Cosmo calculators with tabular output

Jorrie

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.

Attached Thumbnails

 

marcus

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.

marcus

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.)

Jorrie

Yes, I think you are quite right :) Past and future observers would 'freeze frame' the expansion at different stages than us and hence their equivalent definition of comoving distances would yield different values for the same objects/horizons.

It is very interesting that the new Ynow input automatically adjust Ho, Ω_λ and Ω_m. This is an advantage over the usual Ho and Ω input calculators, which usually can take a combination that is invalid (without user knowing it). I must look at a way to adjust S_eq and S_CMB defaults automatically as well and it will be even more convenient for all sorts of cosmo calculations. One can obviously override any of them manually if you want...

marcus

An alternative might be to SUGGEST over on the right what S_CMB the user might like to use, and expect him to type in something different from 1090. For me, it was a learning experience to have to put different stuff in the boxes. A mild "learn by doing" experience, not earth-shaking. But I sense the value of having to do something myself now and then, to get an interesting effect, rather than having the calculator always do it for me.

Basically however, I trust your pedagogical machine design sense. So far all your added features seem like definite improvements and not "too much". It's become a really fine learning machine---someone could write a brief user manual which would suggest things to do with it---cosmological exercise book, things to try on it.

I wish I knew someone who was teaching Introduction to Cosmology at some college or university. I'd like to see TabCosmo tried out for use in a class. I know OF people but I'm not in close enough personal touch with the right ones to be effective.

Does anybody here know of someone teaching Astronomy for Non-Majors or something comparable?

Jorrie

It appears simple, but it turns out to be a rather involved programming change, so it must go to the back burner for now. I will include the steps that you have used somewhere in the info tips in a future update. They are simple enough and as you said, serve some educational purpose. Good work, Marcus.

Jorrie

For completeness of reference,^[1] here is the full compact set of TabCosmo6 equations (added particle horizon from previous).
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, also referred to as H(t)
[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', cosmic event horizon and particle 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}}, \ \ \,
D_{par} = \frac{1}{S}\int_{S}^{\infty}{ \frac{dS}{H}}
[/tex]
To obtain all the values, it 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.

[1] Davis: http://arxiv.org/abs/astro-ph/0402278 (2004), Appendix A. All equations converted to Stretch factor S (in place of t and a in Davis).

Jorrie

Marcus has previously posted many tabular outputs from the TabCosmo calculator, but he had to massage the output considerably in order to make it readable in the [code] tags of the editor. The [tex] array option is available, but that requires a lot more manual work - something that the machine could actually do better. I have added an option for a LaTex compatible output and uploaded it as TabCosmo7.

It requires you to first play around until you have the range of values that you are interested in, tick the radio button for LaTex, Calculate and then copy and paste the code into a LaTex compatible editor. It is optimized for the PF editor, but you can modify any part of the Tex code after copying (obviously at your own risk :-)

Please report any problems/suggestions.

Here is a sample output.

[tex]{\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} (Km/s/Mpc) & \Omega_\Lambda & \Omega_m\\ \hline14&16.5&3280&69.86&0.72&0.28\\ \hline\end{array}}[/tex] [tex]{\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&T_{Hub}(Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.001\\ \hline341.731&0.002926&0.002511&0.003986&44.573&0.130&0.177&0.006\\ \hline107.137&0.009334&0.015296&0.023478&42.386&0.396&0.543&0.040\\ \hline33.589&0.029772&0.089394&0.135218&38.404&1.143&1.614&0.246\\ \hline10.531&0.094961&0.513668&0.772152&31.251&2.968&4.469&1.464\\ \hline3.302&0.302891&2.902232&4.258919&18.588&5.630&10.418&8.506\\ \hline1.035&0.966116&13.274154&13.791148&0.473&0.457&15.728&44.633\\ \hline0.325&3.081570&31.418524&16.391363&-10.476&-32.283&16.428&176.105\\ \hline0.102&9.829121&50.521674&16.496494&-14.143&-139.014&16.496&597.755\\ \hline0.032&31.351430&69.658811&16.499868&-15.295&-479.531&16.500&1942.755\\ \hline0.010&100.000000&88.797170&16.499905&-15.657&-1565.665&16.500&6232.831\\ \hline\end{array}}[/tex]

marcus

It's a beauty, really is the greatest thing since sliced bread!

I bookmarked http://www.einsteins-theory-of-relat...TabCosmo7.html
and will change my signature link.

marcus

I'm continuing to try this version out. Especially the LaTex feature. This is where I checked the "S=1 exactly" box, so the present moment in included in the history. And set it for 29 steps (from 1090 to 1 and then from 1 to 0.05, around year 62 billion in the future.)

I think many of us, perhaps most of the regular posters here, are familiar with the idea that the present expansion rate of distance is 1/140 % per million years.
Can you find when it was in the universe history that the expansion rate was ONE PERCENT per million years? I mean roughly, around what years?

Can you find the FARTHEST DISTANCE a galaxy could have been when it emitted light which is arriving to us today?
At what speed was that galaxy receding when it emitted the light (which we are now receiving)?

Easy questions which may help you get quantitatively engaged with the expansion history (if it is new to you.)

[tex]{\begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14&16.5&3280&69.86&0.72&0.28\\ \hline\end{array}}[/tex] [tex]{\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&T_{Hub}(Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.001\\ \hline856.422&0.001168&0.000566&0.000940&45.550&0.053&0.072&0.001\\ \hline672.897&0.001486&0.000842&0.001381&45.341&0.067&0.091&0.002\\ \hline528.701&0.001891&0.001247&0.002020&45.101&0.085&0.115&0.003\\ \hline415.404&0.002407&0.001839&0.002944&44.825&0.108&0.146&0.004\\ \hline326.387&0.003064&0.002700&0.004279&44.509&0.136&0.185&0.007\\ \hline256.445&0.003899&0.003951&0.006205&44.150&0.172&0.234&0.010\\ \hline201.491&0.004963&0.005761&0.008979&43.740&0.217&0.296&0.015\\ \hline158.313&0.006317&0.008379&0.012973&43.275&0.273&0.373&0.021\\ \hline124.388&0.008039&0.012159&0.018720&42.747&0.344&0.471&0.032\\ \hline97.732&0.010232&0.017610&0.026985&42.149&0.431&0.593&0.046\\ \hline76.789&0.013023&0.025465&0.038867&41.472&0.540&0.746&0.068\\ \hline60.334&0.016574&0.036773&0.055945&40.706&0.675&0.937&0.099\\ \hline47.405&0.021095&0.053047&0.080484&39.840&0.840&1.174&0.144\\ \hline37.246&0.026848&0.076452&0.115738&38.861&1.043&1.468&0.210\\ \hline29.265&0.034171&0.110103&0.166377&37.755&1.290&1.830&0.305\\ \hline22.993&0.043491&0.158470&0.239106&36.507&1.588&2.275&0.442\\ \hline18.066&0.055352&0.227971&0.343537&35.097&1.943&2.818&0.641\\ \hline14.195&0.070449&0.327812&0.493442&33.506&2.360&3.474&0.927\\ \hline11.153&0.089663&0.471192&0.708498&31.711&2.843&4.261&1.341\\ \hline8.763&0.114117&0.677001&1.016667&29.686&3.388&5.192&1.938\\ \hline6.885&0.145241&0.972188&1.457265&27.404&3.980&6.276&2.798\\ \hline5.410&0.184854&1.394848&2.084258&24.837&4.591&7.513&4.036\\ \hline4.250&0.235270&1.998124&2.968150&21.958&5.166&8.885&5.814\\ \hline3.340&0.299437&2.853772&4.190977&18.748&5.614&10.347&8.361\\ \hline2.624&0.381105&4.052600&5.822089&15.215&5.798&11.823&11.988\\ \hline2.062&0.485047&5.694902&7.857010&11.408&5.534&13.201&17.104\\ \hline1.620&0.617337&7.861899&10.128494&7.459&4.605&14.363&24.207\\ \hline1.273&0.785708&10.571513&12.291156&3.574&2.808&15.228&33.862\\ \hline1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686\\ \hline0.786&1.272738&17.277468&15.133799&-3.141&-3.998&16.121&63.399\\ \hline0.715&1.399556&18.729987&15.440794&-4.230&-5.920&16.203&71.239\\ \hline0.650&1.539011&20.208716&15.684266&-5.238&-8.061&16.267&79.889\\ \hline0.591&1.692361&21.707838&15.875269&-6.167&-10.436&16.315&89.422\\ \hline0.537&1.860992&23.223153&16.023472&-7.021&-13.066&16.351&99.921\\ \hline0.489&2.046426&24.750714&16.137834&-7.804&-15.970&16.378&111.480\\ \hline0.444&2.250336&26.287971&16.225336&-8.520&-19.174&16.398&124.201\\ \hline0.404&2.474564&27.832518&16.292069&-9.175&-22.704&16.412&138.196\\ \hline0.367&2.721136&29.382453&16.342940&-9.773&-26.593&16.421&153.593\\ \hline0.334&2.992276&30.936767&16.381374&-10.318&-30.873&16.427&170.527\\ \hline0.304&3.290433&32.494109&16.410600&-10.814&-35.583&16.430&189.153\\ \hline0.276&3.618299&34.054029&16.432542&-11.266&-40.765&16.433&209.637\\ \hline0.251&3.978834&35.615607&16.449246&-11.678&-46.465&16.449&232.164\\ \hline0.229&4.375295&37.178725&16.461699&-12.053&-52.734&16.462&256.937\\ \hline0.208&4.811259&38.742715&16.471229&-12.394&-59.629&16.471&284.179\\ \hline0.189&5.290663&40.307651&16.478264&-12.704&-67.213&16.478&314.137\\ \hline0.172&5.817837&41.873010&16.483706&-12.986&-75.552&16.484&347.080\\ \hline0.156&6.397539&43.438976&16.487660&-13.243&-84.723&16.488&383.307\\ \hline0.142&7.035005&45.005111&16.490781&-13.477&-94.808&16.491&423.143\\ \hline0.129&7.735988&46.571662&16.492987&-13.689&-105.898&16.493&466.950\\ \hline0.118&8.506820&48.138401&16.494627&-13.882&-118.094&16.495&515.121\\ \hline0.107&9.354458&49.705116&16.496007&-14.058&-131.504&16.496&568.092\\ \hline0.097&10.286558&51.272104&16.496902&-14.218&-146.251&16.497&626.342\\ \hline0.088&11.311533&52.839007&16.497721&-14.363&-162.468&16.498&690.396\\ \hline0.080&12.438640&54.406135&16.498195&-14.495&-180.301&16.498&760.833\\ \hline0.073&13.678054&55.973144&16.498697&-14.615&-199.910&16.499&838.287\\ \hline0.066&15.040966&57.540352&16.498931&-14.725&-221.473&16.499&923.460\\ \hline0.060&16.539682&59.107420&16.499254&-14.824&-245.185&16.499&1017.120\\ \hline0.055&18.187733&60.674673&16.499353&-14.914&-271.260&16.499&1120.112\\ \hline0.050&20.000000&62.241776&16.499574&-14.997&-299.933&16.500&1233.366\\ \hline\end{array}}[/tex]

Jorrie

I understand why you prefer the 1/140 % per million years for the present expansion rate, because the value is roughly constant for the next million years or so. I find the use of "Present time needed for 1% growth in cosmic distance" = 140 My (10T_H) slightly easier to remember, although the time may change somewhat over the next 140 My. One can also use "Present time to double all cosmic distances" = 14 Gy, which is directly the present Hubble time. The drawback is that the real time for a doubling in size is much less, because there is a significant (exponential) change in da/dt over the next billion years.

marcus

It's just a layman style of talking and there's no one right or perfect way to express the distance growth rate, I think. As you point out, there are several equally good ways to put it.
I guess I've gotten into a rut of saying "1/140 of a percent per million years". I hope this works, but could try different ways if you want.

To me, the word "per" suggests an instantaneous rate, as when one says the guy is going "miles per hour" even though the guy is only going to drive for 15 minutes. This is important because the instantaneous rate idea is what we need to get across. Plus the idea that it is very slowly changing. Towards 1/165 of a percent.

I really like the fact that in the table you see "dark energy" manifestly there as something real. Namely you see the cosmological constant surface as the limiting expansion rate of 1/165 percent per million years.

You and I have noted that numerous times. But it may still be new to some readers: it jumps out in the table just printed, so clearly. As the eventual 16.5 Gly cosmological horizon and 16.5 Gy Hubbletime. It stares one in the face in two columns, down at the bottom of table, way in future.

One can think of it as a residual built-in expansion rate that cannot go away or as a small residual space-time curvature. We can remind ourselves how that expansion rate or spacetime curvature can converted to a (possibly fictional) "energy" density---basically just converting the curvature into different units using the natural constants G and c.

Put this in the google window: 3c^2/(8 pi G)/(16.5 billion years)^2
when you press the "equals" key you should get 0.593 nanopascals
or in other words 0.593 nanojoules per cubic meter (the energy density that conventionally corresponds to cosmo constant Lambda as currently estimated.)

The constants 3c^2/(8 pi G) are simply what accomplishes the change into units of energy density.

I think it's great that in a table with future like this you get to see the constant Lambda (or its energy density alias 0.6 nanojoules per cubic meter) emerge clearly as something tangible like the distance to a horizon.
==================

The answer to one of the questions a couple of posts back: around year 60 million was when distances were expanding at just 1% per million years.
That was when distances were about 1/40 what they are today. So the stretch factor is in the interval 37 to 47 that one sees in the table.

Can anyone suggest some other questions one could ask as part of practice reading a history table like this? It might be good to have a supply of warm-up exercises.

marcus

Here's another practice question referring to the table a few posts back. Imagine four galaxies that are roughly the same shape and size which are visible today. They are at different distances from us and the light we are receiving today from them was emitted at different times: in year 2 billion, in year 4 billion, in year 6 billion, and in year 8 billion, say.

Call the galaxies A, B, C, and D respectively, if you like. Which one looks the smallest?
In other words which one has the smallest angular width, and makes the smallest angle in the sky?

Maybe instead of 2, 4, 6, 8, I should have said 2.0, 4.0, 5.7, and 7.9 since those times are closer to the times appearing in the table. But mentally interpolating is easy enough. Obviously the one with the smallest angular width is the one which was the farthest away when it emitted the light, and that's not hard to spot.
==============

Another practice question: in what year of the universe history were distances expanding ELEVEN percent per million years? And by what factor have distances and wavelengths expanded since then, up to present day?

Jorrie

For completeness of reference, here is the updated compact set of TabCosmo9 equations.^[1] (changed from Hubble time inputs to Hubble radii and added da/dT). Basic inputs are the Hubble radius [itex]R_{now}[/itex], the long term Hubble radius [itex] R_{\infty}[/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{R_{now}}{R_{\infty}}\right)^2, \ \ \,
\Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \,
\Omega_m = S_{eq}\Omega_r [/tex]

Hubble parameter, also referred to as H(t)
[tex]H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}[/tex]
Hubble radius and Cosmic time (in geometric units, where c=1)
[tex]R = 1/H, \ \ \,
T = \int_{S}^{\infty}{\frac{dS}{S H}}[/tex]
Proper distance 'now', 'then', cosmic event horizon and particle horizon
[tex]D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \,
D_{then} = \frac{D_{now}}{S}, \ \ \,
D_{hor} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}, \ \ \,
D_{par} = \frac{1}{S}\int_{S}^{\infty}{ \frac{dS}{H}}
[/tex]

The expansion rate as a fractional distance per unit time (at time T)
[tex]\frac{da}{dT} = aH = \frac{a}{R} [/tex]

To obtain all the values, it 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.

[1] Davis: http://arxiv.org/abs/astro-ph/0402278 (2004), Appendix A. All equations converted to Stretch factor S (in place of t and a in Davis).

Jorrie

I have experimented a bit and it seems that to multiply da/dT by the present Hubble radius [itex]R_{now}[/itex] gives a more interesting column in the calculator. Its header says [itex]R'_{now}[/itex], for [itex]R_{now}\frac{da}{dT}[/itex], which represents the expansion rate history of an object presently observed exactly at the Hubble radius. Here is a sample table:
[tex]{\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14&16.5&3280&69.86&0.72&0.28\\ \hline \end{array}}[/tex] [tex]{\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&R'_{now}\\ \hline 1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.001&20.164\\ \hline 541.606&0.001846&0.001200&0.001945&45.126&0.083&0.113&0.003&13.292\\ \hline 269.117&0.003716&0.003662&0.005761&44.225&0.164&0.223&0.009&9.029\\ \hline 133.721&0.007478&0.010876&0.016772&42.912&0.321&0.439&0.028&6.242\\ \hline 66.444&0.015050&0.031751&0.048364&41.023&0.617&0.855&0.085&4.357\\ \hline 33.015&0.030289&0.091754&0.138771&38.325&1.161&1.640&0.253&3.056\\ \hline 16.405&0.060958&0.263633&0.397095&34.484&2.102&3.066&0.743&2.149\\ \hline 8.151&0.122680&0.754694&1.132801&29.030&3.561&5.501&2.164&1.516\\ \hline 4.050&0.246896&2.146402&3.182937&21.343&5.269&9.172&6.254&1.086\\ \hline 2.013&0.496887&5.887073&8.078066&11.017&5.474&13.329&17.716&0.861\\ \hline 1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686&1.000\\ \hline 0.631&1.584893&20.670471&15.748412&-5.533&-8.770&16.283&82.739&1.409\\ \hline 0.398&2.511886&28.076314&16.301181&-9.273&-23.293&16.413&140.526&2.157\\ \hline 0.251&3.981072&35.624819&16.449365&-11.680&-46.500&16.449&232.303&3.388\\ \hline 0.158&6.309573&43.210628&16.487217&-13.207&-83.331&16.487&377.810&5.358\\ \hline 0.100&10.000000&50.805908&16.496757&-14.172&-141.718&16.497&608.434&8.487\\ \hline 0.063&15.848932&58.403573&16.499147&-14.781&-234.257&16.499&973.953&13.448\\ \hline 0.040&25.118864&66.001838&16.499740&-15.165&-380.922&16.500&1553.261&21.313\\ \hline 0.025&39.810717&73.600254&16.499880&-15.407&-613.371&16.500&2471.404&33.779\\ \hline 0.016&63.095734&81.198707&16.499907&-15.560&-981.779&16.500&3926.561&53.536\\ \hline 0.010&100.000000&88.797170&16.499905&-15.657&-1565.665&16.500&6232.831&84.849\\ \hline \end{array}}[/tex]
If I interpret this correctly, it means that the object has been outside our Hubble sphere up to around T=3 Gy, then entered the sphere and is leaving it now, to stay outside for as long as accelerated expansion keeps going.

Jorrie

Comparing the following table with the Davis center-panel expansion diagram, it seems that the column for [itex]R'_{now}[/itex] (the expansion rate history of a galaxy that is presently on our Hubble sphere) is valid.

[tex]{\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14&16.5&3280&69.86&0.72&0.28\\ \hline \end{array}}[/tex] [tex]{\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&R'_{now}\\ \hline 3.336&0.299760&2.858302&4.197327&18.733&5.616&10.354&8.375&1.000\\ \hline 3.102&0.322331&3.178963&4.643930&17.702&5.706&10.801&9.338&0.972\\ \hline 2.869&0.348578&3.562786&5.168812&16.558&5.772&11.282&10.497&0.944\\ \hline 2.635&0.379478&4.027752&5.789430&15.280&5.798&11.797&11.912&0.918\\ \hline 2.402&0.416389&4.598945&6.526791&13.844&5.764&12.346&13.669&0.893\\ \hline 2.168&0.461255&5.311204&7.404502&12.220&5.636&12.927&15.891&0.872\\ \hline 1.934&0.516956&6.214226&8.445751&10.372&5.362&13.533&18.766&0.857\\ \hline 1.701&0.587959&7.379324&9.665141&8.260&4.857&14.151&22.584&0.852\\ \hline 1.467&0.681570&8.910486&11.051952&5.843&3.982&14.756&27.828&0.863\\ \hline 1.234&0.810636&10.959447&12.543378&3.088&2.503&15.317&35.330&0.905\\ \hline 1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686&1.000\\ \hline \end{array}}[/tex]

Here is a zoomed portion of the Davis center-panel:



The object presently on the surface of our Hubble sphere will be at redshift z~2.33. It was also on the Hubble sphere at t~2.86 Gyr (the dashed purple lines that I've added) when it first entered our Hubble sphere. Outside the Hubble sphere the recession rate exceed c.

Do you think this experimental column is useful? Or is it just cluttering up the calculator?

Attached Thumbnails

 

marcus

Beautiful graphic! I somehow missed this post yesterday. I am still unclear about the physical meaning of the righthand column quantity, and the example of the object we observe with S=3.336.
I'll keep thinking about it.

I see! You see the dashed line for T=2.86. On the right it does not extend far enough, it should go out to the light cone (where the object is).
But fortunately it does extend out on the left far enough, so it interesects light cone. It shows us that the current distance of the object is around 18.7 Gly just as your calculator says! The comoving distance to the object is around 18.7 Gly, which is pretty much where that T=2.86 line intersects lightcone.

And it also looks to me like the horizontal dashed line intersects lightcone around z=2.336 too, as it should. 2.336 would be say 2/3 of the way from 1 to 3, which it looks like it is. Also since the horizontal z scale is kind of "log-ish" and the "2" mark itself might not be exactly halfway between 1 and 3 but somewhat closer to the 3 mark, in case that matters.

So that all fits with what the top row of your latest table shows, for S=3.336

What is not so fortunate is that the Tamara Davis charts don't have an a(t) curve. The scale factor is used as a vertical scale up the righthand side, sort of as a alternative to time, to mark the stage in history. So we dont have an a(t) curve. Your new column is about the SLOPE of the a(t) curve.
I'm undecided about it, haven't figured out what I think. Somehow it should show a minimum around year 7 billion (you gave it exactly a while back, something like 7.6) Actually it seems to do that! I just looked at the S=1.7 row in the preceding table. That is year 7.4, close enough, and in fact it does look like da/dT is bottoming out right there. I'll get back to this after a while and try to give a coherent opinion
====================

I had another look and I think there are pros and cons about the 9th column. Multiplying by Rnow seems somewhat arbitrary. Doesn't it just scale the numbers up? I thought the notation Rnow' is a bit confusing since it gives the impression it is the derivative of Rnow,and that Rnow is changing. But Rnow is a constant. A fixed parameter of the model. Isn't da/dT what the column is really about? So couldn't you achieve the same effect by making it
100xda/dT, or 1000xda/dT? Some arbitrary multiplicative factor, in other words?
Or perhaps I'm missing something.