194	28.794	3.644	5.623
7.80	0.128	0.811	1.217	28.673	3.675	5.679
7.70	0.130	0.827	1.240	28.550	3.707	5.737
7.60	0.132	0.843	1.265	28.425	3.739	5.796
7.50	0.133	0.860	1.290	28.298	3.772	5.856
7.40	0.135	0.877	1.316	28.168	3.805	5.917
7.30	0.137	0.895	1.343	28.035	3.839	5.980
7.20	0.139	0.914	1.371	27.899	3.873	6.044
7.10	0.141	0.933	1.399	27.761	3.908	6.110
7.00	0.143	0.953	1.429	27.620	3.944	6.177
6.90	0.145	0.974	1.460	27.475	3.980	6.245
6.80	0.147	0.996	1.492	27.328	4.017	6.315
6.70	0.149	1.018	1.525	27.177	4.054	6.387
6.60	0.151	1.041	1.560	27.023	4.092	6.460
6.50	0.154	1.065	1.596	26.866	4.131	6.535
6.40	0.156	1.090	1.633	26.704	4.170	6.612
6.30	0.159	1.116	1.671	26.539	4.210	6.691
6.20	0.161	1.143	1.711	26.370	4.251	6.771
6.10	0.164	1.171	1.753	26.197	4.292	6.854
6.00	0.167	1.201	1.797	26.020	4.334	6.938
5.90	0.169	1.231	1.842	25.838	4.376	7.025
5.80	0.172	1.263	1.889	25.652	4.420	7.114
5.70	0.175	1.296	1.938	25.461	4.463	7.205
5.60	0.178	1.331	1.990	25.265	4.508	7.298
5.50	0.182	1.367	2.043	25.063	4.553	7.394
5.40	0.185	1.405	2.099	24.856	4.599	7.492
5.30	0.189	1.445	2.158	24.644	4.646	7.593
5.20	0.192	1.486	2.219	24.425	4.693	7.697
5.10	0.196	1.530	2.283	24.200	4.741	7.804
5.00	0.200	1.576	2.351	23.969	4.789	7.913
4.91	0.204	1.624	2.421	23.731	4.838	8.026
4.81	0.208	1.675	2.495	23.485	4.887	8.142
4.71	0.213	1.728	2.573	23.232	4.937	8.261
4.61	0.217	1.784	2.655	22.971	4.988	8.383
4.51	0.222	1.843	2.742	22.701	5.039	8.509
4.41	0.227	1.906	2.833	22.423	5.090	8.639
4.31	0.232	1.972	2.929	22.135	5.141	8.773
4.21	0.238	2.042	3.030	21.837	5.192	8.910
4.11	0.244	2.116	3.137	21.529	5.244	9.052
4.01	0.250	2.194	3.251	21.210	5.295	9.198
3.91	0.256	2.278	3.371	20.880	5.345	9.349
3.81	0.263	2.367	3.499	20.537	5.396	9.504
3.71	0.270	2.462	3.634	20.180	5.445	9.664
3.61	0.277	2.563	3.779	19.810	5.493	9.829
3.51	0.285	2.671	3.932	19.425	5.540	9.999
3.41	0.294	2.787	4.095	19.024	5.584	10.175
3.31	0.302	2.912	4.270	18.606	5.627	10.356
3.21	0.312	3.046	4.456	18.171	5.666	10.542
3.11	0.322	3.190	4.656	17.716	5.702	10.735
3.01	0.333	3.345	4.869	17.240	5.733	10.933
2.91	0.344	3.514	5.098	16.742	5.759	11.138
2.81	0.356	3.696	5.344	16.221	5.778	11.349
2.71	0.369	3.895	5.608	15.674	5.789	11.565
2.61	0.384	4.111	5.892	15.100	5.791	11.788
2.51	0.399	4.347	6.198	14.496	5.781	12.017
2.41	0.415	4.606	6.527	13.861	5.757	12.252
2.31	0.433	4.890	6.881	13.191	5.716	12.492
2.21	0.453	5.203	7.262	12.485	5.655	12.737
2.11	0.474	5.548	7.671	11.739	5.569	12.987
2.01	0.498	5.931	8.111	10.951	5.454	13.241
1.91	0.524	6.357	8.583	10.118	5.302	13.497
1.81	0.553	6.832	9.086	9.235	5.107	13.755
1.71	0.585	7.364	9.621	8.301	4.859	14.013
1.61	0.622	7.961	10.185	7.312	4.546	14.268
1.51	0.663	8.633	10.776	6.265	4.153	14.519
1.41	0.710	9.392	11.388	5.158	3.662	14.763
1.31	0.764	10.253	12.014	3.990	3.048	14.997
1.21	0.827	11.233	12.641	2.758	2.282	15.218
1.11	0.902	12.350	13.258	1.464	1.320	15.422
1.01	0.991	13.630	13.849	0.110	0.109	15.606
0.91	1.100	15.104	14.397	-1.301	-1.431	15.769
0.81	1.236	16.809	14.887	-2.765	-3.416	15.908
0.71	1.410	18.800	15.308	-4.273	-6.025	16.021
0.61	1.641	21.152	15.649	-5.820	-9.551	16.109
0.51	1.963	23.979	15.910	-7.397	-14.519	16.172
0.41	2.441	27.473	16.094	-8.997	-21.964	16.212
0.31	3.229	31.991	16.210	-10.611	-34.261	16.230
0.21	4.766	38.318	16.272	-12.233	-58.310	16.272
0.11	9.099	48.849	16.296	-13.860	-126.118 16.296
0.01	100.000	87.918	16.300	-15.489	-1548.864 16.300

Here are nice graphs of most of those columns:

Anyone with means of checking the validity of the outputs? I have a suspicion that D_now is not correct for the future (S < 1), because I think it is supposed to asymptotically approach the S = 0 line, while it appears to be heading for an intercept. The current calculator does not work accurately for S < 0.01, which may actually be the cause of the apparent intercept. Looking into it.

Edit:
The "y-intercept" of the green curve is in fact just an artefact of this thread's definition of D_now for the future: the (negative) distance to an observer that will receive our present signals with a stretch 1/S, i.e. with redshift 1/S + 1. The y-intercept represents the cosmic even horizon (16.3 Gy), where redshift (and time to reach) tends to infinity. Negative S does not have a physical meaning, or does it? One can mathematically extend the curve to the negative domain, but I have no idea what it may mean.

 

[marcus]

Hi Jorrie, I just saw your edit. I think you are right that the curve physically stops at 16.3 where the time for the signal to reach target goes to infinity. If it takes an infinite time for a our signal to reach a galaxy at 16.3 Gly that clearly says it is the limit. I like the clarity.

Can't think of any physical meaning of negative stretch, or negative scale factor.

To me it looks like the calculator does what it has to do, what it should do. reach the axis (where time=) exactly at the right place. It's a really satisfying gadget, you must be be having some proud papa moments these days.

(Or so it seems to me---as a non-expert interested in the subject.)

btw I like the "down-step" feature! It let's me get the size table I want without having to calculate what the step size should be to achieve that. And when I change the upper and lower limits of the table, it stays the desired size. Good (though unconventional) use of the minus sign

EDIT: Hi Jorrie, just saw your next post which wakes me up to the fact that I should have been saying 15.6 here instead of 16.3. The y-intercept of the D_now curve should give the present value of the cosmic event horizon (which is around 15.6 Gly) not the future value.

Here are nice graphs of most of those columns:

Edit:
The "y-intercept" of the green curve is in fact just an artefact of this thread's definition of D_now for the future: the (negative) distance to an observer that will receive our present signals with a stretch 1/S, i.e. with redshift 1/S + 1. The y-intercept represents the cosmic even horizon (16.3 Gy), where redshift (and time to reach) tends to infinity. Negative S does not have a physical meaning, or does it? One can mathematically extend the curve to the negative domain, but I have no idea what it may mean.

 

[Jorrie]

... I have a suspicion that D_now is not correct for the future (S < 1), because I think it is supposed to asymptotically approach the S = 0 line, while it appears to be heading for an intercept. ...

I was wrong; it is actually the future D_then value that diverges to negative infinity, as is clear from this chart.

One can also see the position of the 'equator' of the usual 'teardrop' (or 'onion') shape, where T_Hub and D_then cross over (S = 2.64). This is the maximum value of D_then for all of time (given the standard model and values).

...
Edit:
The "y-intercept" of the green curve is in fact just an artefact of this thread's definition of D_now for the future: the (negative) distance to an observer that will receive our present signals with a stretch 1/S, i.e. with redshift 1/S + 1. The y-intercept represents the cosmic even horizon (16.3 Gy), where redshift (and time to reach) tends to infinity.
...

Wrong again; must have been weekend laziness...

The "y-intercept" of the green curve represents (the negative of) the distance to our or present cosmic event horizon (CEH), at 15.6 Gly. An observer presently at that proper distance will never receive our present signals (and neither will we receive theirs). If accelerated expansion continues as we expect, our future CEH will only reach 16.3 Gly by around 74 Gy from now.

 

[marcus]

That's an especially nice chart with the 5 curves (time=black, horizon=sky blue, ...etc).
I like being able to spot the equatorial bulge on the onionshape lightcone by where the red and purple curves cross, at S=2.64.

Your post alerts me to my having misspoke in post#31, the current CEH being 15.6, I should have been saying that instead of the longterm CEH value of 16.3. The presentday D_now curve should have a y-intercept at the presentday CEH, so at or around -15.6. Which (allowing for the limitations of finite accuracy) it does seem to do!

 

[Jorrie]

Since there may be a change in the generally accepted values for H₀ and Omega_m coming (http://arxiv.org/abs/1208.3281%22), and it may change values for i.a. Cosmic time (age) and Lookback time, I have included Lookback time in the list of compact equations that was listed before.

Given present Hubble time Y_{now}, long term Hubble time Y_{inf} and the redshift for radiation/matter equality z_{eq}. Since the factor z + 1 occurs so often, an extra parameter S = z + 1 = 1/a is defined, making the equations neater.

\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, <br /> \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \,<br /> \Omega_m = S_{eq}\Omega_r

Hubble parameter
H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}

Hubble time, Cosmic time , Lookback time T_look added
Y = 1/H, \ \ \, <br /> T = \int_{S}^{\infty}{\frac{dS}{S H}}, \ \ \,<br /> T_{look} = \int_{1}^{S}{\frac{dS}{S H}}
Proper distance 'now', 'then' and cosmic event horizon
D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \, <br /> D_{then} = \frac{D_{now}}{S}, \ \ \, <br /> D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}

This essentially means integration for S from zero to infinity, but practically it has been limited to 10^{-7} &lt; S &lt; 10^{7} with quasi-logarithmic step sizes, e.g. a small % increase between integration steps.

Using the quoted H₀ = 74.3 km s^-1 Mpc^-1 and Omega_m = 0.278, a rerun of the above numerical integrations gives:

Hubble times: Y_now = 13.3 Gyr, Y_inf = 15.5 Gyr, T_now = 12.96 Gyr and the lookback time to the current most distant galaxy T_z=9.6 = 12.54 Gyr.

Since the change in Omega_m was small, the times essentially changed by the ratio H₀(old)/H₀(new), but this will not hold if Omega_m changes significantly, or the deviation from spatial flatness is significant.

 

[Jorrie]

For anyone who missed the discussion in Marcus' "88 billion year" sticky above, here is the latest version of the "inhouse" tabular cosmo-calculator, as also shown in Marcus' signature (TabCosmo5.html).

The main changes since September last year are: an easy method to get a logarithmic spread of redshifts (actually stretch S = z+1) and that the latest (2013) WMAP9 (combined) maximum likelihood parameters are now used. Please read the info tool-tips of the calculator for clarification of usage.

Here is a sample plot of data generated by the calculator, as copied into a spreadsheet.

Of particular interest from the visuals are the following observations:

1. The max value of D_then ~ 5.8, where D_then crosses T_Hub at S ~ 2.62. You will need 29 S-steps to spot this max precisely on a generated table.
   
2. The correspondence of T_Hubble and D_hor when S < ~0.3, where the cosmological constant completely dominates.
   
3. The 'straight' T-curve into the future (S < 1), with an equation T \approx 13 - 16.5 \ln(S) Gy, with 13 roughly the y-intercept of the linear portion and 16.5 is Y_inf (Hubble time in far future).

 

[marcus]

Part of what Jorrie was just talking about. I.e. stretch factor 2.63 and emission distance 5.8, has to do with the beautiful fact that past lightcones are TEAR-DROP SHAPE.

You can see that at the top level of the "figure 1" in my signature. That is what they look like when you measure in proper distance, the real distance that it actually was at the time, if you could have stopped the expansion process.

Other levels of the "figure 1" show conformal distance---what the distance to that same bit of matter would be today, not what it was back then. So the lightcone is not teardrop, it is some other shape.

the point of S=2.63 is that where the WIDEST bulge of the teardrop comes, in our past light cone. The largest girth. Farther back in time from then, the light cone PULLS IN. Of course that's because distances were smaller back then---and it is what gives it the teardrop or pear shape.

A rather beautiful thing happened around S=2.63 namely when galaxies emitted light then, that was destined to get here today for us to receive with telescopes, that light stayed at the same distance from us for a long time. Making barely if any progress. It stayed at distance 5.8, or more precisely according to the calculator, 5.798. Because its forward motion thru the surrounding space exactly canceled the rate at which the distance 5.798 was growing! So no net headway!

And then after a long time that distance 5.798 had slowed slightly and was not growing at the speed of light and the photons began to make headway towards us. The calculator will give an idea how long they took, all told, to get here. I think it was very nearly 10 billion years.

So you see in the preceding post Jorrie suggests putting 29 into the STEPS box, and also be sure to check the "exactly S=1" box so you get the exact present in your table. Then you will get, among much else, the S=2.62 line in the table, and that 5.8, and the time, what year it was etc.

The widest girth is at a crossing point in the figure which basically says the distance was growing at exactly c. You can see where the two curves cross. Blue and green. Blue for the emission distance, green for the Hubble distance (that distance which is growing at speed c.)

If you click on figure 1 in my signature you will also see a crossing of curves that marks this widest point on the teardrop lightcone. (In the top layer, the version drawn using proper distance. Other layers distort shape.)

 

[Jorrie]

... stretch factor 2.63 and emission distance 5.8, has to do with the beautiful fact that past lightcones are TEAR-DROP SHAPE.

You can see that at the top level of the "figure 1" in my signature. That is what they look like when you measure in proper distance, the real distance that it actually was at the time, if you could have stopped the expansion process.

I have massaged a spreadsheet of the tabular data a little in order to plot a graph that looks somewhat like the top level Davis plot in your sig. In the process I became interested in the relationship between the event horizon and the particle horizon and subsequently have added a column for the particle horizon to TabCosmo5 (saved as TabCosmo6). Graphically it looks like this:

It corresponds (partially) to the Davis diagram turned on its side, with the 'teardrop' the two opposite side D_then distances, crossing and diverging in the future.

Interestingly, there are two other intersections happening simultaneously at another cosmic time, T~4 Gy: (i) the Hubble sphere crossing the past light cone and (ii) the event horizon crossing the particle horizon.

Crossing (i) is as you explained in your prior post, but I'm not sure why crossing (ii) happens at the same time (or at least very closely so, as far as I can tell). The correspondence seems to be independent of the choice of input parameters (Ynow and Yinf).

If I have it right, the cosmic event horizon is the largest proper distance (at time of emission) between an emitter and receiver that light can ever bridge, while the particle horizon is the proper radius of the observable universe at the time of the emission of the signal that is observed at stretch S.

Is it because observed redshift at the event horizon will tend to infinity?

 

[marcus]

Nice!
The present moment is shown in an elegant graphic way as the point joining the past and future lightcones. I'll think about your question shortly, just wanted to respond immediately to the figure

 

[marcus]

Sorry, I got dragged off to lunch and had to prune trees in the garden. I see that simultaneous intersection clearly! I can't explain it. I'll keep thinking about it and may have some luck later.

 

[Chronos]

That is my understanding too, Jorrie. The redshift approaches infinity by the time photons currently emited at the CEH reach us. Of course, the time it takes those photons to reach us also approaches infinity. If you think in terms of scale factor, it all seems to make sense.

 

[marcus]

One thing that occurs to me is that Lineweaver is a talented explainer who devoted his lifetime to cosmology and his figure 1 has THREE bands. Probably you can't get the whole thing into one picture and if you try to, the first picture will start getting complicated and won't communicate as well.

The THIRD band of figure 1 uses comoving distance (each bit of matter is given an unchanging label) and the timescale is adjusted to match that. Then particle horizon is a straight 45 degree line that intersects event horizon which is also a straight 45 degree line and is effectively "the past lightcone at infinity".

The story you can tell about that intersection (P horizon with E horizon) is of a RADAR ECHO. We send out a PING at the start of expansion, and we ask what is the most distant matter that can echo back or send a reply message to us, that we would eventually receive if we could wait arbitrarily long. If we could wait "till infinity" to hear the reply or the echo, then what is the most distant matter we could contact that way. With the whole history of the universal expansion to do it in, to make contact.

And I think your tabular calculator gives the answer to that, and it says WHEN the radar signal bounces, if I recall it is around year 4 billion, which is when the lines intersect. I have to check this.

Yes, I'm just using version 5. It says that the proper distance to that farthest ever ping-able matter is 11.8 Gly, that is at the moment it gets our message (we sent at the start of expansion) and echos it back. And that is at S=2.63. So to find the distance NOW I have to multiply 11.8*2.63 = 31 Gly
And distance now of some particular bit of matter is what they call its "comoving" distance. So that 31 Gly should agree with Lineweaver figure 1.

Actually I don't think this has to do with infinitely redshifting light. It is not what you can practically get a radar ping from it is what you can do IN PRINCIPLE. using arbitrarily large antennas and arbitrarily sensitive receivers etc etc. Let me check and see if Lineweaver puts that intersection at 31 Gly.

Yes, bingo! right on 31 Gly! So I think the analysis is all right.

Now there is still the puzzle Jorrie posed which is why that farthest matter echo event happens right at S = 2.63.
Why should it coincide with...? Have to think some more about that. If somebody else doesn't come up with an explanation I'll think about it tomorrow morning when I'm fresh. We're only just getting started on that one, I think. Intriguing coincidence!

 

[marcus]

This is strange. Using the new calculator version6, I don't actually get a coincidence.
I'm putting in Step=0 so I just get a one-line table, for S=2.632
That is what I am used to using to get the intersection of Hubbleradius and D_then. Or even better: S=2.6321

But that does not give a match between D_hor and particle horizon D_par! It looked on the figure as if they were at the same level so I thought there was an exact coincidence (but couldn't figure out why there would be) and now the table does not give a coincidence.

11.804 ≠ 11.934

Am I missing something? being really dense? Sorry for a possible bungling lapse of competence. Can someone explain this almost but not quite coincidence?

To get D_hor to equal D_par, you have to go to S=2.662
11.736 ≈ 11.735

well, let's still find the comoving (now) distance to the farthest pingable matter: 2.662*11.735 = 31.2 Gly. Yes! that's still good.

I suppose that twice that, namely 62.4 Gly is the distance now of the farthest matter we will ever hear from regardless how long we wait.

 

[Jorrie]

But that does not give a match between D_hor and particle horizon D_par! It looked on the figure as if they were at the same level so I thought there was an exact coincidence (but couldn't figure out why there would be) and now the table does not give a coincidence.

11.804 ≠ 11.934

Am I missing something? being really dense? Sorry for a possible bungling lapse of competence. Can someone explain this almost but not quite coincidence?

To get D_hor to equal D_par, you have to go to S=2.662
11.736 ≈ 11.735

I have also noticed this, but my first reaction was that it is caused by small errors in the numerical integration loops of the various curves. Remember that to get all the values perfect, it requires integration for time (or S) from zero to infinity with an 'infinite number of steps', which is not feasible. Especially D_hor is very susceptible to cut-off errors.

What is intriguing is that the rough correspondence remains when Ynow and Yinf are changed. I'm busy looking at it analytically (not easy) and will report what I find.

 

[marcus]

Because you are doing hard analytical work I should probably be quiet and not distract from that. I had something else I wanted to say, though. It seems to me that the distance 11.735 Gly is somehow UNIVERSAL. It does not know about us, that we are in year 13.7 Gy or so. It depends on sending out a radar ping at the start of the expansion, from wherever you are, and then being able to wait to year infinity to hear back.

The farthest distance, as a proper distance from your matter when the bounceback happens, should be the same for anyone in the universe at any stage in its history.

Is the distance 5.8 comparably universal? It seems strange that it should be roughly HALF of 11.735

But that could be a spurious coincidence. I dimly suspect that the distance 5.8 depends on WHEN in the history of the universe you are. It is the maximum proper distance at emission-time of any light we can detect now. I may be missing something, but that seems to depend on when in the history of the universe we are.

 

[Jorrie]

The redshift approaches infinity by the time photons currently emited at the CEH reach us. Of course, the time it takes those photons to reach us also approaches infinity. If you think in terms of scale factor, it all seems to make sense.

Yes, it is a bit clearer in terms of scale a = 1/S and comoving distances. Working on that.
From Davis http://arxiv.org/abs/astro-ph/0402278 (2004), Eqs. A.19 and A.20, pp. 117, with c=1:

\chi_{par}= \int_{0}^{t}{ \frac{dt}{a}} = \int_{0}^{a}{ \frac{da}{a^2 H}}
\chi_{hor} = \int_{t}^{\infty}{ \frac{dt}{a}} = \int_a^\infty { \frac{da}{a^2 H}}

where H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})} and S = 1/a = 1+z (post #34 above). Further from #34, written in comoving form:

\chi_{Hub}= \frac{1}{a H}
\chi_{then} = \int_{1}^{S}{ \frac{dS}{H}} = \int_a^1 { \frac{da}{a^2 H}}

This looks deceptively easy, but since H is a function of a, I have no idea how to analytically solve for a for either of the two crossings. Maybe Maple software can help? (I do not have it).

Anyone with ideas?

 

[marcus]

...Anyone with ideas?

This is not the type of idea you specifically asked for, but let's explore the idea that the apparent coincidence may be spurious. If that's wrong, and it is a mathematical equality some reader will show up, I trust, and explain. Meanwhile I make the tentative assertion that the maximum girth of the teardrop lightcone (and the time that occurs) depends strongly on where we are in the history of the universe. If we were later the teardrop would be bigger and the bulge would come later. We wouldn't be seeing that time figure of 4 Gy and that maximum emission distance figure of 5.8 Gly. If we were earlier/later in the expansion process those numbers would each be smaller/larger.

So if you want to destroy the spurious coincidence (I assert tentatively) then you don't change the parameters of the universe, you should figure out what numbers we will see later on, or would have seen earlier. Construct our perspective for some time in future.

Because I think the maximum proper distance of a radar bounce is a universal INVARIANT, and so is the year that bounce occurs. It is going to be the same as long as the basic cosmic parameters are the same, whether from the perspective of some one earlier than us or someone far in the future. the reason is that the present expansion age does not enter into the definition.

The greatest proper distance of a radar bounce is always going to be 11.735 Gly and the time that bounce occurs is always going to be year 4 billion. Or 3.97...something billion, to be finicky.

The definition is you imagine sending out a signal right at the start of expansion. And every time it hits something part of the signal bounces back. And at first all those return echos are destined to get back to us eventually. If we wait long enough we will hear the ping.

But there comes a time (year 3.97... billion ) when the signal is at a proper distance of 11.735 Gly, and it makes its LAST BOUNCE that is ever destined to get back to us. Because it has reached a "point of no return", which is the event horizon.

When the particle horizon curve meets the event horizon curve there is no more pingback return from then on. The signal makes the last bounce we can expect to hear.

I'll think about this some more, but it seems obviously independent of when in the expansion history we happen to be at the present time. (which I expect the other numbers aren't independent of, so the coincidence has to be fortuitous even though bizarrely close.)

 

[marcus]

I checked. The coincidence does see merely accidental. I used version 6 and put in S_lower = 1 and Steps=50 (to get nice resolution).

Then I put in Y_now = 12.0 instead of 14.0. That corresponds to an earlier time in the same universe. The age is now only around 10 Gy instead of 13.7 Gy.

Then I looked down to where the TIME was about 3.99 Gy which is when we expect the farthest radar bounce to occur and in fact it did! Both Dhor and Dpar were around 11.7 and roughly equal.

But at that moment in time the other two numbers were NOT roughly equal. Dthen was nowhere near Thub. So people living in Milkyway back in year 10.14 billion would NOT see the coincidence we are talking about.

their maximum teardrop bulge would have occurred around year 2.9 billion and their max pingback bounce would have occurred (as it always does in our universe) at year 4 billion or so.

I didn't bother to adjust the 3250 number for the different perspective because I don't think it would have made any great difference.

I must say I like version 6! Will have to change link in signature.

 

[George Jones]

I checked. The coincidence does see merely accidental.

Here's another way (or the same way from a slightly different perspective) to see this.

The particle and event horizons do not depend on a "now" event, so their intersection does not depend on a "now" event. The Hubble sphere does not depend on "now", but the past lightcone does depend on "now", so their intersection does depend on "now". This is particularly evident in Figure 1 from Davis Lineweaver. As the "now" line shifts up and down, the intersection of the past lightcone and the Hubble sphere changes (for me, especially clear in the bottom panel), but the intersection of the particle and event horizons remains the same.

 

[marcus]

Here's another way (or the same way from a slightly different perspective) to see this.

The particle and event horizons do not depend on a "now" event, so their intersection does not depend on a "now" event. The Hubble sphere does not depend on "now", but the past lightcone does depend on "now", so their intersection does depend on "now". This is particularly evident in Figure 1 from Davis Lineweaver. As the "now" line shifts up and down, the intersection of the past lightcone and the Hubble sphere changes (for me, especially clear in the bottom panel), but the intersection of the particle and event horizons remains the same.

Good! Clear concise way to explain it. Thanks, George.

 

[Jorrie]

This is particularly evident in Figure 1 from Davis Lineweaver. As the "now" line shifts up and down, the intersection of the past lightcone and the Hubble sphere changes (for me, especially clear in the bottom panel), but the intersection of the particle and event horizons remains the same.

Thanks, this gives a clear picture. Like Marcus, I could not find any further empirical or analytical evidence anyway.