
#37
Feb1513, 03:29 PM

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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? 



#38
Feb1513, 03:48 PM

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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 



#39
Feb1513, 05:24 PM

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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.




#40
Feb1513, 05:33 PM

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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.




#41
Feb1513, 10:03 PM

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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 pingable 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! 



#42
Feb1513, 10:54 PM

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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 oneline 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. 



#43
Feb1513, 11:21 PM

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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. 



#44
Feb1513, 11:50 PM

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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 emissiontime 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. 



#45
Feb1613, 12:48 AM

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From Davis http://arxiv.org/abs/astroph/0402278 (2004), Eqs. A.19 and A.20, pp. 117, with c=1: [tex]\chi_{par}= \int_{0}^{t}{ \frac{dt}{a}} = \int_{0}^{a}{ \frac{da}{a^2 H}}[/tex] [tex] \chi_{hor} = \int_{t}^{\infty}{ \frac{dt}{a}} = \int_a^\infty { \frac{da}{a^2 H}}[/tex] where [itex]H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}[/itex] and S = 1/a = 1+z (post #34 above). Further from #34, written in comoving form: [tex]\chi_{Hub}= \frac{1}{a H}[/tex] [tex] \chi_{then} = \int_{1}^{S}{ \frac{dS}{H}} = \int_a^1 { \frac{da}{a^2 H}}[/tex] 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? 



#46
Feb1613, 12:08 PM

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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 in to 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.) 



#47
Feb1613, 01:01 PM

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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. 



#48
Feb1613, 01:25 PM

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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. 



#49
Feb1613, 01:42 PM

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#50
Feb1613, 01:43 PM

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#51
Feb1613, 03:17 PM

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It's bed time in my valley, so I will look at it again some time tomorrow. 



#52
Feb1813, 05:01 AM

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Since the original Davis graphs are so much clearer, I have converted the complete diagram to .jpg and attached it. Since it is now on resident on PF, maybe you should change the link in your sig to this one. It remains pretty clear when zoomed in by means of a browser. 



#53
Feb1813, 07:48 AM

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Hi Jorrie, I neglected to mention something earlier because it wasn't essential to finding proper distances (in the lightcone of someone back in year 10.15 billion).
Their comoving distances are reduced by a factor of 1.318. Because their stretch factors are all reduced by a factor of 1.318. They see recombination (the origin of the CMB) as having occurred not at stretch 1090 but at 1090/1.318. I mentioned earlier I think that I hadn't bothered to change S_eq (because it doesn't make much difference) but that event would have occurred at 3280/1.318 = 2489. So to be more careful, if you want to use your version 6 as a "time machine" then to go back to year 10.15 billion you should put in 12.0 instead of 14.0 2489 instead of 3280 (but that makes very little difference so for a quick and dirty we don't need to change S_eq) I will explain this some more but wanted to send you this right away. 



#54
Feb1813, 08:20 AM

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What you found a couple of posts back was quite consistent. Try dividing our comoving distance 62.3 by the factor 1.318. It should give approximately the right thing.
The basic time machine experiment we did was to change the Hubbletime (Ynow) from 14.0 to 12.0 and that jumps us back into essentially the same universe but at year 10.148 or call it 10.15 billion. But when we go back then, distances are all less by a factor of 1.318. You can check that by staying in our timeframe (Ynow=14.0) and putting in S=1.318 and you will get that Time=10.15 billion. So we know that in our universe, if we go back to year 10.15 billion distances (in that year) are less by that factor. We don't have to worry about that if we are just talking about PROPER distance because that has a kind of independent meaning regardless of what year we are living in. But comoving distances, which are "now" distances at the time we are living in, will be different because we are in a different present. So we have to adjust the S values accordingly and the comoving distances. I could always be wrong about this but I'm pretty sure in this instance that it is right. It's a great calculator! We keep finding more things one can do with it. I suspect that it's an idea whose time is come and we are apt to see other tabular cosmic calculators appear in the next 2 or 3 years. This one will plant a seed in some people's minds and they will talk to other people who talk with other people. And then someone will get the idea and not know where he got it from. the idea will be "in the air". That's how I think it is apt to go. The universe is about continuity and development, so tabular output is natural to it. Thanks for finding the Tamara Davis originals. They are sharp, and colorcoded. I think maybe both Davis and Lineweaver are talented communicators (as well as firstrate cosmologists). I suspect Lineweaver saw a good thing when his Phd student Davis showed him that 3layer "figure 1" and he adopted it straight off the bat. Science progresses not only by people discovering things but also by their finding really good ways to transmit the important ideas. (Or so I thinkjust my two cents as an onlooker.) 


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