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Cosmo calculators with tabular output |
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| Sep4-12, 05:29 PM | #18 |
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Cosmo calculators with tabular output
I am working on flexible rounding of output column data; it's working, but a few issues still prevent it from being released. |
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| Sep4-12, 07:10 PM | #19 |
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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/M...es/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. |
| Sep4-12, 07:18 PM | #20 |
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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:
S=1/a scalefactor a time(Gy) Hubbletime(Gy) Dnow(Gly) Dthen(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 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/M...es/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/M...es/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). |
| Sep6-12, 02:21 PM | #21 |
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The most important differences are:
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| Sep6-12, 05:37 PM | #22 |
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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. |
| Sep6-12, 05:48 PM | #23 |
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| Sep6-12, 05:50 PM | #24 |
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Thanks Jorrie. Much appreciated.
Regards, Noel. |
| Sep7-12, 11:29 AM | #25 |
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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. |
| Sep7-12, 05:03 PM | #26 |
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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. |
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| Sep7-12, 06:49 PM | #27 |
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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 Demit and Dreceive. Or maybe call it Dsend and Dreceive. 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. |
| Sep9-12, 04:15 PM | #28 |
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The basic input parameters are: present Hubble time [itex]Y_{now}[/itex], long term Hubble radius [itex]Y_{inf}[/itex] and the redshift for radiation/matter equality [itex]z_{eq}[/itex]. Since the factor [itex]z + 1[/itex] occurs so often, an extra parameter [itex]S = z + 1[/itex] 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. [tex]\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r [/tex] The 'heart' of any simple cosmological calculator is the time variable Hubble constant [itex]H[/itex], which comes from the Friedman equation as: [tex]H = H_0 \sqrt{\Omega_\Lambda + \Omega_r S^4 + \Omega_m S^3}[/tex] For perfect flatness, it can be expressed as [tex]H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}[/tex] It can be interpreted in terms of the "13.9/16.3 factors" as follows: [itex]\Omega_\Lambda = 0.7272[/itex] and [itex]\Omega_m (1+S/S_{eq})= 0.2728[/itex], 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 [itex]\Omega_m (1+S/S_{eq}) < 0.7272[/itex], when the cosmological constant started to dominate the equation. From H, the following calculator outputs are readily available: Hubble time [tex]Y(a) = 1/H [/tex] Cosmic time [tex]T(S) = \int_{S}^{\infty}{\frac{dS}{S H}}[/tex] Proper distances to a source at stretch S, "now" and "then" respectively, [tex]D_{now} = \int_{1}^{S}{\frac{dS}{H}},\ \ \ \ D_{then} = \frac{D_{now}}{S}[/tex] 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. |
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| Sep14-12, 02:16 AM | #29 |
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For completeness, I'll repeat the prior post's equations together with D_CEH. Given present Hubble time [itex]Y_{now}[/itex], long term Hubble time [itex]Y_{inf}[/itex] and the redshift for radiation/matter equality [itex]z_{eq}[/itex]. Since the factor [itex]z + 1[/itex] occurs so often, an extra parameter [itex]S = z + 1 = 1/a[/itex] is defined, making the equations neater. [tex]\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r [/tex] Hubble parameter [tex]H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}[/tex] Hubble time, Cosmic time [tex]Y = 1/H, \ \ \, T = \int_{S}^{\infty}{\frac{dS}{S H}}[/tex] Proper distance 'now', 'then' and cosmic event horizon [tex]D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \, D_{then} = \frac{D_{now}}{S}, \ \ \, D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}[/tex] This essentially means integration for S from zero to infinity, but practically it has been limited to [itex]10^{-7} < S < 10^{7}[/itex] with quasi-logarithmic step sizes, e.g. a small % increase between integration steps. |
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| Sep16-12, 06:23 AM | #30 |
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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: Code:
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 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. |
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| Sep16-12, 09:54 PM | #31 |
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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 lets 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. 
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| Sep17-12, 12:53 AM | #32 |
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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).
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. |
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| Sep17-12, 06:41 AM | #33 |
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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! |
| Oct10-12, 03:18 AM | #34 |
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Since there may be a change in the generally accepted values for H0 and Omegam 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.
Hubble times: Ynow = 13.3 Gyr, Yinf = 15.5 Gyr, Tnow = 12.96 Gyr and the lookback time to the current most distant galaxy Tz=9.6 = 12.54 Gyr. Since the change in Omegam was small, the times essentially changed by the ratio H0(old)/H0(new), but this will not hold if Omegam changes significantly, or the deviation from spatial flatness is significant. |
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