Past Values of the Hubble Parameter

I have tried to look up past values of the Hubble parameter a number of times now, without luck.  Even finding an equation for past values of the Hubble parameter is challenging.  Below is an effort to provide such an equation.

The Hubble parameter is defined as:

And it is stated that:

 

Where the deceleration parameter is defined as:

 

If a=t, as in a FUGE universe and presumably the flat cosmology model as well, then we would have the following:

If a=t^½, as per the radiation-dominated era in the Standard Model, we would have:

 

If a=t^⅔, as per the matter-dominated era in the Standard Model, we would have:

In the Standard Model, the current era is dominated by Dark Energy and, for that reason, we could say that a=e^H.t, where the Hubble parameter no longer changes as is equal to H₀, in which case:

It is interesting that we have three eras where the equations work perfectly with nice tidy values for the deceleration parameter in those eras.  It would be madness to think of the eras as punctuated by sudden flips to a new value, even if Standard Model does otherwise have some identified issues (see The Problem(s) with the Standard Model).  For this reason, we could look at generalising.

Say that a=t^η, where η is a constant.  Then we would have:

Looking at the equation of state, we can see that it would follow (“(i)f the (perfect) fluid is the dominant form of matter in a flat (and isotropic) universe”) that η=2/(3(1+w).  For a matter-dominated era, w=0, so η=⅔.  For a radiation-dominated era, w=⅓, so η=½.  For a dark-energy-dominated era, w=-1, so there is no valid value of η.  For a FUGE universe, w=-⅓, so η=1.  So we don’t really need η, we can use the standard formulation with the equation of state value, w, except for when it doesn’t work of course (in a dark-energy-dominated era – where we would have a division by zero error).

Note that we are still left with one discontinuity point.  It is unclear and where the scale factor changed from the form a=t^η to a=(e^H_0.t)/B (where B is notionally a constant such that B=e^H_0.t_0, where t₀ notionally means "now", but the value of "now" could be applied to any time since that transition to obtain a(t₀)=1 unless, for some reason, we are in a privileged era).

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To work out the value of H(t) in the past then, we just need to know what t is.  I ask this because it makes sense that t is the age of the universe (and in this context it is for a FUGE universe), but this is not stated explicitly in the documents that I have looked at.  The closest to an explicit statement is that we can compare two values of a(t), one now at time t₀ and another at time t.  This implies that we can use now as the reference and the value of t in the equation is now minus how long ago an event at time t happened.  If we take “now” as being the age of the universe, then t will be the age of the universe at that time.

The most likely reason that people working in this area avoid just saying that t is the age of the universe is that they use the value t_H=1/H₀, which is reported as both 13.79 billion years and 14.4 billion years (with the latter being more common).  This is despite there being the Hubble tension, which means that the Hubble time should be presented with much less certainty, to match the lack of overall certainty associated with the Hubble parameter value (which is very close 76km/s/Mpc [Cepheids], somewhere close to 71km/s/Mpc [Standard Sirens] or very close to 67km/s/Mpc [CMB]).  It’s more accurate to say that the Hubble time, as defined as t_H=1/H₀, falls somewhere in the region of 12.8 to 14.6 billion years (and somewhere around 13.8 billion years according to the Standard Sirens value).

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We can bite the bullet and say that t in H(t) is the age of the universe where the current age of the universe is in the region of t≈14 billion years (13.8 if you like artificial certainty), so H₀≈70km/s/Mpc.

Going further to establish the values that I actually want (for an earlier but recently updated article on redshift), namely the Hubble parameter value for the decoupling/recombination event, we can say that for the Standard Model:

H(380,000 years)=2/3/(380,000 years)=1,720,000km/s/Mpc

And for a FUGE universe (see  FUGE and Redshift):

H(12.5 million years)=(1/12.5 million years)=78,200km/s/Mpc