Avoiding strong coupling problem in the Higgs inflation with R 2 -term. Dmitry Gorbunov

Transcription

1 Avoiding strong coupling problem in the Higgs inflation with R 2 -term Dmitry Gorbunov Institute for Nuclear Research of RAS, Moscow Workshop on the Standard Model and Beyond Corfu Summer Institute Corfu, Greece, Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 1 / 34

2 Outline Outline 1 2 Adding R 2 -term 3 Conclusions Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 2 / 34

3 Outline 1 2 Adding R 2 -term 3 Conclusions Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 3 / 34

4 Big Bang within GR and SM: problems Dark Matter Baryogenesis Dark Energy 0 Λ M 4 Pl, M4 W, Λ4 QCD, etc? Coincidence problems: Ω B Ω DM Ω Λ, η B = n B /n γ (δt /T ) 2, T n d (m n m p ),... ΛCDM tensions: lack of dwarfs? cusps? Horizon, Enthropy, Flatness,... problems l H0 /l H,r (t 0 ) 1 + z r 30 Singularity at the beginning Heavy relics Initial fluctuations δt /T δρ/ρ 10 4, scale-invariant Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 4 / 34

5 2.7 K TODAY 14 by 4.4 K accelerated expansion matter domination 7.7 by 0.26 ev recombination 370 ty e+ p H + γ matter domination 0.8 ev 50 ty radiation domination 50 kev 5 min 3 H + 4 He 7 Li + γ primordial nucleosynthesis 2 H + 2 H n + 3 He 1 MeV 1 s p + p 2 H + γ neutrino decoupling 2.5 MeV 0.1 s 200 MeV QCD transition 10 µs confinement free quarks Electroweak phase transition 100 GeV 0.1 ns baryogenesis hot Universe reheating dark matter production inflation Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 5 / 34

6 Inflationary solution of Hot Big Bang problems no initial singularity in ds space all scales grow exponentially, including the radius of the 3-sphere the Universe becomes exponentially flat any two particles are at exponentially large distances no heavy relics no traces of previous epochs! no particles in post-inflationary Universe to solve entropy problem we need post-inflationary reheating conformal time conformal time space coordinate space coordinate particle horizon casually connected regions inflationary expansion Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 6 / 34

7 Chaotic inflation at large fields in all domains of Planck size each of the form of inflaton energy fluctuates similarly Chaotic inflation, A.Linde (1983), A.Linde (1984) 1 2 φ ( iφ) 2 V (φ) M 4 Pl If V (φ) dominates by chance X φ φ/a 2 +3H φ + V (φ) = 0 for power-law potential at φ > M Pl V const slow roll solution H 2 = 8π 3MP 2 V (φ), a(t) e Ht Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 7 / 34

8 The idea is great, but is not verifiable except the flatness... Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 8 / 34

9 The idea is great, but is not verifiable except the flatness... Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 8 / 34

10 Unexpected bonus: generation of perturbations Quantum fluctuations of wavelength λ of a free massless field ϕ (inflaton and gravitons!!) have 3-momenta q 1/λ and an amplitudes of δϕ λ q Evolution at inflation inside horizon: q > H q 1/a δϕ λ q 1/a inside horizon H(t) outside horizon inside horizon outside horison: q < H q a δϕ λ = const = H infl /2π!!! k q(t)= a(t) got classical fluctuations: δϕ λ = δϕ quantum λ e Ne inflation RD, MD epochs t e t scalar modes δφ λ H infl Later at normal stage tensor modes δg µν h H infl /M Pl H 1/t, q/h, modes enter horizon Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 9 / 34

11 Inflationary solution of Hot Big Bang problems Temperature fluctuations δt /T 10 5 Universe is uniform! conformal time conformal time space coordinate particle horizon casually connected regions inflationary expansion δρ/ρ 10 5 space coordinate Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 10 / 34

12 Probing the matter power spectrum inside horizon H(t) outside horizon inside horizon k q(t)= a(t) inflation t e RD, MD epochs t δφ δρ ρ H2 φ V 3/2 V, h H V 1/2 M Pl ( ) A S V 3/2 V, n S V V, V 2 ( ) V, r A T A S V 2 V Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 11 / 34

13 Chaotic inflation: simple realization S = d 4 x ( ) g M2 P 2 R + ( µ X) 2 βx 4 2 Chaotic inflation, A.Linde (1983) Ẍ+3HẊ + V (X) = 0 H 2 = 1 MP 2 V (X), a(t) e Ht slow roll conditions get satisfied at X e > M Pl MP 2 = M2 Pl /(8π) generation of scale-invariant scalar (and tensor) perturbations from exponentially stretched quantum fluctuations of X X δρ/ρ 10 5 requires = V = βx 4 : β We have scalar in the SM! The Higgs field! ( In a unitary gauge H T = 0,(h + v)/ ) 2 (and neglecting v = 246 GeV) λ S = d 4 x ( ) g M2 P 2 R + ( µ h) 2 λ h4 2 4 Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 12 / 34

14 Chaotic inflation: simple realization S = d 4 x ( ) g M2 P 2 R + ( µ X) 2 βx 4 2 Chaotic inflation, A.Linde (1983) Ẍ+3HẊ + V (X) = 0 H 2 = 1 MP 2 V (X), a(t) e Ht slow roll conditions get satisfied at X e > M Pl MP 2 = M2 Pl /(8π) generation of scale-invariant scalar (and tensor) perturbations from exponentially stretched quantum fluctuations of X X δρ/ρ 10 5 requires = V = βx 4 : β We have scalar in the SM! The Higgs field! ( In a unitary gauge H T = 0,(h + v)/ ) 2 (and neglecting v = 246 GeV) λ S = d 4 x ( ) g M2 P 2 R + ( µ h) 2 λ h4 2 4 Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 12 / 34

15 Planck 2015 favors flat inflaton potentials r = A T A S φ 2 H 2 M 2 Pl ( V ) 2 1 V Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 13 / 34

16 Inflaton potential is apparently concave Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 14 / 34

17 Higgs-driven inflation ( In a unitary gauge H T = 0,(h + v)/ ) 2 F.Bezrukov, M.Shaposhnikov (2007) S JF = d 4 x ( ) g M2 P 2 R ξ H HR + L SM (and neglecting v = 246 GeV) S = d 4 x ( g M2 P + ξ ) h2 R + ( µ h) 2 λ h slow roll behavior due to modified kinetic term even for λ 1 Go to the Einstein frame: (M 2 P + ξ h2 )R JR M 2 P REF gµν JF EF = Ω 2 g µν, Ω 2 = 1 + ξ h2 MP 2 with canonically normalized χ: interval ds 2 changes! dχ dh = M P MP 2 + (6ξ + 1)ξ h2 MP 2 + ξ, U(χ) = h2 λm 4 P h4 (χ) 4(M 2 P + ξ h2 (χ)) 2. we have a flat potential at large fields: U(χ) h M P / ξ Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 15 / 34

18 U(χ) MP λm 4 /ξ 2 /4 log(χ) MP /ξ exact h 3/2ξh 2 /MP 6 log( ξh/mp ) MP / MP /ξ ξ log(h) Reheating by Higgs field after inflation: M P /ξ < h < M P / ξ λm 4 /ξ 2 λ v 4 /4 / v 0 0 χ end χ WMAP χ exponentially flat h M P / ξ : effective dynamics : h 2 χ U(χ) = λm4 P 4ξ 2 ( ( )) 2 2 χ 1 exp 3MP L = 1 2 µ χ µ χ λ 6 M 2 P ξ 2 χ2 Advantage: NO NEW interactions to reheat the Universe inflaton couples to all SM fields! NO NEW d.o.f from WMAP-normalization: ξ λ Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 16 / 34

19 U(χ) MP λm 4 /ξ 2 /4 log(χ) MP /ξ exact h 3/2ξh 2 /MP 6 log( ξh/mp ) MP / MP /ξ ξ log(h) Reheating by Higgs field after inflation: M P /ξ < h < M P / ξ λm 4 /ξ 2 λ v 4 /4 / v 0 0 χ end χ WMAP χ exponentially flat h M P / ξ : effective dynamics : h 2 χ U(χ) = λm4 P 4ξ 2 ( ( )) 2 2 χ 1 exp 3MP L = 1 2 µ χ µ χ λ 6 M 2 P ξ 2 χ2 Advantage: NO NEW interactions to reheat the Universe inflaton couples to all SM fields! NO NEW d.o.f from WMAP-normalization: ξ λ Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 16 / 34

20 F.Bezrukov, D.G., M.Shaposhnikov, similar result J.Garcia-Bellido, D.Figueroa, J.Rubio, log(χ) MP mw 2 (χ) = g2 2 6 m t (χ) = y t M P χ (t) ξ M P χ (t) sign χ(t) 6ξ MP /ξ MP /ξ exact h 3/2ξh 2 /MP 6 log( ξh/mp ) log(h) MP / ξ reheating via W + W, ZZ production at zero crossings then nonrelativistic gauge bosons scatter to light fermions Reheating by Higgs field χ W + W f f after inflation: M P /ξ < h < M P / ξ Hot stage starts almost from T = M P /ξ GeV: effective dynamics : h 2 χ L = 1 2 µ χ µ χ λ 6 M 2 P ξ 2 χ2 Advantage: NO NEW interactions to reheat the Universe inflaton couples to all SM fields! ( ) λ 1/ GeV < T r < GeV n s = 0.967,r = F.Bezrukov, D.G. (2012) Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 17 / 34

21 U(χ) MP λm 4 /ξ 2 /4 log(χ) MP /ξ MP /ξ exact h 3/2ξh 2 /MP 6 log( ξh/mp ) log(h) MP / ξ dχ dh = M P MP 2 + (6ξ + 1)ξ h2 MP 2 + ξ, h2 U(χ) = λm 4 P h4 (χ) 4(M 2 P + ξ h2 (χ)) 2. λm 4 /ξ 2 /16 λ v 4 /4 0 0 v 0 0 χ end χ WMAP χ exponentially flat h M P / ξ : U(χ) = λm4 P 4ξ 2 ( ( )) 2 2 χ 1 exp 3MP renormalizable except gravity frame-dependent renormalization scale strong coupling (φ-dependent) save for inflation but reheating is questionable F.Bezrukov et al (2008) V MPl 4 from WMAP-normalization: ξ λ F.Bezrukov, D.G., M.Shaposhnikov (2008,2011) Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 18 / 34

22 Strong coupling in Higgs-inflation Jordan frame Einstein frame E Strong coupling E Strong coupling Λ Planck Λ g-s Λ Planck M P M P Λ g-s Λ gauge Λ gauge M P /ξ Weak coupling M P /ξ Weak coupling M P /ξ M P / ξ h gravity-scalar sector: M P ξ, for h M P ξ, ξ h Λ g s (h) 2 M, for M P P ξ h M P, ξ ξ h, for h M P. ξ M P /ξ M P / ξ h gravitons: Λ 2 Planck M2 P + ξ h2 gauge interactions: { MP Λ gauge (h) ξ, for h M P ξ, h, for M P ξ h, Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 19 / 34

23 Then we have problems... During inflation and preheating we are always below Λ However, both parametrically and numerically T r M P ξ Λ Recent more detailed studies , of reheating in multi-scalar inflationary models indicate amplification of inflaton decays T r Λ so the reheating may be out of control On top of that: it may be that λ < 0... Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 20 / 34

24 LHC:... Higgs of 125 GeV Λ mh 174 GeV LEPII: m h > 114 GeV fit to EW data: m h 90 < 114 GeV TeVatron: not in 156 < m h < 177 GeV CMS & ATLAS: 0.5 mh GeV Μ,GeV MP1020 m h = 2λυ λ ) 2 (H H υ2 2 λ 4 h4 + λυ 2 h 2 L Y Y f h f f / 2 renormgroup equation m h 125GeV dλ d log µ +#λ 2 #Yt 4 Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 21 / 34

25 Running of the SM couplings m 2 B I g 3 y t 0.5 g 2 g Y 0.0 Λ Β Λ GeV Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 22 / 34

26 Higgs potential with quantum corrections V V φ φ Fermi Planck Fermi Planck Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 23 / 34

27 How weird to live with 125 GeV Higgs... Top pole mass Mt in GeV Instability Meta stability ,2,3 Σ Top pole mass Mt in GeV Instability Meta stability Stability Non perturbativity h Stability Higgs pole mass M h in GeV Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 24 / 34

28 All in all It would be nice to modify the model Introducing as little new physics as possible Keeping cosmological observables determined by the Higgs sector parameters Possibly avoiding the negative selfcoupling for the Higgs Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 25 / 34

29 Outline Adding R 2 -term 1 2 Adding R 2 -term 3 Conclusions Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 26 / 34

30 Adding R 2 -term Natural completion with R 2 D.G., A.Tokareva ξ h 2 R induces R 2 -term S 0 = d 4 x g ( M2 P + ξ h2 R + β 2 4 R2 + ( µh) 2 λ 2 4 (h2 v 2 ) ). 2 hep-th/ introduce a Lagrange multiplier L and auxiliary scalar R S = d 4 x ( ( µ h) 2 g λ 2 4 (h2 v 2 ) 2 M2 P + ξ ) h2 R + β 2 4 R2 LR + LR. integrate out R S = d 4 x ( ) ( µ h) 2 g λ 2 4 (h2 v 2 ) 2 + LR 1 β (L ξ h M2 P )2 with everything here look healthy ξ ξ 2 /β β ξ 2 4π Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 27 / 34

31 Adding R 2 -term Strong coupling: the lowest scale is in the gauge sector Jordan frame Einstein frame E Strong coupling E Strong coupling Λ Planck Λ g-s Λ Planck M P M P Λ g-s Λ gauge Λ gauge M P /ξ Weak coupling M P /ξ Weak coupling M P /ξ M P / ξ h gravity-scalar sector: M P ξ, for h M P ξ, ξ h Λ g s (h) 2 M, for M P P ξ h M P, ξ ξ h, for h M P. ξ M P /ξ M P / ξ h gravitons: Λ 2 Planck M2 P + ξ h2 gauge interactions: { MP Λ gauge (h) ξ, for h M P ξ, h, for M P ξ h, Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 28 / 34

32 Adding R 2 -term Further transformations... D.G., A.Tokareva introducing scalaron φ g µν Ω 2 g µν, Ω 2 2L 2 MP 2, L φ M P 3 logω2. with m = M P / 3β and setting M P = 1/ 6 S = d 4 x g ( R e 2φ ( h) ( φ)2 14 ( e 4φ λh )) 36β (e2φ 1 6ξ h 2 ) 2 both gravity and scalar sector are weakly coupled up to M P with β ξ 2 /(4π) Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 29 / 34

33 Adding R 2 -term And one more... D.G., A.Tokareva The scalar sector becomes h = e Φ tanhh, φ = e Φ /coshh, L = 1 2 cosh2 H( Φ) ( H)2 λ 4 sinh4 H λ 144β (1 e 2Φ cosh 2 H 6ξ sinh 2 H) 2. and the Higgs coupling to gauge bosons, e.g., L gauge = g2 h 2 4 e 2φ W + µ W µ = g2 4 sinh2 H W + µ W µ. W W h W W W W + h + W A g2 p 2 m 2 W W h W + W W W ( ( ) 4 dmw (H) 2 g 2 1) dh W W A p2 M 2 P W p2 2 W M P Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 30 / 34

34 Adding R 2 -term Cosmological spectra D.G., A.Tokareva x 0.14 x X N e=60 H H H Scalar perturbations: β + ξ 2 λ At small β like in the Higgs-inflation heavy scalaron is integrated out ξ 2 4π < β < ξ 2 λ GeV < m < GeV Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 31 / 34

35 Adding R 2 -term Bonus: stable for a bit heavier top-quark β= ξ β= ξ /2 λ μ λ mt μ m t Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 32 / 34

36 Outline Conclusions 1 2 Adding R 2 -term 3 Conclusions Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 33 / 34

37 Conclusions Conclusions Higgs inflation is a viable cosmological model unique in minimality: no new d.o.f., no new interactions to reheat the Universe however it suffers from the stong coupling problem: predictivity δρ/ρ λ is lost R 2 -term with heavy scalaron cures the model: it seems minimal, natural, Higgs-inflation predictions remain intact (the same remedy for Higgs-dilaton) to refine them we must study the reheating and improve the accuracy of Y t, (m h, α s, etc) to convince it works indeed... ILC, FCC, etc Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 34 / 34

38 Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 35 / 34

39 Backup slides Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 36 / 34

40 Horizon problem l H (t) a distance covered by photon emitted at t = 0 size of the causally connected part, that is the visible part of the Universe ( inside horison ) ds 2 = dt 2 a 2 (t)dx 2 = a 2 (η) ( dη 2 dx 2) ds 2 = 0 l H (t) =a(t) dx=a(t) t c dt dη=a(t) 0 a(t ) t a(t) t α, 0 < α < 1, L phys a 1/H(t) l H0 /l H,r (t 0 ) l H0 /l H,r (t r )a(t r )/a 0 H r /H 0 a(t r )/a z r 30 Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 37 / 34

41 Chaotic inflation at large fields: initial conditions Ẍ X/a 2 +3HẊ + V (X) = 0 slow roll: X e > M Pl δρ/ρ 10 5 requires V M 4 Pl inflation starts in a relatively uniform domain of Planck size 1 2 φ ( iφ) 2 V (φ) M 4 Pl X Chaotic inflation, A.Linde (1983), A.Linde (1984) looks rather natural: each of the form of inflaton energy fluctuates similarly Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 38 / 34

42 Is the Higgs-inflation realy unlikely?? Start with the Jordan frame ( S JF = d 4 x g JF M2 P + ξ ) h2 R + ( µ h) 2 λ h D.G., A.Panin (2014) Higgs-scalaron mixture Go to the Einstein frame: g µν = Ω 2 g µν, Ω 2 = 1 + ξ h2 M 2 P, dφ dh = Ω πξ 2 h 2 /MPl 2 Ω 4 (M 2 P + ξ h2 )R M 2 P R chaotic initial conditions effective M Pl Higgs noncanonical kinetic term Ω 2 M 2 Pl RJF ξ ḣ2 ξ ( i h) 2 λh 4 M 4 Pl hence R JF MPl 2 /Ω2 CMB-amplitude V0 EF MPl 4 fixes Ω Hence in the EF 1 2 φ ( i φ) 2 M 4 Pl /Ω M 4 Pl all terms in EF happen to be of the same order!! Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 39 / 34

43 Is the Higgs-inflation realy unlikely?? Start with the Jordan frame ( S JF = d 4 x g JF M2 P + ξ ) h2 R + ( µ h) 2 λ h D.G., A.Panin (2014) Higgs-scalaron mixture Go to the Einstein frame: g µν = Ω 2 g µν, Ω 2 = 1 + ξ h2 M 2 P, dφ dh = Ω πξ 2 h 2 /MPl 2 Ω 4 (M 2 P + ξ h2 )R M 2 P R chaotic initial conditions effective M Pl Higgs noncanonical kinetic term Ω 2 M 2 Pl RJF ξ ḣ2 ξ ( i h) 2 λh 4 M 4 Pl hence R JF MPl 2 /Ω2 CMB-amplitude V0 EF MPl 4 fixes Ω Hence in the EF 1 2 φ ( i φ) 2 M 4 Pl /Ω M 4 Pl all terms in EF happen to be of the same order!! Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 39 / 34

44 The first inflationary model:... from modified gravity! S JF = M2 P 2 g d 4 x Jordan Frame Einstein Frame [ ] R R2 6 µ 2 ( get reed [ ( 1 Q 3 µ 2 1 Q 3 µ 2 ) g JF µν g EF µν = Ω 2 g JF µν, Ω 2 = exp ) ( )] (R Q) + Q Q2 6 µ 2, = Ω 2 A.Starobinsky (1980) ( ) 2/3φ/MP. [ S EF = g EF d 4 x M2 P 2 REF gef µν µ φ ν φ 3 µ2 MP 2 ( 1 1 ) ] 2 4 Ω 2, (φ) 0.20 generation of perturbations requires V M 4 Pl 0.05 Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 40 / 34

45 Inflationary models and quantum corrections inflationary predictions are robust but we cannot test them with low energy particle physics experiment including physics at reheating number of e-foldings N similar observation for many other models: Higgs-inflation, α-attractor, etc large fields: exponentially flat protected by the shift invariance φ φ + const small fields: polynomial potential protected by the renormalizability φ 2 + φ 4 no way to match them at φ M P µ 2 φ 2 /2 ( ) 3MP 2 µ2 /4 1 e #φ/m P Dmitry Gorbunov (INR) Curing Higgs inflation with R 2 -term , CSI 41 / 34