# Cosmo calculators with tabular output

by marcus
Tags: calculators, cosmo, output, tabular
P: 757
 Quote by marcus ... stretch factor 2.63 and emission distance 5.8, has to do with the beautiful fact that past lightcones are TEAR-DROP SHAPE. You can see that at the top level of the "figure 1" in my signature. That is what they look like when you measure in proper distance, the real distance that it actually was at the time, if you could have stopped the expansion process.
I have massaged a spreadsheet of the tabular data a little in order to plot a graph that looks somewhat like the top level Davis plot in your sig. In the process I became interested in the relationship between the event horizon and the particle horizon and subsequently have added a column for the particle horizon to TabCosmo5 (saved as TabCosmo6). Graphically it looks like this:

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

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

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

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

Is it because observed redshift at the event horizon will tend to infinity?
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 Astronomy Sci Advisor PF Gold P: 23,227 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
 Astronomy Sci Advisor PF Gold P: 23,227 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.
 Sci Advisor PF Gold P: 9,445 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.
 Astronomy Sci Advisor PF Gold P: 23,227 This is strange. Using the new calculator version6, I don't actually get a coincidence. I'm putting in Step=0 so I just get a one-line table, for S=2.632 That is what I am used to using to get the intersection of Hubbleradius and D_then. Or even better: S=2.6321 But that does not give a match between D_hor and particle horizon D_par! It looked on the figure as if they were at the same level so I thought there was an exact coincidence (but couldn't figure out why there would be) and now the table does not give a coincidence. 11.804 ≠ 11.934 Am I missing something? being really dense? Sorry for a possible bungling lapse of competence. Can someone explain this almost but not quite coincidence? To get D_hor to equal D_par, you have to go to S=2.662 11.736 ≈ 11.735 well, let's still find the comoving (now) distance to the farthest pingable matter: 2.662*11.735 = 31.2 Gly. Yes! that's still good. I suppose that twice that, namely 62.4 Gly is the distance now of the farthest matter we will ever hear from regardless how long we wait.
P: 757
 Quote by marcus But that does not give a match between D_hor and particle horizon D_par! It looked on the figure as if they were at the same level so I thought there was an exact coincidence (but couldn't figure out why there would be) and now the table does not give a coincidence. 11.804 ≠ 11.934 Am I missing something? being really dense? Sorry for a possible bungling lapse of competence. Can someone explain this almost but not quite coincidence? To get D_hor to equal D_par, you have to go to S=2.662 11.736 ≈ 11.735
I have also noticed this, but my first reaction was that it is caused by small errors in the numerical integration loops of the various curves. Remember that to get all the values perfect, it requires integration for time (or S) from zero to infinity with an 'infinite number of steps', which is not feasible. Especially D_hor is very susceptible to cut-off errors.

What is intriguing is that the rough correspondence remains when Ynow and Yinf are changed. I'm busy looking at it analytically (not easy) and will report what I find.
 Astronomy Sci Advisor PF Gold P: 23,227 Because you are doing hard analytical work I should probably be quiet and not distract from that. I had something else I wanted to say, though. It seems to me that the distance 11.735 Gly is somehow UNIVERSAL. It does not know about us, that we are in year 13.7 Gy or so. It depends on sending out a radar ping at the start of the expansion, from wherever you are, and then being able to wait to year infinity to hear back. The farthest distance, as a proper distance from your matter when the bounceback happens, should be the same for anyone in the universe at any stage in its history. Is the distance 5.8 comparably universal? It seems strange that it should be roughly HALF of 11.735 But that could be a spurious coincidence. I dimly suspect that the distance 5.8 depends on WHEN in the history of the universe you are. It is the maximum proper distance at emission-time of any light we can detect now. I may be missing something, but that seems to depend on when in the history of the universe we are.
P: 757
 Quote by Chronos The redshift approaches infinity by the time photons currently emited at the CEH reach us. Of course, the time it takes those photons to reach us also approaches infinity. If you think in terms of scale factor, it all seems to make sense.
Yes, it is a bit clearer in terms of scale a = 1/S and comoving distances. Working on that.
From Davis http://arxiv.org/abs/astro-ph/0402278 (2004), Eqs. A.19 and A.20, pp. 117, with c=1:

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

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

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

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

Anyone with ideas?
Astronomy
PF Gold
P: 23,227
 Quote by Jorrie ...Anyone with ideas?
This is not the type of idea you specifically asked for, but let's explore the idea that the apparent coincidence may be spurious. If that's wrong, and it is a mathematical equality some reader will show up, I trust, and explain. Meanwhile I make the tentative assertion that the maximum girth of the teardrop lightcone (and the time that occurs) depends strongly on where we are in the history of the universe. If we were later the teardrop would be bigger and the bulge would come later. We wouldn't be seeing that time figure of 4 Gy and that maximum emission distance figure of 5.8 Gly. If we were earlier/later in the expansion process those numbers would each be smaller/larger.

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

Because I think the maximum proper distance of a radar bounce is a universal INVARIANT, and so is the year that bounce occurs. It is going to be the same as long as the basic cosmic parameters are the same, whether from the perspective of some one earlier than us or someone far in the future. the reason is that the present expansion age does not enter 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.)
 Astronomy Sci Advisor PF Gold P: 23,227 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.
Mentor
P: 6,245
 Quote by marcus I checked. The coincidence does see merely accidental.
Here's another way (or the same way from a slightly different perspective) to see this.

The particle and event horizons do not depend on a "now" event, so their intersection does not depend on a "now" event. The Hubble sphere does not depend on "now", but the past lightcone does depend on "now", so their intersection does depend on "now". This is particularly evident in Figure 1 from Davis Lineweaver. As the "now" line shifts up and down, the intersection of the past lightcone and the Hubble sphere changes (for me, especially clear in the bottom panel), but the intersection of the particle and event horizons remains the same.
Astronomy
PF Gold
P: 23,227
 Quote by George Jones Here's another way (or the same way from a slightly different perspective) to see this. The particle and event horizons do not depend on a "now" event, so their intersection does not depend on a "now" event. The Hubble sphere does not depend on "now", but the past lightcone does depend on "now", so their intersection does depend on "now". This is particularly evident in Figure 1 from Davis Lineweaver. As the "now" line shifts up and down, the intersection of the past lightcone and the Hubble sphere changes (for me, especially clear in the bottom panel), but the intersection of the particle and event horizons remains the same.
Good! Clear concise way to explain it. Thanks, George.
P: 757
 Quote by George Jones This is particularly evident in Figure 1 from Davis Lineweaver. As the "now" line shifts up and down, the intersection of the past lightcone and the Hubble sphere changes (for me, especially clear in the bottom panel), but the intersection of the particle and event horizons remains the same.
Thanks, this gives a clear picture. Like Marcus, I could not find any further empirical or analytical evidence anyway.
P: 757
 Quote by marcus I checked. The coincidence does seem 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.
My first reaction was that only the Ynow change would not give valid calculation for an earlier epoch, but to my surprise it works as you have done it. Leaving all the other stuff the same, the calculator calculates the new earlier energy balance and in effect just shrinks the past light cone, while the other outputs remain the same. It essentially just shifts the now-line up and down on the Davis Figure 1. Its a new usage of the tool that you have discovered. :-)

It's bed time in my valley, so I will look at it again some time tomorrow.
P: 757
 Quote by Jorrie My first reaction was that only the Ynow change would not give valid calculation for an earlier epoch, but to my surprise it works as you have done it. Leaving all the other stuff the same, the calculator calculates the new earlier energy balance and in effect just shrinks the past light cone, while the other outputs remain the same. It essentially just shifts the now-line up and down on the Davis Figure 1. Its a new usage of the tool that you have discovered. :-)
I am no longer so sure that this is valid. Although it shifts the now-line up and down, it also changes the convergence on 62.3 Gly (comoving) to some 47.5 Gly. I have checked this convergence on a spreadsheet with Marcus' Y_now = 12 example, leaving the rest the same. This does not seem right. Since D_comoving = S D_proper, and we use the same S, one would expect the 62.3 to stay the same (?). The calculator is designed to work for inputs as at present and it assumes that changing the inputs change the present observed parameters. The past and present values should only be read off the table (or graphs of it).

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.
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 Astronomy Sci Advisor PF Gold P: 23,227 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.
 Astronomy Sci Advisor PF Gold P: 23,227 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 color-coded. I think maybe both Davis and Lineweaver are talented communicators (as well as first-rate cosmologists). I suspect Lineweaver saw a good thing when his Phd student Davis showed him that 3-layer "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 think---just my two cents as an onlooker.)

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