Shape-univ November 2002

shape-univ@cosmo.torun.pl

6 participants
7 discussions

zwicky pages (so far)
by Boud Roukema 02 Dec '02

02 Dec '02

Cześć wszystkim, (1) How to write in latin-2 on emacs: (1a) With a fairly new emacs (e.g. 20.7 or later): - M-x set-language-environment latin-2 - M-x set-input-method latin-2-prefix (also C-x <ret> C-\) - then can toggle input method with: C-\ Examples: `a`e`l's'c'o`z'z'n ąęłśćóżźń to stop ' being used, use, e.g., a space: ' a -> 'a (1b) You can put the following in your ~/.emacs file (custom-set-variables ;; custom-set-variables was added by Custom -- don't edit or cut/paste it! ;; Your init file should contain only one such instance. '(case-fold-search t) '(current-language-environment "latin-2") '(default-input-method "latin-2-prefix")) (2) The zwicky pages at the moment have a verrrrrrrrrry long URL: http://adjani.astro.uni.torun.pl:9673/zwicky/ Please see Tomek (czosnek) or me to create an account. pozd boud

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expected fluctuations in xi(r)
by Gary Mamon 25 Nov '02

25 Nov '02

Dear Boud and other very large scale structure enthusiasts. I've come back to the issue of what amplitude fluctuations in the 2 point correlation function of quasars or galaxies or whatever should we expect given an assumed non-smooth P(k). The enclosed figure shows xi(r)s for 4 choices of P(k), smoothed with a gaussian of sigma=15 h-1 Mpc. The black curves are the CMBFAST prediction for Omega_m = 0.3 Omega_lambda = 0.7 Omega_b = 0.1 h = 0.6 sigma_8 = 0.6 Note that I prefer the CMBFAST predictions of P(k) to the analytical approximations by Bardeen et al. (1989, BBKS) and others, because the approximations cannot reproduce the wiggles in P(k) that generate power in xi(r). The cosmological parameters above were chosen to fit well the P(k) that Hoyle et al. (2002, MNRAS 329, 336) measured from the 2dF-10k release of quasars. Here P(k)s are defined as (2*pi)^3 times the Peebles definition (as was done by Hoyle et al.). I assume (Peebles 1993, eq. [21.40]) xi(r) = 4pi/(2pi)^3 int_0^infty k^2 dk P(k) sin(kr)/(kr) which I compute with Simpson integration as xi(r) = 4pi/(2pi)^3 ln(10) int_{-5}^5 k^3 P(k) sin(kr)/(kr) d log k with a step in log k of 0.01. The red curves use a spline fit to the Hoyle et al. measurements of P(k) within the interval of wavenumbers where they have measurements, and the CMBFAST prediction outside. The green curves use the CMBFAST P(k) everywhere, except that the point at log k/h = -1.162 is multiplied by 3. The blue curves multiply the point at log k/h = -1.162 by 10 instead. The resulting xi(r) curves show very oscillations of much weaker amplitude than found by Roukema, Mamon & Bajtlik (2002, A&A 382, 397), who smoothed their xi(r)'s with the same 15/h Mpc gaussian. Indeed, RMB found oscillations in xi(r) of amplitude 0.05 for Omega_m=0.3,Omega_lambda=0.7, whereas the oscillations predicted here are at best of 0.002 at separations of around 200/h Mpc, hence 25x smaller. Even if the RMB xi(r)s are affected by their incorporation of selection effects in the z-direction, this should affect the normalization of xi, but presumably not the amplitude of the oscillations. Moreover, the Hoyle at al. P(k) produces a minimum of xi(r) at r = 245/h Mpc, whereas RMB predict a maximum at that separation. This makes me worry that the RMB result is caused by noise... Please convince me that I am wrong! I need to think again about the (2pi)^3 factor... cheers Gary

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Re: what's with that transparency
by szajtan odwieczny 15 Nov '02

15 Nov '02

ok....at first thanks - you gave me quite homework to do, it will take a few days until I manage to read that book but I just can't sit like that after what you wrote. firstly I want to remark that I don't know what that damn word manifold mean - I cant find it anywhere - I can just suppose. But this isn't at most important right now. Boud, I have a few remarks to what wrote. Let me quote: > > possible, so we came to the conclusion that the 2-torus in three > > dimensional Euclidean space is not an good example of two > > dimensional closed space of zero curvature. But if we introduced > Agreed. Unless we redefine the metric. For example: > Let's put the 2-torus in the X-Y plane, centred on (0,0,0), with > radius R. > > Then the metric of E^3 (Euclidean 3-space, i.e. R^3 with flat metric) > is > > ds^2 = dx^2 + dy^2 + dz^2 > = (dr^2 + r^2 d\theta^2) + dz^2 where r=\sqrt{x^2+y+2} > >Under this metric, the 2-torus does not, in general, have zero >curvature, and it is inhomogeneous (curvature different on different >points). I assume that theta is an angle between x and y axies and (r^2 = x^2+y^2), right ? If so then this is just an infinitesimal length in E^3 space in cylindrical coordinates. ok, I understand this point. (btw. but even here I have an uncertainty, look at the end, because it's a little different topic) > >So let's just make a new metric on the 2-torus: > >ds^2 = (dr^2 + R^2 d\theta^2) + dz^2 where r=\sqrt{x^2+y+2} > >only difference: "r^2" -> "R^2". ( but I think dr should be = 0 in this metric ) > >(Exercise: Is this a metric on R^3?) as you wrote this is a metric on flat torus. but in fact it's not true. if we use cylindrical sthight-linear coordinates than the equation r=R is an equation of tube, thus this is a metric on a surface of tube not torus. If we go to the coordinates like: r' = r \phi' = \phi z' = \rho arcsin(z/\rho) where \rho is an radius of torus (the one from it's center) then the same equation r' = R describes torus. For the first equation r=R I checked Pythagoras's theorem integrating for each side of chosen triangle, and finally came to the conclusion that 1=1 as expected (meaning it's true). now I wanted to do the same thing with an equation r'=R which would be for torus. In fact in that prime coordinates, tube becomes torus so I thought since ds is invariant, all that should be showed is to check if that transformation is an isometry. Well it is, under condition that \rho = infinity or z=0. And this is good for nothing. I'm still gonna spend some time trying to find such function of the complementary - third side of an triangle on a surface of torus so after integrating along that side and putting it into the Pythagoras's theorem I will find equality. but I am sceptic because... > > >> > the fourth spital dimension than it becomes possible. Like for > >possible: yes; necessary: no. > > > > But the problem now is how to put this idea into the real > > world. All these considerations about the topology of space were > > made from "the higher ground" - I mean from the point of view of a > > space with at lest two dimensions more that the space (earlier a > > transparency or just a piece of paper) we were thinking of. The > > problem appears if we talk about the topology of the Universe > > where there are only three spital dimensions. To make it close > > like 3-torus it is essential to introduce 3 more dimensions (it's > > six) to make it possible. Then I say that, that all topoloy thing > Only if you *assume* that the only spaces which are physically > possible are n-dimensional Euclidean spaces. ... that's just the point - as a basis the first thing was that we were looking among spaces of zero curvature which are Euclidean spaces, so I do assume only Euclidean spaces. The metric tensor g in cylindrical coordinates of torus is (I guess) | R^2 0 | g= | | | 0 1/[1-(z-\rho)^2] | ( and for tube is with g_22 = 1.) It depends on point, and in general det(g) might be negative so it can't be an Euclidean space. Can it ? (and now I am not in some 3-dimensional space but just two). So I bring back my implication about topology science and 3+n - dimensional space. Despite of all what was said, don't we assume k almost zero these days ? ------------------- this is bizzare: Let's imagine tube. From mathematical point of view we can define it's curvature in each of it's points along two perpendicular directions (along the axis of symmetry and the other one). the first radius of curvature will be infinity, while the other one will be R (radius of the tube). Living on a surface of tube I could also measure is's curvature by measuring it's circumference and dividing by 2pi. Now I can imagine a 2-space of shape of this tube. Although I computed the Pythagoras's theorem is satisfied (so the curvature must be zero) how can I deal with the fact, that it is flat in the circumstances I mentioned above. thanks for references. when I finally get satisfied with this problems because if this going to be like this (I working on it for at least week) I won't start with my master thesis - this is quite off my topic ;) nevertheless very interesting, although maybe for you - boring. pozdrawiam, bartek

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Re: what's with that transparency
by szajtan odwieczny 15 Nov '02

15 Nov '02

sorry in that g_22 should be / instead - ;) czyli like this 1/[1-(z/\rho)^2] bart

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Re: what's with that transparency
by Boud Roukema 13 Nov '02

13 Nov '02

Cześć Bartek, Michał, everyone, I'm happy to see this discussion starting up! :) On 12 Nov 2002, Michal Frackowiak wrote: > On Sun, 2002-11-10 at 17:12, szajtan odwieczny wrote: > > questions: > > ---------- > > 1) does the cosmological principle introduce any restrictions on the topology of the universe. This is a *big* question ;). Firstly, it depends what you mean by "cosmological principle" If you mean local homogeneity and local isotropy, which is (more or less) consistent with observations, then it allows all constant curvature 3-manifolds. If you mean global homogeneity and global isotropy, which is not (yet or maybe never?) measured, then many multiply connected 3-manifolds are inconsistent with this strict definition, since some points in space or directions can be "favoured". See http://de.arxiv.org/abs/astro-ph/0101191 > > 2) does the topology science relies on some multidimentional theory - like string theory The physics explaining topology *might* be related to higher dimensional theories, but the mathematics does not require it, and the physics certainly does not *require* a higher dimensional theory. > > now look here, > > > > the problem: > > ------------ > > > > I finally figured it out. It is about that transparency > > problem. The problem was how it is possible to roll up the > > transparency into a torus not stretching it so it's curvature > > remained the same. I was stubborn to say that this was not > > possible, so we came to the conclusion that the 2-torus in three > > dimenitonal eucliean space is not an good example of two > > dimentional closed space of zero curvature. But if we introduced Agreed. Unless we redefine the metric. For example: Let's put the 2-torus in the X-Y plane, centred on (0,0,0), with radius R. Then the metric of E^3 (euclidean 3-space, i.e. R^3 with flat metric) is ds^2 = dx^2 + dy^2 + dz^2 = (dr^2 + r^2 d\theta^2) + dz^2 where r=\sqrt{x^2+y+2} Under this metric, the 2-torus does not, in general, have zero curvature, and it is inhomogeneous (curvature different on different points). So let's just make a new metric on the 2-torus: ds^2 = (dr^2 + R^2 d\theta^2) + dz^2 where r=\sqrt{x^2+y+2} only difference: "r^2" -> "R^2". (Exercise: Is this a metric on R^3?) > > the fourth spital dimention than it becomes possible. Like for possible: yes; necessary: no. > > But the problem now is how to put this idea into the real > > world. All these considerations about the topology of space were > > made from "the higer groud" - I mean from the point of view of a > > spce with at lest two dimentions more that the space (earlier a > > transparency or just a piece of paper) we were thinking of. The > > problem appeares if we talk about the topology of the Universe > > where there are only three spital dimentions. To make it close > > like 3-torus it is essential to introduce 3 more dimentions (it's > > six) to make it possible. Then I say that, that all topoloy thing Only if you *assume* that the only spaces which are physically possible are n-dimensional Euclidean spaces. > > implies that out phisical space is in fact more than three > > dimentional, otherwise we would stand before question: what is > > that thing that out space (universe) lies in. Since this question > > is considered to be sensless, or at least redundant, the only > > solution is to assume the first option. > > It leads to the link with the science of the nature of the space > > itself - like in the string theory, which assumes that our > > spacetime is 10-dimentional - time plus three known spital > > dimentions and 6 other which were reduced to the lenght of Planck > > during the very early stages of evolution of the universe. In that > > case that bending our unverse into a 3-torus around those > > additional axies could be easily (al least theoreticaly) done but > > the "radius" of that bend would be very small. But whatever. > > Is this way of thinking of topology ok ? I think there's still some work to do on your intuition, sorry... > the main problem is that real-world intuition is EXTREMALLY ILLUSIONARY > AND DECEIVING when dealing with spacetime and its curvature. > If you would think in terms of manifolds with given metric (riemann > manifolds) there is ABSOLUTELY NO PROBLEM with 2-torus, 3-torus and so > on. they do not even have to be embedded in other spaces. I would say that ordinary intuition can be used as *one* possible framework for building intuition of other possibilities. But you have to accept that part of your intuition consists of arbitrary assumptions, and consider using other ways of thinking about the space or spacetime which interests you which avoid making these assumptions. > so i strongly recommend http://arxiv.org/abs/gr-qc/9712019 with an > excelent introduction to modern general realtivity Nice :) I added it here: http://www.wikipedia.org/wiki/General_relativity > ps. you will see that you do not have to stretch a transparency to > create a 2-torus, which is a perfect example of a zero-corvature > manifold (PROVIDED that you imply a proper METRIC). > thats the METRIC that is responsible for the curvature, not the SHAPE!!! > Bartek, please read: http://www.wikipedia.org/wiki/Shape_of_the_universe and maybe if you write: http://pl.wikipedia.com/wiki.cgi?action=edit&id=Forma_Wszech%B6wiata in po polsku, then this would help you clarify the question. You could try translating the English version, but rewriting it in the way you understand it and want to explain it. If Michał or I disagree with you, don't worry, we'll correct it sooner or later ;). na razie boud

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what's with that transparency
by szajtan odwieczny 10 Nov '02

10 Nov '02

questions: ---------- 1) does the cosmological principle introduce any restrictions on the topology of the universe. 2) does the topology science relies on some multidimentional theory - like string theory now look here, the problem: ------------ I finally figured it out. It is about that transparency problem. The problem was how it is possible to roll up the transparency into a torus not stretching it so it's curvature remained the same. I was stubborn to say that this was not possible, so we came to the conclusion that the 2-torus in three dimenitonal eucliean space is not an good example of two dimentional closed space of zero curvature. But if we introduced the fourth spital dimention than it becomes possible. Like for example we imagine a string which has one pair of zero dimentional ends, we need at least one more dimentional space (it's two) to make it closed (to joing the ends), so in case of a flat plane (or paper) which has two pairs of one dimentional ends which we have to join, then we need two more dimentions to do that. That would be 4 dimentions. Thus in four dimenitons matematically we can do with the topology of the space whatever we want and it's curvature will still be the same. But the problem now is how to put this idea into the real world. All these considerations about the topology of space were made from "the higer groud" - I mean from the point of view of a spce with at lest two dimentions more that the space (earlier a transparency or just a piece of paper) we were thinking of. The problem appeares if we talk about the topology of the Universe where there are only three spital dimentions. To make it close like 3-torus it is essential to introduce 3 more dimentions (it's six) to make it possible. Then I say that, that all topoloy thing implies that out phisical space is in fact more than three dimentional, otherwise we would stand before question: what is that thing that out space (universe) lies in. Since this question is considered to be sensless, or at least redundant, the only solution is to assume the first option. It leads to the link with the science of the nature of the space itself - like in the string theory, which assumes that our spacetime is 10-dimentional - time plus three known spital dimentions and 6 other which were reduced to the lenght of Planck during the very early stages of evolution of the universe. In that case that bending our unverse into a 3-torus around those additional axies could be easily (al least theoreticaly) done but the "radius" of that bend would be very small. But whatever. Is this way of thinking of topology ok ? further questions: ------------------ For the curvature matter, there is general theory of relativity which connects the energy density with curvature. What about the topology ? What is that force that shapes the topology of space, and can the topology change in time ? pozdrawiam Bartek

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Workshop on Particle Physics...
by Bartosz Lew 03 Nov '02

03 Nov '02

I found some stuff from International Workshop on Particle Physics and the Early Universe (Chicago, Illinois, USA September 18-21, 2002) page with variety of topics concerning cosmology - but there is only the stuff presented on that workshop (no papers). http://pancake.uchicago.edu/~cosmo02/sched/parallel.html ******************************************************************************* Bartosz Lew e-mail: szajtan(a)astri.uni.torun.pl szajtan(a)phys.uni.torun.pl Centrum Astronomii UMK szajtan(a)poczta.onet.pl TORUN, POLAND www: http://www.phys.uni.torun.pl/~szajtan *******************************************************************************

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Shape-univ November 2002