Jorrie,

Again, great post. I was reading about de Sitter space recently:

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

Is this theory somehow related to de Sitter space. In other words, is the space around the inhomogeneities like de Sitter space and away from them in the voids like Anti de Sitter space?

A link to explain why I'm asking about de Sitter and Anti de Sitter spaces.

http://www.bourbaphy.fr/moschella.pdf

Hi Roger,

Nope, Wiltshire's theory is trying to get rid of dark energy (and hence the of cosmological constant) and hence is not de Sitter-like in any way. One must remember that the voids are not devoid of matter, but just less much less dense (matter-wise), possibly of negative local curvature, but not anti-de Sitter-like.

I think in terms of a de Sitter universe as positive cosmological constant (positive energy density/negative pressure, 'anti-gravity'), with matter and radiation negligible. This is postulated for the inflation scenario and also for the long term, accelerated expansion future, when matter and radiation have vanishing density.

An anti-de Sitter universe would be one with a negative cosmological constant (negative energy density/positive pressure, normal gravity), which is not in line with what we observe on the large scale. But, I'm not sure if it may be possible on the small scales that you are more interested in...

-J

Thanks for the response. For a novice, tell me, what is the difference between a "negative local curvature" and a "anti de Sitter space". Mathematically aren't they similar, or is there major differences?

In this case I'm thinking of the large scale (walls and voids of the universe). When I'm saying "Local Inhomogeneities" I mean local as in "100s of millions of Light Years". I'm trying to understand if the positively curved space near the walls would act like de Sitter space or if it would act differently.

Hi Roger, you asked: "what is the difference between a "negative local curvature" and a "anti de Sitter space". Mathematically aren't they similar, or is there major differences?"

AFAIK, they are opposites: anti-de Sitter (adS) space in the cosmic voids would have given them locally positive curvature. We rather think the voids are approaching normal de Sitter (dS) space, because they have negative curvature (expanding too fast for the local energy density). Actually, the whole shebang should be approaching dS space, unless of course, Wiltshire is right and the cosmological constant is near zero.

If I simply reverse the sign of vacuum energy density (Omega_v = -0.73) in my cosmic balloon model spreadsheet, the whole cosmos would eventually collapse, just like if the matter energy density where above critical (Omega_m ~ 3, with Omega_v = 0). This would also be the case in voids with adS space, I guess, which is not what one would expect to happen.

Does it help?

-J

Oh, I see, I had it backwards. Ok. So "negative local curvature" approaches de Sitter Space and "postive local curvature" approaches anti de Sitter Space, is that right?

Also, I noticed you used the word "approaching". Can you explain why you said approaching? (I believe it is the correct term but I think your answer will give me better insights). In other words, what is it about "negative local curvature" that doesn't quite make it de Sitter space. Has it something to do with the curvature?

Hi Roger, you wrote: "So "negative local curvature" approaches de Sitter Space and "postive local curvature" approaches anti de Sitter Space, is that right?"

Yep, that's the way a I think cosmologist would view it. I also spotted the apparent contradiction that S referred to in reply #7, but it is not quite the cosmological view. I'm not sure if that "n-dimensional de Sitter space , denoted dS_n, is the Lorentzian manifold analog of an n-sphere (with its canonical Riemannian metric); it is maximally symmetric, has constant positive curvature, ..." refers to some other situation. It is all a bit confusing, because strictly speaking, a de Sitter universe (or de Sitter cosmic model) refers to a spatially flat cosmos with Omega_v = 1 and Omega_m = 0.

It is also true that if vacuum energy Omega_v > 1, then the resulting cosmos must have positive curvature, but may keep on expanding even faster. It depends a bit on how much matter there is, but that is long story.⁽¹⁾ As it stands, we do not observe that situation, so it is today thought that the present average cosmos is approximately flat, made up of 73% vacuum energy and 27% total matter energy. In the clusters, the vacuum energy is still 73%, but the matter density there is higher than 27% and in the voids it is less. This means that the have clusters have positive spatial curvature and the voids have negative spatial curvature. I called the latter dS space, perhaps wrongly - the 'voids' are not empty of matter. But, they are surely not adS space...

For these reasons, my view is that we must be careful in attaching terms like de Sitter space to any part of the cosmos. The modern cosmological definitions have moved away from that and simply speak of energy densities, mostly as fractions of the critical density that would make make the spatial curvature zero. And zero spatial curvature simply means an expansion rate which makes kinetic energy of expansion plus the energies of mass, radiation and the vacuum equal zero. It has no longer anything to so with whether the cosmos will eventually collapse or expand forever.

"Can you explain why you said approaching?"

I used the words "approaching dS space" a little loosely, intending to say that according to standard ΛCDM cosmology, our cosmos will approach de Sitter space asymptotically in the distant future. Vacuum energy density remains constant, while other forms of energy decrease in density as space expands; hence, practically speaking there will only be vacuum energy density left. It will be negative curvature though, because Omega_total < 1, if vacuum energy density does not change with time

-J

(1) S may remember a discussion we had on this somewhere in one of the the cosmic balloon threads. I recall the numbers Ω_m = 0.5, Ω_v = 2.0, which produced a static cosmos some 25 Gyr after expansion started, albeit probably an unstable equilibrium.

Thanks for the explanation Jorrie.

Jorrie,

I've sort of dragged this conversation off-topic from your original post, so I'm sorry for that, but I'm at the moment interested in the de Sitter universe (or space, whatever we should call it).

Am I correct (or close to correct) when I say that in the de Sitter Universe, the curvature is directly related to the cosmological constant. What I mean is, am I correct when I say that the larger the curvature, the larger the cosmological constant?

Also when I say that the curvature is positive or negative it simply changes the sign of the cosmological constant, is that correct?

I think I've got it right in my comment #11 based on the link provided by StandardsGuy (http://en.wikipedia.org/wiki/De_Sitter_space) where it says:

De Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:

This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by

The scalar curvature of de Sitter space is given by

New Question

I guess my next question is, what is the scalar curvature of the Schwarzschild metric?Is it 48M²/r⁶ or is that something else?

The answer to my last question (in comment #14) I think is what I thought it was based on this link:

http://www.aensionline.com/jasr/jasr/2008/16-31.pdf

See Page 31

Switching Back to De Sitter:

What is n in the equation above?

This blog is magic. Everytime I can't figure something out I post a question here and then immediately afterwards I find an answer. In ancer to my last question about n I believe it means the number of dimensions. so

R=4Λ

I Wrote:"I guess my next question is, what is the scalar curvature of the Schwarzschild metric?Is it 48M²/r⁶ or is that something else?"

That's incorrect. The scalar above is the Kestschman Scalar, not the Ricci Scalar. The Ricci Scalar for the Schwarzchild Metric is zero.

Hi Roger,

You are right. The Schwarzschild curvature is a complex beast, a little off-topic for this thread, so maybe we should open a new one on that.

The issues of de Sitter space and de Sitter cosmic models are however relevant to understanding the cosmic model that Wiltshire is pushing. If he refutes the cosmological constant with his timescape cosmology, then the de Sitter model is also dead. On the other hand, it may well be a mixture of lambda and time differences that cause the observed accelerated expansion. In such a case the long term future may still be tending towards a de Sitter model.

-J

Sorry about the off topic stuff. I'll look for a more appropriate post of yours and migrate the conversation there.

It just seems to me that although the Wiltshire explains our observation of Dark Energy as a result of relativistic effects of the inhomogeneous matter and voids, it doesn't explain why we have such vast voids and matter "walls" in the first place. I conjecture that inhomogeneous spatial expansion might have created a sort of feedback loop during the inflationary period and encouraged these structures to form (encouraged clustering and voids) if the inhomogeneous expansion were dependent (even only slightly) on energy density. Given the high rate of expansion back then, the inhomogeneities in expansion, not measurable today may have been more pronounced in the inflationary expansion.

Back to the Wilshire Model

Have you seen the following article?:

http://www.space.com/8386-huge-chunk-universe-missing-matter.html

I wonder how much of the difference of Wiltshire's effect and the amount of dark energy we actually see is a result of underestimating the amount of matter in the walls of galaxies?

Hi Roger,

Structure formation is on a relatively sound footing, AFAIK. ΛCDM models run on supercomputers actually predicted the inhomogeneity, much like we observe, at least up to as point. The problem with the "missing matter" was that in our broad locality (i.e. late epoch), less matter was observed than what the models said - this is the puzzle now apparently solved. The models seem to have been correct, with the observations lagging. Always the better situation to be in :-)

Wiltshire's TS model simply predicts the effect that the differences in gravitational time dilation in the walls and voids have on the redshift-distance law at large distances. If his sums are correct, his model does not need any dark energy, just the dark matter to help form the structures in the first place.

-J

Hi Jorrie,

You Wrote"ΛCDM models run on supercomputers actually predicted the inhomogeneity"

I think they incorporated quantum fluctuations (uncertainty) in order to do that, is that correct?

You Wrote"If his sums are correct, his model does not need any dark energy, just the dark matter to help form the structures in the first place."

I know, but doesn't his model also improve a little at explaining the observed dark energy now that some of the missing matter has been found in the "matter walls" namely in the spaces between galaxies on those matter walls, or is the missing matter to little to be significant?

Hi again Roger,

Strictly speaking, the ΛCDM model starts after inflation stopped, so I do not think quantum effects are included. They are part of the inflationary BB model that 'delivered' a cosmos that were flat overall (average expansion rate balancing energy content), but with the original quantum fluctuations blown up to macroscopic scale, as we observe in the CMB radiation.

I'm not well versed in structure formation modeling, but one person that you may be interested to read is Martin White of Berkeley. He is a particle physics with an interest in these structure matters... :)

You wrote: "...but doesn't his model also improve a little at explaining the observed dark energy now that some of the missing matter has been found in the "matter walls"..."

Not quite. The predicted amount of observable baryonic matter (overall) is so small that a little of it "missing" in our general vicinity would make no difference. AFAIK, the structure models predict the "missing baryons" anyway. They are just difficult to find...

-J

t h e l o s t g a l ax i e s

James E. Geach

By the latest estimate, the observable universe contains 200 billion galaxies. Astronomers wonder: Why so few?

Forget dark matter: even the supposedly normal matter of the universe is mysterious enough. Why does only a small fraction of it reside in galaxies? Where did the rest go? The current best guess is that the bulk of the normal matter is trapped in giant gaseous filaments. This so-called warm-hot intergalactic medium, or WHIM, is hard to detect directly. Galaxy formation is evidently rather inefficient. As material falls into a galaxy, the galaxy tends to shoot much of it straight back out again-a process known as feedback. The atoms in your body have probably been cycled through intergalactic space. Indeed, galaxies and their contents are not fixed structures but the bright tips of a wider sea of gas.

http://www.astro.umass.edu/~wqd/a191/Scientific_American/scientificamerican0511-46_lost_galaxies.pdf

No problem Roger :)

I would agree that for a de Sitter Universe the curvature is directly related to the cosmological constant, but not quite as you said. Part of the problem is that cosmology sits with a legacy of labeling curvature according to the simple Einstein-de Sitter (matter only) model, where Ω < 1 meant negative curvature (expanding forever), Ω = 1 meant zero curvature (asymptotically zero expansion) and Ω > 1 meant positive curvature (eventual collapse).⁽¹⁾

By these definitions, if you increase the cosmological constant Λ = 8πρ_vac (in geometric units, c=G=1) from zero to above critical, the curvature would go from Minkowski flat, to cosmologically negative, to flat (critical), to positive, but not closed (no "collapse").

It is interesting to play with that balloon spreadsheet: zero the matter density and and enter only vacuum energy, say 0, .7, 1, 1.5, 2.5, 5 etc. Not too high, because my spreadsheet runs into errors - it was not designed for that.

-J

(1) http://en.wikipedia.org/wiki/Shape_of_the_Universe#Open_or_closed: "When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive. These meanings of open and closed, and the mathematical meanings, give rise to ambiguity because the terms can also refer to a closed manifold i.e. compact without boundary, not to be confused with a closed set. With the former definition, an "open universe" may either be an open manifold, i.e. one that is not compact and without boundary,^[8] or a closed manifold, while a "closed universe" is necessarily a closed manifold."

From http://en.wikipedia.org/wiki/De_Sitter_space:

"The n-dimensional de Sitter space...has constant positive curvature, and is simply-connected for n at least 3.

For Anti- de-sitter: http://www.physto.se/~ingemar/Kurs.pdf

Hi S, thanks for the heads-up. As I wrote to Roger, there may be some confusion in the terms, which I originally neglected to mention or clarify.

I may understand de Sitter space wrongly, because I interpreted it as for a de Sitter universe. Scanning the link you gave: http://www.physto.se/~ingemar/Kurs.pdf it seems a little more involved than that. I'll look into it more carefully and let you know what I make of it.

-J

Thank you for the links.

^{"TS cosmic model uses a 'two-curvature' model, where the cluster regions are 'flat' and the voids are 'open' (negative curvature..."}

^{"the TS value corresponds closely to the standard curve (i) at low redshift [O_M=0.25], to (ii) at medium redshift [O_M=0.28], and to (iii) at high redshift [O_M=0.34]."}

___________________________________________________________________________

What would happen, in a two-curvature model, with "structure" regions having closed curvature; and "void" regions having no curvature, i.e. flat ? CMB observations imply, that space is flat; and, "void" regions volumetrically dominate space.

If the "effective matter density", of the two-curvature model, decreases with red-shift, then perhaps that reflects the "void" regions "opening up", i.e. the "void" region scale factor R_v "pulling ahead" of the "structure" region scale factor R_s ??

According to Allen 2002, analysis of galaxy clusters, assuming an all-matter "common-sense cosmology", implies that cluster gas-mass fractions decrease with redshift. Yet, clusters are thought to represent a statistically fair sample, of our cosmos' contents. So, f_gas should be quasi-constant. And, if you assume a "two-Hubble-Constant" model, with h ~ 1/2 "locally", and h ~ 3/4 "globally"; then, their derived f_gas for clusters, in the "common-sense cosmology", become quasi-constant (since f_gas ~ h^1.5).

I partially mis-quoted 'Allen 2002', who actually says "the results, for SCDM, indicate an apparent drop in f_gas, as the redshift increases" (p.L13), i.e. distant, higher-z clusters appear to be gas-poor (implying that stars or DM turns into gas with time!). Thus, for the same SCDM cosmology, increasing h with z would "buoy up" distant clusters' f_gas, to values not less than closer clusters. In fact, f_gas ~ (0.7/0.5)^1.5 ~ 5/3, so that distant clusters' gas fractions would be increased from ~0.13 --> ~0.21, about ~25% higher than closer clusters' gas fractions ~0.17. Perhaps a declining amount of gas, could be construed, as evidence for "cooling flows", i.e. hot intra-cluster plasma radiating in x-rays, cooling, and "condensing out" into the dominant central galaxy, and/or central SMBH ?? I do not understand, if/why cluster observations, could not be construed, as consistent, with an inhomogenous "two-phase" cosmology.

"What would happen, in a two-curvature model, with "structure" regions having closed curvature; and "void" regions having no curvature, i.e. flat ?"

Wiltshire's model fits an overall open universe - too little gravitational and mass-energy for the observed expansion rate (if correct). I guess what you stated would be an overall closed cosmos, but I do not think it will fit the observational curves very well, while Wiltshire's does reasonably well. I do not understand inhomogeneous cosmic models well enough to really offer an opinion.

According to the Friedmann equations for a homogenous universe:

^(eq.1)

Now, please ponder a static-and-closed universe, e.g. near "turn-around", i.e. H = 0, k = 1. Naively, M = (4pi/3) rho a³, and R_s = 2 G M/c². Accordingly, (eq.1) becomes:

K = 1/a² = R_s/a³ _(eq.2)

R_s = a

But, (eq.2) defines the (constant-and-closed) curvature K inside of a spherical, uniform, constant-density distribution of matter, which has "just reached" its Schwarzschild radius, i.e. which has "just become" a black-hole, i.e. R_s = a.

Thus, that first Friedmann equation resembles the "interior solution" of a Schwarzschild space-time (which "wraps around" into complete "self-closure"). Note, that non-static universes (H != 0) must be denser than the corresponding Schwarzschild solution:

If there is a relation, between "clock rates" and "curvature", in the Schwarzschild space-time; then, is there also a similar relation, in (closed) Friedmann space-time ?

ADDENDUM:

In Shwarzschild space-time, the radial coordinate, r = C / 2pi, is defined by measuring (with a standard ruler) "around" the central object; obtaining a circumferential distance C; and, dividing by 2pi; giving the "expected" radial distance, in flat space-time. However, the hyper-spatial "stretch", of the fabric of space-time, around that central object, generates a larger volume, inside a given spherical surface area, than the "expected" amount. Again, you could construct concentric "geodesic domes", around the central object, and measure their physical, metric, areas A_2,1 = 4pi r_2,1² = C_2,1²/pi, "as normal". Thus, the "unsuspecting" might assume, that the physical, metric, distance, between those "geodesic dome onion shells", was dr = r₂-r₁. But, upon rotating their ruler, into the radial direction, the stretch of space-time would make "the ruler come up short", i.e. the actual physical metric distance, would be measured to be dw > dr, and specifically dw = dr / Sqrt(1-(r/R_s)²) _{(vaguely similar to, but not the same, as the "stretch factor", for the "exterior solution" in Schwarzschild space-time)}.

Therefore, the "differential volume" of the "onion shells" is larger, in stretched space-time, by that factor. Thus, the total volume, of the "onion", is larger, according to the integral:

V = Integral(dV) = 4pi R³ Integral(x²dx/Sqrt(1-x²)) = 4pi R³ x pi/4 = V₀ x 3pi/4

Thus, at "critical mass", where-at R_s = R, the metric volume, inside the "event horizon surface" of the central mass, is ~7/3 larger than "expected". Thus, the internal "critical mass" is spread out, over ~7/3 more space. So, the critical internal "Schwarzschild density" rho_s, is ~7/3x lower than "expected", due to the stretch, of curved space, inside the object.

Conversely, that first Friedmann equation "knows nothing" of this effect. Therefore, even at "turn-around" (H=0), the matter-and-energy density, inside of a closed Friedmann space-time fabric, would still be ~7/3x higher, than the "closest corresponding" (interior) Schwarzschild space-time.

_{Disclaimer: I'm trying to clarify these concepts for myself, e.g. crucial distinction between "exterior" & "interior" Schwarzschild space-time; and, am not implying that others didn't already comprehend them. There seems to be recent research associating cosmology with the insides of event horizons.}

I'm not sure if your post fits this thread, but for curvature discussions, we can possibly allow it.

The technical problem is that one cannot use the Schwarzschild metric for any sensible cosmic model, but just for very localized mass concentrations, ignoring rotations, orbits etc. I would not recommend a 'static universe' as a tool for understanding Schwarzschild, or cosmology for that matter.

BTW, I do not have access to the Allen 2002 paper that you quoted in #26, but I read in the abstract:

"... we obtain a tight constraint on the mean total matter density of the Universe, \Omega _m=0.30^+0.04_-0.03, and measure a positive cosmological constant, \Omega _\Lambda=0.95^+0.48_-0.72. Our results are in good agreement with recent, independent findings based on analyses of anisotropies in the cosmic microwave background radiation, the properties of distant supernovae, and the large-scale distribution of galaxies." (My emphasis)

That Ω_Λ=0.95^+0.48_-0.72 is as wide as a barn door!