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ScienceWeek
COSMOLOGY: ON THE SHAPE OF THE UNIVERSE
The following points are made by George F. Ellis (Nature 2003 425:566):
1) What shape is space? In thinking about the large-scale shape of the Universe, three interlinked questions must be confronted. First, what is its spatial curvature? There are three possible answers. Three-dimensional sections of space-time may be "flat" -- in such space sections, parallel lines stay the same distance apart and never meet (as in Euclidean space). Or the space sections may be "negatively curved", such that parallel lines diverge from one another and never meet (the three-dimensional analogue of a "Lobachevsky space"). Finally, they may be "positively curved", such that parallel lines converge and eventually intersect (the three-dimensional analogue of the surface of a sphere). The particular case that exists depends on how well the amount of matter in the Universe, coupled with the driving force of dark energy, balances the Universe's kinetic energy of expansion. This is usually expressed in terms of the normalized density parameter Omega(sub-zero), which is unity for flat space sections; for positive spatial curvature, Omega(sub-zero) is greater than one.
2) The second question is whether the Universe is "open" or "closed" -- that is, is it spatially infinite, containing an infinite amount of matter, or is it spatially finite, containing a finite amount of matter? Positively curved space sections are necessarily closed, but the converse does not necessarily follow: both flat and negatively curved space sections can be finite if their connectivity is more complicated than in Euclidean space, meaning that their topology is quite unusual(3,4) For example, in a flat toroidal space, as you exit right you enter left, and space is finite. So the third issue is, what is the large-scale topology of the Universe?
3) It is worth noting that none of these features is determined by the Einstein gravitational field equations, which are differential equations that govern local, rather than global, properties of space-time(5). Topology and curvature seem to be fixed by the initial conditions at the start of the Universe that have since determined its dynamical evolution. To investigate the topology and curvature of the Universe, we must use astronomical observations; from observed values of the energy densities and the expansion rate in the Universe, the curvature can be deduced using Einstein's field equations.
4) Luminet et al(1) suggest that the topology of the Universe may be a "Poincaré dodecahedral space". They demonstrate that this topology, unlike many others, is supported by data from NASA's Wilkinson Microwave Anisotropy Probe (WMAP), published earlier this year(2).
References (abridged):
1. Luminet, J.-P., Weeks, J. R., Riazuelo, A., Lehoucq, R. & Uzan, J.-P. Nature 425, 593-595 (2003)
2. Bennett, C. L. et al. Astrophys. J. Suppl. Ser. 148, 1-27 (2003)
3. Ellis, G. F. R. Gen. Rel. Grav. 2, 7-21 (1971)
4. Lachieze-Rey, M. & Luminet, J. P. Phys. Rep. 254, 135-214 (1995)
5. Friedmann, A. Z. Phys. 21, 326-332 (1924); transl. Gen. Rel. Grav. 31, 2001-2008 (1999)
Nature http://www.nature.com/nature
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COSMOLOGY: OPEN, CLOSED, OR FLAT UNIVERSE?
The following points are made by Marc Kamionkowski (Science 1998 280:1397):
1) Determination of the geometry of the universe has been a central goal of cosmology ever since Hubble discovered its expansion 75 years ago.
2) The central question is whether the universe is a multi-dimensional equivalent of a 2-dimensional surface ("flat"), a sphere ("closed"), or a saddle ("open"). The geometry, in the context of current theory and observations, determines whether the universe will expand forever or eventually collapse.
3) Until now, most astronomers have pursued the geometry by attempting to measure the mass density of the universe. According to general relativity, if the density is equal to, larger than, or smaller than a critical density fixed by the expansion rate, then the universe is flat, open, or closed, respectively.
4) Another possibility is to look directly at the predicted observational effects of a curved (open or closed) universe versus a flat universe, and in particular at the angular power spectrum of the cosmic microwave background. The author suggests that in the near future a new generation of experiments will provide substantial advances in these observations, enabling more definitive statements about the geometry of the universe, and that these results will in turn provide clues to the new particle physics required to understand the inflation phase following the Big Bang origin of the universe.
Science http://www.sciencemag.org
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ON PRECISION COSMOLOGY
The following points are made by S.L. Bridle et al (Science 2003 299:1532):
1) The recent announcement by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite team of their landmark measurements of the cosmic microwave background (CMB) anisotropy (1-3) has convincingly confirmed important aspects of the current standard cosmological model. The results show with high precision that space is flat (rather than curved) and that most of the energy in the Universe today is "dark energy", which is gravitationally self-repulsive and accelerates the expansion of the universe. The evidence is independent of supernovae results (4,5).
2) The measurements strongly indicate that the amplitudes of spatial variations in density and temperature that seeded the formation of galaxies were roughly independent of length scale, adiabatic (all forms of energy have the same spatial variation), and followed a Gaussian distribution -- just as predicted by the standard Big Bang inflationary model. WMAP heralds a new age of precision cosmology with careful error analysis, tightly constraining many key parameters. For example, the lifetime of the universe has been determined to be 13,400 ± 300 million years. Furthermore, WMAP's new measurement of the CMB polarization as a function of angular scale shows that the epoch of cosmic re-ionization -- associated with the formation of the first stars -- had already occurred when the Universe was several hundred million years old.
3) At the same time we celebrate this triumph, it is important to recognize that important issues remain. For example, it is not yet clear whether the spectrum of temperature fluctuations is truly consistent with inflation. The spectrum is roughly scale-invariant, but there are hints of peculiarities, and a key inflationary prediction -- the presence of gravitational wave effects -- has not yet been observed. We also do not know whether dark energy is due to an unchanging, uniform, and inert "vacuum energy" (also known as a "cosmological constant") or a dynamic cosmic field that changes with time and varies across space (known as "quintessence"). "Dark matter", which is gravitationally self-attractive, also remains mysterious: We do not yet know its nature, nor are we certain about its density or the amplitude of the initial ripples in its distribution.
4) Today's standard theoretical paradigm is the inflationary Big Bang model. According to this picture, the universe began in a state of nearly infinite temperature and density and almost immediately entered a phase of rapid, accelerated expansion ("inflation"). This expansion smoothed out the distribution of energy, flattened any initial warp or curvature in space, and created tiny variations in density. To transform these density variations into the gravitationally collapsed, complex structures we see today, it is essential that there be "dark matter" as well as ordinary (baryonic) matter. Finally, we need dark energy to account for the measured total energy density and to explain the current cosmic acceleration.
References (abridged):
1. C. Bennett et al., in preparation ( http://arXiv.org/abs/astro-ph/0302207 ).
2. G. Hinshaw et. al., in preparation ( http://arXiv.org/abs/astro-ph/0302217 ).
3. A. Kogut et. al., in preparation ( http://arXiv.org/abs/astro-ph/0302213 ).
4. S. Perlmutter et al., Astrophys. J. 517, 565 (1999)
5. A. G. Riess et al., Astron. J. 116, 1009 (1998)
Science http://www.sciencemag.org
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Notes:
WMAP is a NASA Explorer Mission to measure the temperature of the cosmic background radiation
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ON DIMENSIONS AND GEOMETRIES
The following points are made by Lisa Randall (Science 2002 296:1422):
1) We generally take it for granted that we live in a world where there are three infinite spatial dimensions. In fact, we rarely give this fact much thought; we readily refer to left-right, forward-backward, and up-down.
2) Yet the most exciting developments in particle physics in the past few years have involved the recognition that additional dimensions might exist and furthermore might play a role in determining our observable world. New theoretical discoveries are evolving at a very rapid rate. The potential implications range from experimental signatures of extra dimensions, to understanding fundamental questions about the nature of gravity, to new insights into the evolution of our universe.
3) One of the chief motivations for considering additional dimensions came from string theory, which in turn is motivated by the failure of classical gravity to work at very short distance scales or, equivalently, at very high energies, where quantum mechanical effects cannot be neglected. The only known way to consistently reconcile quantum mechanics with Einstein's theory of gravity is string theory, in which the fundamental objects that constitute our universe are not particles but (very tiny) extended objects: "strings". It appears that one can only have a consistent string theory that can describe the known particles if there are many additional spatial dimensions: six or seven, depending on how one looks at it. The question is, then, why don't we see these additional directions? What has become of them? Can they play any role in the physics we see? And is there any chance we will observe them soon?
4) There are two known ways to incorporate additional dimensions of space that are consistent with what we see, or rather what we don't see; namely Kaluza's (1,2) original idea of curling them up into little balls (compactification) or the more recent proposal by Sundrum and Randall of focusing of the gravitational potential in a lower dimensional subspace (localization) (3,4). These ideas are important in and of themselves; one or the other would be the reason we so far haven't observed evidence for extra dimensions, should they exist. Another exciting development in the field of extra dimensions is the fact that even though the dimensions have not yet been seen directly, their existence might explain important features of the observed standard model and will be observed in the near future should these conjectures prove correct.
5) In summary: The field of extra dimensions, as well as the hypothesized sizes of extra dimensions, have grown by leaps and bounds over the past few years. The reasons for the recent activity in this field include the observations that extra dimensions can be macroscopic or even infinite in size. Another new development is the application of extra dimensions to the determination of particle physics parameters and properties.
References (abridged):
1. T. Kaluza, Preuss. Akad. Wiss. 966 (1921)
2. O. Klein, Z. Phys. 37, 895 (1926)
3. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999)
4. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999)
Science http://www.sciencemag.org
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