mm k. So. As lots of other students are at the moment, I am currently preparing for a presentation. My chosen topic: alternatives to the FRLW model.

What I'd appreciate from you guys is simple; I've been coming here long enough to notice that everyone has their own little deviations and interests from the standard - I'd like anyone willing to write a short post about a model that attracts them and why.

Reason being that whilst I have a few specific models i've grown to appreciate over the years, I'm not quite a professional yet. Plus, I think it's more than a little interesting - and very worthwhile - for people to not limit themselves to the 'standard' at the time. If we've learned anything from our past in science it should at least be to drop the common arrogance (perhaps too strong a word) and appreciate that other explanations may exist.

I'm sure most people will agree that in undergraduate level texts and courses, Astronomy and physics (obviously for the reason that it's most important to learn what's accepted at the time..) are limited to just their standard models, which, whilst understandable to a degree, is frustrating in the sense that most lecture courses can't take the time to even mention that there might be other ways out there!

write away!

thanks

-ftj

marcus
Gold Member
Dearly Missed
My chosen topic: alternatives to the FRLW model. ...

write away!

thanks

-ftj
Faster, isn't the standard cosmology model called the "LambdaCDM"?

I'm not sure what actual model cosmology is associated with the names Friedmann, Robertson, Lemaitre, Walker (FRLW).
Friedmann gave us the Friedmann equation that all cosmologists use.
but you can solve that various ways to get various scalefactor functions under various assumptions
Robertson-Walker gave us a particular form of the metric, that depends on the scalefactor.

But you can write a lot of different cosmological models using the Friedmann equations and the R-W metric.

I think another name for "LambdaCDM" is the "concordance" model, because it is the one that, since the 1998 revolution, most working cosmologists have converged on and taken to assuming in their work.

That is the one where the D.E. equation of state, w, is assumed to be equal to minus one. And where spatial flatness is assumed.

marcus, yep you're quite right to point that out. I suppose I should have been more explicit and noting I mean the standard extension/use of Friedmann's equations /RLW work (As above, LambaCDM). By different or alternative, I'm suggesting to things like the inflationary period etc. I'm even interested in perhaps some more whacky proposals, however browsing something like archive isn't often a good idea as one can generally find work claiming to support almost any theory one can imagine.... Your input is, as always, appreciated.

Last edited:
marcus
Gold Member
Dearly Missed
thanks! I'm glad my comment turned out useful to you.
You asked to know our favorites so I will respond to that.

Normally I try to reserve judgment and take a wait-and-see, but I have to confess to being partial to a slight variation of the LambdaCDM, of which the standard version is spatial flat and infinite.

In contrast to that, I'm partial to the version which Ned Wright called the "best fit" universe, in his recent (January 2006) paper----which is nearly flat spatially. Instead of Omega exactly = 1, it has Omega = 1.011. It looks almost flat/infinite but it has a slight positive curvature so that at any given time, space is topologically a 3-sphere-----the 3D analog of a sphere.

in case anyone is curious, Ned Wrights recent paper is:
http://arxiv.org/abs/astro-ph/0701584
Constraints on Dark Energy from Supernovae, Gamma Ray Bursts, Acoustic Oscillations, Nucleosynthesis and Large Scale Structure and the Hubble constant
that is where he talks about the 'best fit' cosmology parameters, as distinct from the flat/infinite concordance version.

Last edited:
marcus
Gold Member
Dearly Missed
...Normally I try to reserve judgment and take a wait-and-see, but ..., I'm partial to the version which Ned Wright called the "best fit" universe, in his recent (January 2006) paper----which is nearly flat spatially. Instead of Omega exactly = 1, it has Omega = 1.011. It looks almost flat/infinite but it has a slight positive curvature so that at any given time, space is topologically a 3-sphere-----the 3D analog of a sphere.

in case anyone is curious, Ned Wrights recent paper is:
http://arxiv.org/abs/astro-ph/0701584
Constraints on Dark Energy from Supernovae, Gamma Ray Bursts, Acoustic Oscillations, Nucleosynthesis and Large Scale Structure and the Hubble constant
that is where he talks about the 'best fit' cosmology parameters, as distinct from the flat/infinite concordance version.
hey FASTERJOAO! If you are still around I'd like to ask you if you like to calculate stuff about cosmology and if so, would you like to calculate the RADIUS OF CURVATURE of the bumpy threesphere which is our space, given the BEST FIT OMEGA of 1.011

HERE IS HOW TO CALCULATE THE 'best fit' RADIUS.

I am using George Smoot (Nobel 2006) lecture notes. He has sometimes taught Physics 139 at Berkeley, an advance undergrad course in Special and General Relativity with some Cosmology. You can get his notes if you google "smoot notes geometry universe".

The notes give a simple formula for the radius of curvature.
You just take the HUBBLE LENGTH which we usually say is 13.8 gly, and you divide it by a factor which is sqrt(Omega - 1). that is all.

If we use the best fit Omega of 1.011, then Omega - 1 is 0.011
and the square root is 0.105

and if you divide 13.8 billion lightyears by 0.105 you get

130 BILLION LIGHT YEARS.

So that is the 'best fit' model's radius of curvature. Fasterjoao let me know if you see any mistake.

That's quite a number, and could obviously be an interesting result. Here's to the wait and see group George Smoot notes were good, i'm looking over them just now.

I have, however, encountered something I'd like someone to take a look at - on:

Considering the equation of motion after substituting the respective Omega's in, then taking a spatially flat cosmology (so Omega(k)=0) - I'm assuming that if that formula was integrated from 0 to t(0) then we'd get a more accurate version of: H0=2/(3t0)

Could someone kindly post a worked integral of this? I've tried by changing the integration variable to a from t0 equals the integral of one from 0 to t0 dt, which seems to work but I'm not getting there. thanks.

Last edited by a moderator:
hellfire
The integral for the look-back time or the age of the universe for a generic cosmological model cannot be solved analytically.

Can I not then integrate:
$$adot^2=H_0^2[\frac{\Omega_0}{a}+(1-\Omega_0)a^2]$$

with a change of variable to a in $$t_0=\int dt$$

Last edited:
hellfire
For the general case, $\Omega_0$ is the total energy density today and it therefore contains the contributions of different components of the energy density in the universe. These different components do not scale in the same way with $a$. In general, you have:

$$\Omega_k = 1 - \Omega = 1 - \Omega_m + \Omega_r + \Omega_{\Lambda}$$

The matter density scales as $1/a^3$, the radiation density as $1/a^4$. You can check by your own that starting from the Friedmann equation:

$$\left( \frac{\dot a}{a} \right)^2 = \frac{8 \pi G}{3 c^2} (\rho_m + \rho_r) - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3}$$

You will arrive at the following equation (subscript 0 for the values today):

$$(\dot a)^2 = H^2_0 \left( \Omega_{k, 0} + \frac{\Omega_{m,0} }{a} + \frac{\Omega_{r,0} }{a^2}+ \Omega_{\Lambda,0} a^2 \right)$$

For a flat space with only matter content, $\Omega_{k, 0} = \Omega_{r, 0} = \Omega_{\Lambda, 0} = 0$ and $\Omega_{m, 0} = 1$ the formula reduces to:

$$(\dot a)^2 = \frac{H^2_0}{a}$$

That you can integrate from a = 0 to a = 1 to get the result you wrote in your previous post for the age of the universe.

Last edited: