The Cosmology of the Nonsymmetric Theory of Gravitation (NGT)

Transcription

1 The Cosmology of the Nonsymmetric Theory of Grvittion () By Tomislv Proopec Bsed on stro-ph/ nd on unpublished wor with Wessel Vlenburg (mster s student) Bonn, 8 Aug 005

2 Introduction: Historicl remrs : metric tensor is not symmetric g =g +B g =g, B () = g = [ ] ( g g νµ ) In 95 Einstein proposed it s unified theory of grvity nd electromgnetism It does not wor since () Geodesic eqution does not reproduce Lorentz force (b) Equtions of motion do not impose divergenceless mgnetic field In 979 Mofft proposed it s generlised theory of grvittion: Nosymmetric Theory of Grvittion () Advntges of - Unlie Einstein s Theory of Generl Reltivity, contins no singulrities? Mofft

3 Introduction: Absence of singulrities 3 E.g. most generl sttic sphericlly symmetric solution m l m = Ω ds dt dr r d r r r From geodesic eqution: when l >m test prticle motion stops t r=l >m before it reches event horizon Problems with Nosymmetric Theory of Grvittion () () When quntised, in its simplest disguise, contins ghosts (b) There is instbility (growing mode) even in Minowsi bcground Fix: me the nonsymmetric field mssive Tht wy one gets rid of ghosts nd unstble mode

4 Introduction: Modified Newton Lw Mofft (00) hs proposed tht cn explin the flt rottion curves of glxies without invoing dr mtter GM M r r 0 0 = exp + + r M r0 r0 such tht t short distnces GM = 0 (r<< r 0) r nd t lrge distnces GM r M M 0 =, G = G (r r + >> 0) 0 Mofft tes r pc, M 0 M to explin flt rottion curves 0 0 Sun 5 To explin light lensing in clusters, Mofft tes different M 0 M Also: Pioneer 0 & ccelertion nomly (Mofft 00) 0 Sun

5 Introduction: Glxy rottion curves Fits to the rottion curves for the glxies: NGC560, NGC 903, NGC 565 nd NGC 5055 (Mofft 00) NGC v(m/s) 50 v(m/s) 0 00 NGC r(pc) NGC r(pc) NGC v(m/s) v(m/s) r 0 M pc, 0 M 0 Sun r(pc) r(pc)

6 Action 6 In the we B-field limit, the ction cn be written s S = S + S + S HE The Hilbert-Einstein ction mtter ( Λ) SHE = d x g R+ 6πGN S = d x g g g g H H m g g B B µα νβ ργ µα νβ ρ αβγ B αβ Field strength H = B + B + B mss is dded for stbility (Kunstter et l 98; Dmour, Deser, McCrthy 993), but it lso rises from cosmologicl term Λ, since The ction of the Nonsymmetric Theory of Grvittion (to nd order) µα ν µα νβ c R B Bα c' R αβb B +..) ρ µ νρ ν ρµ ρ µα g = g B Bµα +.. At linerised order, first term is guge invrint, while the whole ction my be diffeomorphism invrint (with the pproprite choice of constnts c, c ) B B + Λ Λ µ ν ν µ

7 Conforml spce-times 7 The (symmetric prt of) the metric tensor in conforml spce time is g =, = η η dig(,-,-,-) The ction is then = scle fctor S d x η η η H H m η η B B µα νβ ργ µα νβ ρ αβγ B αβ Note tht in the limit -> (lte time infltion), the inetic term drops out, nd the field fluctutions cn grow without limit This is opposite of sclr field ction (inetic term), for which fluctutions get frozen in Ssclr d x ( µ )( α ) µα η φ φ

8 Physicl Modes 8 Consider the electric-mgnetic decomposition of the Klb-Rmond B-field 0 E E E3 B E 0 B3 B = E B3 0 B E B B 0 Ei = spin, prity - Bi = spin, prity + 3 equtions of motion ( + mb) E= 0 ' ( + mb) B ( ηb+ xe) = 0 Lorentz guge (consistency) condition η µ B νρ = 0 implies ηe x B = 0, E = 0 T NB: B= 0 is missing B my be dynmicl NB: B T eqution is not independent (given by the trnsverse electric field) L NB3: E = 0 T T T T NB: From ηe x B = 0 B is function of E Physicl DOFs (mssive cse): pseudovector B (spin =, prity = +) Physicl DOF (mssless cse): longitudinl mgnetic field B L

9 Cnonicl quntistion 9 Impose cnonicl commuttion reltion on B-field nd its cnonicl momentum L 3 d i.x L ( ) * B x ( ) e () B ( )b,η = η ε η + B ( η)b 3 (π) 3 3 b,b ' = ( π) δ ( ') L ε () = 0 Momentum spce eqution of motion for the modes η '' ' + + B ( ) = 0 η NB: Contrry to sclr field (which decys with the scle fctor), the Klb-Rmond B-field grows with the scle fctor

10 Vcuum fluctutions in de Sitter infltion 0 In De Sitter infltion the scle fctor is, '' ' 0 = = ( η < / H I) H η I such tht mode eqution of motion reduces to tht of conforml vcuum η + B ( η) = 0 The conforml rescling of the longitudinl B-mode L L L B(x,η) B(x,η) B(x c,η) = The mode functions re those of conforml vcuum B( η) = e NB: In conforml vcuum there is very little prticle production during infltion ± iη

11 Rdition er In rdition er, = H η ( η>/ H ) such tht the mode eqution I + B ( η) = 0 η η I which is Bessel s eqution of index 3/. The solution is (Bunch-Dvies) i i B( η) = α e + + e β η η iη + iη with the Wronsin condition α β = The mtching coefficients re α H = H H I I I i /HI i e, β H = I NB: For superhorizon modes t the end of infltion (=): << H I

12 Mtter er HI In mtter er, = ( η +η ) eq ( η>ηeq) nd the mode eqution η eq 6 + B ( η) = 0 η (η + η eq) which is Bessel s eqution of index 5/, whose solution is 3i 3 iη 3i 3 + iη B( η) = e e, γ + + ( ) δ ( ) η η η η η = η+η eq with the Wronsin condition nd the mtching coefficients γ δ = γ α i β iηeq iη eq + iηeq e = e e + + η 8( ) eq ηeq 8( ηeq) δ α β i + iηeq iη eq + iηeq e = e + e 8( η ) 8( ) eq ηeq ηeq

13 Spectrum of energy density 3 The stress energy tensor of Klb-Rmond field is stndrd, T δs = g δg The energy density is m 0 B ( ) ( T = B+ E + ( B) + m E B ) η x + ρ B When the only contribution to the spectrum comes from long. B-field 0 d 0 T 0 0 = P (,η ) 3 0 P (, η) = T 0 (, η) π H I L ' L L P (,η) = B( ) B( ) + + ( + m B ) B( η η η η) π NB: this spectrum is relevnt for coupling to (Einstein) grvity NB: for coupling to (CMB) photons, L d 0 B (x,η ) 0 = P' (,η ) my be relevnt

14 Spectrum in rdition er The spectrum in rdition er for mssless Klb-Rmond field H I sin( η) cos( η) P (,η) = + 8π ( ) η η ( η) 8π HI P P P (rdition er) on superhorizon scles (η<) the energy density spectrum scles s P~². on subhorizon scles (η>), P ~ const.+ smll oscilltions η η NB: The energy density in the field, ρ = d P / is dominted by the ultrviolet, where it exhibits log divergence, HI ρ ln ( mxη) 8π To be compred with the yer WMAP CMB spectrum (003)

15 Comprison with grvittionl wve spectrum 5 L mssless field spectrum: d η =,η 0 [B (x, )] 0 P' ( ) H I P' (,η) = sin( ) cos( ) + + ( ) η ( ) η 8π η η η 8π HI P' 0. ln(p GW ) ln(pp NST ) (rdition er) P' ( ) sin ( η) GW,η PP NST η η NB: field is importnt t horizon crossing, grvittionl wves dominte on superhorizon scles

16 Spectrum in mtter er 6 The spectrum in mtter er is shown in figure (log-log plot) 8π HI P 00 0 P, z rec =089) P, z=0 P, z=0 on superhorizon scles (η<) the energy density spectrum scles s P~². P on subhorizon scles (<η</eq), P~/² η 0 00 η on subhorizon scles (η>/eq), P~const. NB: The bump in the power spectrum in mtter er is cused by the modes which re superhorizon t equlity, nd which fter equlity begin scling s nonreltivistic mtter P / NB: The log divergent prt of the spectrum continues scling in mtter er s, such tht the energy density becomes eventully dominted by the bump HI ρ 3 8π 3 eq P

17 Rdition er: mssive B field spectrum 7 L B Hnel function ( de Sitter infltion) Whitter function ( rdition er) L B Lrge B field mss Smll B field mss

18 8 Rdition er: mssive B field spectrum () Smll B field mss Lrge B field mss

19 Discussion 9 Since during mtter er, ll moment with <, corresponding to tody s (comoving) momentum, < H z / ( z 330) phys 0 eq eq scle s nonreltivistic mtter, the mplitude of their perturbtions grow, which begs the question: Cn the nonsymmetric field be DARK MATTER CANDIDATE? η eq ( lphys π/ ~ x50mpc phys π ) ANSWER: PROBABLY NOT (similrly s primoridil grvittionl wves cnnot) Bsiclly, the field lcs power on smll scles) It hs been suggested tht cn provide n explntion for DARK ENERGY. Our results for the energy density scling show tht this is not the cse. The simplest coupling to photons, δl bres guge invrince. EM η µα η νβ FBαβ The relevnt spectrum is in this cse, P' which is scle invrint P /, on superhorizon scles, nd scles s /² on subhorizon scles Introducing smll mss mb (less thn horizon) drmticlly (dditionl power on smll scles) does not lter these conclusions A more complete study of the spectr of geometric s is subject of forthcoming publiction