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

AI Thread Summary
The discussion centers on the dimensions of the cosmic scale factor a(t) in cosmology. The participants clarify that the scale factor can be dimensionless, particularly when defined such that a(now) = 1, or it can have dimensions of length in different conventions. They explore the implications of defining length in terms of mass dimensions, noting that in natural units, time and length can share the same dimensions. The conversation highlights that while a(t) is dimensionless, it remains a function of time, similar to how wave frequencies operate. Ultimately, the choice of notation impacts the interpretation of curvature parameters in cosmological models.
AHSAN MUJTABA
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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.
 
Orodruin said:
The dimension of ##a## depends on how you define it in the metric.
Would you please elaborate a little? Thanks.
 
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.
 
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.
 
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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.
 
AHSAN MUJTABA said:
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.
https://en.wikipedia.org/wiki/Geometrized_unit_system
 
AHSAN MUJTABA said:
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.
 
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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.
 
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Ibix said:
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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To elaborate a little bit, there are two common conventions for the scale factor.

One, which usually uses ##a(t)## as its notation, has the scale factor dimensionless and set so that ##a(now) = 1##. With this notation, the curvature parameter ##k## is a real number with units of inverse length squared.

The second one, which usually uses ##R(t)## as its notation, has a scale factor with units of length. In this notation the curvature parameter ##k## is an integer value equal to either -1, 0, or 1, representing whether the universe is negatively-curved, flat, or positively-curved. This scale factor is the radius of curvature of the universe.

Please note the "usually" in the above: just because a source has one or the other doesn't mean that's the notation they're using.
 
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