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## Look 88 billion years into future with the A20 tabular calculator

One very beautiful thing about this table, as a sample segment of universe history, is that in the distant future one can see the cosmological constant Lambda emerging out of the fog, clearly, as a DISTANCE---a plainly visible concrete thing built into the universe's history.

By convention (going back to before 1920 with Einstein) a small positive Lambda corresponds to a slight negative spacetime curvature---that is a minus one over a large area quantity: the square of a length. So the naturally occurring Lambda constant in the Einstein equation is one over a length squared.

With the usual identification of time and distance, we can simply regard Lambda as a squared growth rate---one over an interval of time, squared. In other words the squared growth rate H2 in the equation a couple of posts back is an ALIAS for the cosmological constant Λ in the Einstein equation. (I'm neglecting a stray factor of 3.)

So when you look at the table and see the time quantity 17.6 Gy emerging at around year 60 billion in the future you are seeing a naked manfest appearance of the cosmological constant.

The same as when you see the distance 17.6 billion lightyears emerge, as the distance to the cosmological event horizon, eventually around year 60 billion in the future.
The reciprocal of that distance, squared, is again essentially the cosmological constant (indicating a slight constant negative space-time curvature) that Einstein wrote down in the equation which is now both our law of gravity and our law of geometry.
 Recognitions: Gold Member Science Advisor Another beautiful thing the cosmic history calculator shows you is the moment when the recession speed (of any chosen galaxy) stopped slowing down and began to pick up. It is an inflection point on the curve showing the distance to the galaxy. With WMAP numbers (pre-Planck mission 14, 16.5, 3280) this comes around year 7.3 billion. Let's choose a galaxy which TODAY is at a distance equal to the Hubble radius: 14 billion lightyears. The table is set to have 26 steps from S=1090 to exact present, and another 26 steps to S=.04. You can see the minimum recession speed (rightmost column!) comes in the S=1.7 row, around year 7.3 billion. You can also see that for the sample case we are tracking, where the distance today is 14 Gly, the current Hubble radius, the slowest recession speed ever attained is 0.8516 c. That is about 85% of the speed of light. At present, because the galaxy is at Hubble radius, the recession speed is exactly c. And as you can also see from the table, in future it will continue to grow. A galaxy at half the distance (now at 7 Gly instead of 14 Gly) would have a proportionally scaled recession speed history---just divide all the speeds by two! So knowing this one sample history lets us get the recession speeds for objects at other distances as well. $${\begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14&16.5&3280&69.86&0.72&0.28\\ \hline \end{array}}$$ $${\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)&v_{rec}\\ \hline 1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.001&20.1636\\ \hline 832.918&0.001201&0.000592&0.000983&45.527&0.055&0.074&0.001&17.0999\\ \hline 636.471&0.001571&0.000922&0.001508&45.289&0.071&0.096&0.002&14.5867\\ \hline 486.356&0.002056&0.001428&0.002302&45.009&0.093&0.125&0.003&12.5044\\ \hline 371.647&0.002691&0.002197&0.003500&44.684&0.120&0.163&0.005&10.7630\\ \hline 283.992&0.003521&0.003365&0.005304&44.307&0.156&0.212&0.008&9.2950\\ \hline 217.011&0.004608&0.005131&0.008015&43.872&0.202&0.275&0.013&8.0485\\ \hline 165.828&0.006030&0.007798&0.012088&43.369&0.262&0.357&0.020&6.9840\\ \hline 126.717&0.007892&0.011817&0.018200&42.790&0.338&0.462&0.031&6.0704\\ \hline 96.830&0.010327&0.017862&0.027367&42.124&0.435&0.598&0.047&5.2831\\ \hline 73.992&0.013515&0.026948&0.041109&41.360&0.559&0.773&0.072&4.6026\\ \hline 56.541&0.017686&0.040590&0.061703&40.483&0.716&0.996&0.109&4.0129\\ \hline 43.205&0.023145&0.061058&0.092556&39.477&0.914&1.280&0.167&3.5009\\ \hline 33.015&0.030289&0.091754&0.138771&38.325&1.161&1.640&0.253&3.0557\\ \hline 25.228&0.039638&0.137768&0.207983&37.005&1.467&2.093&0.383&2.6682\\ \hline 19.278&0.051872&0.206718&0.311611&35.494&1.841&2.661&0.580&2.3305\\ \hline 14.731&0.067883&0.310005&0.466715&33.764&2.292&3.365&0.876&2.0363\\ \hline 11.257&0.088835&0.464670&0.698717&31.784&2.824&4.228&1.323&1.7800\\ \hline 8.602&0.116254&0.696135&1.045272&29.520&3.432&5.269&1.994&1.5571\\ \hline 6.573&0.152136&1.042148&1.561411&26.934&4.098&6.502&3.003&1.3641\\ \hline 5.023&0.199093&1.558281&2.325166&23.985&4.775&7.922&4.517&1.1988\\ \hline 3.838&0.260543&2.324459&3.439363&20.641&5.378&9.496&6.782&1.0605\\ \hline 2.933&0.340960&3.450250&5.016065&16.884&5.757&11.146&10.156&0.9516\\ \hline 2.241&0.446198&5.070303&7.113058&12.751&5.689&12.742&15.136&0.8782\\ \hline 1.713&0.583918&7.312958&9.599448&8.373&4.889&14.119&22.363&0.8516\\ \hline 1.309&0.764145&10.232782&12.059647&4.011&3.065&15.144&32.599&0.8871\\ \hline 1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686&1.0000\\ \hline 0.764&1.308652&17.700005&15.230903&-3.469&-4.539&16.147&65.616&1.2029\\ \hline 0.682&1.465878&19.447858&15.566734&-4.731&-6.935&16.236&75.350&1.3183\\ \hline 0.609&1.641994&21.229081&15.819561&-5.879&-9.654&16.301&86.289&1.4531\\ \hline 0.544&1.839269&23.035135&16.007122&-6.919&-12.726&16.348&98.568&1.6086\\ \hline 0.485&2.060245&24.859344&16.144845&-7.857&-16.187&16.380&112.342&1.7865\\ \hline 0.433&2.307770&26.697095&16.244907&-8.700&-20.077&16.402&127.785&1.9889\\ \hline 0.387&2.585034&28.544549&16.317231&-9.457&-24.446&16.416&145.094&2.2179\\ \hline 0.345&2.895609&30.399001&16.369270&-10.135&-29.346&16.425&164.489&2.4765\\ \hline 0.308&3.243498&32.258319&16.406749&-10.742&-34.841&16.430&186.221&2.7677\\ \hline 0.275&3.633183&34.121403&16.433445&-11.285&-41.000&16.433&210.567&3.0952\\ \hline 0.246&4.069687&35.987064&16.452507&-11.770&-47.901&16.453&237.840&3.4630\\ \hline 0.219&4.558633&37.854565&16.466097&-12.204&-55.634&16.466&268.393&3.8759\\ \hline 0.196&5.106324&39.723214&16.475939&-12.592&-64.297&16.476&302.617&4.3390\\ \hline 0.175&5.719816&41.592963&16.482824&-12.938&-74.001&16.483&340.955&4.8582\\ \hline 0.156&6.407015&43.463378&16.487715&-13.247&-84.873&16.488&383.899&5.4403\\ \hline 0.139&7.176777&45.334268&16.491186&-13.523&-97.051&16.491&432.003&6.0926\\ \hline 0.124&8.039020&47.205331&16.493809&-13.769&-110.692&16.494&485.887&6.8235\\ \hline 0.111&9.004857&49.076799&16.495546&-13.989&-125.973&16.496&546.245&7.6425\\ \hline 0.099&10.086732&50.948438&16.496771&-14.186&-143.090&16.497&613.854&8.5601\\ \hline 0.089&11.298588&52.820200&16.497630&-14.361&-162.263&16.498&689.587&9.5881\\ \hline 0.079&12.656041&54.691883&16.498394&-14.518&-183.740&16.498&774.418&10.7395\\ \hline 0.071&14.176583&56.563793&16.498808&-14.658&-207.797&16.499&869.442&12.0295\\ \hline 0.063&15.879808&58.435746&16.499091&-14.783&-234.745&16.499&975.882&13.4745\\ \hline 0.056&17.787665&60.307731&16.499279&-14.894&-264.931&16.499&1095.110&15.0932\\ \hline 0.050&19.924739&62.179573&16.499566&-14.994&-298.742&16.500&1228.663&16.9063\\ \hline 0.045&22.318568&64.051596&16.499641&-15.082&-336.617&16.500&1378.261&18.9374\\ \hline 0.040&25.000000&65.923630&16.499682&-15.162&-379.041&16.500&1545.833&21.2125\\ \hline \end{array}}$$Time now (at S=1) or present age in billion years: 13.753301 'T' in billion years (Gy) and 'D' in billion light years (Gly), sample recession speed history of matter now at distance R0, shown as multiples of the speed of light ====
 Recognitions: Gold Member Science Advisor Using model parameters from the recent Planck mission report we get nearly the same recession speed history as above. From Planck, combined with earlier data, we get 14.4 Gly, 17.3 Gly, and 3400. Plugging these parameters into the calculator we get that the minimum recession speed comes at S=1.652 and year 7.592 billion. For a galaxy which is now at current Hubble radius Ro = 14.4 Gly from us, the minimum recession speed is 0.87258c . So 87% of the speed of light, instead of 85% (as found with earlier model parameters). I think the difference is mainly due to the longer Hubble radius 14.4 instead of 14.0. The representative galaxy we choose to track is slightly more distant, so its recession speeds are slightly higher throughout history, including the minimum. The minimum is attained somewhat later, namely year 7.6 billion instead of year 7.3 billion which we found in preceding post using 2010 WMAP parameters. One thing that is easy to do with the table calculator is see what happens when you vary parameters slightly. You can find for instance that the increasing the eventual Hubble radius R∞ (keeping the other two the same) will make the minimum speed come later. That makes sense--it delays the onset of "accelerated expansion". A cosmological constant of zero would correspond to infinite Hubble radius, and the expansion speed would continue declining indefinitely and never bottom out. So the longer R∞ is, the longer you have to wait for acceleration to occur. Accordingly, we see the year of the minimum change from 7.3 to 7.6 billion when we adopt Planck mission numbers and increase R∞ from 16.5 to 17.3 Gly.
 Recognitions: Gold Member Science Advisor There might eventually be a "learner's manual" to go with Jorrie's calculator so I'll experiment with a few cosmic history tables that one can find things in to point out and discuss. Here is one that shows the "deja vu" epoch. An earlier time when any galaxy would have the same recession speed that it does right now. This comes around year 3.3 billion. The table also shows a few other points of interest. It runs from S=10 around the time the the first galaxies formed, up to present S=1 and then on to S=0.1 when distances will be ten times what they are now. I used Planck 2013 model parameters and specified 17 steps from start to present.$${\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$'T' in billion years (Gy) and 'D' in billion light years (Gly), a sample recession speed history of matter now at distance Ro is shown in multiples of the speed of light.$${\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)&v_{rec}sample\\ \hline 10.000&0.100&0.545&0.820&30.684&3.068&4.717&1.558&1.76\\ \hline 8.733&0.115&0.668&1.004&29.536&3.382&5.270&1.916&1.64\\ \hline 7.627&0.131&0.819&1.229&28.307&3.711&5.873&2.354&1.54\\ \hline 6.661&0.150&1.004&1.504&26.994&4.053&6.528&2.893&1.44\\ \hline 5.817&0.172&1.229&1.840&25.591&4.399&7.234&3.554&1.35\\ \hline 5.080&0.197&1.505&2.249&24.093&4.743&7.988&4.364&1.26\\ \hline 4.437&0.225&1.842&2.744&22.495&5.070&8.786&5.357&1.18\\ \hline 3.875&0.258&2.253&3.341&20.794&5.367&9.622&6.573&1.11\\ \hline 3.384&0.296&2.753&4.056&18.988&5.611&10.484&8.061&1.05\\ \hline 2.955&0.338&3.358&4.903&17.077&5.779&11.357&9.877&0.99\\ \hline 2.581&0.387&4.088&5.891&15.065&5.837&12.225&12.089&0.95\\ \hline 2.254&0.444&4.960&7.017&12.963&5.751&13.066&14.775&0.91\\ \hline 1.968&0.508&5.994&8.264&10.788&5.481&13.856&18.023&0.89\\ \hline 1.719&0.582&7.203&9.592&8.567&4.983&14.574&21.929&0.87\\ \hline 1.501&0.666&8.593&10.941&6.334&4.219&15.201&26.597&0.88\\ \hline 1.311&0.763&10.164&12.235&4.132&3.151&15.726&32.134&0.90\\ \hline 1.145&0.873&11.902&13.405&2.002&1.749&16.147&38.655&0.94\\ \hline 1.000&1.000&13.787&14.400&0.000&0.000&16.472&46.279&1.00\\ \hline 0.873&1.145&15.794&15.201&-1.890&-2.164&16.714&55.139&1.08\\ \hline 0.763&1.311&17.896&15.814&-3.607&-4.729&16.888&65.388&1.19\\ \hline 0.666&1.501&20.071&16.267&-5.157&-7.743&17.010&77.200&1.33\\ \hline 0.582&1.719&22.297&16.591&-6.544&-11.249&17.093&90.781&1.49\\ \hline 0.508&1.968&24.561&16.818&-7.775&-15.305&17.149&106.372&1.69\\ \hline 0.444&2.254&26.850&16.974&-8.863&-19.976&17.185&124.252&1.91\\ \hline 0.387&2.581&29.157&17.081&-9.820&-25.343&17.207&144.745&2.18\\ \hline 0.338&2.955&31.475&17.153&-10.660&-31.502&17.220&168.223&2.48\\ \hline 0.296&3.384&33.802&17.202&-11.396&-38.563&17.226&195.115&2.83\\ \hline 0.258&3.875&36.135&17.234&-12.041&-46.654&17.234&225.913&3.24\\ \hline 0.225&4.437&38.470&17.256&-12.605&-55.923&17.256&261.183&3.70\\ \hline 0.197&5.080&40.809&17.271&-13.098&-66.538&17.271&301.571&4.24\\ \hline 0.172&5.817&43.149&17.280&-13.528&-78.695&17.280&347.819&4.85\\ \hline 0.150&6.661&45.490&17.287&-13.905&-92.617&17.287&400.776&5.55\\ \hline 0.131&7.627&47.832&17.291&-14.233&-108.558&17.291&461.415&6.35\\ \hline 0.115&8.733&50.174&17.294&-14.521&-126.813&17.294&530.850&7.27\\ \hline 0.100&10.000&52.516&17.296&-14.772&-147.715&17.296&610.357&8.33\\ \hline \end{array}}$$ The sample galaxy's present-day recession speed is 1c, the speed of light. Deja vu is at S=3.00,when the galaxy was also receding at the speed of light. The table comes close enough (S=2.955) so that the speed in that row of the table is 0.99c. Minimum speed occurs around S=1.7. Looking at that row of the table, one can see that for the sample galaxy we've chosen the slowest it ever is, in the whole of cosmic history, is 0.87c, 87% of the speed of light. In its Dthen column the table also shows the radius of the past lightcone. It is the distance of something we are now getting light from at the time it emitted the light. You can see by scanning down the Dthen column that the greatest distance at the time of emission is 5.8 billion light years. An emitter at this maximum remove is receding exactly at speed c, so that the light we are receiving from it at first "stood still" (could not close the distance between us) but later began to make headway. Dthen coincides with Hubble radius R at that moment in time, as the table also shows.
 Recognitions: Gold Member Science Advisor Today I happened to get curious about early times, not the first second or few minutes of the cosmos but something simpler to picture, like year 2000 from the start of expansion. So I put in S=20000. Ooops, have to go to supper. back later, here's the output for that stretch $${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.00005&20000.0&0.00000187&0.00000350&46.177&0.002309&3.21&659.18\\ \hline \end{array}}$$ You can see it is year 1,870. Just a bit before year 2000. I'll think about what it says conditions were like, after supper. Part of this is just learning to read off from the table, and get the decimal point in the right place. You know what the temperature of of the CMB is today, around 2.76 kelvin. To get the temperature of radiation back then I guess you just multiply by 20000, or by whatever S is at the time. So 5.5 x 104 kelvin---i.e. around 55,000 kelvin. And the cube of S is 8 x 1012. So the density of matter was 8 trillion times what it is today. But that isn't all that much because on average it is so scarce today. amounts to only about 0.23 nanojoule per cubic meter. energy equivalent, including dark matter which is the bulk of it. So back then, in year 1870, a cubic meter contained 1840 joules worth of matter 1840 joules/c^2 into Google gives: 2 x 10^-11 grams. I can hardly believe it is so little! Well that is what it seems to be.

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 Quote by marcus $${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.00005&20000.0&0.00000187&0.00000350&46.177&0.002309&3.21&659.18\\ \hline \end{array}}$$ Part of this is just learning to read off from the table, and get the decimal point in the right place. ...
I find it useful to set the decimals to 9 for such small values, because then the digits represent years or light years. For slightly larger minima, six decimal digits obviously represent My and so on. Sadly, one can't change it halfway through a long table...