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

• I
• AHSAN MUJTABA
In summary: Your understanding is correct. The Hubble parameter, ##\frac{\dot{a}}{a}##, does not have any dimension associated with it. It is simply a ratio of the rate of change of the scale factor ##a(t)## to the scale factor itself. The dimension of ##a(t)## will depend on how you define it in the metric, as shown in your approach. However, it is important to note that there are two common conventions for the scale factor, which may use different notations and have different units. It is important to be aware of this when working with the scale factor in your calculations.
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.

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.

AHSAN MUJTABA and PeroK
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.

PeterDonis
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.

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.

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.

AHSAN MUJTABA

## 1. What is the "cosmic scale factor"?

The cosmic scale factor, denoted by ##a(t)##, is a measure of the expansion of the universe over time. It represents the relative size of the universe at a given time compared to its size at some reference time.

## 2. How does the cosmic scale factor change over time?

The cosmic scale factor is not constant and changes over time. It is affected by the amount and distribution of matter and energy in the universe, as well as the curvature of space. In general, the cosmic scale factor increases as the universe expands.

## 3. What is the relationship between the cosmic scale factor and the size of the universe?

The cosmic scale factor is directly related to the size of the universe. As the cosmic scale factor increases, the size of the universe also increases. However, it is important to note that the cosmic scale factor does not represent the absolute size of the universe, but rather its relative size compared to a reference time.

## 4. How is the cosmic scale factor measured?

The cosmic scale factor is typically measured using observations of the redshift of distant galaxies. As the universe expands, the light from these galaxies is stretched, causing a shift towards longer wavelengths. This redshift can be used to calculate the cosmic scale factor at a given time.

## 5. What is the significance of the cosmic scale factor in understanding the universe?

The cosmic scale factor is an important concept in cosmology as it helps us understand the evolution and expansion of the universe. It is a key component in various models of the universe, such as the Big Bang model, and is crucial in determining the age and future fate of the universe.

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