Large number hypothesis and the matter-dominated universe

Abstract

Dirac’s large number hypothesis (LNH), in the formG/G 0=HH 0 −1 , 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 0 −1 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 0 −1 ≥ 2 1 . 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 0 −1 ≥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 0 −1 are more than an order of magnitude above the observational upper limit.