Testing $$R_h=ct$$ R h = c t cosmology from fundamental considerations: Is the Friedmann universe intrinsically flat

Abstract

Recently Melia and Shevchuk (Mon Not R Astron Soc 419:2579,2012) (MS) have proposed the so-called \(R_h =ct\) cosmology where the “Gravitational Horizon” of the universe \(R_h\) is equal to the distance travelled by light since “Big Bang”. Here we would like to see whether the basic claim \(R_h=ct\) is correct or not because MS have not given any cogent derivation for the same. Essentially we will compare the twin expressions for the Einstein energy momentum complex (EMC) of the Friedmann universe obtained by using an appropriate superpotential and also by a direct method. To enable a meaningful comparison of the twin expressions, both are computed by using the same quasi-Cartesian coordinates. We however do not claim that Einstein EMC is superior to many other routes of defining EM of a self-gravitating system. In fact, for static isolated spherical syatems, the idea of a coordinate independent field energy of Lynden-Bell and Katz (Mon Not R Astron Soc 213:21, 1985) might be quite physically significant. Yet, here, we use Einstein EMC because (i) our system is non-static and not isolated one (ii) our primary aim is not find any absolute value of EM, and, finally, (iii) only Einstein pseudo-tensor offers equivalent twin expressions for EM which one can be equated irrespective of any physical significance. Following such comparison of equivalent twin expressions of Einstein energy, we find an exact proof as to why Friedmann universe must be spatially flat even though, mathematically one can conceive of curved spaces in any dimension. Additionally, it follows that, apparently, the scale factor \(S(t) \propto t\) as insisted by \(R_h=ct\) proposition. Nonetheless, because of close similarity of this form, \(S(t) \propto t\), with the (vacuum) Milne metric, and also because of implied unphysical equation of state, \(R_h=ct\) cosmology is unlikely to represent the physical universe.