The problem of the maximum volumes and particle horizon in the Friedmann universe model
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The maximum volume of the closed Friedmann universe is further investigated and is shown to be 2π2R3(t), instead of π2R3(t) as found previously. This discrepancy comes from the incomplete use of the volume formula of 3-dimensional spherical space in the astronomical literature. Mathematically, there exists the maximum volume at any cosmic timet in a 3-dimensional spherical case. However, the Friedmann closed universe in expansion reaches its maximum volume only at the timet m of the maximum scale factorR(t m ). The particle horizon has no limitation for the farthest objects in the closed Friedmann universe if the proper distance of objects is compared with the particle horizon as it should be. It will lead to absurdity if the luminosity distance of objects is compared with the proper distance of the particle horizon.
KeywordsMaximum Volume Universe Model Spherical Case Luminosity Distance Proper Distance
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