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Thanks

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cepheid

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If you're talking about ΩGiven that space is flat or nearly so by all the best data, why does Omega need to be "one" in an accelerating universe.

Not sure what you're getting at here when you talk about the universe not having "run away." Note that, in the absence of dark energy, models with ΩCritical density was a necessity in the Einstein -de Sitter model (q = 1/2) in order to explain why the Hubble universe had not run away or collapsed in 13.7 billion years - we all got sold on the the beautiful mathematical model that had the universe slowing to zero velocity at eternity - now we know that expansion trumps gravity on the large scale

That's the thing. The missing matter- what factors or experiments (other than flatness) now drive the search for the missing matter that makes Omega unity? Is it only geometric flatness?

The dark matter story started as early as the 30's, with astronomer Fritz Zwicky. He observed that velocity dispersions of galaxy clusters appeared to be too high, given the amount of luminous matter that was present. In other words, the individual galaxies in the cluster were moving too fast and ought to escape the cluster. Yet, it was clear that the galaxy clusters were gravitationally-bound systems. The explanation he proposed at the time was the presence of a large amount of non-luminous matter that we could not detect. The same sorts of conclusions were drawn about individual galaxies when observations of their "rotation curves" (plots of rotational speed vs. distance from centre) showed that they were roughly flat (the speeds of the stellar orbits around the galactic centre were roughly the same at all radii). At larger radii, these orbits were much faster than Newtonian gravity would have predicted, and as a result, these galaxies ought to have been flying apart. In both of these situations, the two possible explanations were "there is extra matter present that we cannot detect, but that is providing the necessary gravity to keep the system bound," or, "there is something wrong with our theories of gravity -- maybe gravity behaves differently on large spatial scales or something." My understanding is that, although some modified gravity theories have had limited success in explaining some of the observations, none of them have been able to successfully explain all observations on all spatial scales.

So far I've talked about observational evidence for dark matter at the scales of individual galaxies, and at the scales of clusters of galaxies. What about really really large spatial scales? Well, it turns out that dark matter plays a crucial role in our models of structure formation i.e. in models that describe how tiny density fluctuations in an initially smooth/homogeneous universe grew under gravity to form the

So, we have a fair bit of observational evidence that the vast majority of the matter in the universe is some sort of non-baryonic matter that doesn't interact very strongly with anything else by means of any of the four fundamental forces

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_{total}, then it must be 1, else space would not be flat! If you're talking about Ω_{matter}, then it needn't be 1, and itisn't1.

Not sure what you're getting at here when you talk about the universe not having "run away." Note that, in the absence of dark energy, models with Ω_{tot}< 1 still decelerate, just not as rapidly. The limit is the empty universe, which has constant expansion rate. Acceleration is only possible with dark energy.

Thank you for the scholarly response. My inquiry was concerned primarily with Omega total. I am familiar with the velocity profiles of the spiral nebula and understand some additional matter is ostensibly required to explain the high velocities. But could you clarify your statement regarding deceleration - its my understanding that in the case of an empty universe (e.g., de Sitter) exponential expansion is driven solely by the cosmological constant - the growth rate is not constant - this is also true where the matter content is exactly balanced by the negative pressure - the only left in Einstein's equation is the cosmological constant - so what provokes the deceleration?

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cepheid

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I explicitly stated that I was talking about models with no dark energy i.e. no cosmological constant. Before the late 1990's and the evidence for dark energy/a cosmological constant, it was clear that ΩThank you for the scholarly response. My inquiry was concerned primarily with Omega total. I am familiar with the velocity profiles of the spiral nebula and understand some additional matter is ostensibly required to explain the high velocities. But could you clarify your statement regarding deceleration - its my understanding that in the case of an empty universe (e.g., de Sitter) exponential expansion is driven solely by the cosmological constant - the growth rate is not constant - this is also true where the matter content is exactly balanced by the negative pressure - the only left in Einstein's equation is the cosmological constant - so what provokes the deceleration?

To re-emphasize: the motivation for dark matter never had anything to do with trying to get Ω

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As I understand it, there is no confirmation of either the 25% missing matter nor the 70% missing dark energy - what we perceive is the 5% that is luminous - your post brings me back to my original question - I will re phrase it in the form of another question - why does the cosmological constant necessarily imply dark matter - why cannot space expand exponentially by some other mechanism. Perhaps we have been boxed in by models that may be too restrictiveI explicitly stated that I was talking about models with no dark energy i.e. no cosmological constant. Before the late 1990's and the evidence for dark energy/a cosmological constant, it was clear that Ω_{tot}(if accounting for matter only) was less than 1.

To re-emphasize: the motivation for dark matter never had anything to do with trying to get Ω_{tot}= 1, although I think people were sincerely hoping that an overall inventory (taking into account the inferred amount of dark matter from observations) would show that it was. It didn't.

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cepheid

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As I alluded to before, there are multiple independent sets of observation that all give a value for ΩAs I understand it, there is no confirmation of either the 25% missing matter nor the 70% missing dark energy - what we perceive is the 5% that is luminous -

The same thing is true of Ω

As I've been trying to tell you (twice before) the cosmological constant and the accelerating universe don't imply dark matter. They have nothing to do with it. You are confused. Darkyour post brings me back to my original question - I will re phrase it in the form of another question - why does the cosmological constant necessarily imply dark matter - why cannot space expand exponentially by some other mechanism. Perhaps we have been boxed in by models that may be too restrictive

Dark

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Chronos

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Also, I misspoke myself when typing "your post brings me back to my original question - I will re phrase it in the form of another question - why does the cosmological constant necessarily imply dark matter - why cannot space expand exponentially by some other mechanism. Perhaps we have been boxed in by models that may be too restrictive"

I meant to say "dark energy" rather than "dark matter" So I can understant your frustration

In your post #2 you state omega total must be one for flatness. My follow-up related to a pure de Sitter expansion - where Omega total is zero - Isn't the statement that "Omega must be one for flatness," a model dependent statement?

Thanks for your patience

Yogi

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Is the 27% of dark matter established independently or deduced by subtracting from one. If its confirmed independently - what is the form and is it consituted within the galatic system or scattered throughout space

Yes - the cosmological constant does imply dark energy, not dark matter - as per my correction of a previous post

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1) For a flat universe Omega total must be one

2) Experimental data indicates that dark matter plus visible matter contributes about 27% of the mass necessary to make Omega total equal unity

3) The balance of the cosmological energy is in the form of dark energy which corresponds to the cosmological constant - and this is determined by subtracting 0.27 from one.

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cepheid

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I'm not sure what you mean by model-dependent. First of all, let me address your lingering doubt that flatness requires Ω

Also, I misspoke myself when typing "your post brings me back to my original question - I will re phrase it in the form of another question - why does the cosmological constant necessarily imply dark matter - why cannot space expand exponentially by some other mechanism. Perhaps we have been boxed in by models that may be too restrictive"

I meant to say "dark energy" rather than "dark matter" So I can understant your frustration

In your post #2 you state omega total must be one for flatness. My follow-up related to a pure de Sitter expansion - where Omega total is zero - Isn't the statement that "Omega must be one for flatness," a model dependent statement?

Thanks for your patience

Yogi

[tex] \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho_{\text{tot}} - c^2\kappa [/tex]

where the overdot represents a derivative with respect to time i.e. [itex] \dot{a} \equiv da/dt [/itex]. On the right hand side, [itex] \rho_{\text{tot}} [/itex] is the total energy density of the universe (taking into account all constituents) and it is also a function of time [itex] \rho_{\text{tot}}(t) [/itex]. The second term has [itex]\kappa[/itex], which is the spatial curvature. It is the reciprocal of the radius of curvature. Basically, for κ > 0 (positive curvature) the geometry of the universe is closed, for κ = 0, the geometry of the universe is flat (i.e. Euclidean), and for κ < 0 (negative curvature), the geometry of the universe is open. Now, it can be shown that [itex] \dot{a}/a = H [/itex] where H is the Hubble parameter (also a function of time). So, substituting that in, and rearranging the equation (and also switching to the c = 1 unit system that cosmologists use) we obtain:

[tex] \kappa = \frac{8\pi G}{3}\rho_{\text{tot}} - H^2 [/tex]

Now, let's consider the value of the spatial curvature today. In other words, consider this equation at time t = t

[tex] \kappa = \frac{8\pi G}{3}\rho_{\text{tot}}(t_0) - H_0^2 [/tex]

So the condition for flatness (zero spatial curvature) is:

[tex] 0 = \frac{8\pi G}{3}\rho_{\text{tot}}(t_0) - H_0^2 [/tex]

[tex] H_0^2 = \frac{8\pi G}{3}\rho_{\text{tot}}(t_0) [/tex]

[tex] \rho_{\text{tot}}(t_0) = \frac{3H_0^2}{8\pi G} [/tex]

The density required for flatness is therefore [itex] \rho_{\text{crit}} = 3H_0^2 / 8\pi G [/itex]. Cosmologists call this value the

[tex] \Omega_i \equiv \frac{\rho_i}{\rho_{\text{crit}}} [/tex]

The total density parameter is just the sum of the density parameters for all the individual consituents (baryonic matter, dark matter, radiation (photons), and dark energy).

[tex] \Omega_{\text{tot}} = \frac{\sum_i \rho_i}{\rho_{\text{crit}}} = \sum_i \Omega_i [/tex]

In light of the above, we can rewrite the flatness criterion that I derived above as:

[tex] \rho_{\text{tot}} = \rho_{\text{crit}} [/tex]

[tex]\Rightarrow \frac{\rho_{\text{tot}}}{\rho_{\text{crit}}} = 1[/tex]

[tex]\Rightarrow \Omega_{\text{tot}} = 1 [/tex]

Now, you posed a question about an empty universe (Ω

[tex] \kappa = \frac{8\pi G}{3}\rho_{\text{tot}}(t_0) - H_0^2 [/tex]

[tex]\Rightarrow \kappa = - H_0^2 [/tex]

Apparently, this model actually has maximally negative spatial curvature, and as a result its geometry is open (hyperbolic geometry). This particular Friedmann model, the empty universe, is known as the Milne model:

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

Note: a de Sitter universe (yet another special case of the Friedmann model) is not exactly an empty universe, because it assumes non-zero [itex] \Lambda [/itex] and hence non-zero [itex] \Omega_\Lambda [/itex], which means that [itex] \Omega_{\text{tot}} \neq 0 [/itex]. The de Sitter space is empty of everything

I hope that this addresses some of your concerns.

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Thanks Cepheid, your detailed post nicely clarifies the flatness-density relationship.

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BillSaltLake

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Or would it be de Sitter space -if negative pressure energy cancels positive mass density it would seem to lead to the negative curvature solution that Cepheid derived, and therefore exponential expansion - constant H, and constant R. I wonder if the negative curvature is on a scale not detectable within a finite observation distance

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What if we stick the c^2 term back in Cepheid's result (instead of c = 1) - then the curvature K is H^2/c^2

= 1/R^2 which is the Hubble radius - so maybe the Friedmann equation is not giving an answer to the degree of curvature, but rather this result is implicit in the assumptions made by Friedmann and General relativity that lead to the derivation of the equation i.e., a finite expanding sphere

= 1/R^2 which is the Hubble radius - so maybe the Friedmann equation is not giving an answer to the degree of curvature, but rather this result is implicit in the assumptions made by Friedmann and General relativity that lead to the derivation of the equation i.e., a finite expanding sphere

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