Statistical anisotropy in the inflationary universe

Statistical anisotropy in the inflationary universe Yuri Shtanov Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine Grassmannian Conference, Szczecin (2009)

Outline Primordial spectrum after inflation can be statistically anisotropic Today this would lead to observable statistical anisotropy of the LSS and CMB Quantitative estimates of this effect are connected with the trans-planckian issue in cosmology

Evidence of homogeneity and isotropy T = 2.725 ± 0.002 K δt(ˆn) T 10 5

Statement of the problem δt(ˆn)/t 10 5 as a realization of a random field Is it statistically isotropic? A number of authors raised this issue recently General analysis: Hajian & Souradeep (2003) Gordon, Huterer & Crawford (2005) Armendariz-Picon (2005) Helling, Schupp & Tesileanu (2006) Pullen & Kamionkowski (2007)... Theory of possible origin: Pereira, Pitrou & Uzan (2007) Gümrükçüoğlu, Contaldi & Peloso (2007) Ackerman, Carroll & Wise (2007) Koivisto & Mota (2008) Erickcek, Kamionkowski & Carroll (2008) Yokoyama & Soda (2008)...

Statistical isotropy and anisotropy Statistical isotropy of the CMB means δt T (ˆn 1 ) δt T (ˆn 2 ) = f(ˆn 1 n 2 ) = f(cos θ) δt T (ˆn) = lm a lm Y lm (ˆn), or If it is statistically anisotropic, then a lm a l m = δ ll δ mm C l a lm a l m = δ ll δ mm C l + ζ lml m

Observation of statistical anisotropy Hoftuft, Eriksen, Banday, Górsky, Hansen & Lilje (2009) found evidence of power asymmetry between northern and southern galactic hemispheres in the WMAP data in the range l = 2 600: δt(ˆn) T [ ] δt(ˆn) = [1 + A (ˆn ˆm)] T isotropic with A = 0.072 ± 0.022 for l 64 (l, b)ˆm = (224, 22 ) ± 24

Connection with the primordial perturbations If Φ( k) is statistically anisotropic, so is δt/t(ˆn) : δt T (ˆn) = d 3 k (k, ˆk ˆn)Φ( k) (k, ˆk ˆn) depends on the background cosmology Φ( k) describes the primordial fluctuations General opinion: inflation predicts Gaussian statistically isotropic initial conditions However, this general opinion about inflation may not be entirely correct

Inflation at work Inflation is an accelerated epansion of the universe It makes the universe (locally) homogeneous and isotropic... and also generates quasiclassical inhomogeneities from quantum fluctuations

Inhomogeneities generated during inflation have two independent components ds 2 = a 2 (η) [(1 + 2Φ)dη 2 (1 2Φ)d 2 + h ij d i d j] Scalar (Φ) primary derivation by Mukhanov & Chibisov (1980) in frames of the Starobinsky model (1979); the first quantitatively correct derivation of post-inflationary spectrum by Hawking; Starobinsky; Guth & Pi (1982) By the end of inflation, these inhomogeneities are seeds for the subsequent formation of structure (galaies, clusters, etc.) Tensor (h ij ) the first derivation of post-inflationary spectrum was made by Starobinsky (1979)

The main idea The small-scale (quantum) modes defining the vacuum state are propagating on the background of the large-scale (classical) scalar and tensor inhomogeneities Here the scale division during inflation is set by the Hubble length scale

The basic picture The waves of vacuum quantum fluctuations travel a long distance during inflation before they become primordial seeds for structure formation: D 1 H en, N 10 6 10 12 typically for chaotic inflation Photon last-scattering surface The resulting power spectrum may carry information about the inhomogeneity of the inflationary universe, in a way similar to the Sachs Wolfe effect

Our work The effect of scalar background inhomogeneities was studied by Gennady Chibisov and Yu. S. (1989) The effect of tensor background inhomogeneities was studied by Yu. S. and Hanna Piatkovska (2009) The leading contribution to the effect of statistical anisotropy obtains from tensor background inhomogeneities

A simple model of inflation M P 4 V(ϕ) Scalar field (inflaton) V(ϕ) = 1 2 m2 ϕϕ 2 The ceiling of Planckian energy density: ϕ P M2 P m ϕ M P ϕ P ϕ m ϕ M P 2 10 6 determines the amplitude of primordial perturbations

Generation of quasiclassical inhomogeneities I n f l a t i o n H o t u n i v e r s e time H 1 λ quasiclassical space φ class = φ quant = λ phys = a 2πk k ah d 3 k φ k d 3 k φ k quantum k ah

Propagation of high-frequency modes (k ah) Quasiclassical inhomogeneities are background fields ds 2 = a 2 (η) [(1 + 2Φ B )dη 2 (1 2Φ B )d 2 + hij B d i d j] B δϕ + m 2 ϕ δϕ = 0 B depends on Φ B and h B ij (in linear approimation) δϕ k δϕ k ep [ is k (η, ) ] Φ k Φ k ep [ ic k (η, ) ] S k and C k are linear in Φ B and h B ij

The resulting power spectrum Φ( )Φ( ) d 3 k [ = k 3 P 0(k) 1 + ν( ] k) e i k( ) ˆk i = k i /k anisotropic part ν( k) (n S 1)[ˆk iˆk j Λ ij (k) + ˆk ] i j ˆk ˆk mˆk n Λ ijmn (k) +... n S 1 = d ln P(k) d ln k 0.04 (the scalar spectral inde) Dominated by the quadrupole ˆk iˆk j Λ ij (k) Λ ij (k), Λ ijmn (k),... are almost scale-invariant Proportional to (n S 1)

Magnitude of the statistical anisotropy It depends on the moment of time t i at which one sets the initial conditions ν 2 ( k) (n S 1) 2 H4 i Hk 4 10πmϕ 2M2 P 2 10 3 V i ϕ2 i V k ϕ 2 k M 6 P The inde k refers to the moment of Hubble-radius crossing The applicability limit of the linear theory V i ϕ 2 i M 6 P 1 ν 2 2 10 3 This happens to be the boundary of the self-regenerating inflationary universe

Trans-Planckian issue I n f l a t i o n H o t u n i v e r s e time l P H 1 space λ Hubble-radius crossing N = 14 Planck-radius crossing k ph M P during most part of the evolution Under locally isotropic initial conditions at the Planck-radius crossing: ν 2 10 14

Trans-Planckian issue and black-hole radiation observer Frequencies of the incoming modes tend to infinity as the outgoing modes approach the horizon Several approaches (Unruh, Corley, Jacobson...) but yet no satisfactory avoidance of this problem

Eamples of the trans-planckian safety Electron-positron pair production in a constant electric field if the gauge A = E 0 t is used (Nikishov, 1970). Nothing dangerous takes place as long as ω 2 (k) k 2 M 2 P (Kolb, Starobinsky and Tkachev, 2007) Vacuum stress-energy tensor in the Milne representation of the Minkowski space (Vaudrevange & Kofman, 2007) In both cases, one is obliged to consider high-frequency modes without cut-off to get the usual result The trans-planckian issue is closely related to the issue of Lorentz symmetry

Signatures for the large-scale structure Correlation function and velocity dispersion on large scales: ξ( ) d 3 k P( k)e i k = ξ 0 () + ξ A ( ) ξ A ( ) = p()λ ijˆ iˆ j + q()λ ijmnˆ iˆ jˆ mˆ n +... leading term v i v j = 1 3 v 2 ( δ ij + 2 ) 5 Λ ij

Signatures for the CMB [ P( k) = P 0 (k) 1 + ] g LM (k)y LM (ˆk) LM a lm a l m = δ ll δ mm C l + LM ξlml LM m DLM ll D LM ll dk k 2 P 0 (k)g LM (k)θ l (k)θ l (k) 0 In the case under consideration, this will be dominated by terms with L = 2 The variances with which the quantities g 2M can be measured using the temperature maps of WMAP and Planck are estimated to be, respectively, σ g 1.2 10 2 and 3.8 10 3 (Pullen & Kamionkowski, 2007)

Summary Statistical anisotropy is a nontrivial test of the inflationary theory of the origin of primordial perturbations Testing the predicted signature may be facilitated by its quadrupole-dominated form and scale-invariant character Its connection with the trans-planckian issue is intriguing

THANK YOU

Appendi: scalar cosmological perturbations ds 2 = a 2 (η) [(1 + 2Φ)dη 2 (1 2Φ)d 2] 1 a 2k e ikη δϕ k ϕ A k H iπ ϕ MP 2 Φ k P(k) k 3 A k 2 = 8πm2 ϕ 3M 2 P Ḣ A k H 2 ( n S 1 2 ln a ) 1 fh f 0.96 k ( ) 2 3/2 e ikη k ah k ( ln a ) 2 fh f k n S 1 k m ϕ M P 10 6 k ah

Appendi: tensor cosmological perturbations ds 2 = a 2 (η) [ dη 2 d 2 + h ij d i d j] h 00 = h 0i = h i i = 0, i h i j = 0 h ij (η, ) = d 3 k h k (η)e ij ( k,σ)e ik a k,σ + H.c. σ=+, h k (η) 32π am P k e ikη i 32π k 3/2 H k M P e ikη k k ah k ah

Appendi: Quasiclassical and quantum perturbations φ quant = d 3 k φ k (η) e i k a k + H.c. k ah φ class = d 3 k φ k (η) e i k a k + H.c. k ah φ ( ph ) φ ( yph ) ( ) d 3 1 1 k ph k ph 2 + H2 k 2kph 2 e i k ph( ph y ph) Effective occupation numbers: n k = H2 k 2k 2 ph

Appendi: Inhomogeneous inflationary universe ds 2 = a 2 (η) [(1 + 2Φ B )dη 2 (1 2Φ B )d 2 + hij B d i d j] Φ B = k<ah d 3 k Φ k (η) e i k a k + H.c. hij B = d 3 k h k (η) k<ah σ Φ B ϕ M 2 P e ij ( k,σ) e i k a k + H.c. m ϕ M P 10 6 h B H M P ϕ M P Φ B Φ B during the early period of inflation