The discussion revolves around the values that the variable ##Ω_k## can take in the equation $$Ω_m+Ω_r+Ω_k=1$$. Participants explore the implications of this equation in both mathematical and cosmological contexts, questioning whether it holds true at all times or only at the present moment.
Participants generally agree that ##Ω_k## can take a range of values depending on the conditions set for ##Ω_m## and ##Ω_r##, but there is no consensus on the implications of these values in different cosmological scenarios. Multiple competing views remain regarding the interpretation of the equation and its applicability over time.
There are limitations regarding the assumptions made about the positivity of ##Ω_m## and ##Ω_r##, as well as the dependence on the definitions of curvature in cosmology. The discussion does not resolve the implications of these assumptions.
Arman777 said: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 ?
For flat universe we can take ##Ω_k=0##. In example for positive curvature universe then what could be the value of ##Ω_k## ?PeroK said: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.
##\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.Arman777 said: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 ?
This is accurate.Arman777 said: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 positive value of ##Ω_k## corresponds to ##Ω<1## which its negatively curved universe
where ##Ω=Ω_m+Ω_r##
Thanks, its more clear nowkimbyd said:##\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.