The repulsion between the charges accelerates them. An accelerating charge radiates radio waves, the process which drains energy and slows down the expansion.
But at the distance > 2 R, destructive interference wipes out almost all the energy in the waves. The energy must return to the charges, speeding up the initially slowed down acceleration.
How uneven is the matter content in the universe?
If the matter content in the universe is large-grained enough, the analogous oscillation can substantially alter the speed of the expansion. Can it explain dark energy?
The Boötes Void is one of the largest voids in the universe. It is 330 million light-years across.
The accelerating expansion of the universe is observed in surveys which span several billion light-years.
Gravitational waves
When masses are accelerated in the universe, the uneven distribution produces gravitational waves whose wavelength might be ~ 300 million light years for the largest "pointlike" structures. The waves then travel to various directions, until they are eliminated in destructive interference. The energy in the waves must then be reflected back to the sources which produced the waves.
For the largest structures, the destructive interference may only happen when the waves have traveled ~ 1 billion light-years. Thus, we might have an oscillation whose period is 2 billion years.
If the expanding mass structure is a spherical shell, then the power of its gravitational waves is zero. Only "pointlike" structures produce waves.
The power of gravitational waves from an accelerating gravity monopole
In literature, the power of gravitational waves is always calculated for a quadrupole. The power is 16 times the power of an analogous electric charge.
Let us assume that the power for a pointlike mass M is 16 times the analogous Larmor radiation power:
P = 32/3 G / c³ * M² A²,
where A is the acceleration.
The scale factor a of the universe in the matter-dominated era is
a ~ t^2/3,
and its time derivative is
da / dt ~ t^-1/3.
One billion years ago, the age of the universe was 7% less than now. The expansion velocity was 2% larger than now.
Let us have a galaxy cluster M at the distance of two gigaparsecs from us. It recedes from us at the speed of c / 2. The change in the speed is 2% in a billion years, or 3,000 km/s. The acceleration is
A = 3 * 10⁶ / 3 * 10¹⁶ m/s²
= 10⁻¹⁰ m/s².
The radiation power is
P = 32/3 * 7 / 3³ * 10⁻⁵⁵ * M²
= 3 * 10⁻⁵⁵ M²
watts. Let us have a cluster whose mass is
M = 10⁴⁵ kg.
The energy radiated over 1 billion years is
E = 3 * 10³⁵ * 3 * 10¹⁶
= 10⁵²
joules. Let us compare this to the kinetic energy of the galaxy cluster:
E' = 1/2 M v²
= 10⁴⁵ * 10¹⁶
= 10⁶¹
joules, of which 4% is 4 * 10⁵⁹ joules. We conclude that gravitational waves cannot significantly affect the deceleration.
The result would be completely different if the galaxy cluster would have a mass which is close to the amount of observable matter and dark matter in the universe, 10⁵⁴ kg. Then gravitational waves would have a very large effect.
Conclusions
The uneven distribution of matter and dark matter in the universe cannot explain dark energy. The effect is at least a million times too small.
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