Large number hypothesis and the matter-dominated universe

Abstract

Dirac’s large number hypothesis (LNH), in the formG/G ₀=HH ⁻¹₀ , is applied to the matter-dominated cosmological era, using the framework of the scale covariant theory (Canuto et al., 1977). We obtain explicit expressions forR andβ _a as functions ofR _E , whereR andR _E are the scale factors of the cosmological Robertson-Walker metric, expressed in atomic and gravitational units, respectively, andβ _a is the ratio between the rates of gravitational and atomic clocks. The parameters in these expressions are$\bar q_0$, the deceleration parameter in gravitational units, and$\dot \beta _a$(t ₀)H ⁻¹₀ where$\dot \beta _a$(t ₀) is the present epoch value of the derivative ofβ _a with respect to atomic time. We find that a necessary condition for the LNH to be compatible with a Robertson-Walker model is that$\dot \beta _a$(t ₀)H ⁻¹₀ ≥ ¹₂ . The only experimental values for$\dot \beta _a$(t ₀) available at present are those based on the lengthening of the Moon’s period of revolution around the Earth, suggesting 0.86≥$\dot \beta _a$(t ₀)H ⁻¹₀ ≥0.21; the more promising technique of radar ranging to the inner planets has not yet produced a value for$\dot \beta _a$(t ₀). Using the lunar data, it follows that 0≤$\bar q_0$≲0.42 corresponding to an open universe (k=−1). Closed models (k=1,$\bar q_0$>1/2) are not compatible with the LNH since the required values of$\dot \beta _a$(t ₀)H ⁻¹₀ are more than an order of magnitude above the observational upper limit.