# Universe density types equation

1. Jun 7, 2015

### Mysteryciel

I was searching a cosmology calculator and In one calculator I saw this equation.
$1=Ω_k+Ω_m+Ω_Λ$
is that true true for all type universe's (open,flat,closed) ?
and is here $Ω_k={-k\over H^2_0 /a^2_0}$ isnt it.
(And If you can give me an article about this issue I will be very happy,I need exact proof)
Thanks

Last edited: Jun 7, 2015
2. Jun 7, 2015

### bapowell

It's true for globally isotropic universes with matter and cosmological constant. For proof, consult any introductory cosmology text.

3. Jun 7, 2015

### Chalnoth

That equation is part of the definition of the $\Omega$ parameters. It's true as long as there is no other component of the energy density (e.g. it's true as long as the radiation density is small enough to be ignored). If you have some other component, such as $\Omega_r$ for radiation, then you have to add that to the equation:

$$1 = \Omega_k + \Omega_m + \Omega_\Lambda + \Omega_r$$

And yes, that equation for $\Omega_k$ is correct.

4. Jun 7, 2015

### mathman

Sum = 1 implies a flat universe. > 1 implies closed, < 1 implies open.

5. Jun 7, 2015

### Garth

Not when $\Omega_k$ is included, as Chalnoth said the sum is unity; $\Omega_k$ absorbs the k term.

$\Omega_k$ > 0 implies open, $\Omega_k$ = 0 implies flat and $\Omega_k$< 0 implies closed, (see the OP definition of $\Omega_k$).

Garth