Hubble relation to Scale Factor (2025)

Imagine a Universe where the Hubble parameter is truly a constant, in both space and time.
How much smaller would such a Universe be 14 billion years ago compared to today?

Using the Hubble parameter in terms of scale factor: ##H(t) = \frac{\dot{a}}{a}## leads to
the differential equation: $$\frac{da}{dt}=a~H(t)$$
Solivng for the scale factor yields an exponential growth relation:$$a(t) = a(0)e^{Ht}$$

(##H(t) = H## since H is constant in this Universe and so it's not necessary to use the Hubble parameter
as a function of time)

If we use ##H = 70 kms^-1Mpc^-1##, current time ##t=14\times10^9~y## and initial time 14 Gy ago is ##t_0=0##, then$$\\$$
##Ht = 70~kms^{-1}~Mpc^{-1} * \frac{1}{3.09\times10^{19}}~Mpc~km^{-1} * (3600 * 24 *365.25 * 14\times10^9)~s##

This makes ##Ht## roughly equal to 1 and ##a(14)\approx a(0)e^1##
Therefore $$a(14)\approx 2.72a(0)$$

This, however, seems like a really small number, indicating the Universe was about one third of its current size
14 billion years ago for a "Constant Hubble Universe"
Also a value of 70 for the Hubble parameter corresponds to a Hubble time of 14 billion years. And if H(t) is constant then 14 billion years ago would have the same Hubble time of 14 billion years, it can't be one third of the current size and have the same Hubble time, shouldn't this Hubble time be zero 14 billion years ago?

Something(s) doesn't add upp in my approach and I'm trying to think of where I'm off

Well I think your beginning is wrong. You should derive the scale factor ##a(t)## from the Friedmann Equations. Not by using only $$H = \dot{a} / a$$.

From my knowledge, the only close thing that does look like your solution is universe with lambda only which has a scale factor of
$$a(t) = e^{H_0t}$$ however here $$H_0 = \sqrt{\frac{8 \pi G ε_Λ} {3c^2}} = \sqrt{Λ/3}$$

(Also called de Sitter universe)

TheMercury79 said:

Imagine a Universe where the Hubble parameter is truly a constant, in both space and time.

##H = const## means that the universe is expanding exponentially. It's the case if the Cosmological Constant ##\Lambda > 0## and the matter density ##\rho = 0##, which is expected for the very far future of our universe.