# Dimensions of Cosmic Scale Factor ##a(t)##

AHSAN MUJTABA
TL;DR Summary
I have confusion regarding the dimensions of the cosmic scale factor, ##a(t)##. I have read on the wiki that it is dimensionless, but I wonder about it because it is a function of time, t. I want to use its dimensions to prove the action as non-dimensional.
I know the formula for Hubble's parameter, ##\frac{\dot{a}}{a}##, but I cannot infer any dimension of ##a(t)## from it. Please guide me.
Thanks.

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The dimension of ##a## depends on how you define it in the metric.

AHSAN MUJTABA
The dimension of ##a## depends on how you define it in the metric.
Would you please elaborate a little? Thanks.

AHSAN MUJTABA
Just check my work to find dimensions of ##a(t)##.
I have written the metric as,
##ds^{2}=dt^{2}+a(t)^{2}dx^{2}.##
Now, I am aware of the dimensions of the quantities as:
##ds^{2}=[L]^{2}##, ##dt^{2}=[L]^{2}##( I am defining it in terms of length by L=ct, taking c=1.) and ##dx^{2}=[L]^{2}##.
I need to define everything in terms of mass dimensions. I have ##\lambda=\frac{h}{mc}## and working in natural units, I can define the dimension of length in terms of mass as, ##[M]^{-1}.##
Now, the dimensions of time in terms of mass becomes, ##[M]##.
Now, incorporating these dimensions in the metric we get:
##1=1+a(t)^{2},##
##[a(t)]=0.##
So, from this approach, the cosmic scale factor is coming out to be a dimensionless quantity.
Is it a legal approach, please do have a look.

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As you have defined it, ##a## is dimensionless, yes. That doesn't stop it being a function of time - for example, if a wave with frequency ##f## passes you then the number of complete cycles that have passed you since time zero is ##ft##. That's both dimensionless and a function of time.

• AHSAN MUJTABA and PeroK
AHSAN MUJTABA
Just one little question, does defining length in terms of mass dimensions by using the relation ##L=ct## is legit or not? In natural units i.e. c=1, the time and length would have the same dimensions.

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Just one little question, does defining length in terms of mass dimensions by using the relation ##L=ct## is legit or not? In natural units i.e. c=1, the time and length would have the same dimensions.
That seems to me to be defining time in length units or vice versa. You need an additional choice that ##G=1## to get mass in the same units as length and time. I don't usually do it, and I recall Carroll recommending against it. It isn't wrong, though.

• PeterDonis
AHSAN MUJTABA
According to this article, it is seen that ##L=ct## is right because to convert dimensions of time into length, we set c=1. Secondly, we know Compton's wavelength relation, and from that, we can have ##(mass)^{-1}## dimensions of both time and length.

AHSAN MUJTABA
As you have defined it, ##a## is dimensionless, yes. That doesn't stop it being a function of time - for example, if a wave with frequency ##f## passes you then the number of complete cycles that have passed you since time zero is ##ft##. That's both dimensionless and a function of time.
Yes, now I am clear about that. Thanks.

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