- #1

TheMercury79

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- TL;DR Summary
- Comparing sizes now and then

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

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