G from cosmological measurements

[rbj]

okay, good. actually at another time i was staring at the more general equation and had a few questions regarding it.

This appears to be an incorrect derivation from The Friedmann equations.

The correct expression would be

G = \frac{3 H^2}{8 \pi \rho}

In any event, this equation is not always true - it requires that the cosmological constant \Lambda be zero, that the spatial curvature of the universe K=0, and of course the assumptions that GR is correct and the cosmological principle holds so that the universe is homogenoeus and isotropic.

H here would be Hubble's constant.

so 1/H is the Hubble Time and about the age of the universe (maybe a milli-smidgen longer). this has been measured independently (from astronomical observation), right? and the density of the universe can be sort of guesstimated independently from astronomical observation (from estimating the number of galaxies, about 10¹¹ and star systems per average galaxy 10¹⁰ or 10¹¹ and then some average mass each plus whatever dark matter that i don't know how they measure or estimate and finally divide by the Hubble Volume, (c/H)³ ), right?

that would say that, in whatever system of units, G could be estimated from astronomical observation. but it can also be independently measured (using the same system of units) with a Cavendish-like experiment. if these two values come out differently, is this how they determine the comological constant? someone please illuminate.

 

[yogi]

The problem with measuring G is that it is always tied to some mass - no one seems to have found a way to measure G independently. This is not a problem in some cosmologies, but in Brans-Dicke, SCC and other formalisms, the critial issue turns on the constancy of G. For example if the inertia of an object depends upon cosmological factors such as the position or the amount of distant matter or whatever, then any experiment that measures G by virtue of the constancy of the MG product is going to give a wrong answer to such ultimate questions

Friedmann didn't reach a conclusion about what model best represented the universe - he left the density parameter open and assumed a Big Bang origin where a fixed amount of matter acted to slow subsequent expansion - so if Friedmann models accurately depict the real world, the equations are simple - the problem is that when one tries to estimate the average density from the visual objects, the universe has insufficient mass. Yet there are strong alternative reasons for believing the universe to be flat. So there are questions as to what form non-visual matter most have if the universe is to be flat (critical density or something that makes the universe behave as though it has critical density)