Alexander Friedmann's Cosmic ScenariosFrank Wang, fwang@lagcc.cuny.eduThe Russian mathematician and physicist Alexander Friedmann (1888-1925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits non-static solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. The 2011 Nobel Prize in Physics was awarded to Saul Perlmutter, Brian P. Schmidt and Adam G. Riess for "the discovery of the accelerating expansion of the Universe through observations of distant supernovae." In the Scientific Background on the Nobel Prize in Physics 2011 compiled by the Royal Swedish Academy of Sciences, the Russian mathematician and physicist Alexander Friedmann was said to have "studied the problem of the dynamics of the Universe using essentially the same assumptions as Einstein, and found in 1922 that Einstein's steady state solution was really unstable." As pointed out by Ari Belenkiy in a Physics Today article, this account significantly distorts Friedmann's contributions. The Swedish Academy's oversight is just among the many examples of a wide spread misinformation of Friedmann's paper. Before discussing Friedmann's work, we need to review Einstein's general theory of relativity. In 1915, Einstein obtained the celebrated field equations 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where the spacetime indices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJm51O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== run from 1 to 4 and the constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgma2FwcGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JJm1mcmFjR0YkNigtRiM2KC1JI21uR0YkNiRRIjhGJ0Y3LUY7Nj5RMSZJbnZpc2libGVUaW1lcztGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR0Y2RjQvJSp1bmRlcmxpbmVHRjYvJSpzdWJzY3JpcHRHRjYvJSxzdXBlcnNjcmlwdEdGNi8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y2LyUrZXhlY3V0YWJsZUdGNi8lKXJlYWRvbmx5R0Y2LyUpY29tcG9zZWRHRjYvJSpjb252ZXJ0ZWRHRjYvJStpbXNlbGVjdGVkR0Y2LyUscGxhY2Vob2xkZXJHRjYvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjZGN0Y+RkBGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQRlxxLUYsNiVRJSZwaTtGJ0Y0RjdGWi1GLDYlUSJHRicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGNy1GIzYkLUklbXN1cEdGJDYlLUYsNiVRImNGJ0ZkcUZmcS1GVzYkUSI0RidGNy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGW3MvJSliZXZlbGxlZEdGNkY3RitGNw==. (Here LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is gravitational constant and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= the speed of light.) The spacetime distribution of the energy-momentum tensor 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 determines the local geometry encoded in the metric tensor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiNRIUYnLUYjNiYtRi82JVElJm11O0YnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJy1JI21vR0YkNj5RMSZJbnZpc2libGVUaW1lcztGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR0ZDRkIvJSp1bmRlcmxpbmVHRkMvJSpzdWJzY3JpcHRHRkMvJSxzdXBlcnNjcmlwdEdGQy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0ZDLyUrZXhlY3V0YWJsZUdGQy8lKXJlYWRvbmx5R0ZDLyUpY29tcG9zZWRHRkMvJSpjb252ZXJ0ZWRHRkMvJStpbXNlbGVjdGVkR0ZDLyUscGxhY2Vob2xkZXJHRkMvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRkNGRC8lJmZlbmNlR0ZDLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmhwLUYvNiVRJSZudTtGJ0ZCRkRGREY6RkQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZE, and the Ricci tensor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiNRIUYnLUYjNiYtRi82JVElJm11O0YnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJy1JI21vR0YkNj5RMSZJbnZpc2libGVUaW1lcztGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR0ZDRkIvJSp1bmRlcmxpbmVHRkMvJSpzdWJzY3JpcHRHRkMvJSxzdXBlcnNjcmlwdEdGQy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0ZDLyUrZXhlY3V0YWJsZUdGQy8lKXJlYWRvbmx5R0ZDLyUpY29tcG9zZWRHRkMvJSpjb252ZXJ0ZWRHRkMvJStpbXNlbGVjdGVkR0ZDLyUscGxhY2Vob2xkZXJHRkMvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRkNGRC8lJmZlbmNlR0ZDLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmhwLUYvNiVRJSZudTtGJ0ZCRkRGREY6RkQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZE is determined by 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 and its spacetime derivatives. The Ricci scalar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the local curvature of spacetime. (In the appendix, we will demonstrate how to use Maple's tensor package to evaluate Ricci tensor and scalar for a given metric tensor.) To simplify calculations, the universe can be approximated as a perfect fluid of uniform density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmJnJobztGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= and pressure LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The energy-momentum has the following elements: 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, 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, 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, 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, and all off-diagonal elements are zero. Furthermore, we consider a matter-dominated universe so that the pressure is zero: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. In 1917, Einstein introduced the cosmological constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= in the hope of finding a static cosmological solution of the field equations 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 .The geometry of Einstein's static universe can be regarded as a three-dimensional spherical 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embedded in the four-dimensional Euclidean space (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlRi4tRiM2JC1GOzYkUSIyRidGPkY+RkBGQy1GLDYlRi4tRiM2JC1GOzYkUSIzRidGPkY+RkBGQy1GLDYlRi4tRiM2JC1GOzYkUSI0RidGPkY+RkBGPg==). Friedmann considered a space as a 3D hypersphere similar to that of Einstein's universe, but the radius changes with respect to time. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiotSShtc3Vic3VwR0YkNictRiw2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y9USdub3JtYWxGJ0ZFLUZCNiRRIjJGJ0ZFLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZMLUkjbW9HRiQ2LVEiK0YnRkUvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlUvJSlzdHJldGNoeUdGVS8lKnN5bW1ldHJpY0dGVS8lKGxhcmdlb3BHRlUvJS5tb3ZhYmxlbGltaXRzR0ZVLyUnYWNjZW50R0ZVLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGXm8tRjQ2J0Y2LUYjNiRGR0ZFRkdGSkZNRk8tRjQ2J0Y2LUYjNiQtRkI2JFEiM0YnRkVGRUZHRkpGTUZPLUY0NidGNi1GIzYkLUZCNiRRIjRGJ0ZFRkVGR0ZKRk1GRS1GUDYtUSI9RidGRUZTRlZGWEZaRmZuRmhuRmpuL0Zdb1EsMC4yNzc3Nzc4ZW1GJy9GYG9GZ3AtSSVtc3VwR0YkNiUtRiM2Ji1GLDYlUSJhRidGOUY8LUZQNi1RMCZBcHBseUZ1bmN0aW9uO0YnRkVGU0ZWRlhGWkZmbkZobkZqbi9GXW9RJjAuMGVtRicvRmBvRmVxLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidEYnRjlGPEZFRkVGRUZHRkpGRUYrRkU=Swedish Academy's statement that Friedmann "[used] essentially the same assumptions as Einstein" tremendously undervalued Friedmann's originality. With this non-static model, the time-time component of the Einstein field equations yields a first-order differential equation, known as the Friedmann 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is proportional to the total mass of space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. (To be precise, 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. See appendix for its derivation.) The cosmic radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== in Friedmann's universe is no longer constant, and several scenarios for cosmic evolution can occur. From the Friedmann equation, we can obtain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== by inversion of an elliptic 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in which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjYTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are constants. Let us enter this integral to Maple and see what happens.restart: with(plots):int(sqrt(x/(A-x+lambda*x^3/(3*c^2))), x):Here we end the Maple input with colon to suppress the output, because it is lengthy and not too illuminating. The reader can change the colon to semicolon and see that the result is expressed in terms of EllipticF and EllipticPi. The integral has physical meaning only when the cubic 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under the square-root sign is positive. Friedmann denote 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 by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and consider in the (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY5) plane the family of curves of third degree: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 . With 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, he produced this figure:plot({(x-1/2)/x^3, (x-1)/x^3, 4/27, (x-3/2)/x^3}, x=0..7, -0.5..0.7, labels=[x,y]); For the equation 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 or 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, we can prove that the curve has a maximum 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 , 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 . See below. y := (x-A)/x^3;eq1 := diff(y,x) = 0;solve(eq1, x);eval(y, x=%);Keep in mind that in the cubic denominator 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is proportional to the total mass of space (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), and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is proportional to cosmological constant (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). If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiNEYnRj5GPi1GIzYmRistRiM2Ji1GWjYkUSMyN0YnRj4tRjs2PlExJkludmlzaWJsZVRpbWVzO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHRkIvRjVGQi8lKnVuZGVybGluZUdGQi8lKnN1YnNjcmlwdEdGQi8lLHN1cGVyc2NyaXB0R0ZCLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRkIvJStleGVjdXRhYmxlR0ZCLyUpcmVhZG9ubHlHRkIvJSljb21wb3NlZEdGQi8lKmNvbnZlcnRlZEdGQi8lK2ltc2VsZWN0ZWRHRkIvJSxwbGFjZWhvbGRlckdGQi8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGQkY+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZ3EtSSVtc3VwR0YkNiUtRiw2JVEiQUYnRjRGNy1GWjYkUSIyRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGPkYrRj4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmpyLyUpYmV2ZWxsZWRHRkJGPkYrRj4=, or 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, the cubic equation 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 has two positive roots. If 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, or 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, the equation has no positive roots. Let us set LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= and use 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 as representative numerical examples to plot Friedmann's cubic expression. cd1 := 8/27*x^3 - x + 1;cd2 := 4/27*x^3 - x + 1;cd3 := 2/27*x^3 - x + 1;plot([cd1, cd2, cd3], x=0..3.5, thickness=2, legend=['y'>4/(27*A), 'y'=4/(27*A), 'y'<4/(27*A)], labels=[x, "cubic denominator"], labeldirections=[horizontal, vertical]);We consider the first scenario that 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 has no positive roots by using 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 as an example. The integration of the cosmic radius becomesInt(sqrt(x/(8/27*x^3 - x + 1)), x);i1 := value(%);This is quite a lengthy output. Note that we obtain time as a function of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=). We are more interested in observing the radius as a function of time. We use the parametric form of the plot command to do this trick. plot([i1, x, x=0..7], thickness=2, color=black); pm1 := %:We see that when 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, that is, when the cosmological constant exceeds some critical value that depends on density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmJnJobztGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= or total mass, the cosmos starts at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= from the singularity 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, and it expands forever. Friedmann called this scenario "a monotonic world of the first kind." A careful reader might notice that this curve has an inflection point. To locate the inflection point, we need to find the second derivative 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 and solve the 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 , or 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 . solve(diff(sqrt(x/(A-x+lambda/(3*c^2)*x^3)), x)=0, x);Taking the real solution, we conclude that the expansion rate changes from deceleration to acceleration at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUkmbWZyYWNHRiQ2KC1GIzYoLUkjbW5HRiQ2JFEiM0YnRj4tRjs2PlExJkludmlzaWJsZVRpbWVzO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHRkIvRjVGQi8lKnVuZGVybGluZUdGQi8lKnN1YnNjcmlwdEdGQi8lLHN1cGVyc2NyaXB0R0ZCLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRkIvJStleGVjdXRhYmxlR0ZCLyUpcmVhZG9ubHlHRkIvJSljb21wb3NlZEdGQi8lKmNvbnZlcnRlZEdGQi8lK2ltc2VsZWN0ZWRHRkIvJSxwbGFjZWhvbGRlckdGQi8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGQkY+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZnEtRlU2JS1GLDYlUSJjRidGNEY3LUZqbjYkUSIyRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGXW8tRiw2JVEiQUYnRjRGN0Y+LUYjNiZGKy1GIzYmRl1yRl1vLUYsNiVRKSZsYW1iZGE7RidGaG9GPkY+RitGPi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYnMvJSliZXZlbGxlZEdGQkY+LUZlbjYoLUYjNiQtRmpuNiRGX3NGPkY+LUYjNiRGaW5GPkZdc0Zgc0Zjc0Zlc0ZgckY+RitGPg==. For large LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, we can neglect the constant and linear terms of the cubic denominator:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbWZyYWNHRiQ2KC1GIzYmLUkjbW5HRiQ2JFEiMUYnRj4tRjs2PlExJkludmlzaWJsZVRpbWVzO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHRkIvRjVGQi8lKnVuZGVybGluZUdGQi8lKnN1YnNjcmlwdEdGQi8lLHN1cGVyc2NyaXB0R0ZCLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRkIvJStleGVjdXRhYmxlR0ZCLyUpcmVhZG9ubHlHRkIvJSljb21wb3NlZEdGQi8lKmNvbnZlcnRlZEdGQi8lK2ltc2VsZWN0ZWRHRkIvJSxwbGFjZWhvbGRlckdGQi8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGQkY+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGYHEtSShtZmVuY2VkR0YkNiQtRiM2KS1GOzYtUSYmaW50O0YnRj5GQEZDRkVGR0ZJRktGTUZfcUZhcS1GVTYoLUYjNiRGWUY+LUYjNiYtSSZtc3FydEdGJDYjLUZVNigtRiM2JC1GLDYlUSkmbGFtYmRhO0YnRmJvRj5GPi1GIzYmRistRiM2Ji1GWjYkUSIzRidGPkZnbi1JJW1zdXBHRiQ2JS1GLDYlUSJjRidGNEY3LUZaNiRRIjJGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y+RitGPi8lLmxpbmV0aGlja25lc3NHRmZuLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmF0LyUpYmV2ZWxsZWRHRkJGZ24tRiw2JVEieEYnRjRGN0Y+Rl10Rl90RmJ0RmR0RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGXnUvJSpsaW5lYnJlYWtHUSVhdXRvRictRjs2LVEwJkRpZmZlcmVudGlhbEQ7RidGPkZARkNGRUZHRklGS0ZNRl9xRmFxRmZ0Rj5GPkY+LUYjNiRGZHNGPkZddEZfdEZidEZkdEY+RitGPg== .This is an elementary integration, see the Maple output below.eqasym := t=int(1/sqrt(lambda/3)/x, x);solve(eqasym, x);We conclude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== grows asymptotically like 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. This scenario introduced by Friedmann appears to correspond to reality, as observed by the three Nobel laureates. The second scenario occurs when the cosmological constant is less than some critical value (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRKSZsYW1iZGE7RicvJSdpdGFsaWNHRj5GNUY1RitGNQ== and 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). Here we use 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 as a representative example to plot the cubic denominator.plot(cd3, x=0..3.5, labels=[x, "cubic denominator"], labeldirections=[horizontal, vertical]);In this situation, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== can lie in the intervals (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR0Y7L0YwUSdpdGFsaWNGJy1GIzYkLUYsNiRRIjFGJ0YvRi8vJS9zdWJzY3JpcHRzaGlmdEdGLkYv) and (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLzYlUSgmaW5maW47RidGMkY1Rj4=). For this particular numerical example, we can use Maple to find the two positive roots.sort([solve(cd3=0, x)]);x1 := %[3]; x2 := %%[1];Consider the scenario that 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. Int(sqrt(x/(1-x+2/27*x^3)), x=x2..r);i2 := value(%);plot([i2, r, r=3..5], thickness=2, color=blue); pm2 := %:We see that the expansion of the universe starts from a nonzero value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== and expands forever. Friedmann called it a monotonic world of the second kind. Friedmann cited Weierstrass' work and demonstrated that when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= lies in the interval (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), LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== is a periodic function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The period is given through2*Int(sqrt(x/(1-x+2/27*x^3)), x=0..x1);tp := value(%); evalf(tp);Int(sqrt(x/(1-x+2/27*x^3)), x);ip := value(%);p1 := plot([ip, x, x=0..x1], thickness=2, color=red):p2 := plot([tp-ip, x, x=0..x1], thickness=2, color=red):display([p1, p2]); pp := %:Friedmann called this world "the periodic world": expansion oscillates between 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 and 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. When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= has precisely the critical value 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, then the cubic denominator is degenerate: it has a double positive root. This scenario corresponds to Einstein's static universe.Here is a plot of three possible scenarios that we have discussed.display([pm1, pm2, pp], labels=["time", "cosmic radius"], labeldirections=[horizontal, vertical], labelfont=[TIMES, ROMAN, 14]);Alexander Friedmann's pioneer work is summarized in this plot. He for the first time showed that Einstein's general relativity admits non-static solutions. He called the black curve "a monotonic world of the first kind." The cosmos expands at a decelerating rate from a zero-radius singularity until an inflection point, after which the expansion accelerates. This scenario provides the basis for our current view of the Big Bang and the accelerating universe. The blue curve is named as "a monotonic world of the second kind" in which the universe expands from a nonzero initial radius. The periodic scenario is shown in the red curve: the universe evolves from and back to zero radius. AppendixJohn Wheeler summarized generality in nontechnical terms as "geometry tells matter how to move, and matter tells geometry how to curve." This statement captures the essence of Einstein's field equations 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 .See main text for further elaboration of the meaning of the symbols. The right-hand side of the equation is proportional to the energy-momentum tensor, which is related to matter density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmJnJobztGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=. The left-hand side of the equation (consisting of Ricci tensor and scalar) is the Einstein tensor. The square of the spacetime interval defines the metric 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:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1JJW1zdXBHRiQ2JS1GLDYlUSNkc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRkEvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkwvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZW4tRiM2KC1JJW1zdWJHRiQ2JS1GLDYlUSJnRidGN0Y6LUYjNiZGKy1GIzYmLUYsNiVRJSZtdTtGJy9GOEZMRkEtRkc2PlExJkludmlzaWJsZVRpbWVzO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHRkxGZ28vJSp1bmRlcmxpbmVHRkwvJSpzdWJzY3JpcHRHRkwvJSxzdXBlcnNjcmlwdEdGTC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0ZMLyUrZXhlY3V0YWJsZUdGTC8lKXJlYWRvbmx5R0ZMLyUpY29tcG9zZWRHRkwvJSpjb252ZXJ0ZWRHRkwvJStpbXNlbGVjdGVkR0ZMLyUscGxhY2Vob2xkZXJHRkwvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRkxGQUZKRk1GT0ZRRlNGVUZXL0ZaUSYwLjBlbUYnL0ZnbkZgci1GLDYlUSUmbnU7RidGZ29GQUZBRitGQS8lL3N1YnNjcmlwdHNoaWZ0R0ZFRmhvLUYyNiUtRiw2JVEjZHhGJ0Y3RjpGZG9GQ0Zoby1GMjYlRmlyRmJyRkNGQUYrRkFGK0ZBHere we use Einstein summation convention: whenever a Greek index is repeated in a term, summer over this index from 1 to 4 is implied. The 3D hypersphere considered by Friedmann 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 .(In Einstein's universe, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is independent of time.) It is convenient to transform from Cartesian to polar coordinates: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The spacetime geometry is then described by the 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 to Maple's tensor package, one easily obtains Ricci tensor, Ricci scalar, and Einstein tensor. restart: with(tensor):coords := [chi, theta, phi, t]:
g := array(symmetric, sparse, 1..4, 1..4):
g[4,4] := c^2: g[1,1] := -(a(t))^2: g[2,2] := -(a(t)*sin(chi))^2: g[3,3] := -(a(t)*sin(chi)*sin(theta))^2:
metric := create([-1,-1], eval(g));tensorsGR(coords, metric, contra_metric, det_met, C1, C2, Rm, Rc, R, G, C):displayGR(Einstein, G);Below we simplify the time-time component of the Einstein tensor by copying LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkRzQ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== from Maple's output.Gtt := combine( -3*(-cos(theta)^2*c^2+c^2-cos(chi)^2*c^2+cos(chi)^2*c^2*cos(theta)^2+diff(a(t),t)^2-diff(a(t),t)^2*cos(theta)^2-diff(a(t),t)^2*cos(chi)^2+diff(a(t),t)^2*cos(chi)^2*cos(theta)^2)/a(t)^2/sin(chi)^2/sin(theta)^2);The Friedman equation is the time-time component of the field equations:Eq1 := Gtt + lambda = -kappa*rho*c^2;Eq2 := isolate(Eq1, diff(a(t),t)^2);simplify(rhs(Eq2));The volume of a 3D hypersphere is obtained to the integral below.Int(Int(Int(a(t)^3*sin(chi)^2*sin(theta), chi=0..Pi), theta=0..Pi), phi=0..2*Pi);V := value(%);The total mass is the product of this volume and the density.M := rho*V;With LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= defined as 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, the Fridmann equation is written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJm1mcmFjR0YkNigtRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDY+UTEmSW52aXNpYmxlVGltZXM7RicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRkkvJSp1bmRlcmxpbmVHRkkvJSpzdWJzY3JpcHRHRkkvJSxzdXBlcnNjcmlwdEdGSS8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0ZJLyUrZXhlY3V0YWJsZUdGSS8lKXJlYWRvbmx5R0ZJLyUpY29tcG9zZWRHRkkvJSpjb252ZXJ0ZWRHRkkvJStpbXNlbGVjdGVkR0ZJLyUscGxhY2Vob2xkZXJHRkkvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRklGOi8lJmZlbmNlR0ZJLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmJwLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNictRjI2KC1GIzYkLUY+Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjpGYm9GZG9GZm9GaG9Gam9GXHBGXnBGYHBGY3BGOi1GIzYmRistRiM2JUZhcS1GLDYlUSJ0RicvRktRJXRydWVGJy9GO1EnaXRhbGljRidGOkYrRjovJS5saW5ldGhpY2tuZXNzR0Y5LyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNyLyUpYmV2ZWxsZWRHRklGPS1GIzYmLUYsNiVRImFGJ0ZbckZdci1GPjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y6RmJvRmRvRmZvRmhvRmpvRlxwRl5wRmBwRmNwLUZpcDYkLUYjNiRGaHFGOkY6RjpGK0Y6RjotRjc2JFEiMkYnRjovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjotRiM2JC1GZnA2JS1GLDYlUSJjRidGW3JGXXJGZHNGZ3NGOkZfckZhckZkckZmci1GPjYtUSI9RidGOkZib0Zkb0Zmb0Zob0Zqb0ZccEZecC9GYXBRLDAuMjc3Nzc3OGVtRicvRmRwRmV0LUYyNigtRiM2JkYrLUYjNigtRiw2JVEiQUYnRltyRl1yLUY+Ni1RKCZtaW51cztGJ0Y6RmJvRmRvRmZvRmhvRmpvRlxwRl5wL0ZhcFEsMC4yMjIyMjIyZW1GJy9GZHBGZHVGaHItRj42LVEiK0YnRjpGYm9GZG9GZm9GaG9Gam9GXHBGXnBGY3VGZXUtRjI2KC1GIzYmLUYsNiVRKSZsYW1iZGE7RidGSkY6Rj0tRmZwNiVGaHItRjc2JFEiM0YnRjpGZ3NGOi1GIzYmRistRiM2JkZidkY9Rlx0RjpGK0Y6Rl9yRmFyRmRyRmZyRjpGK0Y6LUYjNiZGK0ZockYrRjpGX3JGYXJGZHJGZnJGOkYrRjo= . Acknowledgements: In December 2014, I revised the worksheet after two readers, Ari Belenkiy and Frank Bensch, posted their comments online. I thank them for their remarks. The solution order from Maple's solve output might vary in different versions, and the sort command was added to address this issue. Thanks to Eithne Murray, Maplesoft's Project Manager, for offering valuable suggestions to me. My statement about the degenerate case was oversimplified. For a more complete analysis, please consult Professor Belenkiy's paper http://arxiv.org/abs/1302.1498, particularly Figure 6 and the paragraph after it. References1. Royal Swedish Academy of Sciences, "The Accelerating Universe," (2011), available at http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/advanced-physicsprize2011.pdf. 2. Ari Belenkiy, "Alexander Friedmann and the origins of modern cosmology," Physics Today, October 2012, 38-43. 3. A. Friedmann, "Ueber die Kruemmung des Raumes," Zeitschrift fuer Physik 10, 377-386 (1922). English translation in A. Friedman, "On the Curvature of Space," General Relativity and Gravitation, 31, 2991-2000 (1999). 4. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, Freeman, 1973.5. Frank Wang, Physics with Maple: The Computer Algebra Resource for Mathematical Methods in Physics, Wiley-VCH, 2006.