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The Dirac large numbers hypothesis refers to an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales. Dirac noted that the ratio of the size of the visible universe, ct with c the speed of light and t the age of the Universe, to the size of a quantum particle r is about ct / r = 10⁴⁰. Hence, in units c = 1 and r = 1, this large number can be taken as the age of the Universe, t = 10⁴⁰.

There is another ratio with this order of magnitude: the ratio of the electrical to the gravitational forces between a proton and an electron, (≈ 4.4 × 10⁻⁴⁰). Hence, taking the charge e of the electron, the mass m_pr/m_e of the proton/electron, and the permittivity factor 4πε₀ as units, the gravitational constant equals G = 10 ^{− 40}. Dirac interpreted this to mean that G varies with time as , and built what remains to this day a largely untested cosmology out of this idea. If correct, the connection between gravity and quantum mechanics would be unmistakable and may point in the direction of a theory of quantum gravity.

Some scientists believe that the hypothesis is the result of a numerological coincidence, and in 1961, Robert Dicke argued that carbon-based life can only arise when the hypothesis is true lest fusion of hydrogen in stars not occur. A few proponents of non-standard cosmologies refer to Dirac's cosmology as a foundational basis for their ideas.

• Dirac, P. A. M. "A New Basis for Cosmology." Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 165, Issue 921, pp. 199-208 (received December 29 1937).
• Dirac, P. A. M. "The Cosmological Constants." Nature 139 (1937) 323.
• Dirac, P. A. M. "Cosmological Models and the Large Numbers Hypothesis." Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 338, No. 1615, (Jul. 16, 1974), pp. 439-446.

• gr-qc/0111034 Guillermo A. Mena Marugan, Saulo Carneiro: Holography and the large number hypothesis
• The Mysterious Eddington-Dirac Number
• Rainer W. Kühne: Time-varying fine-structure constant requires cosmological constant
• A Look at the Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke (arxiv:0708.3518)