Newtonian cosmology with a varying gravitational constant

Abstract

Newtonian cosmology is developed with the assumption that the gravitational constantG diminishes with time. The functional form adopted forG(t), a modification of a suggestion of Dirac, isG=A(k+t) ⁻¹, wheret is the age of the Universe and a small constantk is inserted to avoid a singularity in the two-body problem. IfR is the scale factor, normalized to unity at an epoch time τ, the differential equation is then$R^2 \ddot R = - (4\pi /3)G(t)\rho _0$. Here ρ₀ is the mean density at the epoch time. With the above form forG(t), the solution is reducible to quadratures.

The scale factorR either increases indefinitely or has one and only one maximum. LetH ₀ be the present value of Hubble's ‘constant’$\dot R$/R and ρ_0c the minimum density for a maximum ofR, i.e., for closure of the Universe. The conditions for a maximum lead to a boundary curve of ρ_0c versusH ₀ and the numbers indicate strongly that thisG-variable Newtonian model corresponds to an open universe. An upward estimate of the age of the Universe from 10¹⁰ yr to five times such a value would still lead to the same conclusion.

The present Newtonian cosmology appears to refute the statement, sometimes made, that the Dirac model forG necessarily leads to the conclusion that the age of the Universe is one-third the Hubble time. Appendix B treats this point, explaining that this incorrect conclusion arises from using all the assumptions in Dirac (1938). The present paper uses only Dirac's final result, viz,G∼(k+t)⁻¹, superposing it on the differential equation$R^2 \ddot R = - (4\pi /3)G(t)\rho _0$.