# Friedmann model 'unbounded'

1. May 7, 2007

### da_willem

Does the metric of the Friedmann model yield an unbouded universe? I mean, are the geodesics of this metric closed?

2. May 8, 2007

### Chronos

No. Both bounded and unbounded versions are permitted.

3. May 8, 2007

### pervect

Staff Emeritus
If you do a websearch, you should find mention of a "ciritical density" for the Freidmann universes. If the density is above the ciritical value, the universe is closed. Above the critical value, it's open. Interestingly enough, it appears that the universe is right at the critical value. This is currently explained as a result of inflation. Look up "flatness oldness problem" for more (advanced) detail.

4. May 8, 2007

### marcus

you may want to qualify that. It is currently controversial to claim that Omega is exactly equal to 1.

there are attractive scenarios that make it plausible that Omega = 1.
but lately when they actually measure they often get errorbars all on the positive side of 1.

the practice of assuming Omega = 1 has come under fire from professional cosmologists.

about all one can say, without getting into controversial territory, is that the universe is NEAR the critical value----i.e. spatially NEARLY flat.
(In fact that was the conclusion of David Spergel's team when they presented the results of the WMAP third year data last year)

5. May 8, 2007

### marcus

As long as people say the universe is at or NEAR critical density, and that it is nearly spatially flat, that seems fine. And the data is certainly CONSISTENT with flat.
But AFAIK some qualification is needed to avoid giving the impression that it is known to be EXACTLY of critical density (even though some inflation scenarios make that plausible).

In case anyone is not aware of the background on this there was a paper
by Bruce Bassett et al that appeared recently, citing recent paper(s) by Ned Wright along similar lines.

Bruce Bassett and Ned Wright are both raising the warning flag and saying that one should NOT automatically assume Omega = 1, especially when studying dark energy, BECAUSE IT INTRODUCES SYSTEMATIC ERROR IN THE ANALYSIS. In other words, it is a bad idea to assume more than we actually know from empirical observation, in this case.

Bruce Basset et al
http://arxiv.org/abs/astro-ph/0702670
Dynamical Dark Energy or Simply Cosmic Curvature?
Chris Clarkson, Marina Cortes, Bruce A. Bassett
5 pages, 1 figure
"We show that the assumption of a flat universe induces critically large errors in reconstructing the dark energy equation of state at z>~0.9 even if the true cosmic curvature is very small, O(1%) or less. The spuriously reconstructed w(z) shows a range of unusual behaviour, including crossing of the phantom divide and mimicking of standard tracking quintessence models. For 1% curvature and LCDM, the error in w grows rapidly above z~0.9 reaching (50%,100%) by redshifts of (2.5,2.9) respectively, due to the long cosmological lever arm. Interestingly, the w(z) reconstructed from distance data and Hubble rate measurements have opposite trends due to the asymmetric influence of the curved geodesics. These results show that including curvature as a free parameter is imperative in any future analyses attempting to pin down the dynamics of dark energy, especially at moderate or high redshifts."

Bassett cites Dunkley et al.
Joanna Dunkley et al
http://arxiv.org/abs/astro-ph/0507473
Measuring the geometry of the Universe in the presence of isocurvature modes
J. Dunkley, M. Bucher, P. G. Ferreira, K. Moodley, C. Skordis
4 pages, 5 figs.
Phys.Rev.Lett. 95 (2005) 261303

"The Cosmic Microwave Background (CMB) anisotropy constrains the geometry of the Universe because the positions of the acoustic peaks of the angular power spectrum depend strongly on the curvature of underlying three-dimensional space. In this Letter we exploit current observations to determine the spatial geometry of the Universe in the presence of isocurvature modes. Previous analyses have always assumed that the cosmological perturbations were initially adiabatic. A priori one might expect that allowing additional isocurvature modes would substantially degrade the constraints on the curvature of the Universe. We find, however, that if one considers additional data sets, the geometry remains well constrained. When the most general isocurvature perturbation is allowed, the CMB alone can only poorly constrain the geometry to Omega_0=1.6+-0.3. Including large-scale structure (LSS) data one obtains Omega_0=1.07+-0.03, and Omega_0=1.06+-0.02 when supplemented by the Hubble Space Telescope (HST) Key Project determination of H_0 and SNIa data."

The point is not whether you like flat or don't like flat. The point is we don't know and ASSUMING FLAT INTRODUCES ERRORS.. Assuming flat encourages circular reasoning (according to Ned Wright) and makes what you do unreliable. This is how Basset et al argue, and they cite Ned Wright too:
===quote Basset===
However, we will show that ignoring Omega_k induces errors in the reconstructed dark energy equation of state, w(z), that grow very rapidly with redshift and dominate the w(z) error budget at redshifts (z > 0.9) even if Omega_k is very small. The aim of this paper is to argue that future studies of dark energy, and in particular, of observational data, should include Omega_k as a parameter to be fitted alongside the w(z) parameters.

Looking back, this conclusion should not be unexpected. Firstly the case for flatness at the sub-percent level is not yet compelling: a general CDM analysis [13 the Dunkley paper], allowing for general correlated adiabatic and isocurvature perturbations, found that WMAP, together with largescale structure and HST Hubble constant constraints, yields
Omega_k = −0.06 ± 0.02.
We will show that significantly smaller values of Omega_k lead to large effects at redshifts z ~ 0.9 well within reach of the next generation of surveys.

Secondly, Wright (e.g.[14]) has petitioned hard against the circular logic that one can prove the joint statement (Omega_k = 0,w = −1) by simply proving the two conditional statements (Omega_k = 0 given that w = −1) and (w = −1 given that Omega_k = 0). ...

Given that the constraints on Omega_k evaporate precisely when w deviates most strongly from a cosmological constant, it is clearly inconsistent to assume Omega_k = 0 when deriving constraints on dynamical dark energy...
===endquote===

6. May 8, 2007

### Wallace

The issue is one of model selection though. Inflation is a physical model that makes a prediction that $$\Omega = 1$$. Due to observational uncertainties however, we can never say with infinite certainty that it is measured to be precisely unity.

What we can do is note that this, along with other predictions from inflation (such as Gaussinity (sp?) ) have been observed and therefore it is likely (give current data and theoretical understanding) that inflation occurred and the Universe is flat.

The way cosmology is parametrised makes it easy to forgot that there must be physics going on and the point of finding the best fit parameters is not to find the best fit parameters in a continuous space but to use these fits to discriminate between discreet models.

At heart this is really the problem with dark energy. Using w(a) as a catch all parameter for dark energy lets us find all kinds of constraints depending how we parameterize the w(a) and try all kinds of assumptions about this like perturbations and sound speed. The root problem though is that since we lack a solid set of micro-physical models that predict discreet values for the w(a) parameters all we are doing is parameter fitting but not physical model selection. Say we set $$w(a) = w_0 + w_a(1-a)$$ which is a common parametrisation. If we do some advanced observational campaign that tells us that $$w_0=-0.98[tex] and [tex]w_a=-0.02$$ or something like that, what, if anything, have we really learned about dark energy or cosmology generally, if we don't have a set of discreet models to judge between based on these findings?

The one model selection criteria that we can do is to ask is w=-1 for all time, which is a cosmological constant or vacuum energy (since they are identical as far as GR is concerned). This at least is a definite physical theory (albeit with some problems) that we can test against. We can then ask the question of which model, i.e. w=-1 or w/ne-1 is most supported by the data.

Incidentally on current data overwhelmingly the answer is that w=-1, but the data isn't great. In 5 years things could be very different. The lack of solid micro-physical theories for the infinite range of w(a) values is an issue though, there's really no getting around that.

Last edited: May 8, 2007
7. May 8, 2007

### marcus

Here is some more stuff on this.
Among working professional cosmologists there is a move to abandon the automatice Omega = 1 assumption and allow positive curvature.

http://arxiv.org/abs/astro-ph/0603449
Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
Authors: D. N. Spergel, R. Bean, O. Doré, M. R. Nolta, C. L. Bennett, J. Dunkley, G. Hinshaw, N. Jarosik, E. Komatsu, L. Page, H. V. Peiris, L. Verde, M. Halpern, R. S. Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland, E. Wollack, E. L. Wright
Comments: 91 pgs, 28 figs. Accepted version of the 3-year paper as posted to this http URL in January 2007

NOTE THAT JOANNA DUNKLEY AND NED WRIGHT ARE co-authors with David Spergel of this key WMAP3 results paper. The people involved here are "principal investigator level" WMAP/cosmologist types.

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
Authors: Edward L. Wright (UCLA)
Comments: 17 pages Latex with 8 Postscript figure files. One new Table, one new Figure, and several new references added. Submitted to the ApJ

This paper by Wright attacks the usual practice of assuming flat, although he happily concedes that the data is consistent with flat LCDM.
That is, flat has not been ruled out yet, and neither has curved been.

The WMAP FIRST YEAR data had a confidence interval that inclued 1 as an endpoint. WMAP1 said 1.02 +/- 0.02

The WMAP3 data was the first time I saw a major paper (Spergel et al) with a confidence interval for Omega that did NOT include 1, at least as an endpoint. There was a confidence interval that was all on the positive side of 1.

8. May 8, 2007

### marcus

I'm unsure about a couple of things.
I'm not sure inflation happened, now that we don't need it to solve the horizon problem (QG bounce seems to take care of different parts of the sky having roughly the same CMB temp).

Magueijo and Singh just posted a paper undermining another support for inflation scenarios. So inflation may just be a passing fad.

What counts is what they measure and lately I see errorbars on the positive side of 1 which do not contain 1. So I think things are in flux.

But the main thing I am unsure about, Wallace, is how do you get that inflation (all inflation scenarios) PREDICT that Omega must be EXACTLY equal to 1.

I was under the impression that inflation could provide one possible explanation for why Omega is either one or very very very near one.
How does it work that inflation predicts that Omega has to be exactly 1?

9. May 9, 2007

### Wallace

You can find this in any text on cosmology, but essentially this occurs because curvature goes as 1/R^2 whereas an 'inflaton' mechanism goes as ~1/R. Since inflation increases R by many times, any curvature term is beaten down to an insignificant level.

The mystery is why the inflaton mechanism turned off at some point?

I would recommend having a read of some better source than me though, John Peacock has a good chapter on this in 'Cosmological Physics'.

BTW, could you give a brief description of how 'QG bounce' solves the horizon problem? I know essentially nothing about this theory so a from the basic description would be nice.

10. May 9, 2007

### marcus

Yes! this is how I remembered it, for example as explained in Lineweaver 2003 article!
Curvature is beaten down to an "insignificant level" but not forced to be exactly zero.

The mechanism you are talking about does not, I think, produce an EXACT Omega = 1, only approximately equal to 1 (which I'm comfortable with.)

My question: Is inflation compatible with space being topologically S3

Since you say "insignif" which can mean a positive quantity near zero, I think you may agree after all.

Recent errorbars for Omega are often like (for instance) [1.003, 1.021].
Ned Wright's recent paper gave 1.011 as "best fit".
But these results are not confidence level 95% or 99%, they are only CL 65%, so they are still "consistent" with spatial flat infinite.

Suppose future more accurate observations increase the CL, so we have, say, [1.003, 1.021] with CL 99%. Would this rule out inflation?

Suppose Omega actually IS , say 1.01. Can this have arisen even assuming inflation occured or does it rule out inflation?

Last edited: May 9, 2007
11. May 9, 2007

### marcus

Numerous LQC papers since 2001.
LQC reproduces ordinary cosmology soon after BB, but does not experience a singularity and works smoothly back in time past where the classical breakdown occurred. What it finds is a collapsing classical spacetime.

Coherent quantum states evolve deterministically thru the ex-singularity. A semiclassical state, after collapse and bounce, reexpands to a semiclassical state.

the original paper was Bojowald (2001) "The absence of singularity in LQC"

He has been working on removing the assumptions of homog and isotropy, generalizing the result, by perturbing around an exactly solvable theory. His most recent paper on this (2007) raises questions about whether a bounce occurs in all cases.

Ashtekar's group has been doing numerical simulations of the universe passing thru the bounce, in various cases.

Ashtekar et al (2006) find that bounce happens typically when density reaches about 0.8 planck (80 percent of planck density).
Quantum corrections to gravity make it repellant instead of attractive at planck-levels of density and curvature

Magueijo and Singh (2007) cite references to the effect that the bounce gets rid of the horizon problem because the universe has plenty of time during the prior contracting phase. Plenty of time for distant sectors to communicate and arrive at equilibrium. (That's my understanding of their brief discussion of the horizon problem, before they moved on to structure formation)

Magueijo and Singh's recent paper was about getting rid of a structure formation problem----finding an alternative solution not requiring inflation.

The point about inflation is that nobody really knows how these scenarios work or what an "inflaton" is. The reasons people entertain notions of inflation is because the scenarios address certain problems. If there turn out to be alternative solutions to the problems this undermines the logical support for inflation scenarios.

Last edited: May 9, 2007
12. May 15, 2007

### Chris Hillman

Hi, da willem,

Looks like others gave you some good basic information about FRW models, but for more advanced students there is another issue: the local versus global distinction. All FRW models possess a unique family of spatial hyperslices which are everywhere orthogonal to the world lines of the matter (pressureless perfect fluid or "dust" for "matter dominated" and "radiation fluid" for "radiation dominated"). But it is perfectly possible to write down models in which these slices are each constant negative curvature but compact Riemannian three-manifolds. The easiest way to see why is to read about tilings of the hyperbolic plane and to identify edges of a tile to form a "compact quotient manifold" of H^2. This point was overlooked in classic textbooks (including MTW!) and was only emphasized rather recently by Jeffrey Weeks and Neil Cornish.

13. May 20, 2007

### Chronos

14. May 20, 2007

### da_willem

But open and closed are something different than unbounded and bounded, right?! I know that open and closed versions are possible, depending on the energy content. But I was actually wondering wether an unbounded universe is possible, i.e. with closed geodesics that end on themselves

15. May 20, 2007

### marcus

the words 'open' and 'closed' cause a lot of confusion. I am curious what you mean by them, da_willem.

I also don't understand why you say that "unbounded" means having closed geodesics.
A universe based on a spatial slice which is simply flat infinite R3 (plus a time axis) would have no closed geodesics----and it would be boundaryless.

If all you are asking is whether an universe is possible where the spatial slices are boundaryless then clearly the answer is YES.

the slices could be infinite flat R3
or they could be three-spheres S3
or they could be TOROIDAL (like a cube with opposite faces glued, analogous to Pacman world: a square with opposite edges glued)
and other things too grim to mention that Chris Hillman referred to.

I only take the first two seriously----infinite flat 3D space and the 3D sphere.
Both of them are boundaryless.

The 3D sphere alternative is spatially closed-----and you could imagine spatial geodesics which come around to the starting point. But a universe with that kind of space could still expand forever.
Just to make sure we understand each other, my idea of a spatially closed universe does not necessarily involve a big crunch. That is a separate issue. Do you agree?

16. May 20, 2007

### marcus

My impression of some fairly recent Neil Cornish papers is that he is not proposing these periodic universes as in any sense likely.
He seems to be presenting them as mathematical constructs so that he can use the data to RULE THEM OUT to whatever extent is possible.

That is, he finds a scale like, say, 70 billion LY, within which he can show periodicity does not occur. If there is periodicity, then it must therefore only happen if you go out MORE than 70 billion LY or whatever scale he finds.

This is ultimately the only way to defeat the periodic universe idea whether it is toroidal or soccerball or some exotic tiling of R3, so I see what Cornish has done and needful. Somebody had to do it.

and as more data comes in he can hopefully keep extending that 70 billion LY distance within which periodicity is proven not to occur.

I have a high opinion of Neil Cornish, in part because when I saw his website in 2004 it had a picture of his pet monkey. I hope he still has a pet monkey and that both of them are in good health.

Last edited: May 20, 2007
17. May 20, 2007

### SpaceTiger

Staff Emeritus
If I remember correctly, large-scale inhomogeneities are expected to yield a non-flat observable universe at about the 0.001% level. A universe that is flat to much higher precision than that would, I should think, be inconsistent with inflation (or result in another fine-tuning problem).

I haven't heard anyone making noise about the flatness assumption of late. It is certainly true that higher-precision measurements could yield a non-flat universe at, say, the 0.1% level and act as evidence against inflation. However, nobody I've spoken to thinks that there is any evidence for that just yet. There are only a few cosmology experiments that are precise enough to even distinguish between a flat and non-flat universe and, to my knowledge, these have all allowed the parameter to float.

I don't think you'll hear much controversy on this topic for a while yet, the measurements simply aren't precise enough.

18. May 20, 2007

### marcus

I have, as recently as February 2007:

Bruce Basset et al
http://arxiv.org/abs/astro-ph/0702670
Dynamical Dark Energy or Simply Cosmic Curvature?
Chris Clarkson, Marina Cortes, Bruce A. Bassett
5 pages, 1 figure
"We show that the assumption of a flat universe induces critically large errors in reconstructing the dark energy equation of state at z>~0.9 even if the true cosmic curvature is very small, O(1%) or less. The spuriously reconstructed w(z) shows a range of unusual behaviour, including crossing of the phantom divide and mimicking of standard tracking quintessence models. For 1% curvature and LCDM, the error in w grows rapidly above z~0.9 reaching (50%,100%) by redshifts of (2.5,2.9) respectively, due to the long cosmological lever arm. Interestingly, the w(z) reconstructed from distance data and Hubble rate measurements have opposite trends due to the asymmetric influence of the curved geodesics. These results show that including curvature as a free parameter is imperative in any future analyses attempting to pin down the dynamics of dark energy, especially at moderate or high redshifts."

to the extent that more people stop automatically assuming zero-curvature and start "including curvature as a free parameter" as Bassett advocates, I would expect the noise about it to die down. One way to resolve the problem would be for everybody to stop making that assumption.

Last edited: May 21, 2007
19. May 29, 2007

### hurk4

Nor do I.
To me, as I hesitate to say, it seems that the horizon problem and its solution (inflation) were invented as an attempt to save the idea of a beginning of the BB.
As a non professional even I myself can calculate that our observable universe compressed to Planck-density had a dimension of about 10E-12 m (taking into account 10E-87 baryons + DM + DE). This distance is by far not enough to allow communication between opposite points during Planck-time (10E-43s). The maximum possible distance with the speed of light during that time would only be 10E-34m. Indeed theories giving us plenty time before the BB don’t need a horizon problem and a solution (inflation) for this, ( IMO non existing) problem if there is plenty of time for communication see also post #11.
I ask myself what is wrong with my reasoning where as nowadays it seems to me that inflation is almost accepted by a majority of professional cosmologists as part of a standard-model.

Kind regards hurk4

20. May 29, 2007

### jal

When man discovered the soap bubbles the "inflaton" committed hari-kari.
jal