Will a clock that is isolated and stationary with respect to the CMB report the highest possible value for the age of the universe?

Question

We have a very special clock that has existed since the dawn of time. Its purpose is to measure the age of the universe. It is always very far from any massive body or gravitational field and it is always held stationary with respect to the cosmic background radiation.

A clock in a gravitational field will run more slowly because of the time dilation due to gravitational potential. A clock that had moved at some point in its history would suffer time dilation because of its non-zero velocity.

Is it possible that any other clock could ever run faster than our very special isolated stationary clock? So will our very special clock measure the highest possible value for the age of the universe?

Answer 1

The fastest clock will be one that is at the center of a large void in the universe. This will be slightly faster than a "standard" clock embedded in an extended region of average mass density which will suffer more gravitational time dilation. (this answer just formalizes the exchange of comments with @PM 2Ring)

Assuming the universe is infinite and homogeneous, etc. with mass density, $\rho$, we can take the average gravitational potential as our zero reference. The potential at the surface of a large spherical void is thus $+GM/r$ and the potential at its center is $\Phi = +{3 \over 2}GM/r $. Here $M = \rho.{4 \over 3} \pi r^3$ is the missing mass. This gives $\Phi = +G\rho.2\pi r^2 $ (this seems to involve the surface area of the void - is that a coincidence?)

The ratio of the time-rates is $ {t_{fastest} /t_{standard} } \approx 1+\Phi/c^2 = 1+ (G/c^2) \rho . 2\pi r^2 $

Taking the average density of the universe as $\rho = 6 \times 10^{-27} kg/m^3 $ and using the Giant Void as an example (radius ~0.6 billion light years = $5 \times 10^{24} $ meters), we get $ {t_{fastest} /t_{standard} } \approx 1.0007$.

So the conclusion is that a clock in the middle of the Giant Void would run 0.07% faster than a standard clock. This is 10 million years over the age of the universe (but is still small compared with the uncertainty on that age.)