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What about its value roughly 9 billion years after the Big Bang, when dark energy started to 'take over' and accelerate the expansion of the universe?

Is there a timeline or chart somewhere that shows approximate, theoretical values of Hubble's not-so-constant constant throughout the lifetime of the Universe?

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This answer to the question “Is the Hubble constant dependent on redshift?” gives the formula (a form of the Friedmann equation) for the Hubble parameter $H(z)$ as a function of redshift $z$:

$$ H(z)^2 = H_0^2 \left[ (1+z)^4 \Omega_r + (1+z)^3 \Omega_M + (1+z)^2 \Omega_k + \Omega_\Lambda \right] $$

where the $\Omega$ terms are the fractional densities in radiation, matter, curvature, and dark energy, respectively.

Using that, plus the knowledge that the redshift of the CMB is $z \simeq 1100$, you can plug in values for the densities (I used WMAP values quoted here) and get that $H$ at the time of the CMB was about 22,000 times larger than the current value.

That answer also gives a graph of the value of the Hubble parameter as a function of time.

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