# Hubble relation to Scale Factor

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• TheMercury79
In summary: Well, in summary, the universe would have been one third its current size 14 billion years ago if the Hubble parameter were truly constant.
TheMercury79
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

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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
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.

TheMercury79
I feel kinda stupid now, The Friedmann equations is in the next chapter. I was under the impression this should be solved with contents of the current chapter. But I see now I've read the curriculum wrong.
I have a full time job and taking this course on the side is really stressful. I just knew I was totally off on this.
Tnx for answers.EDIT: Actually I see now I haven't read the curriculum wrong. It's just not very clear, one place says this and that chapter, another place it says only this chapter. Huh

Last edited:
Arman777

## What is the Hubble relation to Scale Factor?

The Hubble relation to Scale Factor is a fundamental principle in cosmology that describes the relationship between the expansion rate of the universe (measured by the Hubble constant) and the scale factor, which represents the size of the universe at a given time.

## How does the Hubble relation to Scale Factor explain the expansion of the universe?

The Hubble relation to Scale Factor is based on the observation that the further away a galaxy is from us, the faster it appears to be moving away. This is due to the fact that the scale factor of the universe increases with time, causing the space between galaxies to expand.

## What is the role of the Hubble constant in the Hubble relation to Scale Factor?

The Hubble constant is a measure of the current expansion rate of the universe and is a key component in the Hubble relation to Scale Factor. It is used to calculate the scale factor at a given time and is essential in understanding the evolution of the universe.

## How does the Hubble relation to Scale Factor support the Big Bang theory?

The Hubble relation to Scale Factor is a key piece of evidence for the Big Bang theory. It demonstrates that the universe is expanding, which is a fundamental prediction of the theory. Additionally, the Hubble constant can be used to estimate the age of the universe, which aligns with the estimated age from the Big Bang model.

## What are some potential implications of the Hubble relation to Scale Factor?

The Hubble relation to Scale Factor has significant implications for our understanding of the universe and its evolution. It provides evidence for the Big Bang theory and supports the idea that the universe is expanding. It also has implications for the future of the universe, as the scale factor suggests that the expansion will continue indefinitely.

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