# Thread: Dark Matter - don't we see all we need? Part 2

1. Originally Posted by Widdekind
If our space-time fabric is topologically 'closed' -- to wit, hyper-spherical -- then, to my understanding, the Radius of Curvature (Rcurv) is the 'hyper-spatial radius', of the 'hyper-sphere', whose 3D 'hyper-surface', is our space,
Try not to think of the universe as embedded in some higher dimensional space. That will lead to all sorts of confusions. For example, if our universe had the shape of a cylinder, that would still be closed but flat (i.e. Eucledian geometry).

You don't have to guess or assume. If you are interested, you can do the calculations yourself and check. Maybe you are right and I'm wrong, but I don't have the energy to do the calculations just for the purpose of a forum post. The calculations are not difficult, they are just very long and tedious, so if you are interested I really invite you to do it yourself.

Friedmann?Lemaître?Robertson?Walker metric - Wikipedia, the free encyclopedia

Use natural units to make the math simpler, so the metric becomes:

ds^2 = - dt^2 + a^2 ( dr^2 / (1 - k r^2) + d theta^2 + r^2 sin(theta)^2 d phi^2 )

Compute the components of the Riemann:

Riemann curvature tensor - Wikipedia, the free encyclopedia
Christoffel symbols - Wikipedia, the free encyclopedia

From here you compute the components of the Ricci tensor and then the Ricci scalar:

Ricci curvature - Wikipedia, the free encyclopedia

The Ricci scalar gives you the curvature. What you want to know is whether it gets very large when t goes to 0.

If you are not practiced doing these calculations it might take you an afternoon, or maybe a day to do them. So you can set aside Saturday and get the answer if it is important to you. Then you can post here the value of the Ricci scalar and we'll see.

But I hope you'll understand if I don't want to do go through all this work just to argue a point in a forum. For one, even if you are right in principle, we'd still have to figure out how small the universe has to be before curvature is significant enough to matter at the scale of nuclear interactions, which is a very small scale. I suspect that what you'll find is that it doesn't matter. I expect you'll find that by the time the universe is cool enough to allow nuclear interactions, it has expanded enough that from the point of view of a quark it might as well be flat.

However, I have one point I would want yet to push -- neutrinos. Our cosmos is submerged in a sea of cold neutrinos, a 'neutrino CMB' as it were. Please ponder this point -- 'way back when', there was an ultra-relativistic, ultra-dense 'neutrino fog', through which all the then-fusing baryons were obliged to propagate. Can I not claim, that modern human particle-accelerator experiments do not mimic such 'neutrino-dense' conditions? And, if not, how would such a 'neutrino soup' backdrop, or stage, have affected the physics??
You cannot claim that because it is a subject you do not understand. You should only make claims that are commensurate with your knowledge. However, you can ask. Asking is always ok.

There is (probably) indeed a neutrino background, analogous to the CMB. This is not unknown or unexpected physics. The neutrino background predates nucleosynthesis by a long while. The universe becomes transparent to neutrinos at about the time the quark-gluon plasma cools enough to produce hadrons. This happens several minutes before the universe is cool enough to even allow nuclear fusion. In other words, by the time the universe is cool enough that nuclear fusion is even possible, it is more than cool enough that neutrinos don't affect the interactions.

But anyway, you touch on this subject in your post, so let's just continue and answer your next question:

I notice that that cited article does say, that the ultra-relativistic, primordial, neutrinos would have cooled out of matter-interactiveness, after two seconds -- several minutes before even PNS. Yet, those same neutrinos, after a further ~400 kyr of cooling, interacted strongly enough with matter, to smooth out fluctuations, ultimately imprinted in the CMBR. I don't understand, how, if neutrinos were matter-interactive, at 400 kyr, why they weren't, at 4 minutes.
It is the difference between the strong nuclear force and gravity. You made a mistake by saying "interaction with matter" without thinking deeper about what interactions you are talking about. The features in the CMB are large scale (i.e. cosmic scale) density fluctuations driven by gravity. Neutrinos produce gravity too, and the amount of neutrinos has detectable effects in cosmic structures.

Incidentally, this is one of the reasons why neutrinos are not the most popular candidate for dark matter. Neutrinos move very fast. If they were the bulk of dark matter, they would make it harder for galaxies to form and galaxies would form top-down, while we can observe that galaxies formed bottom-up.

So as you can see, neutrinos can have perfectly valid effects on the large scale structure of the universe, and none of that implies that they affect nuclear fusion. Fusion is not driven by gravity, it is driven by the strong and weak nuclear forces, and it happens on the nuclear length scale, while the CMB shows features in the scale of the universe.

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3. Daniel, you truly are a paragon of patience.

4. ## The Following 4 Users Say Thank You to DaveW For This Useful Post:

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5. Widdekind: Here is one quick-n-dirty calculation. Let's imagine for the moment that the universe really is analogous to a sphere with a radius. How fast is it expanding at the time of the Big Bang? Pretty fast I assume. Let's say it expands at 50% the speed of light. Nucleosynthesis happens after 3 minutes. How big is this hypothetical universe after 3 minutes?

(50% speed of light) x (3 minutes) = 0.36 AU

That's basically the same as the orbit of Mercury (which is 0.39 AU). Can we agree that a sphere the size of the orbit of Mercury has low enough curvature that from the point of view of a quark it looks flat?

Heck, there used to be people on Earth that thought the Earth was flat, and people are much bigger than Quarks and the Earth is much more curved than the orbit of Mercury.

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7. He he he he... I just came up with some alternate calculations. They came to me when I realized that 3 minutes is actually a very long time.

* The average speed of a cheetah is 70 mph. If the universe is a sphere that expands at cheetach speed, then after 3 minutes it has a radius of 4.5 miles. From the point of view of a quark, that is flat.

Ok, let's go slower...

* The speed of a squirrel is 12 mph. If the sphere-universe expands at squirrel speed, after 3 minutes the sphere has a radius of 1km. Hmm... still pretty flat.

Ok, let's go slower...

* The speed of a chicken is 9 mph. At chicken-speed we get a nucleosynthesis when the universe is larger than two football stadiums.

Ok, let's go slower...

* The speed of a giant tortoise is 0.17 mph. At giant-tortoise-speed we get nucleosynthesis when the universe is ~27m in diameter, or the size of an apartment building.

Ok, let's go slower...

* The speed of a garden snail is 0.03 mph. At garden-snail-speed we get nuclesynthesis when the universe is 2.4 m in radius. That's about the size of my house. Probably looks pretty flat to a quark.

So how slow does the universe have to move so that after 3 minutes its curvature will be noticeable by a quark? I don't want to calculate, but I think most people will agree that that's starting to look like a pretty wimpy "Big Bang".
Last edited by DanielC; 07-15-2011 at 08:23 PM.

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9. Great debate! Love it! Excellent explanations that even I can grasp with some further study. I agree about so called Science Magazines being poorly written. That is partially because they are aimed at people they feel are too incapable of understanding the concepts so they Dumb Them Down and dilute the information. Unfortunately, this tends to lose the meat of the info creating a slurry of poorly worded gibberish.

Lee

10. 'Dim Matter' -- NASA WISE scanner expected to detect dim Brown Dwarves

(source: cmarchesin)

11. I give up. I expect a follow up post next time an astronomer says "hey, that star over there is pretty dim".

12. Originally Posted by DanielC
...How fast is it expanding at the time of the Big Bang? Pretty fast I assume. Let's say it expands at 50% the speed of light. .
Even without inflation, the universe was expanding infinitely fast, at the Big Bang, according to the standard Friedmann equations, H(t=0) --> infinity. I understand, that this is visualizable, as the vertical, 'infinitely steep', asymptote, of the Cosmic Scale Factor, plotted as a function of time, simply in 1D:

or, as a more attention-grabbing 'surface-of-revolution':

Thus, at the "bottom of space-time", back at the Big Bang, the fabric of space-time becomes asymptotically flat, with a flat, planar, tangent surface.

For FRW cosmologies, one chooses the global curvature / topology first ("is k=-1, 0, or 1 ?"), and only then solves for the evolution of the scale factor, as governed by GR, constrained-and-confined to, that prior, "a priori" choice-and-imposition-upon-the-physics-formulae. By construction, the global topological curvature, of the space-time-fabric, is assumed from the start (k=-1,0,1), and never allowed to change thereafter, irrespective of whatever types of matter, radiation, dark energy, etc., that you "throw into the mathematical stew pot", afterwards.

CONCLUSION: The global curvature, of our space-time fabric, as mathematically modeled, in all FRW cosmologies, is, qualitatively ("open, closed, flat"), time-invariant. Again, mathematically, the human physicist-cum-mathematical-modeller, first chooses "willy nilly" an assumed-from-the-first global topology ("open, closed, flat", "k=1,-1,0"); and only then starts to solve any kind of equations, after imposing that metric (describing the topology), into-and-onto, the GR equations. Once you select a topology, there's "no going back". Er go, "once closed, always closed", no matter the math arising from the GR equations:

1. choose global topology ("open, closed, flat", "k=1,-1,0")
2. impose topology into GR, and solve

13. Look up the word topology. "Open" and "Flat" are not different topologies. You don't have to know the meaning of every word, but you should know the meaning of the words you use.

In any case, it looks like you have just made a case against your earlier argument that curvature would have been important at the time of nuclear fusion.

14. Originally Posted by DanielC
I expect you'll find that by the time the universe is cool enough to allow nuclear interactions, it has expanded enough that from the point of view of a quark it might as well be flat.
Yes -- once you know the red-shift of PNS, you know what the (relative) scale-factor was, then. And, assuming that the absolute scale-factor, today, is "of order" the Hubble Distance DH = c T0 = c / H0, then the universal space-time fabric, then, still had a radius-of-curvature comparable in size to a galaxy (~100 thousand light-years).

It is the difference between the strong nuclear force and gravity.... The features in the CMB are large scale (i.e. cosmic scale) density fluctuations driven by gravity. Neutrinos produce gravity too, and the amount of neutrinos has detectable effects in cosmic structures.
That is clear, cogent, & concise, and is the first I have heard that explanation -- so, neutrinos indirectly "smooth out matter fluctuations", at +400 Kyr, by "smoothing out the space-time fabric", which then affects other matter particles. To wit, neutrinos do not produce a "wind" which smooths out other particles, by direct collisional interactions, but by first affecting space-time itself.

That explanation is quite clear, and so I accept it. Would you please tell me where you learned that from ? I read allot, and every source I've seen always has said "neutrinos interacted w/ matter to smooth the CMB", and none said "neutrinos smoothed SPACE-TIME"... (or so I interpreted them to say)... such is an important distinction, what source could I consult that is that clear on the concept??

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