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 twobody problem. IfR is the scale factor, normalized to unity at an epoch time τ, the differential equation is thenR^2 ddot R =  (4π /3)G(t)ρ _0. Here ρ_{0} 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 thisGvariable Newtonian model corresponds to an open universe. An upward estimate of the age of the Universe from 10^{10} 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 onethird 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 equationR^2 ddot R =  (4π /3)G(t)ρ _0.
 Publication:

Celestial Mechanics
 Pub Date:
 December 1977
 DOI:
 10.1007/BF01229283
 Bibcode:
 1977CeMec..16..391V
 Keywords:

 Cosmology;
 Gravitational Constant;
 Newton Theory;
 Closure Law;
 Differential Equations;
 Hubble Diagram;
 Time Functions;
 Astrophysics