Grand Unification & the Early Universe

Transcription

1 Grand Unification & the Early Universe Wilfried Buchmüller DESY, Hamburg Bielefeld, May 2015

2 Reasons for Grand Unified Theories Forces and matter of the Standard Model naturally fit into the unification (GUT) group SU(5) [Georgi, Glashow 74], or larger extensions, SU(3) SU(5) =, SU(2) with quarks and leptons represented by 10, 5 and (1 = c ), u c u c u d d c 0 u c u d d c B 0 u d C Bd c 0 e c A ( c ). 0 e Hope: extrapolation of SM possible up to the GUT energy scale GUT GeV where weak, electromagnetic and strong forces have equal strength.

3 Effective interactions strengths (couplings ) depend on energy and momentum transfer; unification at GUT scale in SM and supersymmetric extension; hint for supersymmetry? Search at LHC, DM,...!! i

4 Grand Unification & Cosmology `First model of inflation [Guth, 80]: inflation driven by false GUT vacuum energy (problem with subsequent phase transition...) GUT baryogenesis: heavy leptoquark decay (gauge couplings too large, not enough departure from thermal equilibrium... [Kolb, Wolfram 80]) Potential problem: formation of topological defects in GUT phase transition, monopoles domain walls, strings,... Theoretical developments: new ways of GUT symmetry breaking, Higgstype phase transitions only for Abelian subgroups (only cosmic strings; SUSY GUTs and extra dimensions: new inflaton & dark matter candidates; `sphalerons, allows for leptogenesis, natural for GUT scale heavy neutrinos Attractive framework: hybrid inflation with with B-L breaking (difference of baryon and lepton number)[linde; Dvali, et al; Binetruy, Dvali; Halyo; all 94]; potential observables: strings, gravity waves, CMB (large field inflation),...

5 Seesaw & Leptogenesis Lepton number can be violated by masses and couplings of heavy Majorana neutrinos to light neutrinos and charged leptons; after electroweak symmetry breaking, 3 light and 3 heavy neutrinos (seesaw mechanism): N ' R + c R, ' V T L + LV c m N ' M, m ' V T m T D 1 M m DV For hierarchical right-handed neutrinos, after electroweak symmetry breaking, light neutrino masses naturally related to the GUT scale: M 3 GUT GeV, m 3 v2 M ev suggests that neutrino physics probes the mass scale of grand unification (but there are also other examples... )

6 CP violating heavy neutrino decays: " 1 = ' (N 1! H + l L ) (N 1! H + l L ) M 1 (hh ) 11 v 2 F N 1! H + l L N 1! H + l L Im h m h 11 Leptogenesis relates successfully neutrino masses and matter-antimatter asymmetry; CP asymmetry is small: suppressed by small Yukawa couplings, and loop effect (quantum interference); natural explanation of very small observed baryon asymmetry

7 c L sl s L t L b L Rough estimate for CP asymmetry (hierarchical heavy neutrinos) and baryon asymmetry: d L Sphaleron b L " M 1m 3 v 2 F d L u L ν e ν µ ν τ B ' 0.1 M 1 M c s f N B L ' 10 2 " 1 apple f Neutrino masses suggest that leptogenesis is process close to thermal equilibrium, i.e. 1 H T =M1 ; in terms of neutrino masses: em = (m Dm D ) 11 m = 16 5/2 M 1 3 p 5 g1/2 v 2 F M P ' 10 3 ev confirmed by solution of Boltzmann equations; good quantitative understanding; significant progress in QFT description

8 Leptogenesis & Supersymmetry Leptogenesis & gravitinos: for (thermal) leptogenesis and typical superparticle masses, thermal production yields observed amount of dark matter: 3/2 h 2 = C 3/2 h is natural value; but why is reheating temperature close to minimal LG temperature? Simple observation: heavy neutrino decay width (for typical LG parameters) N 1 = m 1 8 M1 v F [WB, Domcke, Kamada, Schmitz 13, 14] T R 100 GeV 2 m g, C GeV m 3/2 1TeV GeV, em ev, M GeV yields reheating temperature (for gas of decaying heavy neutrinos) T R 0.2 q 0 N 1 M P GeV wanted for gravitino DM. Intriguing hint or misleading coincidence?

9 Spontaneous B-L breaking & false vacuum decay Supersymmetric SM with right-handed neutrinos: W M = h u ij10 i 10 j H u + h d ij5 i 10 j H d + h ij5 i n c jh u + h n i n c in c is 1 in SU(5) notation: hs 1,2 i = v B L / p 2 10 (q, u c,e c ), 5 (d c,l), n c ( c ) 1 W B L = 2 v2 B L S 1 S 2 yields heavy neutrino masses. ; B-L breaking: Lagrangian is determined by low energy physics: quark, lepton, neutrino masses etc, but it contains all ingredients wanted in cosmology: inflation, leptogenesis, dark matter,..., all related! [F-term hybrid inflation: Copeland et al. 94, Dvali et al. 94]. Technically: Abelian Higgs model in unitary gauge; inflation ends with phase transition ( tachyonic preheating, spinodal decomposition )

10 / V M P 4 / Φ M P S / v B-L H M P time-dependent masses of B-L Higgs, inflaton, heavy neutrinos... (bosons and fermions): m 2 = 1 2 (3v2 (t) v 2 B L), m 2 = v 2 (t), M 2 i =(h n i ) 2 v 2 (t)...

11 1 φ(t) (Tanh) <φ 2 (t)> 1/2 /v (Lattice) n B (x100) (Tanh) n B (x100) (Lattice) Bosons: Lattice Tanh Fermions: Lattice Tanh <φ 2 (t)> 1/2 /v, nb(t) Occupation number: n k e e time: mt 1e k/m Rapid transition from false to true vacuum by fluctuations of waterfall B-L Higgs field; production of low momentum Higgs bosons (contain most energy), also other bosons and fermions coupled to B-L Higgs field [Garcia-Bellido, Morales 02], production of cosmic strings: initial conditions for reheating

12 Schematic view Gauge Higgs + Inflaton Right-handed Neutrinos Radiation + B-L asymmetry Gravitinos tachyonic preheating fast process slow process Comoving number density abs NHaL N 1 th S Inverse temperature M 1 ê T N 1 nt i f a RH a RH a RH R B - L Scale factor a Transition from end of inflation to hot early universe (typical parameters), calculated by means of Boltzmann f X (t, p) = X X C X (Xi 0 j 0.. $ i 0 j 0.. ij.. yields correct baryon asymmetry and dark matter abundance G é

13 Abelian Higgs string: Gµ/c 2 < , consistent with GUT scale strings

14 Gravitational Waves Relic gravitational waves are window to very early universe; contributions from inflation, preheating & cosmic strings (see Maggiore 07, Dufaux et al 10, Hindmarsh 11); cosmological B-L breaking: prediction of GW spectrum with all contributions! Perturbations in flat FRW background: ds 2 = a 2 ( )( µ + h µ )dx µ dx 1, hµ = h µ 2 µ h determined by linearized Einstein equations, h 00 µ (x, )+2 a0 a h 0 µ (x, ) r 2 x h µ (x, ) = 16 GT µ (x, ) yields spectrum of GW background: Z 1 1 GW (k, ) = GW (k, ), ln k d ln GW (k, ) = 1 Dḣij (x, ) (x, ln k 32 G ḣij

15 -1 D AH cosmic strings f PH HsL W GW h HvL f PH preheating inflation f 0 f eq f RH f PH Result: GWs from inflation, preheating and cosmic strings (Abelian Higgs), for typical parameters consistent with leptogenesis and dark matter; similar spectrum from inflation and cosmic strings, but different normalization!

16 -1 D W GW h H1L H2L H4L AH cosmic strings H8L H7L H3L H6L NG cosmic strings H5L inflation preheating Observational prospects (for typical parameters); soon : Advanced Ligo (6) [100 Hz], elisa [0.01 Hz]; later : Einstein Telescope [100 Hz], BBO/DECIGO [0.01 Hz]; eventually determination of reheating temperature (leptogenesis)?!

17 Hybrid Inflation in the Complex Plane Inflationary potential can be strongly affected by supersymmetry breaking (supergravity correction [WB, Covi, Delepine 00]): V ( )=V 0 + V CW ( )+V SUGRA ( )+V 3/2 ( ), V 0 = V CW ( )= V SUGRA ( )= 2 v 4, 4 4 v ln v/ p +..., 2 2 v 4 8M 4 P , V 3/2 ( )= v 2 m 3/2 ( + )+... linear term turns hybrid inflation into two-field model in complex plane; strong effect on inflatonary observables, now dependent on trajectory, i.e. initial conditions! Inflation consistent with Planck data now possible (otherswise very difficult)

18 m 3ê2 = 50 TeV Hs *, t * L Æ V,j < 0 V,j > 0 Æ Æ 1.0 VHs, tl Hs f, t f L N = 50 Æ N = 0 H-v, 0L Hv, 0L F-term hybrid inflation ( 1) with SUSY breaking: resulting linear term in inflaton potential leads to two-field dynamics of complex inflaton in field space, observables depend on inflationary trajectory, range for spectral index consistent with n s ' 0.96

19 100 TeV 1 PeV 10 PeV TeV m 3ê TeV 100 GeV Fine-tuned phase q f close to 0 l 10 GeV q f p ê 32 q f = 0 Cosmic string bound GeV 100 MeV 10-5 m 3ê2 < 10 MeV 10 MeV A s > A s obs Parameter scan: relations between scale of B-L breaking, gravitino mass and scalar spectral index (correct in green band) cosmic string bound automatically fulfilled! Prediction for tensor-scalar ratio: r ; further predictions: characteristic spectrum of gravitational waves, connection between inflation and SUSY searches at the LHC! BUT WHAT ABOUT BICEP2?

20 0.05 BKxBK (BKxBK αbkxp)/(1 α) BB l(l+1)c l /2π [µk 2 ] Multipole L/L peak Fiducial analysis Cleaning analysis Joint BICEP2/Keck/Planck analysis (2015) upper: BICEP2/Keck spectrum before and after subtraction of dust contribution lower: likelihood fit; primordial BB signal almost 2σ effect; favoured value of r about 0.5 More data this year, BB signal not yet excluded r

21 Subcritical Hybrid Inflation Version of supersymmetric D-term inflation; superpotential and Kahler potential for waterfall fields and inflaton (shift symmetry, [Yanagida et al 00]): K = 1 2 ( + ) 2 + S S 2, W = S + S, V D = g2 2 S + 2 S 2 2 Waterfall transition below critical point ' c = g p 2, can be super-planckian for small Yukawa couplings /g 1 (regime used in following discussion, needed for Lyth bound); theoretical question: FI-term [recent work: Wieck, Winkler 14; Domcke, Schmitz, Yanagida 14] [WB, Domcke, Schmitz 14; WB, Ishiwata 14]

22 Inflaton potential above critical point (waterfall fields integrated out): V (') =V 0 1+ g2 (ln x +...), 8 2 with x = 2 ' 2 /(2g 2 ). Below critical point scalar potential for inflaton and waterfall fields ( M P =1): V (', s) = g2 8 (s2 2 ) 2 + Note, potential quadratic in inflaton field (consequence of shift symmetry!) Below critical point waterfall transition (spinodal decomposition; tachyonic preheating [Felder et al 00;...]), rapid transition to global minimum. Following discussion: tachyonic growth of fluctuations for continuing exponential expansion! Inflation can continue after critical point [Clesse 10; Clesse, Garbrecht 12], mostly models with small field inflation]; here: large field inflation after critical point, with final phase of chaotic inflation after tachyonic growth of fluctuations. 2 4 s2 ' 2 + O(s 4 ' 2 ).

23 Number of e-folds V inf HjL ê M GUT j f j * j c Inflaton field pl D ' c Below critical point inflaton potential changes from plateau to quadratic potential of chaotic inflation ; relevant fluctuations are generated at '

24 Tachyonic Growth of Fluctuations Computation of quantum fluctuations beyond critical point requires mode expansion of field operators in de Sitter background: Z ˆ'(t, ~x) ='(t)+e 3 2 Ht d 3 k a (2 ) 3/2 ' ( ~ k)' k (t)e i~ k~x + a '( ~ k)' k(t)e i~ k~x, Z ŝ(t, ~x) =e 3 2 Ht d 3 k a (2 ) 3/2 s ( ~ k)s k (t)e i~ k~x + a s( ~ k)s k(t)e i~ k~x only inflaton field has classical part; close to the critical point ( t(' c )=0): with D 3 = p 2 g ' c '(t) ' ' c + ' c t, V (s; t) ' 1 2 g D3 ts 2 + O(s 4 ; t 2 ), given by velocity of inflaton field at critical point

25 Analytic estimate of variance due to growth of quantum fluctuations [Asaka, WB, Covi 01]: hs 2 (t)i = hs 2 (t)i us ' Z kb (t) 0 ' 32/ hs 2 (0)i us dk k2 2 2 e 3H ct s k (t) 2 D 2 (Dt) 3/2 exp 4 3 (Dt)3/2 + Ht ; for large times, Dt > 1, one has p b (t) >H c, i.e. fluctuations are generated within horizon. For typical parameters ( g 2 =1/2, =5 10 4, p = GeV ), velocity of inflaton field and Hubble parameter ' c 'V 3H 'c = g2 ln 2 4 p 3 2 p. H c H(' c )= GeV, ' c ' 21H 2 c, D ' 1.5 H c.

26 Beyond decoherence time ( s k (t dec ) sk (t dec ) R dec ~ 1), t dec 1 D 3 4 ln(2r dec) 2/3, soft modes ( k<k b (t) ) generate classical waterfall field. Backreaction due to self-interaction of waterfall field can then be taken into account by means of classical field equations: s +3Hṡ g 2 ' +3H ' '2 2 s 2 ' =0, s + g2 2 s3 =0. Classical waterfall field, with matching s(t dec )=hs 2 (t dec )i 1/2, quickly reaches local minimum; approach of global minimum much later, in final stage generation of effective mass term for inflaton.

27 2.5 without backreaction 100 < s 2 HtL > 1ê2 ê s t dec s min shtl with backreaction 0.0 matching Time ê H c D waterfall fields approaches time-dependent minimum due to backreaktion

28 Number of e-folds N e Number of e-folds N e Waterfall field s AM Pl ë 10 3 E Inflaton field Pl D Time ê H c D Time ê H c D at the end of inflation standard picture: inflaton oscillations, reheating, etc...

29 Flattened Chaotic Inflation Effective inflaton potential below critical point depends on only two variables, V inf (') =V (', s min (')) =2Minf 4 ' 2 ' 2 1 c where FI-term and Yukawa coupling are replaced by inflationary energy scale, and relative couplings strength (critical inflaton field), 1 2 ' 2 ' 2 c, ' apple ' c M inf = g p 2 1/2, = p g ; ' 2 c = 2p 2M 2 inf 2 (remark: very similar to chaotic inflation in string models, corrected by moduli effects [WB, Dudas, Hurtier, Westphal, Wieck, Winkler 15])

30 5 40 M 16 GeVD Ne = 60 M GUT Ne = 50 j pl D Ne = 60 Ne = D D Remarkably, the Planck data n s = ± , r < 0.11 (95% CL) determine the energy scale of inflation rather precisely, 1.6 (1.9) apple M inf GeV apple 2.4 (2.2)

31 Tensor-to-Scalar Ratio (r) Subcritical hybrid inflation Ne = 50 Ne = 60 Planck+WP+BAO Planck+WP+highL Planck+WP Natural Inflation Primordial Tilt (ns) 1σ & 2σ regions for different data sets (Planck 2013) compared with natural inflation and subcritical hybrid inflation (green); prediction of subcritical hybrid inflation for bound on r for 60 (50) e-folds: r>0.049 (0.085) Models in their simplest form disfavoured by Planck 2015 data; modifications possible, more data this year...

32 Conclusions Strong theoretical motivation for unification of forces beyond current microscopic theory (SM) Possible cosmic signatures of grand unification: cosmic strings, gravitational waves, B-modes in CMB; several indirect hints: leptogenesis, nonthermal dark matter,... Hybrid inflation attractive framework for describing very early universe (production of entropy, dark matter, leptogenesis,...) Hints for GUT scale already in subcritical hybrid inflation together with Planck data; prediction: lower bound on tensorto-scalar ratio, in reach of upcoming experiments!

33 BACKUP SLIDES

34 [adapted from Schmitz 12] Connection to particle physics in very early universe, before formation of CMB: nucleosynthesis, formation of hadrons, generation of mass (electroweak phase transition), dark matter, leptogenesis... preceeded by inflation (?) which generates initial conditions of hot early universe; important: global picture of very early universe!

35 Inverse temperature M 1 ê T i f a RH a RH a RH Scale factor a Time evolution of temperature: intermediate plateau ( maximal temperature ), determined by neutrino properties! Yields gravitino abundance when combined with standard formula

36 such that WGé h2 = hb < hobs B obs hnt B > hb mgé = 1 TeV vb-l = GeV Wwé > Wobs DM é w é G Whé > Wobs DM é h m é parameter scans: successful leptogenesis and gravitino DM (left) or neutralino DM (right) [non-thermally produced in decays of thermally produced gravitinos] constrains neutrino and superparticle masses; can be tested at LHC! 100

37 Slow-Roll Inflation Inflation is driven by slowly varying potential energy small slow-roll parameters determine spectral indices: V ( ) of inflaton field; = M 2 pl V V 2, = M 2 2 V V n s 1=2 6, n t = 2 Tensor-to-scalar ratio yields energy density during inflation: r = 2 s 2 t = s V ( ) M 4 pl, V 1/4 r 1/ GeV 0.01 Tensor-to-scalar ratios r & require super-planckian field values [Lyth 97; Antusch, Nolde 14]; theoretical challenge, ongoing theory activities...

38 Beyond local spinodal time waterfall field tracks local minimum; asymmetric trajectory in field space: Time ê H c D Waterfall field s AM Pl ë 10 3 E s AM Pl ë 10 4 E Time ê H c D t dec Pl D V Hj, sl Number of e-folds N e Inflaton field Pl D