I What Values Can Ωk Take in the Equation Ωm+Ωr+Ωk=1?

[Arman777]

Let's suppose we have an equation,
$$Ω_m+Ω_r+Ω_k=1$$
In this equation what's the values that $Ω_k$ can take ?
Only 1,0 or -1 ?

Also this equation is true for only "now" or any time in the universe ?

 

[PeroK]

Let's suppose we have an equation,
$$Ω_m+Ω_r+Ω_k=1$$
In this equation what's the values that $Ω_k$ can take ?
Only 1,0 or -1 ?

Also this equation is true for only "now" or any time in the universe ?

Assuming the omegas are positive, then $0 \le \Omega_k \le 1$

If you allow negative values, then $\Omega_k$ can be any real number.

That was the mathematical answer. Rereading your question, I guess you want a cosmological answer. I'll leave that to others.

 

[Arman777]

Assuming the omegas are positive, then $0 \le \Omega_k \le 1$

If you allow negative values, then $\Omega_k$ can be any real number.

For flat universe we can take $Ω_k=0$. In example for positive curvature universe then what could be the value of $Ω_k$ ?
A negative value ?

Cause $Ω_k=-\frac {κ} {a^2H^2}$

Something seems wrong to me. Either this equation only hold for $Ω_k=0$.

Oh okay I understand

 

[Arman777]

For a negative value of $Ω_k$ corresponds to $Ω>1$ which its positively curved universe
For $Ω_k=0$ corresponds to $Ω=1$ which flat universe
For a postive value of $Ω_k$ corresponds to $Ω<1$ which its negatively curved universe

where $Ω=Ω_m+Ω_r$

 

[kimbyd]

Let's suppose we have an equation,
$$Ω_m+Ω_r+Ω_k=1$$
In this equation what's the values that $Ω_k$ can take ?
Only 1,0 or -1 ?

Also this equation is true for only "now" or any time in the universe ?

$\Omega_k$ can be any real number that makes the equation above true. Because $\Omega_m$ and $\Omega_r$ must be positive, $\Omega_k$ cannot be greater than 1. But it can be as negative as you like.

For a negative value of $Ω_k$ corresponds to $Ω>1$ which its positively curved universe
For $Ω_k=0$ corresponds to $Ω=1$ which flat universe
For a postive value of $Ω_k$ corresponds to $Ω<1$ which its negatively curved universe

where $Ω=Ω_m+Ω_r$

This is accurate.

 

[Arman777]

$\Omega_k$ can be any real number that makes the equation above true. Because $\Omega_m$ and $\Omega_r$ must be positive, $\Omega_k$ cannot be greater than 1. But it can be as negative as you like.This is accurate.

Thanks, its more clear now