Higgs Inflation Mikhail Shaposhnikov SEWM, Montreal, 29 June

Higgs Inflation Mikhail Shaposhnikov SEWM, Montreal, 29 June - 2 July 2010 SEWM, June 29 - July 2 2010 p. 1

Original part based on: F. Bezrukov, M. S., Phys. Lett. B 659 (2008) 703 F. Bezrukov, D. Gorbunov, and M. S., JCAP 0906 (2009) 029 F. Bezrukov, A. Magnin, M. S., Phys. Lett. B 675 (2009) 88 F. Bezrukov, M. S., JHEP 0907 (2009) 089 SUGRA extensions: Einhorn, Jones 2009 Ferrara, Kallosh, Linde, Marrani, Van Proeyen 2010 Hyun Min Lee 2010 SEWM, June 29 - July 2 2010 p. 2

Outline Inflation and the inflaton Standard Model Higgs boson as inflaton CMB parameters in Higgs inflation SM as a valid effective theory up to the Planck scale Higgs mass from inflation Comparison with other works Naturalness of Higgs inflation Conclusions SEWM, June 29 - July 2 2010 p. 3

Inflation and the inflaton Important cosmological problems: Horizon problem: Why the universe is so uniform and isotropic? t present horizon recombination horizon at recombination r Expected fluctuations at θ 1 o : δt/t 1. Observed fluctuations: δt/t 1O 5 SEWM, June 29 - July 2 2010 p. 4

Structure formation problem: What is the origin of cosmological perturbations and why their spectrum is almost scale-invariant? SEWM, June 29 - July 2 2010 p. 5

Flatness problem: Why Ω M + Ω Λ + Ω rad is so close to 1 now and was immensely close to 1 in the past? All this requires enormous fine-tuning of initial conditions (at the Planck scale?) if the Universe was dominated by matter or radiation all the time! SEWM, June 29 - July 2 2010 p. 6

Solution: Inflation = accelerated Universe expansion in the past SEWM, June 29 - July 2 2010 p. 7

Solution: Inflation = accelerated Universe expansion in the past Mechanism: scalar field dynamics SEWM, June 29 - July 2 2010 p. 7

Solution: Inflation = accelerated Universe expansion in the past Mechanism: scalar field dynamics Why scalar? SEWM, June 29 - July 2 2010 p. 7

Solution: Inflation = accelerated Universe expansion in the past Mechanism: scalar field dynamics Why scalar? Vector - breaking of Lorentz symmetry SEWM, June 29 - July 2 2010 p. 7

Solution: Inflation = accelerated Universe expansion in the past Mechanism: scalar field dynamics Why scalar? Vector - breaking of Lorentz symmetry Fermion - bilinear combinations are equivalent to scalar fields Uniform scalar condensate has an equation of state of cosmological constant and leads to exponential universe expansion. SEWM, June 29 - July 2 2010 p. 7

Standard chaotic inflation Required for inflation: δt/t 10 5 ) (to get quartic coupling constant λ 10 13 mass m 10 13 GeV, Present in the Standard Model: Higgs boson λ 1, m H 100 GeV δt/t 1 SEWM, June 29 - July 2 2010 p. 8

No - these conclusions are based on a theory with minimal coupling of scalar to gravity!: S = d 4 x g { M2 P 2 R + g µν µ h ν h 2 λ 4 ( h 2 v 2) 2 } Extra term, necessary for renormalizability: non-minimal coupling of scalar to gravity S = d 4 x g { } ξh2 2 R Feynman, Brans, Dicke,... SEWM, June 29 - July 2 2010 p. 9

Standard Model Higgs boson as inflaton Consider large Higgs fields h. Gravity strength: M eff P = M 2 P + ξh2 h All particle masses are h For h > M P ξ (classical) physics is the same (M W /MP eff depend on h)! does not Existence of effective flat direction, necessary for successful inflation. Formalism: go from Jordan frame to Einstein frame with the use of conformal transformation: ĝ µν = Ω 2 g µν, Ω 2 = 1 + ξh2 M 2 P SEWM, June 29 - July 2 2010 p. 10

Redefinition of the Higgs field to make canonical kinetic term dχ dh = Ω 2 + 6ξ 2 h h2 /M 2 P Ω 4 = h χ for h < M P /ξ ( ) h M P χ ξ exp 6MP for h > M P / ξ Resulting action (Einstein frame action) S E = d 4 x ĝ { M2 P 2 ˆR + µχ µ χ 2 1 } λ Ω(χ) 4 4 h(χ)4 Potential: U(χ) = λ 4 χ4 for h < M P /ξ ( ) 2 λm 4 P 1 e 2χ 6MP for h > M 4ξ 2 P /ξ. SEWM, June 29 - July 2 2010 p. 11

Potential in Einstein frame U(χ) λm 4 /ξ 2 /4 Reheating Standard Model λm 4 /ξ 2 λ v 4 /4 /16 0 0 v 0 0 χ end χ COBE χ SEWM, June 29 - July 2 2010 p. 12

Inflaton potential and observations If inflaton potential is known one can make predictions and compare them with observations. δt/t at the WMAP normalization scale 500 Mpc The value of spectral index n s of scalar density perturbations δt(x) T δt(y) T d 3 k k 3 eik(x y) k n s 1 The amplitude of tensor perturbations r = δρ s δρ t These numbers can be extracted from WMAP observations of cosmic microwave background. Higgs inflation: one new parameter, ξ = two predictions. SEWM, June 29 - July 2 2010 p. 13

Slow roll stage COBE normalization U/ǫ = (0.027M P ) 4 gives ξ λ 3 N COBE 0.027 2 49000 λ = 49000 m H 2v Connection of ξ and the Higgs mass! Number of e-folds of inflation at the moment h N is N 6 8 h 2 N h2 end M 2 P /ξ Slow roll ends at χ end M P ; and begins at χ 60 5M P ǫ = M2 P 2 ( du/dχ U n s = 1 6ǫ + 2η, ) 2, η = M 2 P r = 16ǫ d 2 U/dχ 2 U SEWM, June 29 - July 2 2010 p. 14

CMB parameters spectrum and tensor modes 0.4 0.3 WMAP5 50 60 0.2 2 SM+ ξh R 0.1 0.0 0.94 0.96 0.98 1.00 1.02 SEWM, June 29 - July 2 2010 p. 15

Earlier works: non-minimal coupling of scalars in GUTs, etc: B. Spokoiny 84 D. Salopek, J. Bond and J. Bardeen 89 R. Fakir and W. G. Unruh 90 A. O. Barvinsky and A. Y. Kamenshchik 94, 98 E. Komatsu and T. Futamase 99 S. Tsujikawa and B. Gumjudpai 04 Computation of spectral indexes gives the same results in Einstein and Jordan frames. SEWM, June 29 - July 2 2010 p. 16

Life after inflation Two different stages: For scalar field M P > χ > M P the potential for χ is essentially ξ quadratic, m 2 χ λm2 P /ξ2. Exponential expansion of the Universe is changed to the power low, corresponding to matter domination. Particle creation takes place when χ passes through zero. After O(ξ) oscillations the scalar field reaches χ M P ξ. The energy is transferred to other fields of the SM, and the radiation-dominated epoch starts, T r (3.3 8.3) 10 13 GeV. See also J. Garcia-Bellido, D. G. Figueroa, J. Rubio 08 SEWM, June 29 - July 2 2010 p. 17

log U( χ) radiation domination U~ χ 4 χ< M P ξ matter domination M ξ P < χ< M P U~ χ 2 inflation χ>μ P U~const Hot Big Bang log χ SEWM, June 29 - July 2 2010 p. 18

Universe size Higgs inflation a~t 2/3 a~t 1/2 Radiation domination a~t 2/3 Matter Domination a~e h t Dark Energy domination a~e H t Inflation Hot Big Bang No inflation Universe age 10 43 10 36 10 32 recombination 10 17 s SEWM, June 29 - July 2 2010 p. 19

Higgs mass from inflation: qualitative argument Previous consideration tells nothing about the Higgs mass: change λ as ξ 2 - no modifications! However: λ is not a constant, it depends on the energy. Typical scale at inflation M P / ξ. Therefore, SM must be a valid quantum field theory up to the Inflation (or, to be on safe side, up to the Planck scale). m min < m H < m max m min = [126.3 + m t 171.2 2.1 m max = [173.5 + m t 171.2 2.1 4.1 α s 0.1176 0.002 0.6 α s 0.118 0.002 0.6] GeV 0.1] GeV SEWM, June 29 - July 2 2010 p. 20

Behaviour of the scalar self-coupling λ Strong coupling two loop 174 GeV M H 126 GeV M Z zero M Planck µ SEWM, June 29 - July 2 2010 p. 21

For m H > m max : Landau pole for energies E = E Landau < M P quantum field theory is inconsistent for E > E Landau. For m H < m min : Electroweak vacuum is unstable: there is a lower ground state at φ < P P. SEWM, June 29 - July 2 2010 p. 22

Higgs mass from inflation: computation Electroweak theory in the inflationary region, for h M P / ξ, h M P /ξ : Take the SM, freeze the radial mode of the Higgs field, and add to Lagrangian almost massless and almost non-interacting scalar: chiral SM. Why the Higgs decouples? Einstein frame: masses of all the particles M W,..., m t v = h Ω(h) const for h SEWM, June 29 - July 2 2010 p. 23

The procedure for computations of inflationary parameters Compute the effective potential in the inflationary region (tree, one-loop, two-loop,... with the use of the chiral SM Choose the normalization point µ to minimize higher order terms (normally, µ M W or M Z or m t ) To find values of different coupling constants in inflationary region, solve one-loop, two-loop... RG equations, getting initial conditions from tree, one-loop, two-loop,.. relations mapping physical SM parameters to couplings. Find ξ from COBE normalization and compute n s and r as a function of the Higgs mass SEWM, June 29 - July 2 2010 p. 24

Two-loop results Nearly horizontal coloured stripes correspond to the normalization prescription I. Green, red, and blue stripes give the result with normalization prescription II for different m t and α s = 0.1176, two white regions correspond to different α s and m t = 171.2GeV. The width of the stripes corresponds to changing the number of e-foldings between 58 and 60, or approximately one order of magnitude in reheating temperature. SEWM, June 29 - July 2 2010 p. 25

Two-loop results r 0.003 0.0025 0.002 0.0015 0.001 0.0005 130 140 150 160 170 180 190 m H,GeV Tensor-to-scalar ratio r depending on the Higgs mass m H, calculated with the RG enhanced effective potential. Nearly horizontal solid lines correspond to the normalization prescription I. Green, red, and blue dashed lines give the result with normalization prescription II for m t = 169.1,171.2,173.3 GeV. Dependence on the number of e-foldings is very small. SEWM, June 29 - July 2 2010 p. 26

Two-loop results ξ at the scale M P /ξ depending on the Higgs mass m H for m t = 171.2,169.1,173.3 GeV (from upper to lower graph). Solid lines correspond to prescription I, dashed to prescription II. Changing the e-foldings number and error in the WMAP normalization measurement introduce changes invisible on the graph. SEWM, June 29 - July 2 2010 p. 27

Cosmological constraint on the Higgs mass 1 loop computation m min = [136.7 + (m t 171.2) 1.95] GeV m max = [184.5 + (m t 171.2) 0.50] GeV 2 loop computation m min = [126.1 + m t 171.2 2.1 m max = [193.9 + m t 171.2 2.1 4.1 α s 0.1176 0.002 0.6 α s 0.1176 0.002 1.5] GeV, 0.1] GeV. Main effect - the window for the Higgs mass is wider. Theoretical uncertainty ±2.2 GeV. Spectral index behaviour is the same in one and two loops. SEWM, June 29 - July 2 2010 p. 28

Experimental constraints on the Higgs mass Direct searches LEP limit: m H > 114.4 GeV, 95% C.L. Tevatron experiments CDF and D0: the region 160 GeV < m H < 170 GeV is excluded, 95% C.L. The LEP Electroweak Working Group: The preferred value: m H = 90 +36 27 Upper limit: GeV, 68% C.L. m H < 163 GeV, one-sided 95% C.L. Upper limit accounting for LEP result: m H < 191 GeV, one-sided 95% C.L. SEWM, June 29 - July 2 2010 p. 29

Comparison with other works Result of A. Barvinsky, A. Kamenshchik and A. Starobinsky 08: The spectral index n s of scalar fluctuations is only consistent with WMAP observations for a particular value of the Higgs mass m H 230 GeV. Therefore, Higgs inflation is excluded by electroweak precision tests. Running of the coupling constants was not taken into account. Result of A. Barvinsky, A. Kamenshchik, C. Kiefer, A. Starobinsky and C. Steinwachs (one loop) 09: Higgs inflation can take place if 135.6 GeV < m H < 184.5 GeV Their 1-loop numbers agree with our 1-loop numbers Result of A. De Simone, M. Hertzberg and F. Wilczek 08: Higgs inflation works if m > [125.7 + m t 171 2 3.8 α s 0.1176 0.0020 1.4] GeV Our 2-loop computation of m min is in good agreement with their work. SEWM, June 29 - July 2 2010 p. 30

Comparison with other works Simone, M. Hertzberg and F. Wilczek Bezrukov, MS SEWM, June 29 - July 2 2010 p. 31

Comparison with other works r Simone, M. Hertzberg and F. Wilczek 0.003 0.0025 0.002 0.0015 0.001 0.0005 130 140 150 160 170 180 190 m H,GeV Bezrukov, MS SEWM, June 29 - July 2 2010 p. 32

Comparison with other works 0.970 0.965 classical Spectral index ns 0.960 0.955 0.950 M t 169 M t 171 M t 173 0.945 0.940 130 140 150 160 170 180 190 Higgs mass M H GeV Barvinsky et al 09 Bezrukov, MS SEWM, June 29 - July 2 2010 p. 33

Reasons for discrepancies Simone et al : Formalism is not gauge invariant. In particular, Landau gauge was used, but the contribution of Goldstones was not included. Consequences: Running of coupling constants in the inflationary phase is different from ours. Pole-MS matching conditions for coupling constants at the electroweak scale are gauge dependent. Barvinsky et al : The mass of Goldstone bosons in the inflationary phase was taken to be m 2 G = λv2. In fact, it is Consequence: m 2 G = λv 2 (1 + ξv2 M 2 P ) 3 λv2 the Goldstones contributions is largely overestimated (by a factor 50 6 ), what resulted in decrease of n s for large Higgs masses and in larger deviations from the tree value at small Higgs masses. SEWM, June 29 - July 2 2010 p. 34

Comparison with other works Germani - Kehagias 10 : Yet another possibility for Higgs-inflation: action with derivative couplings, S = d 4 x g { R µν µh ν h 2M 2 }, R µν Einstein tensor No new degrees of freedom are introduced as previously. COBE normalisation: M 10 10 GeV. Different prediction of spectral indexes (N 60 is the number of e-foldings): BS : n s 1/(2N) 0.97, r 12/N 2 0.003 GK : n s 5/(3N) 0.97, r 8/(3N) 0.05 In GK case no analysis of reheating and of radiative corrections has been performed. SEWM, June 29 - July 2 2010 p. 35

Naturalness of Higgs inflation SEWM, June 29 - July 2 2010 p. 36

Naturalness of Higgs inflation Standard Model: is the value ξ 10 3 10 4 natural? SEWM, June 29 - July 2 2010 p. 36

Naturalness of Higgs inflation Standard Model: is the value ξ 10 3 10 4 natural? SM: is M P /M W 10 17 natural? SEWM, June 29 - July 2 2010 p. 36

Naturalness of Higgs inflation Standard Model: is the value ξ 10 3 10 4 natural? SM: is M P /M W 10 17 natural? SM: is m t /m u 10 5 natural? SEWM, June 29 - July 2 2010 p. 36

Naturalness of Higgs inflation Standard Model: is the value ξ 10 3 10 4 natural? SM: is M P /M W 10 17 natural? SM: is m t /m u 10 5 natural? SM: is m τ /m e 10 3 natural? SEWM, June 29 - July 2 2010 p. 36

Naturalness of Higgs inflation Standard Model: is the value ξ 10 3 10 4 natural? SM: is M P /M W 10 17 natural? SM: is m t /m u 10 5 natural? SM: is m τ /m e 10 3 natural? Real physics question is not whether this or that theory is natural but whether it is realised in Nature... SEWM, June 29 - July 2 2010 p. 36

What happens at large ξ? Sibiryakov, 08; Burgess, Lee, Trott, 09; Barbon and Espinosa, 09 Tree amplitudes of scattering of scalars above electroweak vacuum hit the unitarity bound at energies E > Λ M P ξ What does it mean? Option 1: The theory fails and must be replaced by a more fundamental one SEWM, June 29 - July 2 2010 p. 37

Option 1 We do not know the more fundamental theory. So, lets add to the SM all sorts of higher dimensional operators suppressed by powers of Λ. Result: good inflation is only possible if the coefficients in front of these operators are small, e.g. β h6 Λ 2, β < 1/ξ2 Since it is not clear why these operators are so much suppressed: the Higgs-inflation is unnatural. Burgess, Lee, Trott, 09; Barbon and Espinosa, 09 SEWM, June 29 - July 2 2010 p. 38

Option 2 Suppose that SM is valid up to the Planck scale. If higher dimensional operators are added, they are suppressed by M P rather than Λ. Perturbation theory for scattering breaks down at E > Λ, meaning than non-perturbative methods are to be used. Computation of the effective potential in inflationary region shows that the higher order corrections are small compared with leading terms. In other words, inflationary dynamics decouples from strong coupling which exist in scattering of the quanta at high energies. Quantum scale invariance - no dangerous operators: the Higgs-inflation is natural. M.S., Zenhausern SEWM, June 29 - July 2 2010 p. 39

Conclusions SEWM, June 29 - July 2 2010 p. 40

Conclusions No new particle inflaton is needed for inflation SEWM, June 29 - July 2 2010 p. 40

Conclusions No new particle inflaton is needed for inflation Higgs boson of the Standard Model can make the universe flat, homogeneous and isotropic, and can lead to primordial perturbations needed for structure formation Inflation is possible in the window of the Higgs boson masses M H [126, 194] GeV (for central values of m t and α s ). This region is somewhat wider that the region of validity of the SM all the way up to the Planck scale M H [126.3, 174] GeV (for central values of m t and α s ). SEWM, June 29 - July 2 2010 p. 40

Open problems SEWM, June 29 - July 2 2010 p. 41

Open problems UV completion of the Standard Model SEWM, June 29 - July 2 2010 p. 41