The units of the cosmological constant

[Safinaz]

TL;DR 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 ?

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[Orodruin]

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]

Because in natural units the dimension of the cosmological constant is energy^4 and it is not known that the dimension is energy^2.

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 $sec^{-2}$

 

[Orodruin]

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 ?

​

​

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.

 

[vanhees71]

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., $\rho_{\Lambda}=u_{\Lambda}/c^2$.