Outline. I. The Why and What of Inflation II. Gauge fields and inflation, generic setup III. Models within Isotropic BG

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2 Outline I. The Why nd Wht of Infltion II. Guge fields nd infltion, generic setup III. Models within Isotropic BG Guge-fltion model Chromo-nturl model IV. Model within Anisotropic BG Infltion with nisotropic hir model Guge-fltion model V. Conclusion

3 Cosmic Microwve Bckground rdition CMB: blckbody rdition with Temperture T=.7 K. This relic rdition the 1 st snpshot of the universe, turns out to be gold mine of cosmologicl informtion!

4 In 199, the Cosmic Bckground Explorer COBE stellite detected cosmologicl fluctutions in the microwve bckground temperture: T 5 ~ 10 T

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6 013 Plnck Plnck

7 Observtion Vs Vs Stndrd Model Model of of Cosmology Horizon Problem: or universe dominted by P w comoving horizon size is H 1 H w 1 3w w 0 0

8 Observtion Vs Vs Stndrd Model Model of of Cosmology Horizon Problem: or universe dominted by P w comoving horizon size is H 1 H w 0 1 3w w 0 0

9 Observtion Vs Stndrd Model of Cosmology ltness Problem: The RW geometry k 3H 1 k H H 1 H w

10 Observtion Vs Stndrd Model of Cosmology ltness Problem: The RW geometry 3H Now, ssuming d k d ln 1 k 0 k H 0 1 3w 0 H 1 H w Observtions of the CMB nd lrge-scle structure find tht ~ 0! k

11 How Infltion cn solve them? The issue in the stndrd big bng cosmology which leds to these problem is tht: lwys 0 H 1 lwys increses with time!

12 How Infltion cn solve them? The problems in the stndrd big bng cosmology Infltion Ide: 0 There is stge in the erly universe with n ccelerted expnsion: 0 H 1 decreses in the infltionry phse!

13 How Infltion cn solve them? The problems in the stndrd big bng cosmology Infltion Ide: 0 There is stge in the erly universe with n ccelerted expnsion: 0 H 1 decreses in the infltionry phse! Infltion must lst enough for end 0 e 60

14 Wht cuses Infltion? RW metric ds riedmnn equtions dt t dr 1 kr H r d d k 3P ccelertion requires: 3P 0!

15 Slow-roll Infltion In order to ensure enough # e-folds: The Hubble prmeter decreses slowly, nd the universe experiences n pproximtely exponentil infltion. To put this qulittively, we use slow-roll prmeters : H H nd H HH Slow-roll conditions

16 Slow-roll Infltion In order to ensure enough # e-folds: The Hubble prmeter decreses slowly, nd the universe experiences n pproximtely exponentil infltion. To put this qulittively, we use slow-roll prmeters : H H nd H HH Slow-roll conditions 1, 1

17 Meet with Observtions Observtions of CMB: t the time of decoupling, the universe ws very nerly homogeneous with smll inhomogeneties t the level. A nturl strtegy: for ll quntities metric nd mtter fields where Liner perturbtions round the homogeneous bckground: Cosmologicl Perturbtions 10 5

18 Meet with Observtions Observtions of CMB: t the time of decoupling, the universe ws very nerly 10 5 homogeneous with smll inhomogeneties t the level. A nturl strtegy: for ll quntities metric nd mtter fields A A A X t, x X t X t, x, where X A X A t, x A t X t, x Liner perturbtions round the homogeneous bckground: G T Cosmologicl Perturbtions

19 Cosmologicl Perturbtions symmetries of sptilly flt, homogeneous nd isotropic bckground llows for decomposition into sclr, divergence-less vector divergence & trce-less tensor g g S g V g T v T T S T V T T v The equtions of ech prt is independent of the other sectors.

20 Cosmologicl Perturbtions An importnt guge-invrint sclr quntity comoving curvture perturbtion H P Usully vector perturbtions re dmping modes nd unimportnt during infltion. R C 0 0 q h Tensor modes hs polriztions h & without ny prity violting term ij h h h

21 A crucil sttisticl mesure of primordil sclr fluctutions Power spectrum of R, The power spectrum of primordil tensor fluctutions 1 * * 3 s n R R R k k k A k P k k k R k t n t t h t k k k A k P k k * * 3 Sttistics of Cosmologicl Perturbtions

22 A crucil sttisticl mesure of primordil sclr fluctutions Power spectrum of R, The power spectrum of primordil tensor fluctutions 1 * * 3 s n R R R k k k A k P k k k R k t n t t h t k k k A k P k k * * 3 Sttistics of Cosmologicl Perturbtions k k r R t

23 CMB Observtions Current observtions of CMB provide vlues for power spectrum of R, spectrl tilt nd impose n upper bound on tensor to sclr rtio: P n r r R R no evidence of non-gussinity ,, 95%, no running 95%, including running P. A. R. Ade et l. Plnck Collbortion, ``Plnck 013 results. XVI. Cosmologicl prmeters, rxiv: [stro-ph.co].

24 Infltion hs mny reliztions. R. Bouchet: CMB nisotropies, Sttus & Properties

25 Guge ields nd Infltion the role nd consequences of guge fields, theoreticl nd observtionl during infltionry er within Einstein GR with minimlly coupled fields hving vector without the guge symmetry, we expect to hve ghost instbility, so we will NOT consider the vector infltion models. B. Himmetoglu, C. R. Contldi nd M. Peloso, Instbility of nisotropic cosmologicl solutions supported by vector fields, Phys. Rev. Lett

26 Guge ields nd Infltion ds Isotropic Bckground dt e t dx ij i dx j ds Anisotropic Bckground dt e e t ij dx i dx j

27 Guge ields nd Infltion ds Isotropic Bckground dt e t dx ij i dx Guge field vlue on BG j A 0 A 0 e t t Scle fctor

28 Guge ields nd Infltion ds Isotropic Bckground dt e t dx ij i dx Guge field vlue on BG j A 0 A 0 non-abelin Guge field in BG Inflton field e.g. guge-fltion Auxiliry field e.g. chromo-nturl

29 Guge ields in Infltion ds Isotropic Bckground dt e t dx ij i dx Guge field vlue on BG j A 0 A 0 Guge field t the perturbtion level

30 perturbtion level: Guge ields nd Infltion ds Isotropic Bckground dt e t dx ij i dx Guge field vlue on BG j A 0 A 0 g g S g V g T v A A S V A T A T T perfect fluid

31 perturbtion level: Guge ields nd Infltion ds Isotropic Bckground dt e t dx ij i dx Guge field vlue on BG j A 0 A 0 g g S g V g T v A A S V A T A T T perfect fluid

32 Guge ields nd Infltion ds Isotropic Bckground dt e t dx ij i dx j

33 Guge ields nd Infltion ds Anisotropic Bckground dt e e t ij dx i dx j e t t Scle fctor t e ij nisotropy fctor

34 Guge ields nd Infltion ds Anisotropic Bckground dt e e t ij dx i dx Guge field vlue on BG j T T perfect fluid A 0

35 Guge ields nd Infltion Anisotropic Bckground ds dt e e t ij dx i dx j Guge field vlue on BG T T perfect fluid A 0 Anisotropic inerti the metric,. ij is the source of the nisotropies of A. M. nd M. M. Sheikh-Jbbri, Revisiting Cosmic No-Hir Theorem for Infltionry Settings, Phys. Rev. D 85 01

36 Guge ields nd Infltion Anisotropic Bckground ds dt e e t ij dx i dx j Guge field vlue on BG T T perfect fluid A 0 Anisotropic inerti is the source of the nisotropies of the metric, ij. if 0, nisotropies of the metric re dmping exponentilly during slow-roll infltion. A. M. nd M. M. Sheikh-Jbbri, Revisiting Cosmic No-Hir Theorem for Infltionry Settings, Phys. Rev. D 85 01

37 Guge ields nd Infltion Anisotropic Bckground ds dt e e t ij dx i dx j Guge field vlue on BG T T perfect fluid A 0 If 0, then nisotropies cn grow during infltion. Infltion puts n upper bound on the vlue of nisotropy: ij H A. M. nd M. M. Sheikh-Jbbri, Revisiting Cosmic No-Hir Theorem for Infltionry Settings, Phys. Rev. D 85 01

38 Guge ields nd Infltion ds Isotropic Bckground dt e t dx ij i dx j ds Anisotropic Bckground dt e e t ij dx i dx j

39 Isotropic Models with Guge ields Isotropic nd homogenous RW bckground ds non-abelin guge field dt G is ny non-abelin compct group t dx ij i dx Lgrngin of the models 1 L R L m I, A A A gf bc A b T A b A c b where [ T, T b j T b G ] if bc T c

40 Isotropic Models with Guge ields Isotropic nd homogenous RW bckground ds dt Lgrngin of the models 1 L R L m I, t dx ij i dx j f bc bc

41 Isotropic Models with Guge ields In the isotropic nd homogenous RW bckground, with Lgrngin of the form L R L m I,, we hve homogenous nd isotropic solution: 1 A t 0 t i 0 i nd I t I A. M. nd M. M. Sheikh-Jbbri, Guge-fltion: Infltion rom Non-Abelin Guge ields, to pper in PRB, rxiv: [hep-ph]. A. M. nd M. M. Sheikh-Jbbri, Non-Abelin Guge ield Infltion, Phys. Rev. D 84, [rxiv: [hep-ph]].

42 Guge-fltion Model n pproprite choice for non-abelin guge field infltion S d 4 x R g Tr A is non-abelin SU guge field A. M. nd M. M. Sheikh-Jbbri, Guge-fltion: Infltion rom Non-Abelin Guge ields, rxiv: [hep-ph], to pper in PLB. A. M. nd M. M. Sheikh-Jbbri, Non-Abelin Guge ield Infltion, Phys. Rev. D 84, [rxiv: [hep-ph]].

43 Guge-fltion Model n pproprite choice for non-abelin guge field infltion S Yng-Mills term Tr d 4 x R g term 1 4 YM 3P YM P 384 Tr YM YM

44 Guge-fltion Model n pproprite choice for non-abelin guge field infltion homogenous nd isotropic nstz reduced Lgrngin density R g x d S H g g H red L i t t i 0 0 A

45 Guge-fltion Model n pproprite choice for non-abelin guge field infltion homogenous nd isotropic nstz reduced Lgrngin density Slow-roll prmeters R g x d S i t t i 0 0 A H g g H red L 1 H g

46 Guge-fltion Model reduced Lgrngin density Lred H g g H is the effective inflton field which evolves slowly during infltion nd fter the end of infltion, it strts oscillting. 1 g H , i i 10, g.510, A. M. nd M. M. Sheikh-Jbbri, Non-Abelin Guge ield Infltion, Phys. Rev. D 84, [rxiv: [hep-ph]].

47 Guge-fltion Model vs. stndrd single sclr model A. M. nd M. M. Sheikh-Jbbri, Non-Abelin Guge ield Infltion, Phys. Rev. D 84, [rxiv: [hep-ph]].

48 Guge-fltion nd Prity Violtion of Grvittionl Wves e P P R R PL P L g H A. M. nd M. M. Sheikh-Jbbri, Non-Abelin Guge ield Infltion, Phys. Rev. D 84, [rxiv: [hep-ph]].

49 Guge-fltion Model & WMAP results A. M. nd M. M. Sheikh-Jbbri, Non-Abelin Guge ield Infltion, Phys. Rev. D 84, [rxiv: [hep-ph]].

50 Guge-fltion Model & Plnck results 95%, no running P. A. R. Ade et l. Plnck Collbortion, ``Plnck 013 results. XVI. Cosmologicl prmeters, rxiv: [stro-ph.co].

51 Guge-fltion Model & Plnck results 95%, including running P. A. R. Ade et l. Plnck Collbortion, ``Plnck 013 results. XVI. Cosmologicl prmeters, rxiv: [stro-ph.co].

52 Chromo-nturl Model Lgrngin density of the Chromo-nturl model: S d x R g cos f 8 f 4 A is n xion field 0, f is non-abelin SU guge field P. Adshed, M. Wymn, Chromo-Nturl Infltion, Phys. Rev. Lett. 108, M. M. Sheikh-Jbbri, Guge-fltion Vs Chromo-Nturl Infltion, Phys. Lett. B P. Adshed, M. Wymn, Guge-fltion trjectories in Chromo-Nturl Infltion, rxiv: [hep-th].

53 Chromo-nturl Model Lgrngin density of the Chromo-nturl model: S d x A R g 1 4 is n xion field cos 0, f is non-abelin SU guge field f 8 f 4 Tr topologicl term P. Adshed, M. Wymn, Chromo-Nturl Infltion, Phys. Rev. Lett. 108, M. M. Sheikh-Jbbri, Guge-fltion Vs Chromo-Nturl Infltion, Phys. Lett. B P. Adshed, M. Wymn, Guge-fltion trjectories in Chromo-Nturl Infltion, rxiv: [hep-th].

54 Chromo-nturl Model Lgrngin density of the Chromo-nturl model: S d x R g 1 4 Yng-Mills term Axion field term cos f 8 f 4 YM 3P YM P 3 YM YM 4 V 1 cos, f YM

55 Lgrngin density of the Chromo-nturl model: Inserting the isotropic nd homogenous nstz The reduced effective ction Chromo-nturl Model 8 cos f f R g x d S i t t i 0 0 A 3 cos H f g f g H red L

56 Chromo-nturl Model L The reduced effective ction: red g H g 1 cos f f H During the slow-roll infltion: H cos 3 f nd is the inflton field, while infltion possible. 3 4 sin 3gH f mkes the slow-roll

57 Chromo-nturl Model 6 4 The clssicl trjectories with g 10, 400, 710,, f 10 strted from different xion initil vlues, χ0. In both pnels, the solid ornge lines, the dshed red lines nd the dotted brown lines correspond to χ0/f vlues equl to 3π/4, π/ nd 0.01, respectively.

58 Guge ields nd Infltion ds Isotropic Bckground dt e t dx ij i dx j ds Anisotropic Bckground dt e e t ij dx i dx j

59 Anisotropic Infltion Model Anisotropic Bckground Lgrngin density of the Anisotropic Infltion model: is n sclr field is n Abelin U1 guge field V f R g x d S A 4 dz dy e dx e e dt ds t t t / c f e

60 Anisotropic Infltion Model H H the time evlution of nisotropy for vrious c with respect to number of e-folds is shown. M. A. Wtnbe, S. Knno nd J. Sod, Infltionry Universe with Anisotropic Hir, Phys. Rev. Lett. 10, [rxiv: [hep-th]].

61 Summry nd Outlook It is possible to hve non-abelin guge fields in the RW bckground s the inflton field or n uxiliry field Guge fields leds to very rich cosmic perturbtion theory e.g. chirl GW nd non-zero power spectrum for sclr nisotropic inerti Anisotropic infltion nd the growth of nisotropies during infltion violtion of cosmic no-hir conjecture

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