Calculating Luminosity Distance in an Empty Cosmology

[jc09]

If had empty cosmology?

What would happen if we had no matter or radiation or dark energyand only had curvature?

How would this effect luminosity distance?

From the friedman equations I get that H²=-k/R². Is this right?

 

[bcrowell]

http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.2

See subsection 8.2.7, "The vacuum-dominated solution."

If you want a simple cosmology with only one component (radiation, dust, or vacuum energy), then this one is actually the one that's the best approximation to the universe we live in right now.

From the friedman equations I get that H²=-k/R². Is this right?

Could you define your notation?

 

[jc09]

H is the Hubble parameter, k is the curvature constant and R is the scale factor.

 

[jc09]

So if I want to get luminosity distance as a function of redshift how would I do that?

 

[bcrowell]

I get $H=R'/R=\sqrt{\Lambda/3}$. See the link in #2.

 

[bcrowell]

So if I want to get luminosity distance as a function of redshift how would I do that?

http://xxx.lanl.gov/abs/gr-qc/9712019

See p. 236.

 

[jc09]

I get $H=R'/R=\sqrt{\Lambda/3}$. See the link in #2.

If Lambda=0 though ??

 

[bcrowell]

If Lambda=0 though ??

http://en.wikipedia.org/wiki/Milne_model

 

[Chalnoth]

What would happen if we had no matter or radiation or dark energyand only had curvature?

How would this effect luminosity distance?

From the friedman equations I get that H²=-k/R². Is this right?

Understand that if you have no matter, radiation, or dark energy, the terms 'k', 'H', and 'R' become meaningless, and you're left with just a reparameterization of flat Minkowski space-time (the space-time described by Special Relativity). This is the Milne model (linked above).

 

[yenchin]

To add on Chalnoth's post, if there is no matter and no radiation, luminosity of what can one measure?

 

[jc09]

What if K the curvature did not equal zero. What if it could be equal to +1 or -1 does this make a difference. Then I get k=-H²R² where we are in units of c equals one. Can we then find a luminosity distance as a function of z even though there is empty cosmology

 

[Chalnoth]

What if K the curvature did not equal zero.

It doesn't make a difference. Either way, it's just a reparameterization of flat Minkowski space-time.

The crucial difference between k and the space-time curvature is that k only includes the spatial components. So in essence what you're doing by setting k to be nonzero is adding some spatial curvature (represented by k) and some curvature in time (represented by H), such that the total space-time curvature remains zero.

Can we then find a luminosity distance as a function of z even though there is empty cosmology

Well, you certainly can. The problem is that it's quite arbitrary, as without any matter there is nothing to sort of "nail down" the cosmology to anything physical. So you can define a wide variety of luminosity distances as a function of z, and they'd all be equally valid as one another (and just as uninformative).

 

[jc09]

Ok I think I understand so I could write luminosity as a function of z but it won't really mean anything to me. So if I was to do that would it be the normal definiton for luminosity:
d_LH₀=z+1/2(1-q₀)z²+...?

 

[Chalnoth]

Ok I think I understand so I could write luminosity as a function of z but it won't really mean anything to me. So if I was to do that would it be the normal definiton for luminosity:
d_LH₀=z+1/2(1-q₀)z²+...?

This expansion isn't often used. Rather we usually use the luminosity distance defined here:
http://arxiv.org/abs/astroph/9905116

But yes, you could certainly make use of the normal luminosity distance in FRW. It's just that it doesn't mean anything because it's not tied to any actual expansion (since there's nothing in the universe to expand, the expansion rate itself is whatever you define it to be).