I Some questions related to the Cosmological Constant

[Arman777]

In Barbara Ryden's introduction to cosmology book its written that
"Introducing $\Lambda$ into the Poisson's equation allows the universe to be static, if you set $\Lambda = 4\pi G\rho$"

Then later on, in the book energy density of the $\Lambda$ defined as $\epsilon_{\Lambda} = \frac{c^2\Lambda}{8\pi G}$

But this not make much sense. Since in this case
$$\epsilon_{\Lambda} = \frac{c^2 4\pi G\rho}{8\pi G} $$
$$\epsilon_{\Lambda} = \rho/2 c^2 $$
Which is not correct.

From other sources I see that actually

$\Lambda = 8\pi G\rho$ and
$\epsilon_{\Lambda} = \frac{c^4\Lambda}{8\pi G}$

And it seems that other sources definiton is the correct one. So the information on the book is wrong ? Or the $\Lambda$ for the static universe case is different then the general $\Lambda$ ?

Also,
$\epsilon_{\Lambda} = \frac{c^4\Lambda}{8\pi G}$ and we know that $\Omega_{\Lambda} = \frac{\epsilon_{\Lambda}}{\epsilon_c}$
and $\epsilon_c = \frac{3H^2c^2}{8 \pi G}$

so we get

$$\Lambda = \frac{3H^2\Omega_{\Lambda}} {c^2}$$

So is this means that $\Lambda$ is actually a time dependent thing since it involves $H(t)$ and $\Omega_{\Lambda}$ ?

 

[timmdeeg]

$$\epsilon_{\Lambda} = \rho/2 c^2 $$
Which is not correct.

Its correct with $c^2$ in the numerator.

$$\Lambda = \frac{3H^2\Omega_{\Lambda}} {c^2}$$

So is this means that $\Lambda$ is actually a time dependent thing since it involves $H(t)$ and $\Omega_{\Lambda}$ ?

It only appears to be. With $\Omega_\Lambda=\frac{8\pi{G}\rho_\Lambda}{3H^2}$ you obtain $\Lambda=8\pi{G}\rho_\Lambda$.

 

[Arman777]

Its correct with c2c2c^2 in the numerator.

Well 2 is dimensionless so it has not much effect on the equation itself. But I don't think its normal to have a 2.

$\Lambda = 8 \pi G\rho_{\Lambda}$

So the equations on the book is wrong ?

It only appears to be.

Hmm I see. Its also interesting. Then $H^2(t)\Omega_{\Lambda}(t) = Constant$ for all times

 

[timmdeeg]

Well 2 is dimensionless so it has not much effect on the equation itself. But I don't think its normal to have a 2.

If you combine the first 2 equations in your OP you get 2 in the denominator. Nothing wrong.
Your first equation follows from the 2. Friedmann Equation with the 2. derivative of the scale factor set zero (because static).

 

[Arman777]

Hmm okay than