LCDM Model best value of $Ω_k$

1. Aug 29, 2015

RyanH42

Whats the best value of $Ω_k$ (Considering planck 2015 results).I asked best value cause If you think our universe can be also hyperbolic so $Ω_k$ can be different then 1.
I remember the relationship between $Ω_k$ and k is
$Ω_k=-k/a(t)h^2$ or some sort of something like that.k is zero I guess then How can
$Ω_k$ can be 1

Thank you

2. Aug 29, 2015

Bandersnatch

$\Omega_k$ is not 1 but 0. It's the total density parameter $\Omega_{total}$ which is equal to 1, not the curvature density parameter.
Planck results show $\Omega_k$ to be 0 with accuracy of +/- 0.005
http://arxiv.org/abs/1502.01589
Since error bars do exits, one can't rule out a hyperbolic nor closed universe, but it's hard not to notice that it's pretty damn flat nonetheless.
Also, the relationship is
$\Omega_k=-\frac{kc}{a^2H^2}$

3. Aug 29, 2015

RyanH42

4. Aug 29, 2015

Bandersnatch

Yeah, sorry - Latex went bonkers there for a while. Should be fine now.

5. Aug 29, 2015

RyanH42

$Ω_T=Ω_M+Ω_Λ+Ω_k$ (T=total),(M here matter BM+DM),(Λ Dark energy)

$Ω_T$ is always 1.It not depend universe curvature (hyperbolic,flat,sphere),or time or anything else.Am I right ?

6. Aug 29, 2015

RyanH42

I see somewhere (I dont remember right know,sorry about that) the error in ΩTotal=1.00±0.02 Is that right ?

7. Aug 29, 2015

Bandersnatch

No, $\Omega_T$ is the total density parameter with contributions from matter, radiation and dark energy densities:
$\Omega_T=\Omega_m+\Omega_r+\Omega_{\Lambda}$
This density is related to the curvature by:
$\Omega_k=1-\Omega_T$
That is, for curvature to be 0, $\Omega_T$ must be 1.

8. Aug 29, 2015

Bandersnatch

This relates to your other thread and my answers there. The measurements from Planck 2015 are consistent with total density being 1.000 +/- 0.005. The The number you refer to is most likely the WMAP 1-year (i.e., earliest, least accurate) result, which was 1.02 +/- 0.02.
You can find a nice breakdown of WMAP results here:
https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe#Measurements_and_discoveries

9. Aug 29, 2015

RyanH42

That site is really good.I see it.

The nine year result says $Ω_k≅-0.037$ (best fit WMAP only) so $Ω_T≅1.037$ which theres error signs of course.It means $Ω_T$ can be 1.020.But theres another number says $Ω_k≅-0.0027$ (WMAP+eCMB+BAO+$H_0$ which here $Ω_T≅1.0027$ which not fits the other result.Here which is the best result.You say $Ω_k≅0.005$ I know that but I want to be sure.Thank you

10. Aug 29, 2015

Bandersnatch

There is no one single precise value, and there'll never be one because that's not how measurements work. You always have uncertainties, and that's the key.

From WMAP-only measurements you've got the 0.037 result with error bars approx. +/- 0.044. That means it could be anything from -0.007 to 0.081 and you've not way of saying which precise value it actually is.
You then combine the result with data from other experiments, each of those having their own range of predicted values.
This lets you narrow the range to -0.0027 +/- 0.0039, which means the actual value can be anything from -0.0066 to 0.0012. That is a much better result. An order of magnitude improvement in accuracy.

Then you've got the other probe: PLANCK. It makes its own measurements, and after combining with all other available data, it ends up with 0.000 +/- 0.005.

All those results are compatible with each other, since the ranges overlap. For example, all three contain the 0 value for the curvature.