Cosmo calculators with tabular output

by marcus
Tags: calculators, cosmo, output, tabular
PF Gold
P: 709
 Quote by 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.
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, Ωλ and Ωm. 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...
Astronomy
PF Gold
P: 22,675
 Quote by Jorrie 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...
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?
PF Gold
P: 709
 Quote by 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.
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.
 PF Gold P: 709 For completeness of reference,[1] here is the full compact set of TabCosmo6 equations (added particle horizon from previous). Given present Hubble time $Y_{now}$, long term Hubble time $Y_{inf}$ 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. $$\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r$$ Hubble parameter, also referred to as H(t) $$H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}$$ Hubble time, Cosmic time $$Y = 1/H, \ \ \, T = \int_{S}^{\infty}{\frac{dS}{S H}}$$ Proper distance 'now', 'then', cosmic event horizon and particle horizon $$D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \, D_{then} = \frac{D_{now}}{S}, \ \ \, D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}, \ \ \, D_{par} = \frac{1}{S}\int_{S}^{\infty}{ \frac{dS}{H}}$$ To obtain all the values, it essentially means integration for S from zero to infinity, but practically it has been limited to $10^{-7}< S <10^{7}$ 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).
 PF Gold P: 709 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 $$array option is available, but that requires a lot more manual work - something that the machine could actually do better. I have added an option for a LaTex compatible output and uploaded it as TabCosmo7. It requires you to first play around until you have the range of values that you are interested in, tick the radio button for LaTex, Calculate and then copy and paste the code into a LaTex compatible editor. It is optimized for the PF editor, but you can modify any part of the Tex code after copying (obviously at your own risk :-) Please report any problems/suggestions. Here is a sample output. [tex]{\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} (Km/s/Mpc) & \Omega_\Lambda & \Omega_m\\ \hline14&16.5&3280&69.86&0.72&0.28\\ \hline\end{array}}$$ $${\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&T_{Hub}(Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.001\\ \hline341.731&0.002926&0.002511&0.003986&44.573&0.130&0.177&0.006\\ \hline107.137&0.009334&0.015296&0.023478&42.386&0.396&0.543&0.040\\ \hline33.589&0.029772&0.089394&0.135218&38.404&1.143&1.614&0.246\\ \hline10.531&0.094961&0.513668&0.772152&31.251&2.968&4.469&1.464\\ \hline3.302&0.302891&2.902232&4.258919&18.588&5.630&10.418&8.506\\ \hline1.035&0.966116&13.274154&13.791148&0.473&0.457&15.728&44.633\\ \hline0.325&3.081570&31.418524&16.391363&-10.476&-32.283&16.428&176.105\\ \hline0.102&9.829121&50.521674&16.496494&-14.143&-139.014&16.496&597.755\\ \hline0.032&31.351430&69.658811&16.499868&-15.295&-479.531&16.500&1942.755\\ \hline0.010&100.000000&88.797170&16.499905&-15.657&-1565.665&16.500&6232.831\\ \hline\end{array}}$$
 Astronomy Sci Advisor PF Gold P: 22,675 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.
 Astronomy Sci Advisor PF Gold P: 22,675 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}}$$
PF Gold
P: 709
 Quote by marcus 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 (10TH) 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.
 Astronomy Sci Advisor PF Gold P: 22,675 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.
 Astronomy Sci Advisor PF Gold P: 22,675 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?
 PF Gold P: 709 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. $$\Omega_\Lambda = \left(\frac{R_{now}}{R_{\infty}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r$$ 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) $$R = 1/H, \ \ \, T = \int_{S}^{\infty}{\frac{dS}{S H}}$$ Proper distance 'now', 'then', cosmic event horizon and particle horizon $$D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \, D_{then} = \frac{D_{now}}{S}, \ \ \, D_{hor} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}, \ \ \, D_{par} = \frac{1}{S}\int_{S}^{\infty}{ \frac{dS}{H}}$$ 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 $10^{-7}< S <10^{7}$ 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).
PF Gold
P: 709
 Quote by 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 $R'_{now}$, 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:
$${\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14&16.5&3280&69.86&0.72&0.28\\ \hline \end{array}}$$ $${\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&R'_{now}\\ \hline 1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.001&20.164\\ \hline 541.606&0.001846&0.001200&0.001945&45.126&0.083&0.113&0.003&13.292\\ \hline 269.117&0.003716&0.003662&0.005761&44.225&0.164&0.223&0.009&9.029\\ \hline 133.721&0.007478&0.010876&0.016772&42.912&0.321&0.439&0.028&6.242\\ \hline 66.444&0.015050&0.031751&0.048364&41.023&0.617&0.855&0.085&4.357\\ \hline 33.015&0.030289&0.091754&0.138771&38.325&1.161&1.640&0.253&3.056\\ \hline 16.405&0.060958&0.263633&0.397095&34.484&2.102&3.066&0.743&2.149\\ \hline 8.151&0.122680&0.754694&1.132801&29.030&3.561&5.501&2.164&1.516\\ \hline 4.050&0.246896&2.146402&3.182937&21.343&5.269&9.172&6.254&1.086\\ \hline 2.013&0.496887&5.887073&8.078066&11.017&5.474&13.329&17.716&0.861\\ \hline 1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686&1.000\\ \hline 0.631&1.584893&20.670471&15.748412&-5.533&-8.770&16.283&82.739&1.409\\ \hline 0.398&2.511886&28.076314&16.301181&-9.273&-23.293&16.413&140.526&2.157\\ \hline 0.251&3.981072&35.624819&16.449365&-11.680&-46.500&16.449&232.303&3.388\\ \hline 0.158&6.309573&43.210628&16.487217&-13.207&-83.331&16.487&377.810&5.358\\ \hline 0.100&10.000000&50.805908&16.496757&-14.172&-141.718&16.497&608.434&8.487\\ \hline 0.063&15.848932&58.403573&16.499147&-14.781&-234.257&16.499&973.953&13.448\\ \hline 0.040&25.118864&66.001838&16.499740&-15.165&-380.922&16.500&1553.261&21.313\\ \hline 0.025&39.810717&73.600254&16.499880&-15.407&-613.371&16.500&2471.404&33.779\\ \hline 0.016&63.095734&81.198707&16.499907&-15.560&-981.779&16.500&3926.561&53.536\\ \hline 0.010&100.000000&88.797170&16.499905&-15.657&-1565.665&16.500&6232.831&84.849\\ \hline \end{array}}$$
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.
PF Gold
P: 709
 Quote by 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 $R'_{now}$ (the expansion rate history of a galaxy that is presently on our Hubble sphere) is valid.

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

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

PF Gold
P: 709
 Quote by marcus 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.

$${\begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14&16.7&3280&69.86&0.703&0.297\\ \hline \end{array}}$$ $${\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&a'R_{0}\\ \hline 3.120&0.320513&3.063831&4.486962&17.510&5.612&10.723&8.992&1.000\\ \hline 2.908&0.343879&3.395474&4.944841&16.512&5.678&11.162&9.991&0.974\\ \hline 2.696&0.370920&3.789680&5.478672&15.409&5.715&11.630&11.186&0.948\\ \hline 2.484&0.402576&4.263660&6.104169&14.183&5.710&12.129&12.634&0.923\\ \hline 2.272&0.440141&4.840610&6.839559&12.813&5.639&12.658&14.416&0.901\\ \hline 2.060&0.485437&5.552535&7.704640&11.273&5.473&13.213&16.647&0.882\\ \hline 1.848&0.541126&6.443855&8.717678&9.535&5.160&13.789&19.497&0.869\\ \hline 1.636&0.611247&7.577281&9.888466&7.566&4.625&14.372&23.228&0.865\\ \hline 1.424&0.702247&9.041571&11.204956&5.332&3.745&14.943&28.254&0.877\\ \hline 1.212&0.825083&10.963724&12.613281&2.809&2.317&15.474&35.279&0.916\\ \hline 1.000&1.000000&13.528145&13.999932&0.000&0.000&15.932&45.581&1.000\\ \hline \end{array}}$$

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.
Attached Thumbnails

Astronomy
PF Gold
P: 22,675
 Quote by Jorrie 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.

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

Using numbers that we were using earlier 14.0, 16.5, 3280 it's more like 2.43 (but about the same.)
$${\begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14&16.5&3280&69.86&0.72&0.28\\ \hline\end{array}}$$ $${\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline2.430&0.411523&4.522759&6.430132&14.028&5.773&12.278&13.434\\ \hline\end{array}}$$
 Astronomy Sci Advisor PF Gold P: 22,675 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. $${\begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14&16.7&3280&69.86&0.703&0.297\\ \hline\end{array}}$$ $${\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline2.6104&0.383083&3.970&5.7192&14.929&5.7192&11.828&11.737\\ \hline\end{array}}$$ 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.
PF Gold
P: 709
 Quote by 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. 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?

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