Early Universe Models Matilde Marcolli Ma148b Spring 2016 Topics in Mathematical Physics Matilde Marcolli Early Universe Models This lecture based on Matilde Marcolli, Elena Pierpaoli, Early universe models from Noncommutative Geometry, Advances in Theoretical and Mathematical Physics, Vol.14 (2010) N.5, 1373{1432. Daniel Kolodrubetz, Matilde Marcolli, Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology, Phys. Lett. B, Vol.693 (2010) 166{174 Re-examine RGE flow; gravitational terms in the asymptotic form of the spectral action; cosmological timeline; running in the very early universe; inflation and other gravitational phenomena Matilde Marcolli Early Universe Models Asymptotic expansion of spectral action for large energies Z Z 1 p 4 p 4 S = 2 R g d x + γ0 g d x 2κ0 Z Z µνρσp 4 ∗ ∗p 4 + α0 Cµνρσ C g d x + τ0 R R g d x 1 Z p Z p + jDHj2 g d 4x − µ2 jHj2 g d 4x 2 0 Z Z 2 p 4 4 p 4 − ξ0 R jHj g d x + λ0 jHj g d x 1 Z p + (G i G µνi + F α F µνα + B Bµν ) g d 4x 4 µν µν µν A modified gravity model with non minimal coupling to Higgs and minimal coupling to gauge theories Matilde Marcolli Early Universe Models Coefficients and parameters: 2 1 96f2Λ − f0c 1 4 2 f0 2κ2 = 2 γ0 = 2 (48f4Λ − f2Λ c + d) 0 24π π 4 3f 11f α = − 0 τ = 0 0 10π2 0 60π2 2 2 2 f2Λ e 1 π b µ0 = 2 − ξ0 = 12 λ0 = 2 f0 a 2f0a f0, f2, f4 free parameters, f0 = f (0) and, for k > 0, Z 1 k−1 fk = f (v)v dv: 0 a; b; c; d; e functions of Yukawa parameters of νMSM y y y y a = Tr(Yν Yν + Ye Ye + 3(Yu Yu + Yd Yd )) y 2 y 2 y 2 y 2 b = Tr((Yν Yν) + (Ye Ye ) + 3(Yu Yu) + 3(Yd Yd ) ) c = Tr(MMy) d = Tr((MMy)2) y y e = Tr(MM Yν Yν): Matilde Marcolli Early Universe Models Two different perspectives on running: parameters a; b; c; d; e run with RGE The relation between coefficients κ0, γ0, α0, τ0, µ0, ξ0, λ0 and parameters a; b; c; d; e hold only at Λunif : constraint on initial conditions; independent running There is a range of energies Λmin ≤ Λ ≤ Λunif where the running of coefficients κ0, γ0, α0, τ0, µ0, ξ0, λ0 determined by running of a; b; c; d; e and relation continues to hold (very early universe only) The first perspective requires independent running of gravitational parameters with given condition at unification; the second perspective allows for interesting gravitational phenomena in the very early universe specific to NCG model only Matilde Marcolli Early Universe Models The first approach followed in Ali Chamseddine, Alain Connes, Matilde Marcolli, Gravity and the Standard Model with neutrino mixing, ATMP 11 (2007) 991{1090, arXiv:hep-th/0610241 For modified gravity models Z 1 ! θ p C C µνρσ − R2 + R∗R∗ g d4x 2η µνρσ 3η η Running of gravitational parameters (Avramidi, Codello{Percacci, Donoghue) by 1 133 β = − η2 η (4π)2 10 1 25 + 1098 ! + 200 !2 β = − η ! (4π)2 60 1 7(56 − 171 θ) β = η: θ (4π)2 90 Matilde Marcolli Early Universe Models With relations at unification get running like log10 Μ GeV 5 10 15 log10 Μ GeV 4 6 8 10 12 14 16 "0.4 ! " # ! " # "0.005 " 0.6 Ω log Μ MZ "0.8 "0.010 ! ! " ## Η log Μ MZ "1.0 ! ! " ## "0.015 "1.2 log10 Μ GeV 5 10 15 ! " # "0.2 Θ log Μ MZ "0.4 ! ! " ## "0.6 Plausible values in low energy limit (near fixed points) Look then at second point of view: M.M.-E.Pierpaoli (2009) Matilde Marcolli Early Universe Models Renormalization group equations for SM with right handed neutrinos and Majorana mass terms, from unification energy (2 × 1016 GeV) down to the electroweak scale (102 GeV) AKLRS S. Antusch, J. Kersten, M. Lindner, M. Ratz, M.A. Schmidt Running neutrino mass parameters in see-saw scenarios, JHEP 03 (2005) 024. Remark: RGE analysis in [CCM] only done using minimal SM Matilde Marcolli Early Universe Models df 1-loop RGE equations: Λ dΛ = βf (Λ) 19 41 16π2 β = b g 3 with (b ; b ; b ) = (−7; − ; ) gi i i SU(3) SU(2) U(1) 6 10 3 3 17 9 16π2 β = Y ( Y yY − Y yY + a − g 2 − g 2 − 8g 2) Yu u 2 u u 2 d d 20 1 4 2 3 3 3 1 9 16π2 β = Y ( Y yY − Y yY + a − g 2 − g 2 − 8g 2) Yd d 2 d d 2 u u 4 1 4 2 3 3 3 9 9 16π2 β = Y ( Y yY − Y yY + a − g 2 − g 2) Yν ν 2 ν ν 2 e e 20 1 4 2 3 3 9 9 16π2 β = Y ( Y yY − Y yY + a − g 2 − g 2) Ye e 2 e e 2 ν ν 4 1 4 2 2 y y T 16π βM = Yν Yν M + M(Yν Yν ) 3 3 3 16π2 β = 6λ2 − 3λ(3g 2 + g 2) + 3g 4 + ( g 2 + g 2)2 + 4λa − 8b λ 2 5 1 2 2 5 1 2 2 Note: different normalization from [CCM] and 5=3 factor included in g1 Matilde Marcolli Early Universe Models Method of AKLRS: non-degenerate spectrum of Majorana masses, different effective field theories in between the three see-saw scales: RGE from unification Λunif down to first see-saw scale (largest eigenvalue of M) (3) Introduce Yν removing last row of Yν in basis where M diagonal and M(3) removing last row and column. Induced RGE down to second see-saw scale (2) (2) Introduce Yν and M , matching boundary conditions Induced RGE down to first see-saw scale (1) (1) Introduce Yν and M , matching boundary conditions Induced RGE down to electoweak energy Λew (N) (N) Use effective field theories Yν and M between see-saw scales Matilde Marcolli Early Universe Models Running of coefficients a; b with RGE 1.07 1.90 1.06 1.89 1.05 1.88 1.04 2´1014 4´1014 6´1014 8´1014 1´1015 2´1014 4´1014 6´1014 8´1014 1´1015 Coefficients a and b near the top see-saw scale Matilde Marcolli Early Universe Models Running of coefficient c with RGE 3.45383´1029 3.49´1029 3.45392´1029 29 3.45390´1029 3.45383´10 3.48´1029 29 3.45388´10 3.45383´1029 3.47´1029 3.45386´1029 4.1´1012 4.2´1012 4.3´1012 4.4´1012 4.5´1012 2´1014 4´1014 6´1014 8´1014 1´1015 1.05´10141.10´10141.15´10141.20´10141.25´10141.30´1014 Running of coefficient d with RGE 1.10558´1059 59 1.10560´1059 1.10558´10 1.10558´1059 1.10560´1059 1.10558´1059 1.10558´1059 1.10560 1059 ´ 1.10558´1059 1.10558´1059 1.10559´1059 1.10558´1059 59 1.10558´1059 1.10558´10 1.05´10141.10´10141.15´10141.20´10141.25´10141.30´1014 1´1014 2´1014 3´1014 4´1014 5´1014 3.5´1012 4.0´1012 4.5´1012 5.0´1012 Effect of the three see-saw scales Matilde Marcolli Early Universe Models Running of coefficient e with RGE 3.0´1029 6.215´1027 2.5´1029 6.210´1027 6.1865´1027 2.0´1029 6.205´1027 29 1.5´10 6.200´1027 6.1860´1027 29 1.0´10 6.195´1027 5.0´1028 6.190´1027 2´1012 4´1012 6´1012 8´1012 1´1013 2´1014 4´1014 6´1014 8´1014 1´1015 5.0´1013 1.0´1014 1.5´1014 2.0´1014 Effect of the three see-saw scales With default boundary conditions at unification of AKLRS ...but sensitive dependence on the initial conditions ) fine tuning Matilde Marcolli Early Universe Models Changing initial conditions: maximal mixing conditions at unification ζ = exp(2πi=3) 0 1 ζ ζ2 1 1 U (Λ ) = ζ 1 ζ PMNS unif 3 @ A ζ2 ζ 1 0 12:2 × 10−9 0 0 1 1 δ = 0 170 × 10−6 0 ("1) 246 @ A 0 0 15:5 × 10−3 y Yν = UPMNS δ("1)UPMNS Daniel Kolodrubetz, Matilde Marcolli, Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology, arXiv:1006.4000 Matilde Marcolli Early Universe Models Effect on coefficients running Matilde Marcolli Early Universe Models Matilde Marcolli Early Universe Models Further evidence of sensitive dependence: changing only one parameter in diagonal matrix Yν get running of top term: Matilde Marcolli Early Universe Models Geometric constraints at unification energy λ parameter constraint 2 π b(Λunif ) λ(Λunif ) = 2 2f0 a(Λunif ) Higgs vacuum constraint p af 2M 0 = W π g See-saw mechanism and c constraint 2 2 2f2Λunif 6f2Λunif ≤ c(Λunif ) ≤ f0 f0 Mass relation at unification X 2 2 2 2 2 (mν + me + 3mu + 3md )jΛ=Λunif = 8MW jΛ=Λunif generations Matilde Marcolli Early Universe Models Choice of initial conditions at unification: Compatibility with low energy values: experimental constraints Compatibility with geometric constraints at unification It is possible to modify boundary conditions to achieve both compatibilities Example: using maximal mixing conditions but modify parameters in the Majorana mass matrix and initial condition of Higgs parameter to satisfy geometric constraints Matilde Marcolli Early Universe Models Cosmology timeline Planck epoch: t ≤ 10−43 s after the Big Bang (unification of forces with gravity, quantum gravity) Grand Unification epoch: 10−43 s ≤ t ≤ 10−36 s (electroweak and strong forces unified; Higgs) Electroweak epoch: 10−36 s ≤ t ≤ 10−12 s (strong and electroweak forces separated) Inflationary epoch: possibly 10−36 s ≤ t ≤ 10−32 s - NCG SM preferred scale at unification; RGE running between unification and electroweak ) info on inflationary epoch.