- #1
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Here's a sample history from fairly far back in the past, going up to the present (S = 1) in 20 expansion ratio steps, and then in another 20 expansion steps, going out a good stretch into the future, when distances will be 25 times what they are today.
I could have asked for a wider expanse of time and a greater overall expansion to be covered, but this seemed ample for test-driving the new Planck parameters.{\begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.56&17.6&3400&67.17&0.684&0.316\\ \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)\\ \hline45.000&0.022&0.056&0.085&39.362&0.875&1.247&0.153\\ \hline37.201&0.027&0.075&0.114&38.595&1.037&1.488&0.206\\ \hline30.753&0.033&0.100&0.151&37.750&1.228&1.772&0.277\\ \hline25.423&0.039&0.133&0.201&36.821&1.448&2.107&0.372\\ \hline21.017&0.048&0.178&0.268&35.798&1.703&2.500&0.498\\ \hline17.374&0.058&0.237&0.357&34.672&1.996&2.959&0.667\\ \hline14.363&0.070&0.316&0.475&33.434&2.328&3.494&0.893\\ \hline11.874&0.084&0.420&0.632&32.071&2.701&4.111&1.196\\ \hline9.816&0.102&0.559&0.841&30.573&3.115&4.821&1.599\\ \hline8.115&0.123&0.745&1.118&28.926&3.565&5.628&2.137\\ \hline6.708&0.149&0.991&1.485&27.115&4.042&6.538&2.855\\ \hline5.546&0.180&1.317&1.971&25.129&4.531&7.551&3.812\\ \hline4.584&0.218&1.751&2.610&22.951&5.006&8.659&5.086\\ \hline3.790&0.264&2.323&3.443&20.571&5.428&9.846&6.780\\ \hline3.133&0.319&3.076&4.516&17.982&5.740&11.084&9.028\\ \hline2.590&0.386&4.060&5.861&15.189&5.865&12.330&11.999\\ \hline2.141&0.467&5.325&7.485&12.215&5.705&13.526&15.903\\ \hline1.770&0.565&6.923&9.331&9.111&5.148&14.608&20.991\\ \hline1.463&0.683&8.883&11.256&5.963&4.075&15.518&27.544\\ \hline1.210&0.827&11.200&13.059&2.882&2.382&16.225&35.865\\ \hline1.000&1.000&13.834&14.560&0.000&0.000&16.730&46.281\\ \hline0.851&1.175&16.259&15.529&-2.253&-2.646&17.023&56.991\\ \hline0.725&1.380&18.819&16.232&-4.264&-5.883&17.220&69.718\\ \hline0.617&1.621&21.473&16.718&-6.040&-9.789&17.348&84.771\\ \hline0.525&1.904&24.191&17.040&-7.589&-14.446&17.429&102.522\\ \hline0.447&2.236&26.951&17.248&-8.928&-19.964&17.478&123.419\\ \hline0.381&2.627&29.739&17.380&-10.079&-26.473&17.507&147.994\\ \hline0.324&3.085&32.543&17.463&-11.065&-34.139&17.521&176.879\\ \hline0.276&3.624&35.358&17.515&-11.908&-43.154&17.526&210.819\\ \hline0.235&4.257&38.180&17.548&-12.627&-53.751&17.548&250.694\\ \hline0.200&5.000&41.006&17.568&-13.241&-66.203&17.568&297.535\\ \hline0.170&5.873&43.834&17.580&-13.763&-80.832&17.580&352.560\\ \hline0.145&6.899&46.664&17.588&-14.208&-98.017&17.588&417.194\\ \hline0.123&8.103&49.495&17.592&-14.587&-118.205&17.592&493.115\\ \hline0.105&9.518&52.327&17.595&-14.910&-141.917&17.595&582.294\\ \hline0.089&11.180&55.159&17.597&-15.185&-169.772&17.597&687.047\\ \hline0.076&13.133&57.991&17.598&-15.419&-202.490&17.598&810.091\\ \hline0.065&15.426&60.823&17.599&-15.618&-240.921&17.599&954.621\\ \hline0.055&18.119&63.656&17.599&-15.788&-286.064&17.599&1124.389\\ \hline0.047&21.283&66.488&17.600&-15.932&-339.089&17.600&1323.801\\ \hline0.040&25.000&69.321&17.600&-16.055&-401.373&17.600&1558.036\\ \hline\end{array}}Time now (at S=1) or present age in billion years:13.834
'T' in billion years (Gy) and 'D' in billion light years (Gly)
There are lots of things to recognize in the new history table. The widest point in the teardrop shaped lightcone comes around year 4 billion in the S=2.59 row. A source that emitted light then which we are receiving now would have been receding at the speed of light. Before that objects we see today were receding faster than c and you can calculate the multiple simply by dividing Dthen/THub
I understand the inflection point in the scalefactor curve comes around year 7.7 billion---that's been determined separately (it does not jump out at you from the table.) I haven't included enough rows in this table to capture it. The moment of inflection in distance growth comes between two of my rows.
The eventual growth rate of 1/176 percent per million years is already in effect by the end of the table. You can see its reciprocal clearly in the Cosmic Event Horizon distance DHor by that time, and in the convergence the Hubble time to the corresponding eventual constant value. The square of that growth rate is an alias for the cosmological constant Lambda, so the table makes Lambda visible.
Those are just some of the things recognizable in this sample history.
I could have asked for a wider expanse of time and a greater overall expansion to be covered, but this seemed ample for test-driving the new Planck parameters.{\begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.56&17.6&3400&67.17&0.684&0.316\\ \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)\\ \hline45.000&0.022&0.056&0.085&39.362&0.875&1.247&0.153\\ \hline37.201&0.027&0.075&0.114&38.595&1.037&1.488&0.206\\ \hline30.753&0.033&0.100&0.151&37.750&1.228&1.772&0.277\\ \hline25.423&0.039&0.133&0.201&36.821&1.448&2.107&0.372\\ \hline21.017&0.048&0.178&0.268&35.798&1.703&2.500&0.498\\ \hline17.374&0.058&0.237&0.357&34.672&1.996&2.959&0.667\\ \hline14.363&0.070&0.316&0.475&33.434&2.328&3.494&0.893\\ \hline11.874&0.084&0.420&0.632&32.071&2.701&4.111&1.196\\ \hline9.816&0.102&0.559&0.841&30.573&3.115&4.821&1.599\\ \hline8.115&0.123&0.745&1.118&28.926&3.565&5.628&2.137\\ \hline6.708&0.149&0.991&1.485&27.115&4.042&6.538&2.855\\ \hline5.546&0.180&1.317&1.971&25.129&4.531&7.551&3.812\\ \hline4.584&0.218&1.751&2.610&22.951&5.006&8.659&5.086\\ \hline3.790&0.264&2.323&3.443&20.571&5.428&9.846&6.780\\ \hline3.133&0.319&3.076&4.516&17.982&5.740&11.084&9.028\\ \hline2.590&0.386&4.060&5.861&15.189&5.865&12.330&11.999\\ \hline2.141&0.467&5.325&7.485&12.215&5.705&13.526&15.903\\ \hline1.770&0.565&6.923&9.331&9.111&5.148&14.608&20.991\\ \hline1.463&0.683&8.883&11.256&5.963&4.075&15.518&27.544\\ \hline1.210&0.827&11.200&13.059&2.882&2.382&16.225&35.865\\ \hline1.000&1.000&13.834&14.560&0.000&0.000&16.730&46.281\\ \hline0.851&1.175&16.259&15.529&-2.253&-2.646&17.023&56.991\\ \hline0.725&1.380&18.819&16.232&-4.264&-5.883&17.220&69.718\\ \hline0.617&1.621&21.473&16.718&-6.040&-9.789&17.348&84.771\\ \hline0.525&1.904&24.191&17.040&-7.589&-14.446&17.429&102.522\\ \hline0.447&2.236&26.951&17.248&-8.928&-19.964&17.478&123.419\\ \hline0.381&2.627&29.739&17.380&-10.079&-26.473&17.507&147.994\\ \hline0.324&3.085&32.543&17.463&-11.065&-34.139&17.521&176.879\\ \hline0.276&3.624&35.358&17.515&-11.908&-43.154&17.526&210.819\\ \hline0.235&4.257&38.180&17.548&-12.627&-53.751&17.548&250.694\\ \hline0.200&5.000&41.006&17.568&-13.241&-66.203&17.568&297.535\\ \hline0.170&5.873&43.834&17.580&-13.763&-80.832&17.580&352.560\\ \hline0.145&6.899&46.664&17.588&-14.208&-98.017&17.588&417.194\\ \hline0.123&8.103&49.495&17.592&-14.587&-118.205&17.592&493.115\\ \hline0.105&9.518&52.327&17.595&-14.910&-141.917&17.595&582.294\\ \hline0.089&11.180&55.159&17.597&-15.185&-169.772&17.597&687.047\\ \hline0.076&13.133&57.991&17.598&-15.419&-202.490&17.598&810.091\\ \hline0.065&15.426&60.823&17.599&-15.618&-240.921&17.599&954.621\\ \hline0.055&18.119&63.656&17.599&-15.788&-286.064&17.599&1124.389\\ \hline0.047&21.283&66.488&17.600&-15.932&-339.089&17.600&1323.801\\ \hline0.040&25.000&69.321&17.600&-16.055&-401.373&17.600&1558.036\\ \hline\end{array}}Time now (at S=1) or present age in billion years:13.834
'T' in billion years (Gy) and 'D' in billion light years (Gly)
There are lots of things to recognize in the new history table. The widest point in the teardrop shaped lightcone comes around year 4 billion in the S=2.59 row. A source that emitted light then which we are receiving now would have been receding at the speed of light. Before that objects we see today were receding faster than c and you can calculate the multiple simply by dividing Dthen/THub
I understand the inflection point in the scalefactor curve comes around year 7.7 billion---that's been determined separately (it does not jump out at you from the table.) I haven't included enough rows in this table to capture it. The moment of inflection in distance growth comes between two of my rows.
The eventual growth rate of 1/176 percent per million years is already in effect by the end of the table. You can see its reciprocal clearly in the Cosmic Event Horizon distance DHor by that time, and in the convergence the Hubble time to the corresponding eventual constant value. The square of that growth rate is an alias for the cosmological constant Lambda, so the table makes Lambda visible.
Those are just some of the things recognizable in this sample history.
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