I want to compare eigenmodes of spherical space with CMBr using Mathematica, but do not know where to start. Is there anybody who can help? Thanks
http://arxiv.org/abs/gr-qc/0205009
The anisotropies of the CMB are a function defined on the celestial sphere. They can be decomposed in spherical eigenmodes (also called spherical harmonics) in much the same way that a signal of time can be decomposed in Fourier modes. The spherical harmonics are just the solutions of the spherical part of the Laplace operator. An orthogonal set of solutions is of the form
Y^m_l(\phi, \theta) = e^{im\phi) P^m_l(\cos(\theta)).
Where P^m_l is the Legendre Polynomial, l= 0,1,2,3,... and -l
@Rogier:thank you so much for your answer, perhaps do you have a paper on this topic? Thanks
@Henk: yes it could be that the universe resembles a cube, or dodecahedron or other simple forms.
@Rogier: i just found a paper discussing the sound of unusual drum, but still trying to figure out how to compare it with CMBR. Do you perhaps other idea? Thanks
What I described is simply the theory of spherical harmonics and all I am saying is that since the temperature fluctuations on the CMB are a signal on the celestial sphere, they can be decomposed in spherical harmonics
In the context of the CMB, have a look at this series of lectures of Ruth Durrer
http://arxiv.org/pdf/astro-ph/0109522.pdf
in particular section 4.3 (she also has a book on the subject of the CMB).
The idea that the universe has a non trivial topology was popularised by Jeff Weeks who wrote a nice article about it in the notices of the AMS http://www.ams.org/notices/200406/fea-weeks.pdf. One of the ideas is that a non trivial topology allows compact (i.e. finite!) spaces with negative curvature. None of that seems to be observationally confirmed and that the visible universe is within the errors of measurement flat. My personal take on that is that is better to think of that as saying that the visible universe is small compared to the curvature scale of the universe proper (there is no reason that it is not say spherical with a radius of curvature 10^20 times the radius of the visible universe) and that we may never find out what the real geometry and topology of the universe is. See this discussion https://www.researchgate.net/post/The_flatness_problem_or_why_do_we_think_the_observable_universe_is_big (which I have to read up on)
You may also want to have a look at figure 19 in the Planck paper
http://arxiv.org/pdf/1303.5062v1.pdf
which I got pointed at through this nice stack exchange post
http://physics.stackexchange.com/questions/66496/recent-planck-probe-results-how-unexpected
@Rogier: thank you, i have tried to check figure 19 at the Planck report, but it seems too confusing. What i am trying to figure out is how to compare calculation of eigenmodes of spherical space and CMBR anisotropy. Can you suggest a paper on this issue?
@Mohamed: yes, Prof. Luminet from OBSPM popularized the idea that Poincare's dodecahedron may reflect the topology of the universe. You can find some of his papers on this topic at arxiv.org
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