
#1
Nov211, 07:52 PM

P: 1,444

Given that space is flat or nearly so by all the best data, why does Omega need to be "one" in an accelerating universe. Critical 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  what factors or experiments (other than flatness) now drive the search for the missing matter that makes Omega unity? Is it only geometric flatness?
Thanks 



#2
Nov311, 01:02 AM

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PF Gold
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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 gravitationallybound systems. The explanation he proposed at the time was the presence of a large amount of nonluminous 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 large scale structure of the universe that we see today. If you assume that there was no dark matter and only baryonic matter, you run into a problem. The basic problem has to do with the temperature fluctuations that we see in the Cosmic Microwave Background (CMB) radiation. These fluctuations in temperature are, roughly speaking, caused by (and have the same order of magnitude as) the density perturbations in the primordial plasma that existed in the very early universe (the soup of charged particles and photons). Baryonic theories of structure formation predict that, at the time, these fluctuations would have had to have been at a level of around 10^{3} (i.e. that's the factor by which there would by an overdensity or underdensity relative to the mean density level). If the fluctuations were any smaller, then under the (no dark matter) models of structure growth, there would not have been sufficient time for the overdensities to have reached a large enough level for there to be galaxies, stars etc. at the present day. The problem is, these initial perturbations are much smaller. The temperature fluctuations in the CMB are observed to be a level of ~10^{5}. This is a problem! Theories of structure formation without dark matter predict that we shouldn't exist, because the matter density perturbations shouldn't have grown large enough to form the structures that we see. In other words, the universe should still be much smoother and less clumpy than it is. However, if you include nonbaryonic dark matter in the models of structure formation, this problem goes away (I'm getting a bit too tired of typing to explain the details of how that works). In fact, the standard lambdaCDM (cold dark matter) cosmology has had tremendous success in explaining (with great precision) how the large scale structure that we see in the universe today developed. Furthermore, detailed observations of the fluctuations in the CMB have allowed us to figure out the "recipe" for the constituents of the universe that you hear quoted all over the place: ~73% dark energy, ~22% dark matter, and ~5% ordinary (baryonic) matter  i.e. stuff that is made out of atoms. These values are in agreement with those obtained from other nonCMB observations. So, we have a fair bit of observational evidence that the vast majority of the matter in the universe is some sort of nonbaryonic matter that doesn't interact very strongly with anything else by means of any of the four fundamental forces except gravity. I would go even further and say that the observational evidence for dark matter is nowadays considered to be fairly conclusive. We can "see" it, in the sense that we can use gravitational lensing to map out how it is distributed spatially. This works well if you have a very dense region (such as a galaxy cluster) that lenses background objects very strongly. One of the best examples of this is the observations of the bullet cluster, a collision between two galaxy clusters that is considered to be a sort of "smokinggun" for the presence of dark matter and nail in the coffin for alternative explanations such as modified gravity. Again, I've typed enough and would rather not explain why the bullet cluster is such strong evidence for DM. Instead, here's a link: http://apod.nasa.gov/apod/ap060824.html 



#3
Nov311, 11:43 AM

P: 1,444

[QUOTE=cepheid;3595689]If you're talking about Ω_{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 it isn't 1.
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? 



#4
Nov311, 01:40 PM

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Missing Matter  Is anything really MissingTo reemphasize: 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. 



#5
Nov311, 07:48 PM

P: 1,444





#6
Nov311, 10:41 PM

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P: 5,198

The same thing is true of Ω_{matter} (in total) as well as its two constituents Ω_{CDM} and Ω_{B} (for ordinary Baryonic matter). CMB and other observations give what have come to be known as "concordance" values for these parameters (values that are in agreement with multiple data sets). Again, we have fairly strong observational measurements that these two are 0.22 and 0.05 respectively. Dark matter, on the other hand, is a much more developed idea. Although there have been no direct experimental detections of it to date, efforts to do so are underway, and theoretically, at least, we have clear candidates (from the standard model of particle physics and its extensions) for what the dark matter particle might be. The purpose of my really long first post was to make it clear what things do imply dark matter i.e. to give you a clear, comprehensive, and compelling explanation for why the existence of dark matter is so readily and widely accepted in the astrophysics community. I had hoped that my post would have conveyed to you the wealth of observational evidence that exists in that regard. 



#7
Nov311, 10:52 PM

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P: 9,182

The WMAP data, among other studies, assures us the baryonic content of the universe is much larger than it appears to be. Our models of the universe are very well behaved and predictive when we inject enough dark matter to fit observation. Unless something very strange is going on that has thus far eluded detection, the ~27% matter contribution to the total energy of the universe is well established. Furthermore, the cosmological constant implies dark energy, not dark matter. It is derived by subtracting the difference between Omegematter and 1 to result in a flat universe  which is what we observe.




#8
Nov411, 01:38 PM

P: 1,444

Cepheid, Thanks for the info on the BAQs  I wasn't aware of these experiments
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 followup 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 



#9
Nov411, 01:50 PM

P: 1,444

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



#10
Nov511, 01:31 AM

P: 1,444

From the posts I draw the following conclusions:
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. 



#11
Nov611, 12:14 AM

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[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_{0}, where t_{0} is the present age of the universe. The value of the Hubble parameter today, H(t_{0}), is just H_{0}, what we call the Hubble constant. So we get: [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 critical density. For the universe to be flat, the total energy density today must be equal to the critical density. I'm going to omit the t_{0} argument to the density functions henceforth, and it will just be understood that all densities referred to are present values. Anyway, by definition, the density parameter [itex] \Omega_i [/itex] of the "i^{th}" constituent of the universe is given by: [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] In conclusion, that's why flatness implies that Ω_{tot} = 1. It comes straight from the Friedmann equation. Now, you posed a question about an empty universe (Ω_{tot} = 0). I can certainly understand your confusion. We all know that General Relativity says that mass/energy curves spacetime, so we'd expect an empty universe to have zero spatial curvature and to just reduce to the flat Minkowski spacetime of special relativity. However, if you look at the Friedmann equation, you can see that that's not what General Relativity actually says will happen. Instead, there is some intrinsic curvature to the spacetime of the Friedmann world model even with no energy content (ρ_{tot} = 0): [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 nonzero [itex] \Lambda [/itex] and hence nonzero [itex] \Omega_\Lambda [/itex], which means that [itex] \Omega_{\text{tot}} \neq 0 [/itex]. The de Sitter space is empty of everything except dark energy. I hope that this addresses some of your concerns. 



#12
Nov611, 06:21 PM

P: 1,444

Thanks Cepheid, your detailed post nicely clarifies the flatnessdensity relationship.




#13
Nov611, 08:21 PM

PF Gold
P: 184

Note that a trivial solution, in which a = constant, will also solve the equations. In this case H = 0 identically, so κ and ρ are both zero. This would be a true Minkowski flat spacetime. Curvature is not necessarily required if ρ = 0.




#14
Nov711, 03:46 PM

P: 695

In the case where gravity is stronger and causes the expansion to slow down and even reverse, does that really happen? If the universe is infinite homogenous and isotropic how does it collapse if the forces are more or less equal everywhere and in all directions?




#15
Nov711, 11:24 PM

P: 1,444





#16
Nov711, 11:38 PM

P: 1,444

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 


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