Cosmo calculators with tabular output

[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.)

{\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}} {\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}}

 

[Jorrie]

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?

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 R_{now}, the long term Hubble radius R_{\infty} and the redshift for radiation/matter equality z_{eq}.

Since the factor z + 1 occurs so often, an extra parameter S = z + 1 = 1/a is defined, making the equations neater.
\Omega_\Lambda = \left(\frac{R_{now}}{R_{\infty}}\right)^2, \ \ \, <br /> \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \,<br /> \Omega_m = S_{eq}\Omega_r

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

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

To obtain all the values, it essentially means integration for S from zero to infinity, but practically it has been limited to 10^{-7}&lt; S &lt;10^{7} 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]

For completeness of reference, here is the updated compact set of TabCosmo9 equations. (changed from Hubble time inputs to Hubble radii and added da/dT).

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

I have experimented a bit and it seems that to multiply da/dT by the present Hubble radius R_{now} gives a more interesting column in the calculator. Its header says R&#039;_{now}, for R_{now}\frac{da}{dT}, which represents the expansion rate history of an object presently observed exactly at the Hubble radius. Here is a sample table:
{\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) &amp; R_{∞} (Gly) &amp; S_{eq} &amp; H_{0} &amp; \Omega_\Lambda &amp; \Omega_m\\ \hline14&amp;16.5&amp;3280&amp;69.86&amp;0.72&amp;0.28\\ \hline \end{array}} {\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&amp;a=1/S&amp;T (Gy)&amp;R (Gly)&amp;D (Gly)&amp;D_{then}(Gly)&amp;D_{hor}(Gly)&amp;D_{par}(Gly)&amp;R&#039;_{now}\\ \hline 1090.000&amp;0.000917&amp;0.000378&amp;0.000637&amp;45.731&amp;0.042&amp;0.056&amp;0.001&amp;20.164\\ \hline 541.606&amp;0.001846&amp;0.001200&amp;0.001945&amp;45.126&amp;0.083&amp;0.113&amp;0.003&amp;13.292\\ \hline 269.117&amp;0.003716&amp;0.003662&amp;0.005761&amp;44.225&amp;0.164&amp;0.223&amp;0.009&amp;9.029\\ \hline 133.721&amp;0.007478&amp;0.010876&amp;0.016772&amp;42.912&amp;0.321&amp;0.439&amp;0.028&amp;6.242\\ \hline 66.444&amp;0.015050&amp;0.031751&amp;0.048364&amp;41.023&amp;0.617&amp;0.855&amp;0.085&amp;4.357\\ \hline 33.015&amp;0.030289&amp;0.091754&amp;0.138771&amp;38.325&amp;1.161&amp;1.640&amp;0.253&amp;3.056\\ \hline 16.405&amp;0.060958&amp;0.263633&amp;0.397095&amp;34.484&amp;2.102&amp;3.066&amp;0.743&amp;2.149\\ \hline 8.151&amp;0.122680&amp;0.754694&amp;1.132801&amp;29.030&amp;3.561&amp;5.501&amp;2.164&amp;1.516\\ \hline 4.050&amp;0.246896&amp;2.146402&amp;3.182937&amp;21.343&amp;5.269&amp;9.172&amp;6.254&amp;1.086\\ \hline 2.013&amp;0.496887&amp;5.887073&amp;8.078066&amp;11.017&amp;5.474&amp;13.329&amp;17.716&amp;0.861\\ \hline 1.000&amp;1.000000&amp;13.753303&amp;13.999929&amp;0.000&amp;0.000&amp;15.793&amp;46.686&amp;1.000\\ \hline 0.631&amp;1.584893&amp;20.670471&amp;15.748412&amp;-5.533&amp;-8.770&amp;16.283&amp;82.739&amp;1.409\\ \hline 0.398&amp;2.511886&amp;28.076314&amp;16.301181&amp;-9.273&amp;-23.293&amp;16.413&amp;140.526&amp;2.157\\ \hline 0.251&amp;3.981072&amp;35.624819&amp;16.449365&amp;-11.680&amp;-46.500&amp;16.449&amp;232.303&amp;3.388\\ \hline 0.158&amp;6.309573&amp;43.210628&amp;16.487217&amp;-13.207&amp;-83.331&amp;16.487&amp;377.810&amp;5.358\\ \hline 0.100&amp;10.000000&amp;50.805908&amp;16.496757&amp;-14.172&amp;-141.718&amp;16.497&amp;608.434&amp;8.487\\ \hline 0.063&amp;15.848932&amp;58.403573&amp;16.499147&amp;-14.781&amp;-234.257&amp;16.499&amp;973.953&amp;13.448\\ \hline 0.040&amp;25.118864&amp;66.001838&amp;16.499740&amp;-15.165&amp;-380.922&amp;16.500&amp;1553.261&amp;21.313\\ \hline 0.025&amp;39.810717&amp;73.600254&amp;16.499880&amp;-15.407&amp;-613.371&amp;16.500&amp;2471.404&amp;33.779\\ \hline 0.016&amp;63.095734&amp;81.198707&amp;16.499907&amp;-15.560&amp;-981.779&amp;16.500&amp;3926.561&amp;53.536\\ \hline 0.010&amp;100.000000&amp;88.797170&amp;16.499905&amp;-15.657&amp;-1565.665&amp;16.500&amp;6232.831&amp;84.849\\ \hline \end{array}}
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]

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.

Comparing the following table with the Davis center-panel expansion diagram, it seems that the column for R&#039;_{now} (the expansion rate history of a galaxy that is presently on our Hubble sphere) is valid.

{\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) &amp; R_{∞} (Gly) &amp; S_{eq} &amp; H_{0} &amp; \Omega_\Lambda &amp; \Omega_m\\ \hline14&amp;16.5&amp;3280&amp;69.86&amp;0.72&amp;0.28\\ \hline \end{array}} {\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&amp;a=1/S&amp;T (Gy)&amp;R (Gly)&amp;D (Gly)&amp;D_{then}(Gly)&amp;D_{hor}(Gly)&amp;D_{par}(Gly)&amp;R&#039;_{now}\\ \hline 3.336&amp;0.299760&amp;2.858302&amp;4.197327&amp;18.733&amp;5.616&amp;10.354&amp;8.375&amp;1.000\\ \hline 3.102&amp;0.322331&amp;3.178963&amp;4.643930&amp;17.702&amp;5.706&amp;10.801&amp;9.338&amp;0.972\\ \hline 2.869&amp;0.348578&amp;3.562786&amp;5.168812&amp;16.558&amp;5.772&amp;11.282&amp;10.497&amp;0.944\\ \hline 2.635&amp;0.379478&amp;4.027752&amp;5.789430&amp;15.280&amp;5.798&amp;11.797&amp;11.912&amp;0.918\\ \hline 2.402&amp;0.416389&amp;4.598945&amp;6.526791&amp;13.844&amp;5.764&amp;12.346&amp;13.669&amp;0.893\\ \hline 2.168&amp;0.461255&amp;5.311204&amp;7.404502&amp;12.220&amp;5.636&amp;12.927&amp;15.891&amp;0.872\\ \hline 1.934&amp;0.516956&amp;6.214226&amp;8.445751&amp;10.372&amp;5.362&amp;13.533&amp;18.766&amp;0.857\\ \hline 1.701&amp;0.587959&amp;7.379324&amp;9.665141&amp;8.260&amp;4.857&amp;14.151&amp;22.584&amp;0.852\\ \hline 1.467&amp;0.681570&amp;8.910486&amp;11.051952&amp;5.843&amp;3.982&amp;14.756&amp;27.828&amp;0.863\\ \hline 1.234&amp;0.810636&amp;10.959447&amp;12.543378&amp;3.088&amp;2.503&amp;15.317&amp;35.330&amp;0.905\\ \hline 1.000&amp;1.000000&amp;13.753303&amp;13.999929&amp;0.000&amp;0.000&amp;15.793&amp;46.686&amp;1.000\\ \hline \end{array}}

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?

 

[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 don't 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.

 

[Jorrie]

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.

The Hubble radius is a 'characteristic' size of the universe, so I thought multiplying by it should scale da/dT to something interesting, and it did. The problem is that the column becomes a little confusing in the context of the calculator, because it gives the recession rate (in units c) at a specific redsift (a source presently at the Hubble radius). The rest of the columns represent objects at different redshifts, detracting from the appeal of such a column.

The table below complies closely with Tamara Davids' panels (she used H₀ = 70 km/s per Mpc and then 0.7 and 0.3 for the Omegas.

{\begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) &amp; R_{∞} (Gly) &amp; S_{eq} &amp; H_{0} &amp; \Omega_\Lambda &amp; \Omega_m\\ \hline 14&amp;16.7&amp;3280&amp;69.86&amp;0.703&amp;0.297\\ \hline \end{array}} {\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&amp;a=1/S&amp;T (Gy)&amp;R (Gly)&amp;D (Gly)&amp;D_{then}(Gly)&amp;D_{hor}(Gly)&amp;D_{par}(Gly)&amp;a&#039;R_{0}\\ \hline 3.120&amp;0.320513&amp;3.063831&amp;4.486962&amp;17.510&amp;5.612&amp;10.723&amp;8.992&amp;1.000\\ \hline 2.908&amp;0.343879&amp;3.395474&amp;4.944841&amp;16.512&amp;5.678&amp;11.162&amp;9.991&amp;0.974\\ \hline 2.696&amp;0.370920&amp;3.789680&amp;5.478672&amp;15.409&amp;5.715&amp;11.630&amp;11.186&amp;0.948\\ \hline 2.484&amp;0.402576&amp;4.263660&amp;6.104169&amp;14.183&amp;5.710&amp;12.129&amp;12.634&amp;0.923\\ \hline 2.272&amp;0.440141&amp;4.840610&amp;6.839559&amp;12.813&amp;5.639&amp;12.658&amp;14.416&amp;0.901\\ \hline 2.060&amp;0.485437&amp;5.552535&amp;7.704640&amp;11.273&amp;5.473&amp;13.213&amp;16.647&amp;0.882\\ \hline 1.848&amp;0.541126&amp;6.443855&amp;8.717678&amp;9.535&amp;5.160&amp;13.789&amp;19.497&amp;0.869\\ \hline 1.636&amp;0.611247&amp;7.577281&amp;9.888466&amp;7.566&amp;4.625&amp;14.372&amp;23.228&amp;0.865\\ \hline 1.424&amp;0.702247&amp;9.041571&amp;11.204956&amp;5.332&amp;3.745&amp;14.943&amp;28.254&amp;0.877\\ \hline 1.212&amp;0.825083&amp;10.963724&amp;12.613281&amp;2.809&amp;2.317&amp;15.474&amp;35.279&amp;0.916\\ \hline 1.000&amp;1.000000&amp;13.528145&amp;13.999932&amp;0.000&amp;0.000&amp;15.932&amp;45.581&amp;1.000\\ \hline \end{array}}

I have changed the 9th column header to be more sensible dot{a}R_0. This corresponds with the values shown on the zoomed center panel below. The redshift of an object that is on the Hubble sphere now is actually z~1.45 or S~2.45. I got that from my old Cosmocalc_2013, with Tamara's values. The z=2.1 represents a more distant galaxy, permanently outside the Hubble sphere, but whose photons managed to reach the Hubble sphere, and hence also to reach us.

Does this make sense?

Edit: Thanks Marcus, I have corrected the z=1.45.

 

[marcus]

I have changed the 9th column header to be more sensible dot{a}R_0. This corresponds with the values shown on the zoomed center panel below. The redshift of an object that is on the Hubble sphere now is actually z=1.67 or S=2.67. I got that from my old Cosmocalc_2013, with Tamara's values. The z=2.1 represents a more distant galaxy, permanently outside the Hubble sphere, but whose photons managed to reach the Hubble sphere, and hence also to reach us.
...
Does this make sense?

It makes better sense with the new header!
You should probably check that the number S=2.67 is right. You might have intended, say, S=2.47, and simply misremembered. That's easy to do, memory glitch at one digit and the rest right. We should both check.

I will check using your parameters 14.0, 16.7, 3280. Let me see what I get when I put those in and look for an S that will give me the present distance D = 14.0.

I get S=2.454 using your numbers.

{\begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) &amp; R_{∞} (Gly) &amp; S_{eq} &amp; H_{0} &amp; \Omega_\Lambda &amp; \Omega_m\\ \hline14&amp;16.7&amp;3280&amp;69.86&amp;0.703&amp;0.297\\ \hline\end{array}} {\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&amp;a=1/S&amp;T (Gy)&amp;R (Gly)&amp;D (Gly)&amp;D_{then}(Gly)&amp;D_{hor}(Gly)&amp;D_{par}(Gly)\\ \hline2.454&amp;0.407498&amp;4.338413&amp;6.201108&amp;13.998&amp;5.704&amp;12.202&amp;12.864\\ \hline\end{array}}

Using numbers that we were using earlier 14.0, 16.5, 3280 it's more like 2.43 (but about the same.)
{\begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) &amp; R_{∞} (Gly) &amp; S_{eq} &amp; H_{0} &amp; \Omega_\Lambda &amp; \Omega_m\\ \hline14&amp;16.5&amp;3280&amp;69.86&amp;0.72&amp;0.28\\ \hline\end{array}} {\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&amp;a=1/S&amp;T (Gy)&amp;R (Gly)&amp;D (Gly)&amp;D_{then}(Gly)&amp;D_{hor}(Gly)&amp;D_{par}(Gly)\\ \hline2.430&amp;0.411523&amp;4.522759&amp;6.430132&amp;14.028&amp;5.773&amp;12.278&amp;13.434\\ \hline\end{array}}

 

[marcus]

I think I know now what the vertical dashed line labeled z=1.67 is supposed to be. With your numbers 14.0, 16.7, 3280, we get S=2.61 for the intersection of lightcone with Hubble radius.
That is, a galaxy we are observing today which was receding at c in the past when it emitted the light.

THAT is a galaxy which was subsequently inside the Hubble sphere, and then later was again outside.{\begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) &amp; R_{∞} (Gly) &amp; S_{eq} &amp; H_{0} &amp; \Omega_\Lambda &amp; \Omega_m\\ \hline14&amp;16.7&amp;3280&amp;69.86&amp;0.703&amp;0.297\\ \hline\end{array}} {\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&amp;a=1/S&amp;T (Gy)&amp;R (Gly)&amp;D (Gly)&amp;D_{then}(Gly)&amp;D_{hor}(Gly)&amp;D_{par}(Gly)\\ \hline2.6104&amp;0.383083&amp;3.970&amp;5.7192&amp;14.929&amp;5.7192&amp;11.828&amp;11.737\\ \hline\end{array}}

So the vertical line for that galaxy does slice off a bit of the side bulge of the Hubble radius curve, just the way it appears in the figure. First it is outside Hubble sphere, then the sphere expands more rapidly than the galaxy is receding, and takes it in (for a while). Then its recession begins to dominate and it exits.

But that galaxy is not NOW at the Hubble radius. Your calculator says that its current distance is 14.929 Gly, not 14.0 Gly.

So instead of being labeled "z=1.67" the vertical dashed line probably wants to be labeled "z=1.61"
or S=2.61, and to be moved slightly over to the right so that it passes exactly thru the intersection of lightcone with Hubble radius. It will still slice off some of the bulge, on its way up, though slightly less of it.

OOPS! EDIT EDIT EDIT!
I see you relabeled that to say z=1.45. Now it makes sense, talking about a galaxy which is at comoving distance (now distance) Rnow = 14.0 Gly.

So multiplying that by the scale factor a(t) we get the past distance history of that galaxy
D(t) = Rnow a(t)

OK so that is a sample proper distance history. And you are going to take the slope of that.
And the slope should decline at first and then start increasing---the distance growth curve should have an inflection point where the slope is at a minimum. Which, as I recall, it does.

Yes! I checked on your table. S=1.636 is where the table minimum of the slope comes. Which is around year 7.6 billion. So that looks quite good. So I can see a real pedagogical benefit.

This is making a lot of sense now. I still don't have a definite opinion whether the 9th column pedagogical benefits outweigh the cost of having a more elaborate table. Probably it depends on who one expects to be the user.

 

[Jorrie]

I think I know now what the vertical dashed line labeled z=1.67 is supposed to be. With your numbers 14.0, 16.7, 3280, we get S=2.61 for the intersection of lightcone with Hubble radius.
That is, a galaxy we are observing today which was receding at c in the past when it emitted the light.

THAT is a galaxy which was subsequently inside the Hubble sphere, and then later was again outside.

So the vertical line for that galaxy does slice off a bit of the side bulge of the Hubble radius curve. First it is outside Hubble sphere, then the sphere expands more rapidly than the galaxy is receding, and takes it in (for a while). Then its recession begins to dominate and it exits.

But that galaxy is not NOW at the Hubble radius. Your calculator says that its current distance is 14.929 Gly, not 14.0 Gly.

I'll have to think about this a little more. A dotted vertical line represents a constant co-moving distance and, I think, a constant redshift over time. Galaxies below z ~ 1.67 must have entered the Hubble radius of the time and later exited it again. Now if the recession speed "then" must have been c when the galaxy entered the Hubble distance and again when it leaves it, there must be a single redshift that satisfies this condition for such galaxies. I could not find such a solution through the calculator, so now I'm a little confused.

What am I missing?

 

[marcus]

this is not a criticism of your Rnow a(t) column based on the vertical line labeled z = 1.45.
that made sense to me (and I edited my post) as soon as I saw you had relabeled it z = 1.45.

However there is a general comment to make. I think we need a notation for the maximum Dthen.

Dthen is the outline of the light cone, the galaxies we can be getting light from today. We've talked about its teardrop shape and its maximum girth, before.

If I call that maximum value of Dthen by the name "D_max", D_max = 5.7 or 5.8 depending on the parameters.
And the corresponding S_max = 2.61
And 2.61 x 5.7 = 14.9 billion light years, which is the comoving or now distance of a galaxy which emitted the light at the instant when it was receding at speed c, so that the light "stood still" at first, for a while and did not make any headway. this is a unique distance.

14.9 Gly is the unique comoving distance with that property.
====================

There's a slight possibility of confusion associated with plotting R_oda/dT in that it tracks the distance to something that is NOT ON THE LIGHTCONE.
Always in the past when we pick some S like S = 2.45 we are talking about a galaxy which we are getting light from today stretched by factor 2.45, and the distances in that row of the table tell us about the distance to that galaxy. So it's breaking with that precedent (for better or worse.)

You see the intersection of the horizontal line year 3.1 billion and the vertical S=2.45 is not on the red light cone curve. So we aren't getting any light from that galaxy that it emitted in years 3.1 and we aren't getting any light from it redshifted z=1.45. So the story with that galaxy is not LIKE the other stories we may be telling ourselves, habitually, about rows of the table. there is an "anomaly" in how we have to think about it.

But if you get back on the light cone, by using S=2.61, then your 9th column will be slightly different. The slope will start off at 1, at 2.61, and then it will decline as expansion slows, and then it will inflect and start increasing, and then it will reach 1 slightly BEFORE the present day, and then it will already be faster than light at the present. It will be greater than 1 at the present day. Which might not be a bad thing to show.
And you will be following the increasing distance of a real galaxy which we can see today, that is on our light cone. Because you start the vertical dash line at the INTERSECTION of the Hubble radius with the light cone.

I think that is pedagogically better, except that we have no NAME for SmaxDmax the comoving distance of the galaxy. Have to go, back later.

Back now. I guess one could fantasize teaching with this concept included in the kitbag. Explain that the past lightcone is onion-shape and the maximum proper radius we are going to call Dmax.
And then say that the COMOVING radius of the light cone (at its fattest) is going to be called Rcone. And we going to plot the recession speed history of a galaxy at R_cone.

this is a galaxy which, when it emitted the light we are getting, was RECEDING AT c!
So the speed number is going to be 1.
And that will be at S=2.61 and at a certain time, when it emitted the light, and when distance to it was increasing at c. So we picture that.
Smax x Dmax = 2.61 x 5.7 = Rcone = 14.9 Gly.
The thing which sent us photons that at first stood still is now 14.9 Gly from here, and we are getting the photons today.

And the 9th column record of R_coneda/dT starts at 1, when it emitted the light and was receding at c, and then it sags down because the thing's recession was slowing, and then it bottoms out and starts rising, and then it GETS TO ONE again, but it isn't the present yet. And by the time we come to present day it is actually receding a little bit faster than c. Good! That seems to work pedagogically.

However the cost is that one has to introduce a new concept R_cone the comoving radius of the past lightcone at its widest girth. So one has to weigh the cost. I'm interested enough I would like to see the 9th column used that way to get a sense of what it looks like.

I realize I haven't thought enough about this. The idea may have obvious flaws that I will only see later. But the 9th column (in units of the speed of light) does seem like an interesting idea. In future it would presumably show high multiples of the speed of light. And in past, before S=2.61.

 

[marcus]

I don't know why it's taking me such a long time to catch on. We were discussing Jorrie's idea of a 9th column that takes some distance as an example and watches it expand, during some interval of time. Column 9 would log the speed of the receding galaxy (as a multiple of c) and for earlier part of history this is slowing down while for later it is increasing. So we'd get to see this.

All cosmological distances (between CMB stationary pairs of observers) grow proportionally to a(t), which is a dimensionless number normalized to a(now) = 1.

If you multiply da/dT by the present-day Hubble radius, you get the recession speed history of a galaxy which is located at comoving distance R_now, and in the units we are using the speed comes out = 1. So the speeds are being expressed as multiples of c, i.e. in units of the speed of light.

What I'm undecided about (and periodically confused about) is whether one should allow optional flexibility about what one multiplies by. If you multiplied by HALF the Hubble radius instead, the speed numbers would come out half as big. And it would be a history of a galaxy only half as far away. So that seems consistent. Or you might multiply da/dT by 4/3 the Hubble radius and the speed numbers would be different accordingly, but they would be correct for a galaxy that is now 4/3 as far away.

The speed is always going to be expressed as a multiple of c, because of the units being used. Gy for time and Gly for distance. Maybe there should be a box where you put in a number like .5 or 1.333 and it says "da/dT will be multiplied by [box] times R_o the current Hubble radius, to give the recession speeds shown in column 9". And a tooltip says the speeds are given in units of c.

Still undecided about the desirable degree of flexibility.

 

[Jorrie]

A Wiki for Tabular Cosmo calculator user manual

With a complete overhaul of TabCosmoX taking shape and a draft user manual already posted by Mordred (to be updated for new 'release'), I was looking for a suitable Wiki-hosting site. Wikidot.com seems to be a good option for the manual. It allows collaboration with some control options and sports very good features, including Latex.

What do you think?

PS: WikiDot (or alike) also seems to be a good place for the calculator to be hosted, getting it off my private website, to where it may have more longevity...

 

[Jorrie]

New Look Tabular Calculator (LightCone)

The "complete overhaul of TabCosmoX" is completed and the new link is in my signature. It is now named "LightCone", proposed by Marcus. A sample screenshot is attached.

The main differences from TabCosmoX are the flexibility of selectable columns and a choice of default data sets (only WMAP and Planck at this time). More can be inserted if useful.

The main change is the column selector:

More columns can be added to the selection list with relative ease now.

Please report any usage issue or bugs detected.

 

[Mordred]

The "complete overhaul of TabCosmoX" is completed and the new link is in my signature. It is now named "LightCone", proposed by Marcus. A sample screenshot is attached.

The main differences from TabCosmoX are the flexibility of selectable columns and a choice of default data sets (only WMAP and Planck at this time). More can be inserted if useful.

The main change is the column selector:

More columns can be added to the selection list with relative ease now.

Please report any usage issue or bugs detected.

If you using IE 8, and see all the boxes in the column selector overrun each other instead of the view above. Check and make sure you have compatibility view turned off. Some IE 8 browsers may experience " a script is causing your browser to run slower than normal" error state no each time it asks to turn off script.
The script error appears to only occur on IE 8 and not other browsers. Jorrie is working on this issue.