Breaking statistical isotropy Marco Peloso, University of Minnesota Gumrukcuoglu, Contaldi, M.P, JCAP 0711:005,2007 Gumrukcuoglu, Kofman, MP, PRD 78:103525,2008 Himmetoglu, Contaldi, M.P, PRL 102:111301,2009 Himmetoglu, Contaldi, M.P, PRD 79:063517,2009
For FRW a lm a l m = C l δ ll δmm Claims of violation of statistical isotropy of the CMB perturbations. A number of 2 3 σ effects, significance susceptible to statistics used. Some of these claims concern the largest scale modes, for which additional problems due to galaxy contamination Larger modes (low l) probe earlier inflationary stages! H!1 Inflation Matter / Radiation
Cleaned CMB map by Tegmark et al 03. WMAP1 channels combined to eliminate foregrounds (depending on l and latitude) Axis from maxˆn m 2 a lm (ˆn) 2 m Low quadrupole Planar octupole Alignment, θ 10 0 1/20 1/20 1/62 Plane of alignment, 30 0 apart from galactic plane Anomalous lack of power at large scales outside galaxy Slightly greater quadrupole than WMAP1 (using galaxy cut)
Land, Magueijo, 05, l = 2 5 aligned max m, ˆn a lm (ˆn) 2 WMAP1 and WMAP+ data agree at large scales. treatment l = 2 l = 3 l = 4 l = 5 Mean Map (b, l) m (b, l) m (b, l) m (b, l) m inter-θ Land, Magueijo, 06 LILC1 (0.9, 156.7) 0 (63.0, -126.9) 3 (56.7, -163.7) 2 (48.6, -94.7) 3 51.4 TOH1 (58.5, -102.9) 2 (62.1, -120.6) 3 (57.6, -163.3) 2 (48.6, -93.4) 3 22.4 TOH3 (76.5, -134.0) 2 (27.0, 51.9) 1 (35.1, -130.6) 1 (47.7, -94.7) 3 53.8 WMAP3 (2.7, -26.5) 0 (62.1, -122.6) 3 (34.2, -131.2) 1 (47.7, -96.0) 3 53.7 0.1% (expect 57 0 ) Model comparison (Bayesian) S. M. vs. a lm 2 (ˆn) 2 = C l ɛ C l if m = l if m = l 10, 000 Data-set (b l) ɛ H f H AIC H BIC ln B LILC1 63-120.042 6.51 2.01 2.78 1.36 TOH1 61-113.032 7.48 2.98 3.75 1.85 2% TOH3 74-129.018 6.97 2.47 3.24 1.27 WMAP3 64-123.043 6.49 1.99 2.76 1.32
North-south asymmetry Eriksen et al 04 WMAP1, Q, V, W bands with Kp0 (-25%) Ecliptic axis 164 circles, draw 82 axis, and compute power in the separate hemispheres, in C l = 2 63 Dark = high C l,n /C l,s when north points inside big circle Power in that disk > in 80% of isotropic simulations Power in that disk < in 20% of isotropic simulations In the frame in which is maximal, asymmetry > than for 99.7% of isotropic realizations
Hoftuft et al 09, model comparison SM vs. [1 + A ˆn ˆv] s (ˆn) WMAP5 downgraded to 4.5 0 KQ85 ( 16.3%); KQ85e ( 26.9%) dipole modulation Gordon et al 05 random gaussian Data Mask l mod (l bf, b bf ) A bf Significance (σ) log L log E ILC KQ85 64 (224, 22 ) ± 24 0.072 ± 0.022 3.3 7.3 2.6 V -band KQ85 64 (232, 22 ) ± 23 0.080 ± 0.021 3.8 V -band KQ85 40 (224, 22 ) ± 24 0.119 ± 0.034 3.5 V -band KQ85 80 (235, 17 ) ± 22 0.070 ± 0.019 3.7 W -band KQ85 64 (232, 22 ) ± 24 0.074 ± 0.021 3.5 ILC KQ85e 64 (215, 19 ) ± 28 0.066 ± 0.025 2.6 Q-band KQ85e 64 (245, 21 ) ± 23 0.088 ± 0.022 3.9 V -band KQ85e 64 (228, 18 ) ± 28 0.067 ± 0.025 2.7 W -band KQ85e 64 (226, 19 ) ± 31 0.061 ± 0.025 2.5 ILC a Kp2 40 (225, 27 ) 0.11 ± 0.04 2.8 6.1 1.8 WMAP3 1/2 resolution
( ) [[ (( )) ]] Groeneboom and Eriksen 08 studied the ACW model ( ) ( ) ] 2 P k = P (k) [1 + g k v Ackerman, Carroll, Wise 07 Note: Taylor expansion that may be expected for axis (x y) & planar (z z) symmetric geometry. ACW model is unstable ± g = 0.10 0.04 g = 0.15 ± 0.04 (3.8σ) Increase of significance with l Pullen, Kamionkowski 07
Models? Likely, accepted only if it explains N > 1 effects Perhaps we should learn to compute cosmological perturbations beyond FRW [ ] Simplest: Bianchi-I with residual 2d isotropy ds 2 = dt 2 + a (t) 2 dx 2 + b (t) 2 [ dy 2 + dz 2] Standard formalism for FRW Bardeen 80; Mukhanov 85 δg µν, δφ v, h +, h
Anisotropic background How many physical modes? Still 3 δg µν, δφ 4 δg 0µ non dynamical 4 Gen. coord. transf. + 11 modes How do they behave? Signatures? Coupled level (due to less symmetric background) Decoupled to each other already at the linearized UV regime a lm a l m δ ll δ mm limit of isotropic background Nonstandard amplitude for gravity waves ( = results for the 2 polarizations)
g µν = ( ) ( ) ( ) () ) ( (1 + 2Φ) a χ b ( i B + Bi T ) a 2 (1 2Ψ) [ [ a b i B + B i T b 2 (1 2Σ) [ δ ij + i j E + (i E T j) ] seven 2d scalars + three 2d vectors, decoupled at linearized level Choose a gauge preserving all δg 0µ, since they are the nondynamical fields (ADM formalism) Harder to indentify nondynamical modes in standard gauges Can be promoted to gauge invariant formulation ( ˆΦ Φ + ( Σ H b ) ; ˆB = B Σ b H b + b Ė ;... )
S = Dynamical Y i and nondynamical N i fields d 3 k dt [ a ij Ẏ i Ẏ j + ( b ij Ni + ( d ij Ẏ i ) Ẏ j + h.c. + c ij Ni N j ) ( ) ) ] Y j + h.c. + e ij Yi Y j + (f ij Ni Y j + h.c.) Coefficients background-dependent Solving for N, δs δn i = 0 c ij N j = b ij Ẏ j f ij Y j Action for the dynamical (propagating) modes ) S d 3 k dt [Ẏ i K ij Ẏ j + Λ ij Y j + h.c. (Ẏ i Y i Ω 2 ij Y j ] K ij a ij ( b ) ik Λ ij d ij ( b ) ik Ω 2 ij e ij ( f ) ik ( c 1 ) km b mj ( c 1 ) km f mj ( c 1 ) km f mj Eigenvalues indicate the nature of a mode δs δy i = 0 K ij Ÿ j + [ K ij + (Λ ij + h.c.)] Ẏj + ( Λij + Ω 2 ij ) Y j = 0 Expect divergency if K is noninvertible
Simplest example L = M 2 p 2 R 1 2 ( φ)2 V (φ) Asymptotic Kasner (vacuum) solution in the past ds 2 = dt 2 + t 2α dx 2 + t 2β ( dy 2 + dz 2) α + 2β = 1, α 2 + 2 β 2 = 1 a t iso M p Vin b a b t t α = 1, β = 0 α = 1 3, β = 2 3
S (2) = 1 2 [ [ [ ( ( dη d 3 k H 2 ω 2 H 2 + H + 2 + V 2 H+, V ) ( )] [ Ω 2 H+ V GW polarizations Scalar (density) mode [ ( ) [ ( Initial conditions from early time frequencies ( )? ) ( Pbm: ω 2, Ω2 p 2 + f ( H) 2), and H 1 t, p x t 1/3, p y,z 1 ( ) H p in the asymptotic past (mode in long wavelength regime) t 2/3 [ [ ω 2 a 2 5 ] ] 9t + 2 p2 y + p 2 z [ ] 4, Ω 2 ij a 2 9t + 2 p2 y + p 2 z δ ij No adiabatic evolution; ω 2 < 0
1e+15 1e+10 P H p 3 H 2 P HX /P 0 100000 1 1e-05 1e-10 Large growth 1e-15 1e-20 1e-25 1e-30 p = 0.01 H 0 p = 10 H 0 p = 10 4 H 0 1e-08 1e-06 0.0001 0.01 1 t H 0 Strong scale dependency Analogous to instability of contracting Kasner Belinsky, Khalatnikov, Lifshitz 70, 82 Potentially detectable GW, even if V 1/4 0 < 10 16 GeV Tuned duration of inflation. If N 60, effect blown away
( ( ) Search for a longer / controllable anisotropic stage (contrast Wald s theorem on isotropization of Bianchi spaces) Higer curvature terms Barrow, Hervik 05 Kalb Ramond axion Kaloper 91 Vector field, A z = 0 Potential term V (A µ ( A µ ) ( Ford ) 89 ) [ Fixed norm Ackerman, Carroll, Wise 07 Slow roll due to A µ A µ R Golovnev, Mukhanov, Vanchurin 08 Kanno, Kirma, Soda, Yokoyama 08 Yokoyama, Soda 08 Chiba 08 Kovisto, Mota 08
S = d 4 x g M p 2 2 R F 2 4 + ξ 2 R A2 Nonminimal coupling g µν = diag ( 1, a 2, b 2, b 2) H = 1 3 [H a + 2 H b ] h = 1 3 [H b H a ] A µ = (0, a B M p, 0, 0) A 2 = M 2 p B2 { { ( )} B+3 H Ḃ+ 2 H h 5 h 2 2ḣ + (1 6 ξ) 2H 2 + h 2 + Ḣ B = 0 ξ = 1/6 used for Primordial magnetic fields Turner, Widrow 88 Vector inflation Golovnev, Mukhanov, Vanchurin 08 Vector curvaton Dimopoulos, Lyth, Rodriguez 08
L = 1 4 F 2 + 1 12 R A2 = 1 4 F 2 + H 2 A 2 + sign leads to a ghost (not a tachyon!) Stückelberg: A µ B T µ + 1 H µφ H 2 A 2 H 2 B T µ B µt + µ φ µ φ (signature + ++) Does not require anisotropy! Cf. PF mass m 2 h µν h µν + m 2 ( h µ µ) 2 to avoid ghost
Alternatively, L = 1 4 F 2 M 2 P µν = η µν + k µ k ν /M 2 k 2 + M 2 2 A2 = 1 2 Aµ P 1 µν Aν ( M ) M( 2 > 0. Go in the rest ) frame, k µ = k µ = 2, 0, 0, 0 ( η µν + k µ k ν /M 2) = diag (0, 1, 1, 1) M 2 < 0. Frame with no energy, k µ = k µ = ( ) (0, 0, 0, M 2 ) ( η µν + k µ k ν /M 2) = diag (1, 1, 1, 0) Exhaustive computation if Aµ has no VEV (no δa µ δg µν linearized mixing)
Vector inflation Golovnev, Mukhanov, Vanchurin 08 ( ) Kanno, Kimura, Soda, Yokoyama 08 L = a 1 4 F (a) µν F (a)µν 1 2 ( m 2 R ) A (a) µ A (a)µ 6 A (a) = a M p B (a) δg µν B + 3 H Ḃ + m 2 B = 0, 10 4 = 2 dyn + 4 non dyn h H 1 N δa (a) µ 4 N = 3N dyn + N non dyn Simplest case, 3 mutually orthogonal vectors with equal vev 18 coupled modes δ 2 S a 2 M p [ (Ḃ + H B ) δf (i) 0j + ( m 2 2H 2 Ḣ ) ] B δa (i) j h T ij T
18 coupled modes, 11 dynamical and 7 non dynamical We computed the kinetic matrix for the dynamical modes δ 2 S = d 3 k dt [ Ẏi K ij Ẏj +... ] det K 0.004 0.002 1.0 0.5 0.0 0.5 H Log 10 p 0.002 0.004 Two negative eigenvalues Singularity expected k 1 : k 2 : k 3 = 100 : 80 : 60
Simplified computation: concentrate on one vector field Collective effect of the remaining( ones ) cosmological constant L = M 2 p 2 R V 0 1 4 F µνf µν 1 2 ( m 2 R ) A µ A µ 6 A µ = (0, a B M p, 0, 0)+δA µ H = V0 3 Mp +O ( B 2), h = H 3 B2 +O ( B 4) B slowly rolling for m H vev along x only, can do 2d decomposition 1 dyn + 3 non dyn 2ds 2 dyn + 1 non dyn 2ds δg µν 1 dyn + 1 non dyn 2dv δa µ 1 dyn 2dv 7 coupled modes rather than 18
Parametrically, det K = B 2 H2 p 2 +... Perturbations diverge when it vanishes 1.0 Λ 1 Λ 1,0, Λ 2 Λ 2,0, Λ 3 Λ 3,0 0.5 Mp V Α 0 in 0 0.0 0.5 1.0 0.005 0.010 0.015 0.020 0.025 H p
Fixed norm vector fields S = Kostelecky, Samuel 89; Jacobson, Mattingly 04; Carroll, Lim 04 d 4 x [ M 2 p g 2 R 1 4 F µν F µν + λ ( ] A 2 m 2) V 0 Ackerman, Carroll, Wise 07 A x = m ( ds 2 = dt 2 + e 2 H a t dx 2 + e 2 H b t dy 2 + dz 2) ) H b = ( 1 + m2 M 2 p H b Test field χ " P δχ = P ( k ) ( 1 + g k 2 x ) " #! t ~! $ Assumed δχ δg µν through modulated pertrbations Dvali, Gruzinov, Zaldarriaga 03 Kofman 03 " r! " t
Complete study (δg µν, δa µ ) shows that this model is unstable 3 2! " 0! # 1 0-1 -2 ( ) -3-3 -2.5-2 -1.5-1 -0.5 0 t H a Divergency at horizon crossing p 2 x = ( 2 + m2 M 2 p ) H a H b One mode becomes a ghost at that point Kinetic term vanishes perturbations diverge Problems in theories with fixed A 2 also pointed out by Clayton 01
So what? Linearized computation blows up; maybe nonlinear evolution ok Linearized computation CMB Assume singularity cured, any other problem? nonlinear interactions: 0 ghost-nonghost; UV For a gravitationally coupled ghost today, Λ < 3 MeV Cline et al 03 U(1) hard breaking, A L interactions p/m enhanced Quantum theory out of control at E > m H whole sub-horizon regime Unclear UV completion ± DH 2 ± m 2 A 2 Ghost condensation?
Scalar-Vector Coupling supports anisotropic expansion due to vector field. Isotropic expansion Power Spectrum for
Conclusions Some evidence of broken statistical isotropy Full computations in simplest non FRW scalar-tensor coupling; P + P ; Nondiagonal C ll mm Easy to extend further Problems with specific realizations