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Cosmo calculators with tabular output

  1. Sep 2, 2012 #1

    marcus

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    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?
     
    Last edited: Sep 2, 2012
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  3. Sep 2, 2012 #2

    marcus

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

    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!)
     
    Last edited by a moderator: May 6, 2017
  4. Sep 2, 2012 #3

    Jorrie

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    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. :smile:
     

    Attached Files:

  5. Sep 2, 2012 #4

    marcus

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    YAY!!!
    AFAICS tt's turning out to be a beauty. It'll be fun to share.
     
  6. Sep 2, 2012 #5

    marcus

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    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:


    Code (Text):

    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 DS,then is just equal to the Hubble radius cYS

    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?"
     
    Last edited: Sep 2, 2012
  7. Sep 2, 2012 #6

    marcus

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    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 Dnow/Ynow.
    The present is S=1, so Ynow 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 DS,then > cYS 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 DS,then > YS
     
    Last edited: Sep 2, 2012
  8. Sep 2, 2012 #7

    marcus

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    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.
    Code (Text):

    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.
    Dnow/Ynow = 43.97 Gly/13.9 Gy = 3.16 c
    Dthen/Ythen = 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.
     
    Last edited: Sep 3, 2012
  9. Sep 3, 2012 #8

    Jorrie

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    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...

    Code (Text):

    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...
     
    Last edited: Sep 3, 2012
  10. Sep 3, 2012 #9

    Jorrie

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    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.
     
  11. Sep 3, 2012 #10

    marcus

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    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.
     
  12. Sep 4, 2012 #11

    marcus

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    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 :biggrin:
    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/ [Broken]
     
    Last edited by a moderator: May 6, 2017
  13. Sep 4, 2012 #12

    marcus

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    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:
    Code (Text):

    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.
     
    Last edited: Sep 4, 2012
  14. Sep 4, 2012 #13

    Jorrie

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    This is a cool idea, but unfortunately not one guaranteed to work...:devil:
    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...
     
  15. Sep 4, 2012 #14

    marcus

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    I edited my previous post in conformance. Is this OK?
    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.
     
    Last edited: Sep 4, 2012
  16. Sep 4, 2012 #15

    Jorrie

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    Yes, that will work fine.:smile:

    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.
     
  17. Sep 4, 2012 #16

    Jorrie

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    Neither does Firefox with those inputs. It is when end - start = step (with large S) that hangups may occur.
     
  18. Sep 4, 2012 #17

    marcus

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    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.
     
    Last edited: Sep 4, 2012
  19. Sep 4, 2012 #18

    Jorrie

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    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.
     
  20. Sep 4, 2012 #19

    marcus

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    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 Dnow column corresponds to the LIGHTCONE curve. That spreads farther and farther out as you go back further to smaller scalefactor. that is because Dnow 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 Dthen 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.
     
    Last edited: Sep 4, 2012
  21. Sep 4, 2012 #20

    marcus

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    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:
    Code (Text):

    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).
     
    Last edited: Sep 4, 2012
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