Cosmology and the Poisson summation formula

Transcription

1 Bonn, June 2010

2 This talk is based on: MPT M. Marcolli, E. Pierpaoli, K. Teh, The spectral action and cosmic topology, arxiv: The 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, MP M. Marcolli, E. Pierpaoli, Early universe models from noncommutative geometry, arxiv: KM D. Kolodrubetz, M. Marcolli, Boundary conditions of the RGE flow in noncommutative cosmology, arxiv:

3 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?

4 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 More refined topological properties? (coupling to matter)

5 The noncommutative space X F extra dimensions product of 4-dim spacetime and finite NC space The spectral action functional Tr(f (D A /Λ)) J ξ, D A ξ D A = D + A + ε J A J 1 Dirac operator with inner fluctuations A = A = k a k[d, b k ] Action functional for gravity on X (modified gravity) Gravity on X F = gravity coupled to matter on X

6 Spectral triples (A, H, D): involutive algebra A representation π : A L(H) self adjoint operator D on H compact resolvent (1 + D 2 ) 1/2 K [a, D] bounded a A even Z/2-grading [γ, a] = 0 and Dγ = γd real structure: antilinear isom J : H H with J 2 = ε, JD = ε DJ, and Jγ = ε γj n ε ε ε bimodule: [a, b 0 ] = 0 for b 0 = Jb J 1 order one condition: [[D, a], b 0 ] = 0

7 Asymptotic formula for the spectral action (Chamseddine Connes) Tr(f (D/Λ)) f k Λ k D k + f (0)ζ D (0) + o(1) k DimSp for large Λ with f k = 0 f (v)v k 1 dv and integration given by residues of zeta function ζ D (s) = Tr( D s ); DimSp poles of zeta functions Asymptotic expansion Effective Lagrangian (modified gravity + matter) At low energies: only nonperturbative form of the spectral action Tr(f (D A /Λ)) Need explicit information on the Dirac spectrum!

8 Product geometry (C (X ), L 2 (X, S), D X ) (A F, H F, D F ) A = C (X ) A F = C (X, A F ) H = L 2 (X, S) H F = L 2 (X, S H F ) D = D X 1 + γ 5 D F Inner fluctuations of the Dirac operator A self-adjoint operator D D A = D + A + ε J A J 1 A = a j [D, b j ], a j, b j A boson fields from inner fluctuations, fermions from H F

9 Get realistic particle physics models [CCM] Need Ansatz for the NC space F A LR = C H L H R M 3 (C) everything else follows by computation Representation: M F sum of all inequiv irred odd A LR -bimodules (fix N generations) H F = N M F fermions Algebra A F = C H M 3 (C): order one condition F zero dimensional but KO-dim 6 J F = matter/antimatter, γ F = L/R chirality Classification of Dirac operators (moduli spaces)

10 Dirac operators and Majorana mass terms ( ) S T D(Y ) =, S = S T S 1 (S ), T = Y R : ν R J F ν R 0 0 Y( 1) 0 S 1 = Y( 1) Y ( 1) Y ( 1) Y( 3) 0 S 3 = Y ( 3) Y ( 3) Y ( 3) 0 0 Yukawa matrices: Dirac masses and mixing angles in GL N=3 (C) Y e = Y ( 1) (charged leptons) Y ν = Y ( 1) (neutrinos) Y d = Y ( 3) (d/s/b quarks) Y u = Y ( 3) (u/c/t quarks) M = YR t Majorana mass terms symm matrix

11 Moduli space of Dirac operators on finite NC space F C 3 C 1 C 3 = pairs (Y ( 3), Y ( 3) ) modulo W j unitary matrices: Y ( 3) = W 1 Y ( 3) W 3, Y ( 3) = W 2 Y ( 3) W 3 G = GL 3 (C) and K = U(3): C 3 = (K K)\(G G)/K dim R C 3 = 10 = (eigenval, coset 3 angles 1 phase) C 1 = triplets (Y ( 1), Y ( 1), Y R ) with Y R symmetric modulo Y ( 1) = V 1 Y ( 1) V 3, Y ( 1) = V 2 Y ( 1) V 3, Y R = V 2 Y R V 2 π : C 1 C 3 surjection forgets Y R fiber symm matrices mod Y R λ 2 Y R dim R (C 3 C 1 ) = 31 (dim fiber 12-1=11)

12 Parameters of νmsm - 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 - QCD vacuum angle Moduli space of Dirac operators on F geometric form of all the Yukawa and Majorana parameters

13 Fields content of the model Bosons: inner fluctuations A = j a j[d, b j ] - In M direction: U(1), SU(2), and SU(3) gauge bosons - In F direction: Higgs field H = ϕ 1 + ϕ 2 j Fermions: basis of H F 3 0, 3 0, 1 0, 1 0 Gauge group SU(A F ) = U(1) SU(2) SU(3) (up to fin abelian group) Hypercharges: adjoint action of U(1) (in powers of λ U(1)) L R Correct hypercharges to the fermions

14 Action functional Tr(f (D A /Λ)) J ξ, D A ξ Fermion part: antisymmetric bilinear form A( ξ) on H + = {ξ H γξ = ξ} nonzero on Grassmann variables Euclidean functional integral Pfaffian Pf (A) = e 1 2 A( ξ) D[ ξ] (avoids Fermion doubling problem of previous models based on symmetric ξ, D A ξ for NC space with KO-dim=0) Explicit computation gives part of SM Larangian with L Hf = coupling of Higgs to fermions L gf = coupling of gauge bosons to fermions L f = fermion terms

15 The asymptotic expansion of the spectral action from [CCM] S = 1 π 2 (48 f 4 Λ 4 f 2 Λ 2 c + f 0 g 4 d) d 4 x + 96 f 2 Λ 2 f 0 c 24π 2 R g d 4 x f π 2 ( 11 6 R R 3 C µνρσ C µνρσ ) g d 4 x + ( 2 a f 2 Λ 2 + e f 0 ) π 2 ϕ 2 g d 4 x + f 0a 2 π 2 D µ ϕ 2 g d 4 x f 0 a 12 π 2 R ϕ 2 g d 4 x + f 0b 2 π 2 ϕ 4 g d 4 x f π 2 (g3 2 Gµν i G µνi + g2 2 Fµν α F µνα g 1 2 B µν B µν ) g d 4 x,

16 Parameters: f 0, f 2, f 4 free parameters, f 0 = f (0) and, for k > 0, f k = 0 f (v)v k 1 dv. a, b, c, d, e functions of Yukawa parameters of SM+r.h.ν a = Tr(Y ν Y ν + Y e Y e + 3(Y u Y u + Y d Y d)) b = Tr((Y ν Y ν ) 2 + (Y e Y e ) 2 + 3(Y u Y u ) 2 + 3(Y d Y d) 2 ) c = Tr(MM ) d = Tr((MM ) 2 ) e = Tr(MM Y ν Y ν ).

17 Normalization and coefficients 1 S = 2κ 2 R g g d 4 x + γ 0 d 4 x 0 + α 0 C µνρσ C µνρσ g d 4 x + τ 0 R R g d 4 x + 1 DH 2 g d 4 x µ 2 0 H 2 g d 4 x 2 ξ 0 R H 2 g d 4 x + λ 0 H 4 g d 4 x + 1 (Gµν i G µνi + Fµν α F µνα + B µν B µν ) g d 4 x, 4 Energy scale: Unification ( GeV) g 2 f 0 2π 2 = 1 4 Preferred energy scale, unification of coupling constants

18 Coefficients 1 = 96f 2Λ 2 f 0 c 2κ π 2 γ 0 = 1 π 2 (48f 4Λ 4 f 2 Λ 2 c + f 0 4 d) α 0 = 3f 0 10π 2 τ 0 = 11f 0 60π 2 µ 2 0 = 2f 2Λ 2 e a f 0 λ 0 = π2 b 2f 0 a 2 ξ 0 = 1 12 In [MP] [KM]: running coefficients with RGE flow of particle physics content from unification energy down to electroweak. Very early universe models! (10 36 s < t < s)

19 Effective gravitational constant G eff = κ2 0 8π = Effective cosmological constant 3π 192f 2 Λ 2 2f 0 c(λ) γ 0 = 1 4π 2 (192f 4Λ 4 4f 2 Λ 2 c(λ) + f 0 d(λ)) Conformal non-minimal coupling of Higgs and gravity 1 R gd 4 x 1 R H 2 gd 4 x 16πG eff 12 Conformal gravity 3f 0 10π 2 C µνρσ C µνρσ gd 4 x C µνρσ = Weyl curvature tensor (trace free part of Riemann tensor)

20 Cosmological implications of the NCG SM Linde s hypothesis (antigravity in the early universe) Primordial black holes and gravitational memory Gravitational waves in modified gravity Gravity balls Varying effective cosmological constant Higgs based slow-roll inflation Spontaneously arising Hoyle-Narlikar in EH backgrounds Effects in the very early universe: inflation mechanisms - Remark: Cannot extrapolate to modern universe, nonperturbative effects in the spectral action: requires nonperturbative spectral action

21 Cosmological models for the not-so-early-universe? Need to work with non-perturbative form of the spectral action Can to for specially symmetric geometries! Concentrate on pure gravity part: X instead of X F The spectral action and the question of cosmic topology (with E. Pierpaoli and K. Teh) Spatial sections of spacetime closed 3-manifolds S 3? - Cosmologists search for signatures of topology in the CMB - Model based on NCG distinguishes cosmic topologies? Yes! the non-perturbative spectral action predicts different models of slow-roll inflation

22 ( ) 1 δρ 4 ρ + Φ (ηlss) ˆn.ve(ηLSS) + η0 ηlss ( Φ + Ψ)dη (22) Cosmic topology he quantities Φ and Ψ are the usual Bardeen potentials, Fig.1. Temperaturemap for a Poincaré dodecahedralspace with s the velocity within the electron fluid; overdots denote Ωtot = 1.02, Ωmat = 0.27 and h = 0.70 (using modes up to rivatives. The first terms represent the Sachs-Wolfe and k = 230 for a resolution of 6 ). contributions, evaluated at the LSS. The last term is effect. This formula is independent of the spatial topolis valid in the limit of an infinitely thin LSS, neglecting tion. temperature distribution is calculated with a CMBFast ware developed by one of us 1, under the form of temperctuation maps at the LSS. One such realization is shown, where the modes up to k = 230 give an angular resof about 6 (i.e. roughly comparable to the resolution E map), thus without as fine details as in WMAP data. r, this suffices for a study of topological effects, which inant at larger scales. maps are the starting point for topological analysis: or noise analysis in the search for matched circle pairs, ibedin Sect. 3.2; secondly, throughtheirdecompositions erical harmonics, which predict the power spectrum, as d in Sect. 4. In these two ways, the maps allow direct ison between observational data and theory. cles in the sky Fig. 2. The last scattering surface seen from outside in the universal covering space of the Poincaré dodecahedral space with Ωtot = 1.02, Ωmat = 0.27 and h = 0.70 (using modes up to k = 230 for a resolution of 6 ). Since the volume of the physical (Luminet, Lehoucq, Riazuelo, Weeks, et al.: simulated CMB sky) Best candidates: Poincaré homology 3-sphere and other spherical -connected space can be seen as a cell (called the funl domain), copies of which tile the universal cover. If us of the LSS is greater than the typical radius of the LSS wraps all the way around the universe and interelf along circles. Each circle of self-intersection appears bserver as two different circles on different parts of the with the same OSW components in their temperature ons, because the two different circles on the sky are resame circle in space. If the LSS is not too much bigger space is about 80% of the volume of the last scattering surface, fundamental cell, each circle pair lies in the planes of the latter intersects itself along six pairs of matching circles. ching faces of the fundamental cell. Figure 2 shows the tion of the various translates of the LSS in the universal s seen by an observer sitting inside one of them. forms (quaternionic space), These circles are flat generated tori by a pure Sachs-Wolfe effect; in iazuelo developed the program CMBSlow to take into account reality additional contributions to the CMB temperature fluctuations (Doppler and ISW effects) blur the topological signal. s fine effects, in particular topological ones. Two Testable Cosmological predictions? (in various gravity models)

23 Poisson summation formula h(x + λn) = 1 ( ) 2πinx exp λ λ n Z n Z ĥ( n λ ) λ R + and x R with ĥ(x) = R h(u) e 2πiux du Idea: write Tr(f (D/Λ)) as sums over lattices - Need explicit spectrum of D with multiplicities - Need to write as a union of arithmetic progressions λ n,i, n Z - Multiplicities polynomial functions m λn,i = P i (λ n,i ) Tr(f (D/Λ)) = i P i (λ n,i )f (λ n,i /Λ) n Z

24 The standard topology S 3 (Chamseddine Connes) Dirac spectrum ±a 1 ( n) for n Z, with multiplicity n(n + 1) Tr(f (D/Λ)) = (Λa) 3 f (2) (0) 1 4 (Λa) f (0) + O((Λa) k ) with f (2) Fourier transform of v 2 f (v) 4-dimensional Euclidean S 3 S 1 Tr(h(D 2 /Λ 2 )) = πλ 4 a 3 β 0 u h(u) du 1 2 πλaβ h(u) du+o(λ k ) g(u, v) = 2P(u) h(u 2 (Λa) 2 + v 2 (Λβ) 2 ) ĝ(n, m) = g(u, v)e 2πi(xu+yv) du dv R 2 0

25 A slow roll potential from non-perturbative effects perturbation D 2 D 2 + φ 2 gives potential V (φ) scalar field coupled to gravity Tr(h((D 2 +φ 2 )/Λ 2 ))) = πλ 4 βa 3 uh(u)du π 2 Λ2 βa V(x) = +πλ 4 βa 3 V(φ 2 /Λ 2 ) Λ2 βa W(φ 2 /Λ 2 ) 0 0 u(h(u + x) h(u))du, W(x) = x 0 0 h(u)du h(u)du

26 Slow roll parameters Minkowskian Friedmann metric on S R ds 2 = a(t) 2 ds 2 S dt2 accelerated expansion ä a = H2 (1 ɛ) Hubble parameter H 2 (φ) (1 13 ) ɛ(φ) = 8π 3mPl 2 V (φ) m Pl Planck mass inflation phase ɛ(φ) < 1 η(φ) = m2 Pl 8π ɛ(φ) = m2 Pl 16π ( V ) (φ) 2 V (φ) ( V ) (φ) m2 Pl V (φ) 16π ( V ) (φ) 2 V (φ) second slow-roll parameter measurable quantities n s = 1 6ɛ + 2η r = 16ɛ spectral index and tensor-to-scalar ratio

27 Slow-roll parameters from spectral action S = S 3 ɛ(x) = m2 Pl 16π ( η(x) = m2 Pl 8π ( m2 Pl 16π h(x) 2π(Λa) 2 x h(u)du x 0 h(u)du + 2π(Λa)2 0 u(h(u + x) h(u))du h (x) + 2π(Λa) 2 h(x) x 0 h(u)du + 2π(Λa)2 0 u(h(u + x) h(u))du h(x) 2π(Λa) 2 x h(u)du 0 u(h(u + x) h(u))du x 0 h(u)du + 2π(Λa)2 In Minkowskian Friedmann metric Λ(t) 1/a(t) Also independent of β (artificial Euclidean compactification) ) 2 ) 2

28 The quaternionic space SU(2)/Q8 (quaternion units ±1, ±σ k ) Dirac spectrum (Ginoux) 3 + 4k with multiplicity 2(k + 1)(2k + 1) k + 2 with multiplicity 4k(k + 1) 2 Polynomial interpolation of multiplicities P 1 (u) = 1 4 u u Spectral action P 2 (u) = 1 4 u2 3 4 u 7 16 Tr(f (D/Λ)) = 1 8 (Λa)3 f (2) (0) 1 32 (Λa) f (0) + O(Λ k ) (1/8 of action for S 3 ) with g i (u) = P i (u)f (u/λ): Tr(f (D/Λ)) = 1 4 (ĝ 1(0) + ĝ 2 (0)) + O(Λ k ) from Poisson summation Same slow-roll parameters

29 The dodecahedral space Poincaré homology sphere S 3 /Γ binary icosahedral group 120 elements Dirac spectrum: eigenvalues of S 3 different multiplicities generating function (Bär) F + (z) = m( k, D)zk F (z) = m( ( 3 + k), D)zk 2 k=0 k=0 F + (z) = 16( )G + (z) ( ) 3 ( )H + (z) G + (z) = 6z z z z 17 2z 19 6z 21 2z 23 +2z 25 +4z 27 +3z 29 +z 31 H + (z) = 1 3z 2 4z 4 2z 6 +2z 8 +6z 10 +9z 12 +9z 14 +4z 16 4z 18 9z 20 9z 22 6z 24 2z z z z 32 + z 34 F (z) = 1024( )G (z) ( ) 8 ( )H (z) G (z) = 1+3z 2 +4z 4 +2z 6 2z 8 6z 10 2z z z z 18 +6z 20 H (z) = 1 3z 2 4z 4 2z 6 +2z 8 +6z 10 +9z 12 +9z 14 +4z 16 4z 18 9z 20 9z 22 6z 24 2z z z z 32 + z 34

30 Polynomial interpolation of multiplicities: 60 polynomials P i (u) 59 j=0 P j (u) = 1 2 u2 1 8 Spectral action: functions g j (u) = P j (u)f (u/λ) Tr(f (D/Λ)) = 1 60 = 1 60 R 59 j=0 ĝ j (0) + O(Λ k ) P j (u)f (u/λ)du + O(Λ k ) by Poisson summation 1/120 of action for S 3 Same slow-roll parameters j

31 The flat tori Dirac spectrum (Bär) ± 2π (m, n, p) + (m 0, n 0, p 0 ), (1) (m, n, p) Z 3 multiplicity 1 and constant vector (m 0, n 0, p 0 ) depending on spin structure Tr(f (D3 2 /Λ 2 )) = ( 4π 2 ((m + m 0 ) 2 + (n + n 0 ) 2 + (p + p 0 ) 2 ) ) (m,n,p) Z 3 2f Poisson summation Z 3 g(m, n, p) = Z 3 ĝ(m, n, p) ĝ(m, n, p) = g(u, v, w)e 2πi(mu+nv+pw) dudvdw R ( 3 4π 2 ((m + m 0 ) 2 + (n + n 0 ) 2 + (p + p 0 ) 2 ) ) g(m, n, p) = f Λ 2 Λ 2

32 Spectral action for the flat tori Tr(f (D3/Λ 2 2 )) = Λ3 4π 3 f (u 2 + v 2 + w 2 )du dv dw + O(Λ k ) R 3 X = T 3 S 1 β : Tr(h(D 2 X /Λ2 )) = Λ4 βl 3 4π 0 uh(u)du + O(Λ k ) using ( 4π 2 2 h (Λl) 2 ((m + m 0) 2 + (n + n 0 ) 2 + (p + p 0 ) 2 ) + 1 (Λβ) 2 (r + 1 ) 2 )2 (m,n,p,r) Z 4 ( 4π 2 g(u, v, w, y) = 2 h Λ 2 (u2 + v 2 + w 2 ) + y 2 ) (Λβ) 2 (m,n,p,r) Z 4 g(m+m 0, n+n 0, p +p 0, r ) = (m,n,p,r) Z 4 ( 1) r ĝ(m, n, p, r)

33 Different slow-roll potential and parameters Introducing the perturbation D 2 D 2 + φ 2 : Tr(h((D 2 X + φ2 )/Λ 2 )) = Tr(h(D 2 X /Λ2 )) + Λ4 βl 3 slow-roll potential V(x) = V (φ) = Λ4 βl 3 4π V(φ2 /Λ 2 ) 0 u (h(u + x) h(u)) du 4π V(φ2 /Λ 2 ) Slow-roll parameters (different from spherical cases) ( ) 2 ɛ = m2 h(u)du Pl x 16π u(h(u + x) h(u))du 0 ( η = m2 Pl h(x) 8π u(h(u + x) h(u))du 1 ) 2 h(u)du x 2 u(h(u + x) h(u))du 0 0

34 Conclusion (for now) A modified gravity model based on the spectral action cannot rule out most likely cosmic topology candidates (dodecahedral, quaternionic) but can distinguish the spherical candidates from the flat ones on the basis of different inflation scenarios!