| Abstract: Anthropic arguments of Carter, Carr, and Rees give two approximate power-law relations between the elementary charge e, the mass of the proton m_p, and the mass of the electron m_e in Planck units: m_p ~ e^{18}, m_e ~ e^{21}. A renormalization group argument of mine gives e^{-2} ~ -(10/pi)ln{m_p}. Combining this with the Carter-Carr-Rees relations gives e^2 ln{e} ~ -pi/180. Taking the exact solutions of these approximate equations gives values for e, m_p, and m_e whose logarithms have relative errors only 0.7%, 1.3%, and 1.0% respectively, without using as input the observed values of any parameters with potentially continuous ranges. One can then get anthropic estimates for the masses, sizes, luminosities, times, velocities, etc. for many other structures in physics and astronomy, from atoms to giraffes to the universe, as simple powers of the anthropic estimate for the elementary charge. For example, one gets an anthropic estimate for the real part of the logarithm of the dark energy density with a relative error of only 0.2%. |