1. Assume E=MC^2 is true. That is, energy is equal to mass times the (speed of light squared).
2. The Casimir Effect can add relativistic mass to an atom (very little, and for a short time, but true). See: The Casimir effect as a possible source of cosmic energy. Basically, the Casimir Effect shows that photons can appear out of the vacuum, and normally just disappear right back into the vacuum. But in high density areas of space, those photons can be absorbed by the electrons of atoms, raising the energy of the atom, and therefore raising the relativistic mass of the atoms (assumes the speed of light is a constant). Now as it happens, those electrons will emit that energy right back out very quickly, but in high mass density areas (stars, planetary cores) that emitted energy is immediately absorbed by an electron of an adjacent atom. Over time, the entire core is increasing in energy, and thus is increasing in relativistic mass.
3. What would we expect to see in the universe if this was occurring? Well for one thing, it provides an alternative viewpoint regarding what is known as the "redshift" related to distant galaxies. Current cosmological theory holds that the universe is expanding, because more distant galaxies are "redshifted". See: Cosmologist claims Universe may not be expanding.
Astronomers measure whether objects are moving away from or towards Earth by analysing the light that their atoms emit or absorb, which comes in characteristic colours, or frequencies. When matter is moving away from us, these frequencies appear shifted towards the red, or lower-frequency, part of the spectrum, in the same way that we hear the pitch of an ambulance siren drop as it speeds past....
But, as Wetterich points out, the characteristic light emitted by atoms is also governed by the masses of the atoms' elementary particles, and in particular of their electrons. If an atom were to grow in mass, the photons it emits would become more energetic. Because higher energies correspond to higher frequencies, the emission and absorption frequencies would move towards the blue part of the spectrum. Conversely, if the particles were to become lighter, the frequencies would become redshifted.
Because the speed of light is finite, when we look at distant galaxies we are looking backwards in time — seeing them as they would have been when they emitted the light that we observe. If all masses were once lower, and had been constantly increasing, the colours of old galaxies would look redshifted in comparison to current frequencies, and the amount of redshift would be proportionate to their distances from Earth. Thus, the redshift would make galaxies seem to be receding even if they were not.
4. So I put these 3 concepts together and conclude - if the gravitational constant is not constant, but every so slowly increasing (because the Casimir Effect is adding to relativistic mass in high density areas of space such as stellar and planetary cores), then we would see redshifting of distant galaxies even if the universe were not expanding (or not expanding as fast as current cosmology believes).
5. The most distant galaxies appear to be so redshifted that they should be escaping their galactic clusters, since they are not cosmologists invented the concept of "dark matter" that prevents them from doing so. But if the redshifting is due to the fact that in the past those galaxies were just much lower in mass, then the redshifting is due to that and not their speed, and therefore there is no need to pull "dark matter" out of a hat to explain it.
6. Now if mass is increasing, the speed of the orbit of earth (and all other planets) around the sun will increase (very slowly). Basically this is because the orbital speed of a body at a given distance from a larger one is directly proportional to the mass of the larger body. A secondary effect is the rotation of the smaller body slows down. And as it happens, earth's rotation has slowed down. (If everything is held constant, in about another 150 million years earth will have a 25 hour day, and 250 million years ago the earth had less than a 23 hour day). Now the slowing of the earth's rotation has been asserted to be due to tidal effects from the moon. And this is true. But the gravitational force of all the planets on each other, and especially with the sun, slows down their rotations. This is why the planet mercury rotates so slowly.
Summary: due to the Casimir Effect, high density regions slowly gain in relativistic mass. The gain in mass explains the redshifting, and no need for "dark matter". However, I know of no way to measure the effect with modern instruments due to how slowly this occurs. Possibly one could measure the rotation of a planet like Venus (which has no moon) and determine if the rotation is slowing more rapidly than would be case if mass (and thus the gravitational constant) is unchanging. This would require satellites in orbit over Venus with very accurate clocks measuring the rotation, and a long number of years to determine the truth.