That is, if we put ourselves at the origin of the spatial coordinates, a typical baryon B has been receding from us at a constant velocity v for the past ~ 14 billion years.
Let us check. The Hubble constant is either 73 km/s/Mpc (standard candles) or 68 km/s/Mpc (CMB). The two different estimation methods give conflicting values.
A megaparsec is 3.09 * 10²² meters.
t₁ = 1 megaparsec / (73 km/s)
= 13.4 billion years,
t₂ = 1 megaparsec / (68 km/s)
= 14.4 billion years.
The values agree with the current estimate
t = 13.8 billion years.
Observation of the inertia of small shells of mass implies that something must cancel the extra inertia in a linear movement
In the previous blog post we argued that the inertia of the masses in a small expanding and contracting shell (= radial movement) should be smaller than in a linear movement if the system is deep in the potential well caused by neighboring masses.
The masses in the observable universe might make the inertia of a linear movement 10% larger than for a radial movement.
We do not know how precisely people have measured the inertia in various synchronous movements, but the difference has to be much less than 10%. Someone would have noticed it, if the difference would be that large.
Our conclusion is that negative energy or negative pressure must almost exactly cancel the inertia which would be caused by the mass of distant galaxies. There is a problem of fine-tuning in this.
Or, the maximum range of gravity is of the order 200 million light-years. Then there is no fine-tuning, but it does not look nice to assume an arbitrary maximum range.
A negative cosmological constant
In the standard ΛCDM model of the universe, dark energy is the cosmological constant Λ.
In the above equations k = 0 if the spatial metric is flat. Observations suggest that it is flat. The Hubble constant is H and a is the scale factor.
For H to stay approximately constant, a negative value of Λ must contribute considerably to the mass-energy density ρ. A negative Λ produces negative pressure.
A dust ball explosion in the Minkowski space
The spatial metric of a uniform expanding dust ball probably is flat if the particles will reach the infinity and their velocity is zero there. Here we assume that gravity is the only interaction.
1. A negative cosmological constant would correspond to an ad hoc scalar field which has positive energy density and which for an unknown reason expands with the dust ball. This hypothesis is ugly.
2. Suppose that there are particles which repel ordinary matter. The particles are dark matter and are spread quite evenly inside the dust ball. Their repulsion could be fine-tuned to cancel the gravity of distant parts of the dust ball. The repulsion cannot be between ordinary matter particles, because then we would have no gravity and no stars.
A possible way to fine-tune: we demand that the sum of the "gravity charges" of all particles in the universe must be zero. This would be analogous to the fact that electric charges seem to cancel each other out in the universe.
The spatial metric of comoving coordinates would be hyperbolical. The large-scale geometry for static observers would be Minkowski. The velocity of the expansion stays constant.
3. If the maximum range of gravity is ~ 200 million light-years, it would keep clusters of galaxies together, but allow the universe at a large scale to expand at a constant velocity. The spatial metric at a large scale would be hyperbolical, since that is the geometry of comoving coordinates for an explosion with no gravity in the Minkowski space. The ad hoc parameter 200 million light-years is ugly.
Of these alternatives, 2 looks the least ugly.
Negative gravity charge particles
If the universe is teeming with dark matter particles with negative gravity charges, why we have not noticed them? Maybe negative gravity charge particles do not clump together, but stay free.
Negative gravity charge particles can not be used to create negative gravity in the sense that they would reduce the inertia of an ordinary particle and make clocks to tick faster. All interactions add to the inertia of a particle. It does not matter if the interaction is repulsive or attractive.
The inertia of a negative gravity charge particle is positive. It is just like an ordinary particle, but repels all particles of ordinary matter.
The kinetic energy of a negative gravity charge particle would add its negative gravity charge, just like kinetic energy adds to the mass-energy of an ordinary particle.
A negative gravity charge particle does not annihilate easily since it repels positive gravity particles.
Negative gravity charge particles are another step in reducing gravity to just an ordinary force.
Conclusions
We need a mechanism which explains why distant galaxies do not considerably increase the inertia of a linear motion.
It turns out that the same mechanism causes the universe to expand surprisingly fast - the mechanism is equivalent to dark energy.
We still have the mystery of why the ordinary matter and dark matter densities of the universe are relatively close to the density which makes the spatial metric flat. The matter content is estimated to be 30% of what is required for a flat metric. In ΛCDM the equivalent problem is why the universe entered the "dark energy dominated" period quite recently.
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