Decelerated Expansion

The currently understood chronology of the universe has two periods of deceleration after inflation, the radiation-dominated era and the matter-dominated era.

The radiation-dominated era was very short at about 47,000 years, during which the value of the equation of state parameter w=1/3.  Using the equations discussed in Equations for Expansion, we find that dH_RE/dt=-2H² and H=1/(2t).

The matter-dominated era extended from 47,000 after the Big Bang to about 4 billion years ago, during which the equation of state parameter w=0.  Using the equations discussed in Equations for Expansion again, we find that dH_ME/dt=-3H²/2 and H=2/(3t).

In Inflation, I conclude that having an expansion in the order of 10²⁶ over the 10^-32s available would require a Hubble parameter value of H=~10⁶¹km/s/Mpc.  This is about the same magnitude of Hubble parameter as in the first unit of Planck time.  So, if the deceleration was from that value, it would be as if the clock had been turned back to t=t_PL, so by about 10^-32s.  After that, the expansion of the universe during the radiation-dominated era would be at half pace, H=1/(2t), expanding by half the speed of light, for 47,000 years, reaching about 23,500 light years in radius in addition to the approximately 10^-10 light seconds it was at the end of the inflationary era.  If everything went back to “normal” then (with H=1/t), then the universe would be 23,500 light years smaller than we would have expected, but this is a small fraction of the expected 13.77 billion light years (~0.00017% smaller, or basically indistinguishably smaller).

Then, in the matter-dominated era, the expansion of the universe would be at a rate of 2/3 of the speed of light, for a period of approximately 9.8 billion years, resulting in a radius of about 6.6 billion light years (swamping any delta caused by inflation for 10^-32s or slower expansion during 47,000 years of radiation-dominated era).

The combined effect would look like this, using the same notion as per the Inflation post (approximately):

The H value at the end of that process would be approximately that expected at about 14.7 billion years (assuming a constant H=1/t), or 66.5km/s/Mpc.

To get to where we would expect to be today, assuming a constant H=1/t over 13.77 billion years, it would look like this:

This would imply that a radius at 13.77 billion light years would be receding away from us at about 1.8 times the speed of light.  However, that if objects at (r_H=)13.77 billion light years remove were receding away at 1.8 times the speed of light, that is equivalent to H=128km/s/Mpc, which is not what we measure.  Note however, that this here illustrates a period of faster expansion, not of expansion that accelerates.  I will look at the dark-matter-dominated era with what appears to be accelerated expansion next.