The equations of General Relativity with scale factor of the form:

and the other symbols have their usual meanings.

Solution:

Interpretation of the solution

The value of the gravitational constant is determined by the rescaling process and the surrounding matter distribution, but gravity does not change the rate of rescaling.

The cosmological constant is interpreted as a constant. It is not interpreted as the energy density of the vacuum and also does not affect the rate of the rescaling.

The scale factor a is interpreted as applying to the size of material objects as well as the distance between objects. H is regarded as a fundamental constant of nature.

The speed of light can be regarded as constant, and the universe static, as the rescaling is undetectable in principle.

The measurable motion of matter, is then determined by the values of G at different points, Einsteins theory of gravitation and the laws of thermodynamics.

Flatness problem

With this model the universe is always near critical density, because the value of G depends on the surrounding matter distribution.

According to

For a derivation see www.rescalingsymmetry.com

The density parameter

where is the critical density, and noting that the in the above equation is half of Hubbles constant. = Hubbles constant.

So although this model universe can be regarded as static, the traditional calculation for the density parameter gives

The horizon problem

This model universe appears static (scale-size) to those living in it. Although there could be motion of matter, distant parts of the universe would have time to communicate with each other and hence major temperature and density variations would be smoothed out.

The supernovae problem

Recent measurements on supernovae show an accelerating universe.
With this model

so the deceleration parameter is

i.e. the model represents a static universe (scale-size) with an apparently accelerating expansion. There can also be motion of matter ‘on top’ of this and even an expansion or contraction of the matter density. The value of the deceleration parameter is consistent with the supernovae measurements which show an apparently accelerating universe.

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