Strange anomaly in the calculation of the cosmological constant

Question

I was trying to calculate the cosmological constant with two different methods proposed on the internet, that are apparently equivalent but they give different results and different dimensional analysis.

We know that: $$\displaystyle \Lambda = \frac{3 H_0^2}{c^2} \, \Omega_{\Lambda_0}\tag1$$ but WolframAlpha and Wikipedia page suggest that $$\displaystyle \Lambda = \frac{8 \pi G}{c^2} \, \rho_{vac}\tag2$$ where $\rho_{vac}$ is the cosmological vacuum energy density.

$\Omega_{\Lambda_0}$ is defined as $\displaystyle\frac{\rho_{vac}}{\rho_{c}}$, where $\rho_{c}$ is by definition $\displaystyle\frac{3 H_0^2}{8 \pi G}\ (3)$; so when we substitute eq. 3 into 1, it's easy to see that we obtain eq.2.

Let's do a dimensional analysis:

$[\Lambda]=m^{-2}$, $[H_0]=s^{-1}$, $[c^{2}]=m^2 s^{-2}$, $[G]=m^3kg^{-1}s^{-2}$, $[\rho_{vac}]=J/m^{3}=kg\space m^{-1}s^{-2}$

for eq. 1 we have: $$\displaystyle \Lambda =\frac{s^{-2}}{m^2s^{-2}}=m^{-2}$$ which is correct. But for eq. 2 we have: $$\displaystyle \Lambda =\frac{m^3kg^{-1}s^{-2}}{m^2s^{-2}}\space kg \space m^{-1}s^{-2}=s^{-2} \neq m^{-2}.$$

How can this be possible? It's the same equation.

In addition WolframAlpha states that $\rho_{vac}\approx 5.36 \times 10^{-10}$, while if taking $\Lambda=1.106 \times 10^{-52}$, $G=6.6743 \times 10^{-11}$ and $c=299\space792\space458$, and inverting eq. 2, we get $\rho_{vac}\approx 5.92585 \times 10^{-27}$, which is a value 17 orders of magnitude smaller!

Answer 1

Equation (2) should have $c^4$ in the denominator, and equation (3) should have a $c^2$ in the numerator. Whenever you are off by 17 orders of magnitude, think $c^2$. There is a lot of inconsistency in the cosmological literature over whether $\rho$ means a mass density or an energy density.