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Could a cosmo tabulator give hands-on experience with cosmic Event Horizon?

  1. Sep 8, 2012 #1

    marcus

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    Cosmo calculators and tabulators a primarily about the PAST expansion history and they give learners hands-on understanding by being able to vary the model parameters and see change. That's good. I sometimes notice a difference here at PF between how posters with mainly verbal understanding think and how those do who also have some of the quantitative feel you get from playing with an equation model of the cosmos or with one of the online interactive models.

    However the murky talk about "dark energy" and all that is essentially about a FUTURE leveling off of the declining Hubble expansion rate, to about 60 km/s per Mpc. Or if you like about the plateauing of its reciprocal, the Hubble time, at about 16.3 billion years. And this adds a very interesting feature to our universe, which has to do with the FUTURE--namely the CEH (cosmic event horizon).

    I'm not sure of its exact estimate right now, somewhere between 15 and 16 Gly. It is gradually converging to 16.3 Gly. So let's say it is now "around 16" billion lightyears.

    This is a very interesting thing. If a galaxy is NOW less than ~16 Gly from us and we send them some light TODAY the light will eventually get there. Though if the galaxy is very close to the 16 Gly limit it could take a very very long time to reach them. The closer it is to the limit the longer it will take, because given the expansion of distances going on it can just barely make it, traveling at the speed of light.

    However if the galaxy is NOW beyond the ~16 limit and we send them a flash today it will never get there.

    And it works both ways. We see lots of galaxies out there which are farther than 16 Gly. In fact most of the galaxies you can see with a telescope are farther than that. But if something happens TODAY in one of those galaxies we will never see it. They are beyond the current event horizon and they cannot now send us a message that will ever reach us.

    So that is an interesting feature of the universe which basically has to do with this plateauing of the Hubbletime at 16.3 Gly and involves future time. Can a cosmo tabulator give us hands-on experience with future expansion as well as experience with our universe's past expansion history?

    You'd have to be able to play around with the Hubbletime parameters (ie. with the two expansion rate parameters) and see how the distances change. Namely the distance NOW to the galaxy we want to send light to, and the distance THEN to the galaxy when the light finally gets there (despite all the expansion that will have been happening).
    Does anyone know of an online resource like this, or have any ideas? It could be helpful for people learning about cosmology, and kind of neat actually :biggrin:
     
    Last edited: Sep 8, 2012
  2. jcsd
  3. Sep 9, 2012 #2

    Drakkith

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    I could see a simple flash animation with some slider bars for the variables such as distance being useful.
     
  4. Sep 9, 2012 #3

    marcus

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    It's a suggestive idea. It got me started playing around with the two main parameters of the mode---the two Hubbletimes (call them for lack of any neater notation Ynow and Y). Reciprocal growth rates, now and at infinity.
    If you multiply them by log(2) you get what would be the distance doubling times at that constant rate of distance growth.
    If you divide a distance by the current Y you get the speed it is increasing. (distance divided by time equals speed and this is the time to divide by).

    So the Hubbletime at any epoch is a handy package of information about expansion, and the two Hubbletimes--now and at infinity--are the two crucial parameters that we get everything else from (given the flat Friedman equation and the observed proportion of light to matter.)

    I tried varying the two Hubble times and noticed that they have opposite effects on an interesting distance: the farthest distance anything we can see was when it emitted the light.

    We never see anything that was farther than 5.8 billion lightyears when it emitted the light and this maximum "as-we-see-it" was achieved in the S=2.64 era (when distances were 1/2.64 what they are today).

    You can see the max stand out clearly if you use the calculator with upper limit S=3.0
    lower limit S=2.5 and step=0.01
    It's quite pronounced, and a bit curious.

    So I was prompted by your post to try varying the two Hubbletimes 13.9 and 16.3 Gly to see how that would change.

    INCREASING Ynow makes this interesting max distance bigger and come sooner.
    Decreasing Ynow makes it occur later and smaller.

    But changing the longterm limit Hubbletime has the opposite effect.

    INCREASING Y makes this max as-we-see-it distance later and smaller.
    Decreasing Y makes it earlier and bigger.

    I didn't try varying both at once but it's pretty clear that if you bring 13.9 and 16.3 closer together by increasing the former and decreasing the latter both at once, you get that maximum distance earlier (i.e. with a larger S) and bigger.
    And increasing the Hubbletime separation by simultaneously decreasing 13.9 and increasing 16.3 would have the opposite effect.

    Anyway Drakkith, this is an example of using the online cosmic model as if it had slider bars.

    You just set:
    upper S=3.0
    lower S=2.5
    step=0.01
    and play around with the two Hubbletimes, varying Ynow and Y.

    this maximum as-we-see-it distance still intrigues me and I'd like to get more intuition about it. It always occurs just when the THEN distance is exactly equal to the Hubble distance which is cY---that is the then Hubble time multiplied by the speed of light.

    I'd like to understand better why that happens.
     
    Last edited: Sep 9, 2012
  5. Sep 9, 2012 #4

    Jorrie

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    When you set the two Ys closer together, you effectively increase the cosmological constant relative to matter and radiation. Accelerating expansion will hence start earlier in time. Will think about other puzzles tomorrow - past mid-nite here...
     
  6. Sep 9, 2012 #5

    marcus

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    The capability of the new A20 version to go into the future is GREAT!!!!

    I tried a sample and in confirms what the toy model says that the time when distances will be 100 times what they are today will be about 88 billion years in the future.
    That is the S=0.01 era. The S number is always a comparison of OUR lengths and distances with those of the other era.

    So S=10 is the era in the past when distances were 1/10 those of today.
    S=0.1 is the era in the future when distances will be 1/.1 of those today (in other words 10 times distances now.)

    In effect the S tells what our present distances look like compared with theirs (whether they are in the past or in the future.) I'm telling myself this to get used to the terminology.

    I think it is really great to be able to look into the future and actually SEE THE COSMIC EVENT HORIZON looming up in front of us at around 15.5 to 16 billion light years.

    this is the distance limit on the galaxies NOW that we can expect to send light to that they will eventually receive. The calculator is exploring our forward light cone, by numerical integration, in an expanding universe. I can see Dnow beginning to converge.
     
  7. Sep 10, 2012 #6

    Jorrie

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    I have used Tamara's equation for comoving event horizon distance:
    [tex]\chi_c(t) = c \int_{t}^{t_{end}} \frac{dt'}{R(t')}[/tex]
    to approximate the proper CEH and it gives a present value of 15.64 Gly.

    As far as can see, in our present calculator parlance, the proper distance to the horizon will translate to:
    [tex]D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}[/tex]

    where [itex]H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}[/itex],
    as per this post.

    I guess this could perhaps easily be incorporated into a calculator.
     
    Last edited: Sep 10, 2012
  8. Sep 11, 2012 #7

    George Jones

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    marcus, I think that you have explained the intuition behind this :biggrin: in other, related posts about our particular expanding and accelerating universe.

    The top panel of Figure 3 is an even better, expanded diagram that shows the Hubble sphere crossing our past lightcone at the lightcone's maximum "bulge" of 5.8 billion ly. Now move to pages 18 and 19.

    Supposes that a galaxy on our past lightcone emits a photon towards us when it: 1) is a distance of 4 billion ly from us; 2) is at time earlier than the bulge. The photon moves up our past lightcone until it reaches us. Differentiating equation (16) gives equations (19) and (20) along the worldline (our past lightcone) of the photon, with [itex]v_{pec} = -c[/itex] (negative, since the photon is directed towards us). Consequently, for the photon, all along the photon's worldline,

    [tex]v_{tot} = v_{rec} - c,[/tex]

    where (using the paper's notation)

    [tex]v_{rec} = \dot{R} \chi = \frac{\dot{R}}{R}R \chi = HD[/tex]

    Initially, [itex]v_{rec}[/itex] is greater than [itex]c[/itex], so [itex]v_{tot}[/itex] is positive, and the photon moves away from us, i.e., [itex]D[/itex] increases. In our universe, the Hubble parameter [itex]H[/itex] decreases as time increases. For this photon, [itex]H[/itex] decreases faster than [itex]D[/itex] increases, so [itex]v_{rec} = HD[/itex] and [itex]v_{tot} = v_{rec} - c[/itex] both decrease. The photon's worldline curves upwards until [itex]v_{rec}[/itex] has decreased to the value of [itex]c[/itex], at which time 1) the photon is on the Hubble sphere by definition; 2) the photon's worldline is vertical; 3) the photon's proper distance is maximum (5.8 billion ly). After, this [itex]v_{rec}[/itex] keeps decreasing, but now the photon moves with decreasing proper distance, since [itex]v_{rec}[/itex] gives that [itex]v_{tot}[/itex] is less than zero. The photon's worlline is curving downwards.

    This is a subtle example of a first-year calculus local maximum problem: [itex]0 = \dot{D} = v_{tot}[/itex] is a necessary condition to have a local maximum for [itex]D[/itex]. In this eaxmple, [itex]0 = v_{tot}[/itex] is the condition to be on the Hubble sphere.

    Also, look at the last sentence of the paragraph that contains equation (19) and (20):

    "Nevertheless we can eventually receive photons that initially were receding from us because the Hubble sphere expands and overtakes the receding photons so the photons find themselves in a region with vrec < c (Section 3.3)."
     
    Last edited by a moderator: May 6, 2017
  9. Sep 14, 2012 #8

    marcus

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    Thanks G.J. this is a helpful explanation. Also the "bulge" is a good intuitive term for the midriff of the pearshaped lightcone :biggrin:
    We should keep a link to this post handy. I or someone else will save it in the stickythread.
     
  10. Sep 14, 2012 #9

    Jorrie

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    I have incorporated the CEH into CosmoLean_A22, together with a few minor upgrades. The numerical integration now has to run for scale factor 10-7 to 107, so I had to compromise on the number of steps to get an acceptable execution time (some 3 million steps). Nevertheless, the accuracy seems to be at least 3 significant digits.

    Here is a "copy & paste friendly" output sample (a check-box to this effect is one of the minor upgrades):
    Code (Text):

    S   a   T   T_Hub   D_now   D_then  D_hor
    1.090   0.917   12.579  13.372  1.213   1.112   15.458
    1.000   1.000   13.755  13.900  0.000   0.000   15.622
    0.910   1.099   15.090  14.392  -1.288  -1.416  15.768
    0.820   1.220   16.612  14.838  -2.604  -3.176  15.894
    0.730   1.370   18.361  15.227  -3.957  -5.421  15.999
    0.640   1.562   20.387  15.553  -5.343  -8.348  16.084
    0.550   1.818   22.765  15.814  -6.755  -12.282 16.149
    0.460   2.174   25.610  16.010  -8.187  -17.799 16.195
    0.370   2.703   29.112  16.147  -9.635  -26.040 16.222
    0.280   3.571   33.626  16.233  -11.092 -39.616 16.233
    0.190   5.263   39.932  16.279  -12.556 -66.083 16.279
    0.100   10.000  50.388  16.297  -14.022 -140.21 16.297
    0.010   100.000 87.918  16.300  -15.489 -1548.8 16.300
     
    A small amount of manual formatting may still be required. I had to remove some trailing digits from the last two lines. I'm not sure if there is a HTML table tag available in the forum editor, because that would have made it even easier to paste.
     
  11. Sep 14, 2012 #10

    marcus

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    Works really well! I just now asked for a table with 110 rows, from distant past (first galaxies) to over 70 billion years into future (year 88 billion), and the table appeared instantly. I then asked for the copy-paste version and that appeared instantly. Here it is, let's see what it it looks like. this is without any aligning by hand:
    Code (Text):



    S   a   T   T_Hub   D_now   D_then  D_hor
    10.000  0.100   0.559   0.839   30.890  3.089   4.653
    9.910   0.101   0.566   0.851   30.814  3.109   4.687
    9.820   0.102   0.574   0.862   30.737  3.130   4.722
    9.730   0.103   0.582   0.874   30.659  3.151   4.758
    9.640   0.104   0.590   0.887   30.579  3.172   4.794
    9.550   0.105   0.599   0.899   30.499  3.194   4.831
    9.460   0.106   0.607   0.912   30.418  3.215   4.868
    9.370   0.107   0.616   0.925   30.335  3.237   4.906
    9.280   0.108   0.625   0.939   30.251  3.260   4.945
    9.190   0.109   0.634   0.952   30.166  3.282   4.984
    9.100   0.110   0.644   0.967   30.080  3.305   5.024
    9.010   0.111   0.653   0.981   29.992  3.329   5.064
    8.920   0.112   0.663   0.996   29.903  3.352   5.105
    8.830   0.113   0.673   1.011   29.813  3.376   5.147
    8.740   0.114   0.684   1.027   29.721  3.401   5.190
    8.650   0.116   0.695   1.043   29.628  3.425   5.233
    8.560   0.117   0.705   1.059   29.533  3.450   5.277
    8.470   0.118   0.717   1.076   29.437  3.475   5.322
    8.380   0.119   0.728   1.093   29.339  3.501   5.367
    8.290   0.121   0.740   1.111   29.240  3.527   5.413
    8.200   0.122   0.752   1.129   29.139  3.554   5.461
    8.110   0.123   0.765   1.148   29.037  3.580   5.508
    8.020   0.125   0.778   1.168   28.933  3.608   5.557
    7.930   0.126   0.791   1.187   28.827  3.635   5.607
    7.840   0.128   0.805   1.208   28.719  3.663   5.658
    7.750   0.129   0.819   1.229   28.609  3.692   5.709
    7.660   0.131   0.833   1.250   28.498  3.720   5.762
    7.570   0.132   0.848   1.273   28.384  3.750   5.815
    7.480   0.134   0.864   1.296   28.269  3.779   5.870
    7.390   0.135   0.880   1.319   28.151  3.809   5.925
    7.300   0.137   0.896   1.343   28.031  3.840   5.982
    7.210   0.139   0.913   1.369   27.909  3.871   6.040
    7.120   0.140   0.930   1.394   27.785  3.902   6.099
    7.030   0.142   0.948   1.421   27.658  3.934   6.159
    6.940   0.144   0.966   1.449   27.529  3.967   6.220
    6.850   0.146   0.986   1.477   27.397  4.000   6.282
    6.760   0.148   1.005   1.506   27.263  4.033   6.346
    6.670   0.150   1.026   1.537   27.126  4.067   6.411
    6.580   0.152   1.047   1.568   26.987  4.101   6.478
    6.490   0.154   1.069   1.601   26.844  4.136   6.546
    6.400   0.156   1.091   1.634   26.698  4.172   6.615
    6.310   0.158   1.115   1.669   26.550  4.208   6.686
    6.220   0.161   1.139   1.705   26.398  4.244   6.758
    6.130   0.163   1.164   1.742   26.243  4.281   6.832
    6.040   0.166   1.190   1.781   26.084  4.319   6.908
    5.950   0.168   1.217   1.821   25.922  4.357   6.985
    5.860   0.171   1.245   1.862   25.757  4.395   7.064
    5.770   0.173   1.274   1.906   25.587  4.434   7.145
    5.680   0.176   1.305   1.950   25.414  4.474   7.227
    5.590   0.179   1.336   1.997   25.236  4.514   7.312
    5.500   0.182   1.369   2.046   25.054  4.555   7.398
    5.410   0.185   1.403   2.096   24.868  4.597   7.487
    5.320   0.188   1.439   2.149   24.677  4.638   7.578
    5.230   0.191   1.476   2.203   24.481  4.681   7.671
    5.140   0.195   1.514   2.260   24.280  4.724   7.766
    5.050   0.198   1.555   2.320   24.074  4.767   7.864
    4.960   0.202   1.597   2.382   23.862  4.811   7.964
    4.870   0.205   1.641   2.447   23.645  4.855   8.066
    4.780   0.209   1.688   2.515   23.422  4.900   8.171
    4.690   0.213   1.736   2.586   23.192  4.945   8.279
    4.600   0.217   1.787   2.660   22.956  4.991   8.390
    4.510   0.222   1.840   2.738   22.714  5.036   8.503
    4.420   0.226   1.896   2.819   22.463  5.082   8.620
    4.330   0.231   1.955   2.905   22.206  5.128   8.740
    4.240   0.236   2.017   2.995   21.940  5.175   8.863
    4.150   0.241   2.082   3.089   21.667  5.221   8.989
    4.060   0.246   2.151   3.189   21.384  5.267   9.119
    3.970   0.252   2.224   3.294   21.093  5.313   9.252
    3.880   0.258   2.301   3.404   20.791  5.359   9.389
    3.790   0.264   2.382   3.520   20.480  5.404   9.529
    3.700   0.270   2.468   3.643   20.157  5.448   9.674
    3.610   0.277   2.559   3.773   19.824  5.491   9.823
    3.520   0.284   2.656   3.911   19.478  5.534   9.976
    3.430   0.292   2.759   4.056   19.120  5.574   10.133
    3.340   0.299   2.869   4.210   18.748  5.613   10.295
    3.250   0.308   2.987   4.374   18.361  5.650   10.461
    3.160   0.316   3.112   4.548   17.960  5.684   10.632
    3.070   0.326   3.246   4.733   17.542  5.714   10.808
    2.980   0.336   3.390   4.930   17.108  5.741   10.988
    2.890   0.346   3.544   5.139   16.655  5.763   11.174
    2.800   0.357   3.710   5.362   16.182  5.779   11.364
    2.710   0.369   3.889   5.601   15.689  5.789   11.559
    2.620   0.382   4.083   5.855   15.174  5.791   11.760
    2.530   0.395   4.292   6.127   14.635  5.784   11.965
    2.440   0.410   4.519   6.417   14.070  5.767   12.175
    2.350   0.426   4.766   6.727   13.479  5.736   12.390
    2.260   0.442   5.035   7.059   12.859  5.690   12.609
    2.170   0.461   5.329   7.413   12.208  5.626   12.832
    2.080   0.481   5.651   7.791   11.524  5.540   13.058
    1.990   0.503   6.005   8.194   10.804  5.429   13.287
    1.900   0.526   6.394   8.622   10.048  5.288   13.518
    1.810   0.552   6.823   9.077   9.252   5.111   13.751
    1.720   0.581   7.299   9.556   8.413   4.891   13.983
    1.630   0.613   7.826   10.061  7.531   4.620   14.213
    1.540   0.649   8.412   10.587  6.602   4.287   14.441
    1.450   0.690   9.066   11.133  5.625   3.879   14.663
    1.360   0.735   9.798   11.691  4.598   3.381   14.878
    1.270   0.787   10.618  12.257  3.520   2.772   15.084
    1.180   0.847   11.540  12.821  2.391   2.027   15.278
    1.090   0.917   12.579  13.372  1.213   1.112   15.458
    1.000   1.000   13.755  13.900  0.000   0.000   15.622
    0.910   1.099   15.090  14.392  -1.288  -1.416  15.768
    0.820   1.220   16.612  14.838  -2.604  -3.176  15.894
    0.730   1.370   18.361  15.227  -3.957  -5.421  15.999
    0.640   1.562   20.387  15.553  -5.343  -8.348  16.084
    0.550   1.818   22.765  15.814  -6.755  -12.282 16.149
    0.460   2.174   25.610  16.010  -8.187  -17.799 16.195
    0.370   2.703   29.112  16.147  -9.635  -26.040 16.222
    0.280   3.571   33.626  16.233  -11.092 -39.616 16.233
    0.190   5.263   39.932  16.279  -12.556 -66.083 16.279
    0.100   10.000  50.388  16.297  -14.022 -140.21 16.297
    0.010   100.000 87.918  16.300  -15.489 -1548.8 16.300
     
    Nice! Quite legible! that will be a big time saver for people who use tablulator to illustrate a point.
     
    Last edited: Sep 14, 2012
  12. Sep 16, 2012 #11

    Jorrie

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    For better visualization, here is a spreadsheet graph of your table over a useful range:

    attachment.php?attachmentid=50877&stc=1&d=1347782210.jpg

    The visual relationships between T, T_Hubble and D_horizon are particularly interesting. I guess a lot can be written about it.

    PS: I have re-uploaded CosmoLean_A22, fixing some practical problems with the input-alert pop-ups; I have removed them completely. In their place I am changing the color of the 'range-text' to red when the values are out of range. This is less annoying and compatible with pop-up blockers. I have not changed designation, so no need to change your sig.

    I have also used the opportunity to add one useful piece of info: the Hubble constant corresponding the Ynow input, so that we can get a feel for what our inputs are doing to the conventional values.

    I am contemplating the addition of two other corresponding conventional values, i.e. [itex]\Omega_\Lambda[/itex] and [itex]\Omega_m[/itex]. What do you think?
     

    Attached Files:

    Last edited: Sep 16, 2012
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