Missing Matter - Is anything really Missing

yogi

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

cepheid

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.

That's the thing. The missing matter doesn't make Ω unity. Including dark matter, Ω_matter ≈ 0.25. My understanding is that, throughout the 80's and 90's, observations kept resulting in that sort of value. This was kind of a problem in the sense that theorists were hoping for Ω_tot = 1, and we assumed that the universe was matter-dominated and didn't know anything about any other sort of constituent (like dark energy). However, it was not a problem in the sense that you seem to be implying. You seem to be under the impression that the motivation for introducing dark matter was to get Ω_tot to be 1. This is not correct. The motivation for introducing dark matter has always been to explain other anomalous observations, most of which have to do with gravitationally-bound systems (on various spatial scales).

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 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 non-baryonic 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 lambda-CDM (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 non-CMB observations.

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 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 "smoking-gun" 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

yogi

[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?

cepheid

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 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.

yogi

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 restrictive

cepheid

As I alluded to before, there are multiple independent sets of observation that all give a value for Ω_Λ of ~ 0.73. There are measurements made of the CMB anisotropy, there are observations of luminosity distance vs. redshift made using Type Ia supernovae, and there are measurements of "Baryon Acoustic Oscillations" (BAOs) which rough speaking are just specific frequency modes in the density structure of matter on large scales. If you combine the constraints on Ω_Λ from these three different sets of observations, you actually end up with a fairly strong (i.e. tight) constraint on the value of this parameter, which expresses the present energy density of the dark energy relative to the critical density. If you're interested in what specifically the error bars are, there are numerous papers on the subject. I think that the WMAP seven-year results paper on cosmological parameters would be a good place to start.

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.

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. Dark energy, some sort of mysterious substance with a constant energy density (in spite of expansion) and negative pressure is what is thought to be responsible for the (observationally-supported) accelerated expansion and hence the presence of a non-zero cosmological constant term in the Friedmann equation. Granted, nobody has the slightest idea of what dark energy could be. There is some idea that this could be the quantum mechanical "vacuum energy" (energy associated with empty space), but there are problems with this idea. I should also point out that the "cosmological constant" variety of dark energy is only the simplest version, and there have been proposed other more exotic forms that have non-constant energy density (as a result they correspond to extra term in the Friedmann equation, but not a constant one). In any case, since nobody has much of an idea of what dark energy could be, it's admittedly on much shakier theoretical ground than dark matter.

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.

Chronos

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 Omege-matter and 1 to result in a flat universe - which is what we observe.

yogi

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 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

yogi

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

yogi

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.

cepheid

I'm not sure what you mean by model-dependent. First of all, let me address your lingering doubt that flatness requires Ω_tot = 1. To do that, let's take a look at the model (specifically the Friedmann world models). First we need to wrap our heads around the idea of the scale factor, 'a'. The scale factor is a dimensionless quantity, and it is a function of time: a(t). Consider any two objects in the universe. Basically, you can think of a(t) as the ratio: (separation of objects at time t)/(separation of objects now). So, right now, a = 1, and at any time in the past, a < 1, with it being smaller the farther back you go (because the universe has been expanding with time, so if you go back to earlier times, separations between objects decrease). So all of the information about the expansion history of the universe is encoded in this function a(t). This function is determined by a differential equation called the Friedmann equation, which is in turn derived from the Einstein Field Equations of General Relativity. The Friedmann equation is as follows:

[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 = t₀, where t₀ is the present age of the universe. The value of the Hubble parameter today, H(t₀), is just H₀, 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 t₀ 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 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 except dark energy.

I hope that this addresses some of your concerns.

yogi

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

BillSaltLake

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.

Tanelorn

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?

yogi

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

yogi

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