Relationship betwen ##Ω_k## and ##k##

[Quarlep]

$H^2(t)-8ρπG/3=-k/a^2(t)$ (Lets suppose a(t)=1) then
$H^2(t)-8ρπG/3=-k$ In equations $k$ can be -1,0,1.
Then If $k<0$ then $-k>0$ then
$H^2(t)-8ρπG/3>0$→Hyperbolic Universe. This means $Ω_k>0$ I mean If $k$ negative $Ω_k$ must be positive isn't it ? I am confused here.
In cosmology calculator says $Ω_k=1-Ω_m-Ω_Λ$

(Its look like hyperbolic universe I am trying to understand)

ThanksΩ

 

[Chalnoth]

Yes, $\Omega_k$ has a sign opposite to $k$. Specifically, $\Omega_k = -k/H_0^2$ (note: this assumes the we're using the convention where $a(now) = 1$. For the convention where $k = {-1, 0, 1}$, $\Omega_k = -k/H_0^2a^2(now)$).

 

[Quarlep]

Thanks

 

[Quarlep]

I made a mistake soory