Noncommutative geometry and cosmology Matilde Marcolli Beijing, August 2013 Matilde Marcolli Noncommutative geometry and cosmology This talk is based on: MPT M. Marcolli, E. Pierpaoli, K. Teh, The spectral action and cosmic topology, arXiv:1005.2256, CMP 304 (2011) 125-174 MPT2 M. Marcolli, E. Pierpaoli, K. Teh, The coupling of topology and inflation in Noncommutative Cosmology, arXiv:1012.0780, CMP 309 (2012) N.2, 341-369 CMTB. Ca´ci´c,M.´ Marcolli, K. Teh, Topological coupling of gravity to matter, spectral action and cosmic topology, arXiv:1106.5473, to appear in JNCG and refs on NCG standard model and cosmology: CCM A. Chamseddine, A. Connes, M. Marcolli, Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007), no. 6, 991{1089. MP M. Marcolli, E. Pierpaoli, Early universe models from noncommutative geometry, arXiv:0908.3683, Adv. Theor. Math. Phys. 14 (2010) 1373-1432 KM D. Kolodrubetz, M. Marcolli, Boundary conditions of the RGE flow in noncommutative cosmology, arXiv:1006.4000, Phys. Lett. B 693 (2010) 166-174. Matilde Marcolli Noncommutative geometry and cosmology Two topics of current interest to cosmologists: • Modified Gravity models in cosmology: Einstein-Hilbert action (+cosmological term) replaced or extended with other gravity terms (conformal gravity, higher derivative terms) ) cosmological predictions • The question of Cosmic Topology: Nontrivial (non-simply-connected) spatial sections of spacetime, homogeneous spherical or flat spaces: how can this be detected from cosmological observations? Matilde Marcolli Noncommutative geometry and cosmology The question of Cosmic Topology: Nontrivial (non-simply-connected) spatial sections of spacetime, homogeneous spherical or flat spaces: how can this be detected from cosmological observations? Matilde Marcolli Noncommutative geometry and cosmology Cosmic Microwave Background best source of cosmological data on which to test theoretical models (modified gravity models, cosmic topology hypothesis, particle physics models) COBE satellite (1989) WMAP satellite (2001) Planck satellite (2009): new data available now! Matilde Marcolli Noncommutative geometry and cosmology Cosmic topology and the CMB Einstein equations determine geometry not topology (don't distinguish S3 from S3=Γ with round metric) Cosmological data (BOOMERanG experiment 1998, WMAP data 2003): spatial geometry of the universe is flat or slightly positively curved Homogeneous and isotropic compact case: spherical space forms S3=Γ or Bieberbach manifolds T 3=Γ Is cosmic topology detected by the Cosmic Microwave Background (CMB)? Search for signatures of multiconnected topologies Matilde Marcolli Noncommutative geometry and cosmology CMB sky and spherical harmonics temperature fluctuations 1 ` ∆T X X = a Y T `m `m `=0 m=−` Y`m spherical harmonics Methods to address cosmic topology problem Statistical search for matching circles in the CMB sky: identify a nontrivial fundamental domain Anomalies of the CMB: quadrupole suppression, the small value of the two- point temperature correlation function at angles above 60 degrees, and the anomalous alignment of the quadrupole and octupole Residual gravity acceleration: gravitational effects from other fundamental domains Bayesian analysis of different models of CMB sky for different candidate topologies Results: no conclusive evidence of a non-simply connected topology Matilde Marcolli Noncommutative geometry and cosmology Our approach: NCG provides a modified gravity model through the spectral action The nonperturbative form of the spectral action determines a slow-roll inflation potential The underlying geometry (spherical/flat) affects the shape of the potential (possible models of inflation) Different inflation scenarios depending on geometry and topology of the cosmos More refined topological properties from coupling to matter Matilde Marcolli Noncommutative geometry and cosmology The noncommutative space X × F extra dimensions product of 4-dim spacetime and finite NC space The spectral action functional 1 Tr(f (D =Λ)) + h J ξ;~ D ξ~i A 2 A 0 −1 DA = D + A + " JAJ Dirac operator with inner fluctuations ∗ P A = A = k ak [D; bk ] Action functional for gravity on X (modified gravity) Gravity on X × F = gravity coupled to matter on X Matilde Marcolli Noncommutative geometry and cosmology Spectral triples( A; H; D): • involutive algebra A • representation π : A!L(H) • self adjoint operator D on H • compact resolvent (1 + D2)−1=2 2 K • [a; D] bounded 8a 2 A • even Z=2-grading [γ; a] = 0 and Dγ = −γD • real structure: antilinear isom J : H!H with J2 = ", JD = "0DJ, and Jγ = "00γJ n 0 1 2 3 4 5 6 7 " 1 1 -1 -1 -1 -1 1 1 "0 1 -1 1 1 1 -1 1 1 "00 1 -1 1 -1 • bimodule: [a; b0] = 0 for b0 = Jb∗J−1 • order one condition: [[D; a]; b0] = 0 Matilde Marcolli Noncommutative geometry and cosmology Asymptotic formula for the spectral action (Chamseddine{Connes) Z X k −k Tr(f (D=Λ)) ∼ fk Λ −|Dj + f (0)ζD (0) + o(1) k2DimSp R 1 k−1 for large Λ with fk = 0 f (v)v dv and integration given by residues of −s zeta function ζD (s) = Tr(jDj ); DimSp poles of zeta functions Asymptotic expansion ) Effective Lagrangian (modified gravity + matter) At low energies: only nonperturbative form of the spectral action Tr(f (DA=Λ)) Need explicit information on the Dirac spectrum! Matilde Marcolli Noncommutative geometry and cosmology 1 2 Product geometry( C (X ); L (X ; S); DX ) [ (AF ; HF ; DF ) 1 1 A = C (X ) ⊗ AF = C (X ; AF ) 2 2 H = L (X ; S) ⊗ HF = L (X ; S ⊗ HF ) D = DX ⊗ 1 + γ5 ⊗ DF Inner fluctuations of the Dirac operator 0 −1 D ! DA = D + A + " JAJ A self-adjoint operator X A = aj [D; bj ] ; aj ; bj 2 A ) boson fields from inner fluctuations, fermions from HF Matilde Marcolli Noncommutative geometry and cosmology Get realistic particle physics models [CCM] Need Ansatz for the NC space F ALR = C ⊕ HL ⊕ HR ⊕ M3(C) ) everything else follows by computation Representation: MF sum of all inequiv irred odd N ALR -bimodules (fix N generations) HF = ⊕ MF fermions Algebra AF = C ⊕ H ⊕ M3(C): order one condition F zero dimensional but KO-dim 6 JF = matter/antimatter, γF = L/R chirality Classification of Dirac operators (moduli spaces) Matilde Marcolli Noncommutative geometry and cosmology Parameters of νMSM Get an extension of minimal Standard Model with right handed neutrinos, Majorana mass terms, and lepton mixing - three coupling constants - 6 quark masses, 3 mixing angles, 1 complex phase - 3 charged lepton masses, 3 lepton mixing angles, 1 complex phase - 3 neutrino masses - 11 Majorana mass matrix parameters • Bosons from inner fluctuations of D (gauge: manifold direction, Higgs: NC direction) • Fermions from basis of representation space HF • Correct gauge group and hypercharges of particles • Action functional: spectral action on X × F , asymptotic expansion gives full Lagrangian of νMSM with non-minimal coupling to (modified) gravity. Matilde Marcolli Noncommutative geometry and cosmology The asymptotic expansion of the spectral action from [CCM] 1 f Z p S = (48 f Λ4 − f Λ2 c + 0 d) g d 4x π2 4 2 4 96 f Λ2 − f c Z p + 2 0 R g d 4x 24π2 f Z 11 p + 0 ( R∗R∗ − 3 C C µνρσ) g d 4x 10 π2 6 µνρσ (−2 a f Λ2 + e f ) Z p + 2 0 j'j2 g d 4x π2 f a Z p + 0 jD 'j2 g d 4x 2 π2 µ f a Z p − 0 R j'j2 g d 4x 12 π2 f b Z p + 0 j'j4 g d 4x 2 π2 f Z 5 p + 0 (g 2 G i G µνi + g 2 F α F µνα + g 2 B Bµν ) g d 4x; 2 π2 3 µν 2 µν 3 1 µν Matilde Marcolli Noncommutative geometry and cosmology Parameters: 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 SM+r.h.ν 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 Noncommutative geometry and cosmology Normalization and coefficients 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 Energy scale: Unification (1015 { 1017 GeV) g 2f 1 0 = 2π2 4 Preferred energy scale, unification of coupling constants Matilde Marcolli Noncommutative geometry and cosmology Coefficients 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 f2Λ e 1 µ0 = 2 − ξ0 = 12 f0 a π2b λ0 = 2 2f0a Matilde Marcolli Noncommutative geometry and cosmology Renormalization group running of parameters Used to obtain physical predictions from the particle physics sector of the model: - [CCM] RGE for particle sector to estimate Higgs mass: 170 GeV... too large! - Recent methods for correcting to realistic Higgs mass ∼125 GeV: • A. Chamseddine, A. Connes, arXiv:1208.1030 (additional scalar field coupled to Higgs) • C. Estrada, M. Marcolli, arXiv:1208.5023 (gravity correction terms, asymptotic safety, anomalous dimensions) - In [MP] [KM]: running coefficients with RGE flow of particle physics content from unification energy down to electroweak.