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# College Geometry MATH 355

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This 3 page Study Guide was uploaded by Jeromy Torphy on Monday October 5, 2015. The Study Guide belongs to MATH 355 at California State University - Long Beach taught by Scott Crass in Fall. Since its upload, it has received 46 views. For similar materials see /class/218787/math-355-california-state-university-long-beach in Applied Math And Statistics at California State University - Long Beach.

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Date Created: 10/05/15

Dodecahedral space topology as an explanation for weak wideangle temperature correlations in the cosmic microwave background JeanPierre Luminell Je rey R Weeksz Alain Riazuelo3 Roland Lehoucnl 3 81 JeanPhilippe Uzanquot IObservatoire de Paris 92195 Meudon Cedex France 215 Farmer Street Canton New York 13617 1120 USA 3CEASaclay 91191 Gif sur Yvette Cedex France 4Laboratoire de Physique Theorique Universite Paris XI 91405 Orsay Cedex France The current standard model of cosmology posits an in nite at universe forever expanding under the pressure of dark energy First year data from the Wilkinson Microwave Anisotropy Probe WMAP con rm this model to spectacular precision on all but the largest scalesquot Temperature correlations across the micro wave sky match expectations on angular scales narrower than 60 but contrary to predictions vanish on scales wider than 60 4 Several explanations have been proposedMA One natural approach questions the underlying geometry of space namely its curvature5 and topology l In an in nite at space waves from the Big Bang would ll the universe on all length scales The observed lack of temperature correlations on scales beyond 60 means that the broadest waves are missing perhaps because space itself is not big enough to support theme Here we present a simple geometrical model of a nite space the Poincare dode cahedral space which accounts for WMAP s observations with no ne tuning required The predicted density is 20 s 14013 gt 1 and the model also predicts temperature correlations in match ing circles on the 4 Temperature uctuations on the microwave sky may be expressed as a sum of spherical harmonics just as music and other sounds may be expressed as a sum ofordinary harmonics A musical note is the sum ofa fundamental a second harmonic a third harmonic and so on The relative strengths of the harmonics the note s spectrum determines the tone quality distinguishing say a sustained middle C played on a ute from t e same note played on a clarinet Analogously the temperature map on the microwave sky is the sum of spherical harmonics The relative strengths of the harmo nics the power s ectrum is a signature of the physics and geo metry of the Universe Indeed the power spectrum is the primary tool researchers use to test their models predictions against observed realit The in nite universe model gets into trouble at the low end of the 2 3 letters to nature power spectrum Fig 1 The lowest harmonic the dipole with wavenumber I 1 is unobservable because the Doppler effect of the Solar System s motion through space creates a dipole 100 times stronger swamping out the underlying cosmological dipole The as would be expected in an in nite at space The probability that this could happen by mere chance has been estimated at about 02 ref 2 The octopole term with wavenumber l 3 is also weak at 72 of the expected value but not nearly so dramatic or signi cant as the quadrupole For large values of l ranging up to l 900 and corresponding to small scale temperature uctuations the spec trum tracks the in nite universe predictions exceedingly well Cosmologists thus face the challenge of nding a model that accounts for the weak quadrupole while maintaining the success of the in nite at universe model on small scales high I The weak wide angle temperature correlations discussed in the introductory paragraph correspond directly to the weak quadrupole Microwave background temperature uctuations arise primarily but not exclusively from density uctuations in the early Universe because photons travelling from denser regions do alittle extra work against gravity and therefore arrive cooler while photons from less dense regions do less work against gravity and arrive warmer The density uctuations across space split into a sum of three dimen sional harmonics in effect the vibrational overtones of space itself just as temperature uctuations on the sky split into a sum of two dimensional spherical harmonics and a musical note splits at space because it de nes a preferred length scale in an otherwise scale invariant space A more natural explanation invokes a nite universe where the size of space itself imposes a cut off on the wavelengths Fig 2 Just as the vibrations of a bell cannot be larger than the bell itself the density uctuations in space cannot be larger than space itself Whereas most potential spatial topologies fail to t the WMAP results the Poincare dodecahedral space ts them very well The pacei aJ J J blockofspace with opposite faces abstractly glued together so objects passing out of the dodecahedron across any face return from the opposite face Light travels across the faces in the same Way so if We sit inside the Figure 2 Wavelengths of density fluctuations are limited by the size of a flnite wraparound39 universe a Atwo dimensional creature living on the surface of a cylinder travels due east eventually going all the way around the cylinder and returning to her starting point h lfwe cut the cylinder open and flatten it into a square the creature39s path goes out of the squares right side and returns from the left side c A flat torus is like a cylinder only nowthetop and bottom sides connect as well as the left and right It Waves in a torus universe may have wavelengths no longer than the width ofthe square itself To construct a multiconnected three dimensional space start with a solid polyhedron for Figure1 Comparison of the WMAP power spectrum to that of l 39 39 d d k d 39 space and an inflnite flat universe Atthe low end of the power spectrum WMAP39s results plack pars match the Poincare dodecahedral space light grey better than they match the expectations for an inflnite flat universe dark grey Computed for 0m 0 28 and 0A 0 734 with Poincare space data normalized to the I 4 term NATURE lVDL 425 l 9 OCTOBER 2003 l WMnaturecomnature example a cube and identify its faces in pairs so that any object leaving the polyhedron t at oh ok39 a m k l w l l how the faces are identifled Nevertheless the same principle applies that letters to nature dodecahedron and look outward across a face our line of sight re enters the dodecahedron from the opposite face We have the illusion of looking into an adjacent copy of the dodecahedron If we take the original dodecahedral block of space not as a euclidean dodecahedron with edge angles 117 but as a spherical dodeca hedron with edge angles exactly 120 then adjacent images of the dodecahedron t together snugly to tile the hypersphere Fig 3b analogously to the way adjacent images of spherical pentagons with perfect 120 angles t snugly to tile an ordinary sphere Fig 3a The power spectrum of the Poincare dodecahedral space depends strongly on the assumed mass energy density parameter 20 Fig 4 The octopole term I 3 matches WMAP s octopole best when 1010 lt 20 lt 1014 Encouragingly in the subinterval 1012 lt 20 lt 1014 the quadrupole l 2 also matches the WMAP value More encouragingly still this subinterval agrees well with observations falling comfortably within WMAP s best t range of 20 102 i 002 ref 1 The excellent agreement with WMAP s results is all the more striking because the Poincare dodecahedral space offers no free Figure 3 Sphericai pentagons and dodecahedra fit snugiy uniike their euciidean counterparts a 12 sphericai pentagons tiie the surface of an ordinary sphere They fit together snugiy because their corner angies are exactiy 120 Note that each sphericai pentagon iS iust a pentagonai piece of a sphere b 120 sphericai dodecahedra tie the surface of a hypersphere A hypersphere iS the three dimensionai surface of a four dimensionai baii Note that each sphericai dodecahedron isiust a dodecahedrai piece of a hypersphere The sphericai dodecahedra fit together snugiy because their edge angies are exactiy 120 In the construction of the Poincare dodecahedrai space the dodecahedron39s 30 edges come together in ten groups of three edges each forcing the dihedrai angies to be 120 and requiring a sphericai dodecahedron rather than a euciidean one Software for VisuaiiZing sphericai dodecahedra and the Poincare dodecahedrai space iS avaiiabie at httpwwwgeometrygamesorgCurvedSpaces 594 parameters in its construction The Poincare space is rigid meaning that geometrical considerations require a completely regular dode cahedron By contrast a 3 torus which is nominally made by gluing opposite faces of a cube but may be freely deformed to any parallelepiped has six degrees of freedom in its geometrical con struction Furthermore the Poincare space is globally homo geneous meaning that its geometry and therefore its power spectrum looks statistically the same to all observers within it By contrast a typical nite space looks different to observers sitting at different locations Con rmation of a positively curved universe 20 gt 1 would require revisions to current theories of in ation but it is not certain how severe those changes would be Some researchers argue that positive curvature would not disrupt the overall mechanism and effects of in ation but only limit the factor by which space expands during the in ationary epoch to about a factor of teng Others claim that such models require ne tuning and are less natural than the in nite at space model9 Having accounted for the weak observed quadrupole the Poin care dodecahedral space will face two more experimental tests in the next few years 1 The Cornish Spergel Starkman circles in the sky method7 predicts temperature correlations along matching circles in small multiconnected spaces such as this one When 20 at 1013 the horizon radius is about 038 in units of the curvature radius while the dodecahedron s inradius and outradius are 031 and 039 respectively in the same units In this case the horizon sphere self intersects in six pairs of circles of angular radius about 35 making the dodecahedral space a good candidate for circle detection if technical problems galactic foreground removal inte grated Sachs Wolfe effect Doppler effect of plasma motion can be overcome Indeed the Poincare dodecahedral space makes circle searching easier than in the general case because the six pairs of matching circles must a priori lie in a symmetrical pattern like the faces of a dodecahedron thus allowing the searcher to slightly relax the noise tolerances without increasing the danger of a false positive 2 The Poincare dodecahedral space predicts 20 at 1013 gt 1 The upcoming Planck Surveyor data or possibly even the existing WMAP data in conjunction with other data sets should determine 20 to within 1 Finding 20 lt 101 would refute the Poincare space as a cosmological model while 20 gt 101 would provide strong evidence in its favour Figure 4 Vaiues of the mass energy density parameter 00 for which the Poincare dodecahedrai space agrees with WMAP39s resuits The Poincare dodecahedrai space duadrupoie trace 2 and octopoie trace 4 fit the WMAP duadrupoie trace 1 and octopoie trace 3 when 1012 lt 00 lt 1014 Larger vaiues of 00 predict an unreaiisticaiiy weak octopoie To obtain these predicted vaiues we first computed the eigenmodes of the Poincare dodecahedrai space using the 39ghost method39 of ref 10 with two of the matrix generators computed in Appendix B of ref 11 and then appiied the method of ref 12 using 0m 028 and 0A no 7 028 to obtain a powerspectrum and to simuiate sky maps Numericai iimitations restricted our set of three dimensionai eigenmodes to wavenumbers klt 30 which in turn restricted the reiiabie portion of the power spectrum to I 2 3 4 We set the overaii normaiization factorto match the WMAP data at I 4 and then examined the predictions for I 2 3 NATUREVOL 425 9 OCTOBER 2003 Wwwnaturecomnature nit nr 1n observational Since antiquity humans have wondered whether our Universe is nit N l 39 39 l 39 i Rzrnvzd 23 lune arrrptrd 2t ruly zoos der to IOSXnaturro 1944 Batman c 1 zlal noryear Wilkerson Miuowavamwuopy Prob lwm l oaremaonr d bane ramks Aslmply r 5qu Ma 127 am it year w neon Miaowave Anisouopy Prob lwtvm l observauom indiecm letters to nature In the investigation of anomalous metallic properties often referred to as non Fermi liquid phenomena complications of aiiou 39 l 4 39 39 39 39 These can be related for example to band structure anomalies to W T L magneto to disorder effects In other materials the emergence of superoon ductivity may make it difficult to probe the fundamental nature of the normal state of the underlying electronic system In order to r r r r r conclusively assess and clarify our ideas about non Fermi liquid amso a tee so 01 Amopart y 0700le L 1 v 1 n 4 39 L 39 39 purity simplicity and convenience The itiner 00 M d 0 of m M ant electron ferromagnet nSi appears to be one such example 5 m tootsie erroneous nip aasiylamx spa umpires anNaLRAslmrtSae mums W Un er ambient conditions ofpressure and applied magnetic field 5 n on A 1 r1 1r nSi trrta mall rs WM v r a a t i i i 7 Emir er mm 5 Evd m huw mmsww mwh 391 mm temperature Tc of295 K ref 11 At low temperatures it exhibit backyoundraduum czar Quarava reassreamlieet H d 8 r1 1 v v Li V1 11 anze soe tin rite pree lmda a canwe have in ation With 2 gt 1 Carmel Aslmpam ergo 0200212003 daemon E kmmma39t er ragrrn er a memorial gy ciao QuartL va 19468347081200 mam E oeq 1 Lumma39t r r Um H amber topologeal lmsmg in spherical spaces ciao uartLGmu lasissesitelmv a r in 1 v 1 us rnulaeomeeted spaces were Rev 1 tin rhepree pxapzmlal 39lnpHarmvotyamorphDZIZDD lam Acknowledgements litw thanlo the MacArthur Feundanen far mpperr Cnmnelin interests statement Th autherr drriarr thatthry have no remprnng ananrial mtnzlll Femt39 39quid breakdown 39n the paramagnetic phase oi a pure metal Dnlmn Leyralltl I R Walker L 39I39allleferl Ill J Slelner S R JIIllan 8r Ii Ii Lumrlch Cavcndtrh Labmamvy Unwev ry efCambndge Cambridge CB3 0H K Dc parrcmcnrdc Physique Unwev f dc Shevbmaki Quebec1K 2R1 Canada Fermi liquid theory1 the standard model of metals has been challenged by the discovery of anomalous properties in an increasinglylarge number otmetals The anomalies often occur near a quantum critical point a continuous phase transition in and paramagnetic phasesr Although not understood in detail was anticipated nearly three decades ago by theories going beyond the standard model2 5i Here we report electrical resis tivity measurements of the 3d metal Mnsi indicating an un expected breakdown oftbe Fermi liquid model it crossover region close to a quantum criti normally expected to tail but over a wide re on of the p diagram ne der corrections to the Fermi liquid model are expected to be small T e range in pressure temperature and app ed magnetic field the electrical resistivity in Mnsi is not consistent with the 4 4 4 44 1 war This may suggest the emergence of a well de ned but enigmatic quantum phase of matterr protum V0 42519 OCJ39OEER zoos lwwwnamrr remnature r F0 Fermi liquid dominated by moderately renormalized 3d bands With its full three dimensional cubic crystal structure th e tronic and magnetic properties of Mnsi are essentially isotropic except for the effects ofa well understood long wavelength helical twist of the ferromagnetic order characteristic of crystalline struc tures lacking inversion symmetry such as the B20 lattice oanSi ref 12 The possibility ofproducing ultra pure single crystals oanSi t i i i i t I p I 113 kblr e g 40 km 3 Ambient 3 pronun pltpc 50 100 150 7 K I39 a 25 E 150 m1 3 E 2 3 21 5m 5 n r m so 2 275 khlr rm D n 5 1n 1 7 tK39quot Figure 1 Dependence on temperature We electrrcal resrstwrty of MnSl lne temperature dependent part Ag crtne resrstwrty above me resrddal resrstrvrty a p a N1 a mwlmtlr Mam rne 11 N1 a calenr rnecrrrrcal nre nre rnarr r mnm mnnrn rnenre nre are lEJJ l8l l 96 253 and 275 kbar The Clear depamlre from a quadratic behaviour is Shown in the msetfor lEJJ l96 and 275 kbar 595

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