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Ab initio estimates of the size of the observable universe

When one combines multiverse predictions by Bousso, Hall, and Nomura for the observed age and size of the universe in terms of the proton and electron charge and masses with anthropic predictions of Carter, Carr, and Rees for these masses in terms of the charge, one gets that the age of the universe should be roughly the inverse 64th power, and the cosmological constant should be around the 128th power, of the proton charge. Combining these with a further renormalization group argument gives a single approximate equation for the proton charge, with no continuous adjustable or observed parameters, and with a solution that is within 8% of the observed value. Using this solution gives large logarithms for the age and size of the universe and for the cosmological constant that agree with the observed values within 17%.