Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Transcription

1 Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies Ma148b Spring 2016 Topics in Mathematical Physics

2 Based on: Adam Ball,, Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology, arxiv:

3 Homogeneity versus Isotropy in Cosmology Homogeneous and isotropic: Friedmann universe R S 3 ±dt 2 + a(t) 2 ( σ1 2 + σ2 2 + σ3 2 ) with round metric on S 3 with SU(2) invariant 1-forms {σ i } satisfying relations dσ i = σ j σ k for all cyclic permutations (i, j, k) of (1, 2, 3)

4 Homogeneous but not isotropic: Bianchi IX mixmaster models R S 3 F (t) (±dt 2 + σ2 1 W 2 1 (t) + σ2 2 W 2 2 (t) + σ2 3 W 3 3 (t)) with a conformal factor F (t) W 1 (t)w 2 (t)w 3 (t) Isotropic but not homogeneous? Swiss Cheese Models

5 Main Idea: M.J. Rees, D.W. Sciama, Large-scale density inhomogeneities in the universe, Nature, Vol.217 (1968) Cut off 4-balls from a FRW spacetime and replace with different density smaller region outside/inside patched across boundary with vanishing Weyl curvature tensor (isotropy preserved)

6 Packed Swiss Cheese Cosmology Iterate construction removing more and more balls Apollonian sphere packing of 3-dimensional spheres Residual set of sphere packing is fractal Proposed as explanation for possible fractal distribution of matter in galaxies, clusters, and superclusters F. Sylos Labini, M. Montuori, L. Pietroneo, Scale-invariance of galaxy clustering, Phys. Rep. Vol. 293 (1998) N. 2-4, J.R. Mureika, C.C. Dyer, Multifractal analysis of Packed Swiss Cheese Cosmologies, General Relativity and Gravitation, Vol.36 (2004) N.1,

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8 Apollonian sphere packings best known and understood case: Apollonian circle packing Configurations of mutually tanget circles in the plane, iterated on smaller scales filling a full volume region in the unit 2D ball: residual set volume zero fractal of Hausdorff dimension

9 Many results (geometric, arithmetic, analytic) known about Apollonian circle packings: see for example R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks, C.H.Yan, Apollonian circle packings: number theory, J. Number Theory 100 (2003) 1 45 A. Kontorovich, H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, Journal of AMS, Vol 24 (2011) Higher dimensional analogs of Apollonian packings: much more delicate and complicated geometry R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks, C.H.Yan, Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions, Discrete Comput. Geom. 35 (2006)

10 Some known facts on Apollonian sphere packings Descartes configuration in D dimensions: D + 2 mutually tangent (D 1)-dimensional spheres Example: start with D + 1 equal size mutually tangent S D 1 centered at the vertices of D-simplex and one more smaller sphere in the center tangent to all 4-dimensional simplex

11 Quadratic Soddy Gosset relation between radii a k ( D+2 a k k=1 ) 2 1 = D D+2 k=1 ( 1 curvature-center coordinates: (D + 2)-vector a k ) 2 w = ( x 2 a 2, 1 a a, 1 a x 1,..., 1 a x D) (first coordinate curvature after inversion in the unit sphere) Configuration space M D of all Descartes configuration in D dimensions = all solutions W to equation W t Q D W = I D with left and a right action of Lorentz group O(D + 1, 1)

12 Dual Apollonian group GD generated by reflections: inversion with respect to the j-th sphere S j = I D D+2 e t j 4 e j e t j e j = j-th unit coordinate vector D 3: only relations in G D are (S j ) 2 = 1 GD discrete subgroup of GL(D + 2, R) Apollonian packing P D = an orbit of GD on M D iterative construction: at n-th step add spheres obtained from initial Descartes configuration via all possible S j 1 S j 2 S j n, j k j k+1, k there are N n spheres in the n-th level N n = (D + 2)(D + 1) n 1

13 iterative construction of sphere packings

14 Length spectrum: radii of spheres in packing P D L = L(P D ) = {a n,k : n N, 1 k (D + 2)(D + 1) n 1 } radii of spheres Sa D 1 n,k Melzak s packing constant σ D (P D ) exponent of convergence of series ζ L (s) = n=1 (D+2)(D+1) n 1 k=1 a s n,k Residual set: R(P D ) = B D n,k Ba D n,k with Ba D n,k = Sa D 1 n,k P D Packing Vol D (R(P D )) = 0 L ad n,k < σ D(P D ) D packing constant and Hausdorff dimension: dim H (R(P D )) σ D (P D ) for Apollonian circles known to be same

15 Sphere counting function: spheres with given curvature bound N α (P D ) = #{S D 1 a n,k P D : a n,k α} curvatures c n,k = a 1 n,k α 1 for Apollonian circles power law (Kontorovich Oh) N α (P 2 ) α 0 α dim H(R(P 2 )) for higher dimensions (Boyd): packing constant lim sup α 0 log N α(p D ) log α = σ D (P D ) if limit exists N α (P D ) α 0 α (σ D(P D )+o(1))

16 Screens and Windows in general ζ LD (s) need have analytic continuation to meromorphic on whole C screen S: curve S(t) + it with S : R (, σ D (P D )] window W = region to the right of screen S where analytic continuation M.L. Lapidus, M. van Frankenhuijsen, Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings, Second edition. Springer Monographs in Mathematics. Springer, 2013.

17 Screens and windows

18 Some additional assumptions Definition: Apollonian packing P D of (D 1)-spheres is analytic if 1 ζ L (s) has analytic to meromorphic function on a region W containing R + 2 ζ L (s) has only one pole on R + at s = σ D (P D ). 3 pole at s = σ D (P D ) is simple Also assume: lim α 0 log Nα(P D) log α = σ D (P D ) Question: in general when are these satisfied for packings P D? focus on D = 4 cases with these conditions

19 Rough estimate of the packing constant P = P 4 Apollonian packing of 3-spheres S 3 a n,k at level n: average curvature γ n N n = n n 1 k=1 1 a n,k estimate σ 4 (P 4 ) with averaged version: n N n( γn N n ) s σ 4,av (P) = lim n log(6 5 n 1 ) log ( γ n 6 5 n 1 ) generating function of the γ n known (Mallows) G D=4 = γ n x n = n=1 (1 x)(1 4x)u x 5x 2 u = sum of the curvatures of initial Descartes configuration

20 obtain explicitly (u = 1 case) γ n = ( ) n ( ) + (11 166) n ( ) 3 n this gives a value σ 4,av (P) = in Apollonian circle case where σ(p) known this method gives larger value, so expect σ 4 (P) < σ 4,av (P) constraints on the packing constant: 3 < dim H (R(P)) σ 4 (P) < σ 4,av (P) =

21 Models of (Euclidean, compactified) spacetimes 1 Homogeneous Isotropic cases: S 1 β S 3 a 2 Cosmic Topology cases: Sβ 1 Y with Y a spherical space form S 3 /Γ or a flat Bieberbach manifold T 3 /Γ (modulo finite groups of isometries) 3 Packed Swiss Cheese: Sβ 1 P with Apollonian packing of 3-spheres Sa 3 n,k 4 Fractal arrangements with cosmic topology

22 Fractal arrangements with cosmic topology Example: Poincaré homology sphere, dodecahedral space S 3 /I 120, fundamental domain dodecahedron

23 considered a likely candidate for cosmic topology S. Caillerie, M. Lachièze-Rey, J.P. Luminet, R. Lehoucq, A. Riazuelo, J. Weeks, A new analysis of the Poincaré dodecahedral space model, Astron. and Astrophys. 476 (2007) N.2, build a fractal model based on dodecahedral space

24 Fractal configurations of dodecahedra (Sierpinski dodecahedra)

25 spherical dodecahedron has Vol(Y ) = Vol(S 3 a /I 120 ) = π2 60 a3 simpler than sphere packings because uniform scaling at each step: 20 n new dodecahedra, each scaled by a factor of (2 + φ) n dim H (P I120 ) = log(20) log(2 + φ) = close up all dodecahedra in the fractal identifying edges with I 120 : get fractal arrangement of Poincaré spheres Y a(2+φ) n zeta function has analytic continuation to all C ζ L (s) = n 20 n (2 + φ) ns = (2 + φ) s exponent of convergence σ = dim H (P I120 ) = log(20) log(2+φ) and poles σ + 2πim log(2 + φ), m Z

26 Spectral action models of gravity (modified gravity) Spectral triple: (A, H, D) 1 unital associative algebra A 2 represented as bounded operators on a Hilbert space H 3 Dirac operator: self-adjoint D = D with compact resolvent, with bounded commutators [D, a] prototype: (C (M), L 2 (M, S), /D M ) extends to non smooth objects (fractals) and noncommutative (NC tori, quantum groups, NC deformations, etc.)

27 Action functional Suppose finitely summable ST = (A, H, D) ζ D (s) = Tr( D s ) <, R(s) >> 0 Spectral action (Chamseddine Connes) S ST (Λ) = Tr(f (D/Λ)) = Mult(λ)f (λ/λ) λ Spec(D) f = smooth approximation to (even) cutoff

28 Asymptotic expansion (Chamseddine Connes) for (almost) commutative geometries: Tr(f (D/Λ)) β Σ + ST f β Λ β D β + f (0) ζ D (0) Residues D β = 1 2 Res s=β ζ D (s) Momenta f β = 0 f (v) v β 1 dv Dimension Spectrum Σ ST poles of zeta functions ζ a,d (s) = Tr(a D s ) positive dimension spectrum Σ + ST = Σ ST R + Warning: for fractal spaces also oscillatory terms coming from part of Σ ST off the real line

29 Zeta function and heat kernel (manifolds) Mellin transform heat kernel expansion D s = 1 Γ(s/2) 0 e td2 t s 2 1 dt Tr(e td2 ) = α t α c α for t 0 zeta function expansion ζ D (s) = Tr( D s ) = α c α Γ(s/2)(α + s/2) + holomorphic taking residues Res s= 2α ζ D (s) = 2c α Γ( α)

30 Example spectral action of the round 3-sphere S 3 S S 3(Λ) = Tr(f (D S 3/Λ)) = n Z n(n + 1)f ((n )/Λ) zeta function ζ (s) = 2ζ(s 2, 3 DS 3 2 ) 1 2 ζ(s, 3 2 ) ζ(s, q) = Hurwitz zeta function by asymptotic expansion S S 3(Λ) Λ 3 f Λf 1 can also compute using Poisson summation formula (Chamseddine Connes): estimate error term O(Λ )

31 Example: round 3-sphere S 3 a radius a ζ DS 3 a (s) = a s (2ζ(s 2, 3 2 ) 1 2 ζ(s, 3 2 )) S S 3 a (Λ) (Λa) 3 f (Λa)f 1 Example: spherical space form Y = S 3 a /Γ (Ćaćić, Marcolli, Teh) S Y (Λ) 1 #Γ S S 3 a (Λ)

32 Why a model of (Euclidean) Gravity? M compact Riemannian 4-manifold Tr(f (D/Λ)) 2Λ 4 f 4 a 0 + 2Λ 2 f 2 a 2 + f 0 a 4 coefficients a 0, a 2 and a 4 : cosmological term f 4 Λ 4 D 4 = 48f 4Λ 4 π 2 Einstein Hilbert term f 2 Λ 2 D 2 = 96 f 2Λ 2 24π 2 g d 4 x R g d 4 x modified gravity terms (Weyl curvature and Gauss Bonnet) f (0) ζ D (0) = f 0 10π 2 ( 11 6 R R 3C µνρσ C µνρσ ) g d 4 x C µνρσ = Weyl curvature and R R = 1 4 ɛµνρσ ɛ αβγδ R αβ momenta: (effective) gravitational and cosmological constant µνr γδ ρσ

33 Spectral action on a fractal spacetime: S 1 β P: Apollonian packing S 1 β P Y : fractal dodecahedral space 1 Construct a spectral triple for the geometries P and P Y 2 Compute the zeta function 3 Compute the asymptotic form of the spectral action 4 Effect of product with Sβ 1 look for new terms in the spectral action (in additional to usual gravitational terms) that detect presence of fractality

34 The spectral triple of a fractal geometry case of Sierpinski gasket: Christensen, Ivan, Lapidus similar case for P and P Y for D-dim packing P D = {S D 1 a n,k : n N, 1 k (D + 2)(D + 1) n 1 } (A PD, H PD, D PD ) = n,k (A PD, H S D 1 a n,k, D S D 1 a n,k ) for P Y with Y a = S 3 /I 120 : with a n = a(2 + φ) n (A PY, H PY, D PY ) = (A PY, n H Yan, n D Yan )

35 Zeta functions for Apollonian packing of 3-spheres: Lengths zeta function (fractal string) ζ L (s) := n N 6 5 n 1 k=1 a s n,k with L = L 4 = {a n,k n N, k {1,..., 6 5 n 1 }} zeta function of Dirac operator of the spectral triple Tr( D P s ) = n=1 6 5 n 1 k=1 Tr( D S 3 an,k s ) each term Tr( D S 3 an,k s ) = an,k s (2ζ(s 2, 3 2 ) 1 2 ζ(s, 3 2 )) gives ( Tr( D P s ) = 2ζ(s 2, 3 2 ) 1 2 ζ(s, 3 ) 2 ) an,k s n,k = (2ζ(s 2, 32 ) 12 ζ(s, 32 ) ) ζ L (s)

36 Spectral action for Apollonian packing of 3-spheres: (under good conditions on ζ L (s)) Positive Dimension Spectrum: Σ + ST PSC asymptotic spectral action = {1, 3, σ 4 (P)} Tr(f (D P /Λ)) Λ 3 ζ L (3) f 3 Λ 1 4 ζ L(1) f 1 +Λ σ (ζ(σ 2, 3 2 ) 1 4 ζ(σ, 3 2 )) R σ f σ + S osc Λ σ = σ 4 (P) packing constant; residue R σ = Res s=σ ζ L (s), and momenta f β = 0 v β 1 f (v)dv additional term SΛ osc coming from series of contributions of poles of zeta function off the real line: oscillatory terms

37 Oscillatory terms (fractals) zeta function ζ L (s) on fractals in general has additional poles off the real line (position depends on Hausdorff and spectral dimension: depending on how homogeneous the fractal) best case exact self-similarity: s = σ + 2πim log l, m Z heat kernel on fractals has additional log-oscillatory terms in expansion C 2π (1 + A cos( log t + φ)) + tσ log l for constants C, A, φ: series of terms for each complex pole

38 Log-oscillatory terms in expansion of the spectral action: G.V. Dunne, Heat kernels and zeta functions on fractals, J. Phys. A 45 (2012) [22p] M. Eckstein, B. Iochum, A. Sitarz, Heat kernel and spectral action on the standard Podlés sphere, Comm. Math. Phys. 332 (2014) M. Eckstein, A. Zajaç, Asymptotic and exact expansion of heat traces, arxiv:

39 effect of product with Sβ 1 (leading term without oscillations) case of Sβ 1 S a 3 (Chamseddine Connes) ( ) D S 1 β Sa 3 = 0 D S 3 a 1 + i D S 1 β D S 3 a 1 i D S 1 0 β Spectral action Tr(h(D 2 S /Λ)) β 1 S3 2βΛTr(κ(D2 a S /Λ)), a 3 test function h(x), and test function κ(x 2 ) = h(x 2 + y 2 )dy R

40 Case of S 1 β P: with momenta S S 1 β P (Λ) 2β ( Λ 4 ζ L (3) h 3 Λ ζ L(1) h 1 ) +2β Λ σ+1 ( ζ(σ 2, 3 2 ) 1 4 ζ(σ, 3 2 ) ) R σ h σ h 3 := π 0 h(ρ 2 )ρ 3 dρ, h σ = 2 0 h 1 := 2π h(ρ 2 )ρ σ dρ 0 h(ρ 2 )ρdρ

41 Interpretation: Term 2Λ 4 βa 3 h Λ2 βah 1, cosmological and Einstein Hilbert terms, replaced by 2Λ 4 βζ L (3)h Λ2 βζ L (1)h 1 zeta regularization of divergent series of spectral actions of 3-spheres of packing Additional term in gravity action functional: corrections to gravity from fractality 2β Λ σ+1 ( ζ(σ 2, 3 2 ) 1 4 ζ(σ, 3 2 ) ) R σ h σ

42 Case of fractal dodecahedral space P Y Zeta functions ζ L(PY )(s) = 20 n (2 + φ) ns n 0 ζ DPY (s) = (2ζ(s as 2, ) 12 ζ(s, 32 ) ) ζ L(PY )(s) Spectral action: Tr(f (D PY /Λ)) (Λa) 3 ζ L(P Y )(3) 120 f 3 Λa ζ L(P Y )(1) f (Λa) σ ζ(σ 2, 3 2 ) 1 4 ζ(σ, 3 2 ) f σ + SY osc,λ 120 log(2 + φ) σ = dim H (P Y ) = log(20) log(2+φ) =

43 on product geometry S 1 β P Y S S 1 β P Y (Λ) 2β ( Λ 4 a3 ζ L(PY )(3) β Λ σ+1 aσ (ζ(σ 2, 3 2 ) 1 4 ζ(σ, 3 2 )) 120 log(2 + φ) h 3 Λ 2 aζ ) L(P Y )(1) h h σ + S osc Sβ 1 Y,Λ Note: correction term now at different σ than Apollonian P oscillatory terms SY osc,λ more explicit than in the Apollonian case

44 Oscillatory terms: dodecahedral case zeros of zeta function ζ L (s) s m = σ + 2πim log(2 + φ), m Z with σ = log(20)/ log(2 + φ) contribution to heat kernel expansion of non-real zeros: C t σ (a 0 + 2R(a 1 t 2πi/ log(2+φ) ) + ) with coefficients a m proportional to Γ(s m ): for fixed real part σ decays exponentially fast along vertical line oscillatory terms are small

45 Slow-roll inflation potential from the spectral action perturb the Dirac operator by a scalar field D 2 + φ 2 spectral action gives potential V (φ) shape of V (φ) distinguishes most cosmic topologies: spherical forms and Bieberbach manifolds (Marcolli, Pierpaoli, Teh)

46 Fractality corrections to potential V (φ) additional term in potential U σ (x) = 0 u (σ 1)/2 (h(u + x) h(u))du depends on σ fractal dimension size of correction depends on (leading term) (ζ(σ 2, 3 2 ) 1 4 ζ(σ, 3 2 ))R σ further corrections to U σ come from the oscillatory terms presence of fractality (in this spectral action model of gravity) can be read off the slow-roll potential (hence the slow-roll coefficients, which depend on V, V, V )