The Units of the Cosmological Constant: eV^2

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Discussion Overview

The discussion revolves around the units of the cosmological constant, specifically the claim that it should be measured in ##eV^2## rather than ##eV^4##. Participants explore the implications of different interpretations of the cosmological constant and its relation to energy density.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant asserts that in natural units, the cosmological constant is known to have units of ##eV^2##, questioning a paper that states it as ##meV^4##.
  • Another participant clarifies that the dimension of the cosmological constant is energy^4, not energy^2, suggesting a misunderstanding of the units.
  • A later reply distinguishes between the cosmological constant ##\Lambda## and the energy density ##\rho_\Lambda##, indicating that the paper in question discusses energy density, which is related but different from the constant itself.
  • One participant suggests reverting to SI units to clarify the dimensions of ##\Lambda##, explaining that it corresponds to ##1/L^2## in natural units, thus leading to the conclusion that it has dimensions of ##eV^2##.
  • Another participant notes that the relationship between ##\Lambda## and energy density involves constants like ##8\pi G##, which also influences the dimensional analysis.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate units for the cosmological constant, with some asserting it should be ##eV^2## while others maintain it is ##eV^4##. The discussion remains unresolved regarding the interpretation of these units and their implications.

Contextual Notes

The discussion highlights potential confusion between the cosmological constant and its associated energy density, as well as the dependence on the definitions and contexts in which these terms are used.

Safinaz
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A question about the unit and the value of the cosmological constant
In natural units, it’s known that the unit of the cosmological constant is ##eV^2##.
I don‘t get why in this paper :

https://arxiv.org/pdf/2201.09016.pdf

page (1), it says the value of ##\Lambda \sim meV^4##, this means ##\Lambda \sim (10^6 ~ eV)^4 \sim 10^{24} eV^4 ##, shoud not the unit ##eV ^2 ## instead ?

 
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Because in natural units the dimension of the cosmological constant is energy^4 and it is not known that the dimension is energy^2.
 
Safinaz said:
Summary: A question about the unit and the value of the cosmological constant

In natural units, it’s known that the unit of the cosmological constant is ##eV^2##.
I don‘t get why in this paper :

https://arxiv.org/pdf/2201.09016.pdf

page (1), it says the value of ##\Lambda \sim meV^4##, this means ##\Lambda \sim (10^6 ~ eV)^4 \sim 10^{24} eV^4 ##, shoud not the unit ##eV ^2 ## instead ?

Safinaz said:
See for instance the discussion here:

https://www.quora.com/Why-is-the-cosmological-constant-without-units

Or this paper : https://arxiv.org/pdf/hep-th/0012253.pdf, equation (2)

They say the units of ##\Lambda## is ##eV^2 ## or equivalently in natural units ##cm^{-2} ## or ##sec^{-2}##
Ok, so the confusion is regarding ##\Lambda## vs ##\rho_\Lambda##. I was referring to ##\rho_\Lambda##, the energy density of the cosmological constant. The first paper you cite in the OP is discussing the scale of the energy density, the others discuss the constant appearing in front of the metric in the Einstein field equations. These differ by a constant ##8\pi G## and ##G## has dimensions energy^-2 in natural units.
 
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In such cases it's always good to go back to SI units first. From the Einstein equations you read off that ##\Lambda## has the same units as ##[G E]/(V^4 L^3)## (##G## gravitational constant, ##V## dimension of velocity, ##L## dimension of length). The gravitational constant itself has the dimension of ##[E] L/M^2## and thus ##[\Lambda]=1/L^2##. In "natural units" with ##\hbar=c=1## that means it dimension ##\text{eV}^2##.

A different way to see it is to realize that ##\Lambda c^4/(8 \pi G)=u_{\Lambda}## is an energy density (density of "dark energy"). Often you see also ##\rho_{\Lambda}=\Lambda c^2/(8 \pi G)##, which is the corresponding "mass density", i.e., ##\rho_{\Lambda}=u_{\Lambda}/c^2##.
 
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