CMB two- and three-point correlation functions from Alfvén waves
aa r X i v : . [ a s t r o - ph . C O ] S e p CMB two- and three-point correlation functions from Alfv´en waves
Tina Kahniashvili
1, 2, 3, ∗ and George Lavrelashvili † McWilliams Center for Cosmology and Department of Physics,Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA Department of Physics, Laurentian University, Ramsey Lake Road, Sudbury, ON P3E 2C,Canada Abastumani Astrophysical Observatory, Ilia State University, 2A Kazbegi Ave, Tbilisi, GE-0160, Georgia Department of Theoretical Physics, A.Razmadze Mathematical Institute,I.Javakhishvili Tbilisi State University, GE-0177 Tbilisi, Georgia (Dated: November 14, 2018)We study the cosmic microwave background (CMB) temperature fluctuations non-gaussianity dueto the vector mode perturbations (Alfv´en waves) supported by a stochastic cosmological magneticfield. We present detailed derivations of the statistical properties, two and three point correlationfunctions of the vorticity perturbations and corresponding CMB temperature fluctuations.
PACS numbers: 98.70.Vc, 98.80.-k
I. INTRODUCTION
In the framework of the standard cosmological scenario the cosmic microwave background (CMB) temperaturefluctuations are gaussian and are fully determined by the CMB temperature fluctuation two-point correlation functionswhile higher order odd correlation functions (for example three-point correlations) are identically zero (for a reviewon CMB fluctuations and possible non-gaussianity see Ref. [1] and references therein). This is a consequence of theinflationary scenario that predicts the gaussian initial perturbations, and even at the level of perturbations there isno violation of rotational symmetry . Several observations indicate that the CMB temperature map could be slightlynon-gaussian [3] and thus to adequately describe the CMB fluctuations one must study higher order correlationsfunctions. Furthermore, some modifications of standard inflationary models lead to a slightly non-gaussian CMB map[4].A common way to characterize the CMB temperature fluctuations non-gaussianity is to introduce the f NL parameter,which in fact determines the relation between the two-point and three-point correlation functions [5]. The currentCMB data limits f NL = 32 ±
21 [3]. PLANCK mission will be able to give us with stronger limits (to improve currentlimit by an order or a few). If the nearest future CMB measurements, for example PLANCK data will confirm that | f NL | is not significantly less then one, the standard cosmological scenario must be revised substantially. There areseveral different ways for such a revision. For example the inflation could be driven by multiple fields, or the Universeisotropy has been violated at very early epochs [1]. Recent studies [6, 7] address the magnetic field induced densityperturbations as a source of the CMB temperature fluctuations non-gaussianity. The physical meaning of this effectis as follows: the temperature anisotropies caused by the magnetic field are proportional to the magnetic field energydensity parameter, ∆ T /T ∝ B /ρ cr [6], where ρ cr is the critical density today and B is the comoving value of aneffective magnetic field. Accounting that the square of the magnetic field B is a non-linear form [8], the corresponding∆ T /T fluctuations must be non-gaussian. The limits for the scale invariant magnetic field amplitude from the CMBnon-gaussianity test is of order of 10 − Gauss [9].In this paper we investigate the CMB non-gaussianity due to the vector mode of perturbations induced by astochastic magnetic magnetic field, (see Refs. [10–12] for details of the vector magnetized mode). If the magnetic fieldpresence is a reason for the CMB non-gaussianity, this magnetic field must satisfy several conditions: (i) the magneticfield must be generated in the early Universe, prior to recombination; There are different mechanisms to generatemagnetic fields in the early Universe, such as inflation, phase transitions, see for reviews [13]; (ii) the correlationlength of the cosmological magnetic field must satisfy the requirement of causality [14], and thus to have a fieldcorrelated over the horizon scale or even larger this field should be generated during the inflation and have a scaleinvariant spectrum [15]; (iii) the amplitude of this magnetic field should be small enough to preserve the isotropy of ∗ Electronic address: [email protected] † Electronic address: [email protected] The generation of the vector mode, which involves a preferred direction, is not excluded during the inflation, but the exponentialexpansion washes out the vector (vorticity) perturbations if no supporting external source is present [2]. the background Friedman-Lemaˆıtre-Robertson-Walker (FLRW) metric (so the energy density of the magnetic fieldshould be the first order of perturbations), be below the upper bounds (few nGauss) given by observations [16]. Onthe other hand the amplitude of the magnetic field should be large enough to leave observational traces on CMB.Recently it was argued that non-observation of blazars in TeV range by Fermi mission indicates the presence of largescale correlated intergalactic magnetic field with a lower bound of order of 10 − Gauss [17]. The existence of thelower bound of order of 10 − Gauss magnetic field favors the magnetic field of cosmological origin [18], and thus themagnetic field amplitude at 1 Mpc is squeezed between 10 − and 10 − Gauss.Vector and tensor modes of magnetized perturbations are much more complicated then the scalar one. The firstpaper to address the CMB bispectrum induced by the vector and tensor mode of perturbations has been Ref. [19]where the analytical expressions were derived. A natural extension of that approach, namely numerical estimation ofthe vector and tensor modes induced non-gaussianity were presented in Refs. [19–22]. In our study we give detailedderivations of the two and three point correlation functions of the CMB temperature fluctuations induced by themagnetized vector mode. In this sense at this stage this article has a methodological nature.The outline of the rest of the paper is as follows: In Sec. II we define the magnetized vector mode of perturbationsand compute the Lorentz force two- (Sec. IIA) and three- (Sec. IIB) point correlation functions. We explicitlydiscuss in details the difference of the vorticity perturbations two-point correlations functions, and show that thenon-gaussianity of the vector field (vorticity) can be seen already from the two-point correlation function. In Sec. IIIwe address the CMB temperature fluctuations induced by the magnetized vector mode perturbations. We presentanalytical expressions for the two- (Sec. IIIA) and three- (Sec. IIIB) point correlations of the CMB temperatureanisotropies. We briefly discuss our results and conclude in Sec. IV. Useful mathematical formulae and details ofcomputations are given in Appendix.
II. MAGNETIZED PERTURBATIONS VECTOR MODE
To study the dynamics of linear magnetic vector perturbations about a spatially-flat FLRW homogeneous cosmolog-ical spacetime background (described by the metric tensor ¯ g µν = a η µν , where η µν = diag( − , , ,
1) is the Minkowskimetric tensor and a ( η ) the scale factor) we follow the standard procedure and decompose the metric tensor into aspatially homogeneous background part (¯ g µν ) and a perturbation part, g µν = ¯ g µν + δg µν , where µ, ν ∈ (0 , , ,
3) arespacetime indices. Vector perturbations δg µν can be described by two three-dimensional divergence-free vector fields A and H [2], where δg i = δg i = a A i , δg ij = a ( H i,j + H j,i ) . (1)Here a comma denotes the usual spatial derivative, i, j ∈ (1 , ,
3) are spatial indices, and A and H vanish at spatialinfinity. Studying the behavior of these variables under infinitesimal general coordinate transformations one finds that V = A − ˙H is gauge-invariant (the overdot represents a derivative with respect to conformal time). V is a vectorperturbation of the extrinsic curvature [23]. Exploiting the gauge freedom we choose H to be constant in time. Thenthe vector metric perturbation may be described in terms of two divergenceless three-dimensional gauge-invariantvector fields, the vector potential V and a vector representing the transverse peculiar velocity of the plasma, thevorticity Ω = v − V , where v is the spatial part of the four-velocity perturbation of a stationary fluid element [11]. As we noted in the Introduction, in the absence of a source the vector perturbation V decays with time (this followsfrom the equation for ˙V , ˙V + 2( ˙ a/a ) V = 0) and therefore can be ignored.The residual ionization of the primordial plasma is large enough to ensure that magnetic field lines are frozen intothe plasma. Neglecting fluid back-reaction onto the magnetic field, the spatial and temporal dependence of the fieldseparates, B ( t, x ) = B ( x ) /a [25]. Since the fluid velocity is small the displacement current in Amp`ere’s law maybe neglected; this implies the current J is determined by the magnetic field via J = ∇ × B / (4 π ). Accounting fora frozen-in magnetic field lines the induction law takes the form ˙ B = ∇ × ( v × B ). As a result the baryon Eulerequation for v has the Lorentz force L ( x ) = − B ( x ) × [ ∇ × B ( x )] / (4 π ) as a source term. The photons are neutral sothe photon Euler equation does not have a Lorentz force source term. The Euler equations for photons and baryonsare [11, 12] ˙ Ω γ + ˙ τ ( v γ − v b ) = 0 , (2) Given the general coordinate transformation properties of the velocity field v , two gauge-invariant quantities can be constructed, theshear s = v − ˙H and the vorticity Ω = v − A [23]. In the gauge ˙H = 0 (i.e., V = A ) we get Ω = v − V [24]. ˙ Ω b + ˙ aa Ω b − ˙ τR ( v γ − v b ) = L ( V ) ( x ) a ( ρ b + p b ) , (3)where the subscripts γ and b refer to the photon and baryon fluids, and ρ and p are energy density and pressure.Here ˙ τ = n e σ T a is the differential optical depth, n e is the free electron density, σ T is the Thomson cross section, R = ( ρ b + p b ) / ( ρ γ + p γ ) ≃ ρ b / ρ γ is the momentum density ratio between baryons and photons, and L ( V ) i is thetransverse vector (divergenceless) part of the Lorentz force. In the tight-coupling limit v γ ≃ v b , so we introduce thephoton-baryon fluid divergenceless vorticity Ω (= Ω γ = Ω b ) that satisfies(1 + R ) ˙ Ω + R ˙ aa Ω = L ( V ) ( x ) a ( ρ γ + p γ ) . (4)The average Lorentz force h L ( x ) i = −h B × [ ∇ × B ] i / (4 π ) vanishes, while the r.m.s. Lorentz force h L ( x ) · L ( x ) i / isnon-zero and acts as a source in the vector perturbation equation.To proceed one needs to obtain an expression for the Lorentz force in terms of the magnetic field characteristics.We assume that the magnetic field is Gaussian and satisfy , h B i ( k ) B ⋆j ( k ′ ) i = (2 π ) δ ( k − k ′ ) P ij ( ˆk ) M ( k ) for k ≤ k D , (6)and vanishes for k > k D . Here a star denotes complex conjugation, and δ ( k − k ′ ) is the usual 3-dimensional Diracdelta function, and k D is the magnetic field damping scale defined through the Alfv´en waves damping [10, 26]. Weapproximate power spectrum M ( k ) by simple power laws.Following Ref. [11] it can be shown that the Fourier transform of the Lorentz force L ( V ) i ( k ) ≡ k Π i ( k ) is related tothe Fourier transform of spacial part of magnetic field energy momentum tensor τ ( B ) mj ( k ) τ ( B ) ij ( k ) = 14 π π ) Z d q [ B i ( q ) B j ( k − q ) − δ ij B m ( q ) B m ( k − q )] , (7)as Π i ( k ) = 12 ( P ij ( ˆk )ˆ k m + P im ( ˆk )ˆ k j ) τ ( B ) mj ( k ) , (8)where P ij ( ˆk ) = δ ij − ˆ k i ˆ k j is the transverse plane projector with unit wavenumber components ˆ k i = k i /k .Next we shall solve the Eq. (4). It can be solved in two different regimes, for length scales larger and smaller thencomoving Silk scale λ S . These two solutions are [11]: ⋆ For the scales λ > λ S , ( k < k S ) Ω ( k , η ) = k Π ( k ) η (1 + R )( ρ γ + p γ ) , (9)where ρ γ and p γ are photon energy and pressure today. ⋆ For smaller scales with λ < λ S , ( k > k S ) Ω ( k , η ) = Π ( k )( kL γ / ρ γ + p γ ) , (10)where L γ is the comoving photon mean-free path.Eqs. (9) and (10) define the vorticity through the magnetic source, Π ( k ), Eq. (8). As we can see the vorticityperturbations at scales below the Silk damping stay constant, while the vorticity perturbations above the Silk dampingare increasing linearly with the time. In order to compute the magnetized vector mode induced effects we need tocompute the vorticity perturbations two- and three-point correlations.Below we present our computations. It must be stressed that the form of the magnetic field two-point correlationfunction, Eq. (6) presumes the following properties of the field: (i) transverse, divergence free field, ∇ · B = 0, andin the fourier space this is insured by the projector P ij ( ˆk ); (ii) isotropy (no preferred direction) insured by δ ( k − k ′ )and axis-symmetry of the field, the two point correlation is symmetric under under i and j indices exchanges; (iii)The magnetic field is a gaussianly distributed field. For a vector field F we use F j ( k ) = Z d x e i k · x F j ( x ) , F j ( x ) = Z d k (2 π ) e − i k · x F j ( k ) , (5)when Fourier transforming between real and wavenumber spaces; we assume flat spatial hypersurfaces. A. Lorentz Force Two-Point Correlation Function
For the two-point correlation function of the Π ( k ) magnetic vector source we have ζ (12) i i ( k , k ′ ) ≡ h Π ⋆i ( k )Π i ( k ′ ) i . (11)Using results of Sect. A 2 one finds for it ζ (12) i i ( k , k ′ ) = δ ( k − k ′ ) ψ (12) i i ( k ) , (12)with ψ (12) i i ( k ) = 1(8 π ) Z d q M ( | q | ) M ( | k − q | ) h P i a ( ˆk )ˆ k b + P i b ( ˆk )ˆ k a i h P i c ( ˆk )ˆ k d + P i d ( ˆk )ˆ k c i h P ac ( ˆq ) P bd ( ˆk − q ) + P ad ( ˆq ) P bc ( ˆk − q ) i . (13)Note that as usual we assume summation over the repeated indices. Using P ij ( ˆk ) projector symmetry properties anddefining γ = ˆk · ˆq , β = ˆk · ( ˆk − q ), and µ = ˆq · ( ˆk − q ), after long but simple computations we obtain for the Lorentzforce two-point correlation function h L ( V ) ⋆i ( k ) L ( V ) j ( k ′ ) i = k P ij ( ˆk )(8 π ) δ ( k − k ′ ) Z d q M ( | q | ) M ( | k − q | )(2 − β − γ ) ++ k i k j (8 π ) δ ( k − k ′ ) Z d q M ( | q | ) M ( | k − q | ) (cid:2) γ β − γ (1 − β ) − β (1 − γ ) (cid:3) − k (8 π ) δ ( k − k ′ ) Z d q M ( | q | ) M ( | k − q | ) n ˆ q i ˆ q j (1 − β ) + ( ˆk − q ) i ( ˆk − q ) j (1 − γ ) − γ (ˆ k i ˆ q j + ˆ k j ˆ q i )(1 − β ) − β h ˆ k i ( ˆk − q ) j + ˆ k j ( ˆk − q ) i i (1 − γ ) − γβ h ˆ q i ( ˆk − q ) j + ˆ q j ( ˆk − q ) i i } . (14)Note that the form of the Lorentz force two-point correlation function, Eq. (14), has several symmetries: (i) symmetryunder i and j index exchange; (ii) the symmetry under ˆq and ˆk − q exchange (i.e. γ and β angles exchange symmetry).It must be underlined that the trace of Eq. (14) is given by h L ( V ) ⋆i ( k ) L ( V ) i ( k ′ ) i = k π ) δ ( k − k ′ ) Z d q M ( | q | ) M ( | k − q | )(1 + µγβ − β γ ) , (15)which totally agrees with the result of Refs. [8, 11, 12]. Further simplification of Eq. (14) gives following result forthe Lorentz force two-point correlation function: h L ( V ) ⋆i ( k ) L ( V ) j ( k ′ ) i = k (8 π ) P ij ( ˆk ) δ ( k − k ′ ) Z d q M ( | q | ) M ( | k − q | )(1 + µγβ − β γ ) . (16)The ensemble averaging procedure insures that rotational isotropy is preserved. Note that without ensemble averaging in one realization the isotropy of the two point correlation can be violated and only the averaging leads to cancelationof anisotropic component and restoration of isotropy. B. Lorentz Force Three-Point Correlation Function
The three-point correlation function of the vector magnetic source is given by ζ (123) i i i ( k , k , k ) = h Π i ( k )Π i ( k )Π i ( k ) i . (17) It is appropriate to define the three point correlation function as an average of three Π i ’s without the complex conjugation. Using results of Sec. A 2 one finds ζ (123) i i i ( k , k , k ) = δ ( k + k + k ) ψ (123) i i i ( k , k ) , (18)with ψ (123) i i i ( k , k ) = 18 π Z d q M ( | q | ) M ( | k − q | ) M ( | k + q | ) P i j ( ˆk ) P j s ( ˆ k − q )( ˆk + k ) s P i j ( ˆk ) P j j ( ˆ k + q ) P i j ( ˆk + k )ˆ k s P s s ( ˆq )ˆ k s . (19)The Lorentz force three point correlation function is given as h L ( V ) i ( k ) L ( V ) i ( k ) L ( V ) i ( k ) i = 18 π δ ( k + k + k ) Z d q M ( | q | ) M ( | k − q | ) M ( | k + q | ) P i j ( ˆk ) P i j ( ˆk ) P i j ( ˆk + k ) P j s ( ˆ k − q )( k + k ) s P j j ( ˆ k + q )[( k × ˆq )( k × ˆq )] . (20)We see that the r.h.s. of Eq. (20) is symmetric under exchange k and k . It is obvious that this symmetry isreflected in the CMB temperature three point correlation function, see below. III. TEMPERATURE FLUCTUATIONS FROM THE MAGNETIZED VECTOR MODE
Vector perturbations induce CMB temperature anisotropies via the Doppler and integrated Sachs-Wolfe effects[27, 28], ∆ TT ( x , n , η ) = − v · n | η η dec + Z η η dec dη ˙ V · n , (21)where n is the unit vector in the light propagation direction, η dec is the conformal time at decoupling. x denotesthe observer position, x dec = x + n ( η − η dec ). Due to the spherical symmetry the temperature fluctuations aredecomposed using the spherical harmonics as,∆ TT ( x , n , η ) = ∞ X l =0 l X m = − l a lm ( x , η ) · Y lm ( n ) . (22)Accounting for the spherical harmonics property, see Chapter 5 of Ref. [29], we obtain a lm ( x , η ) = R d Ω n Y ⋆lm ( n )∆ T /T ( x , n , η ). The decaying nature of the vector potential V implies that most of its contribu-tion toward the integrated Sachs-Wolfe term comes from near η dec . Neglecting a possible dipole contribution due to v today, from Eq. (21) we obtain,∆ TT ( n ) ≃ v ( η dec ) · n − V ( η dec ) · n = Ω ( η dec ) · n . (23)Here we placed observer at the origin x = 0 and we skip η denoting ∆ T /T ( n ) ≡ ∆ T /T ( x = 0 , n , η = η ) and a lm ≡ a lm ( x = 0 , η ). Since x dec = n ( η − η dec ) Fourier transform of Eq. (23) results in∆ TT ( k , n ) = Ω ( k , η dec ) · n ) e i ( k · n )∆ η , (24)where Ω ( k , η dec ) is the Fourier amplitude of vorticity perturbations at η = η dec , wave vector k = k ˆk labels theresulting Fourier mode after transforming from the coordinate representation x to the momentum representation byusing e i kx , and ∆ η = η − η dec ≈ η is the conformal time from decoupling until today.For the multipole coefficient Fourier transform we obtain, a lm ( k ) = Z d n ( Ω ( k , η dec ) · n ) e i k · n η Y ⋆lm (ˆ n ) , (25)where we use a lm ( k ) ≡ a lm ( k , η = η ). A. Two-point correlation function
The two-point correlation of the temperature fluctuations is given by C ( n , n ) ≡ h ∆ TT ( n ) ∆ TT ( n ) i = X l m X l m h a ⋆l m a l m i Y ⋆l m ( n ) Y l m ( n ) , (26)here P lm ≡ P ∞ l =0 P lm = − l . In the case when the spatial isotropy and rotational invariance is preserved the multipoleswith different l and m do not correlate and h a ⋆l m a l m i = C l δ l l δ m m . It is obvious that C ( n , n ) is the functionof the angle between n and n vectors, and we can easy get C ( n , n ) = X lm l + 14 π C l P l ( n · n ) . (27)The sound waves (density perturbations) sourced by the magnetic field keep the rotational invariance unchangedand thus the multipole correlation matrix has a diagonal form, and the two-point correlation function depends onone angle, n · n . The same time the temperature fluctuations are non-gaussian [6, 7] due to the non-linearity ofthe magnetic field energy density with respect to the magnetic field. Note, that the magnetic field which has anon-gaussian energy momentum tensor [8] can be itself field with Gaussian distribution.As soon as the vector mode is present in the Universe there is an additional direction inserted through the vorticityfield Ω ( η dec ). Before ensemble averaging procedure the two-point correlation is not rotationally invariant and thecross correlation function between the multipoles has off-diagonal components [27, 28]. The measurement of theseoff-diagonal terms might serve as a tool to constraint the primordial homogeneous magnetic field.We define the multipole two point correlation function as usual: D m m l l ≡ h a ⋆l m a l m i = Z d k (2 π ) d k (2 π ) h a ⋆l m ( k ) a l m ( k ) i , (28)and the simple calculation leads to h a ⋆l m a l m i = Z d k (2 π ) d k (2 π ) d n d n e i ( − k · n + k · n ) η Y ⋆l m ( n ) Y l m ( n ) h [ n · Ω ⋆ ( k , η dec )][ n · Ω ( k , η dec )] i . (29)We also decompose the vorticity perturbation plane wave over the vector spherical harmonics as, Ω ( k ) e i k · n η = P λ,l,m A ( λ ) lm Y ( λ ) lm ( n ), with λ = − , , ∇ · Ω = 0 condition, leading to Ω ( k ) · Y ( − lm ( ˆk ) = 0, see also Appendixof Ref. [28]. We use also the definition n · Y ⋆lm ( n ) = Y ( − ⋆lm ( n ). Then we have D m m l l = Z d k (2 π ) d k (2 π ) d n d n Y ⋆l m ( n ) Y l m ( n ) X t r X t r Y ⋆t r ( n ) Y t r ( n ) h A ( − ⋆t r A ( − t r i , (30)with A ( − lm = 4 πi l − p l ( l + 1) j l ( kη ) kη ( Ω ( k ) · Y (+1) lm ( ˆk )) . (31)The integration over n in Eq. (30) gives us δ l t , δ l t , δ m r , δ m r , and the sums are eliminated and we get D m m l l = ( − l − l i l + l − p l ( l + 1) l ( l + 1)(2 π ) Z d k d k j l ( k η ) j l ( k η ) k k η h Y (+1) ⋆l m ( ˆk ) i i h Y (+1) l m ( ˆk ) i i h Ω ⋆i ( k )Ω i ( k ) i . (32)Next we have to consider separately case of large scale approximation Eq. (9) and case of small scale approximationEq. (10). Using Eq. (16) we obtain: D m m l l = ( − l − l i l + l − p l ( l + 1) l ( l + 1)(2 π ) [2(1 + R dec )( ρ γ + p γ )] (cid:18) η dec η (cid:19) Z d k d q M ( | q | ) M ( | k − q | ) j l ( kη ) j l ( kη )(1 + µγβ − γ β ) (cid:16) Y (+1) ⋆l m ( ˆk ) · Y (+1) l m ( ˆk ) (cid:17) (33)for the scales larger then the Silk damping scale. For the scale smaller than the Silk damping scale we have D m m l l = ( − l − l i l + l − p l ( l + 1) l ( l + 1)(2 π ) [2( L γ / ρ γ + p γ )] Z d k d q M ( | q | ) M ( | k − q | ) j l ( kη ) j l ( kη )( kη ) (1 + µγβ − γ β ) (cid:16) Y (+1) ⋆l m ( ˆk ) · Y (+1) l m ( ˆk ) (cid:17) . (34)Recall that we assume that the magnetic field power spectrum is given by a simple power law, M ( | q | ) ∝ q n . Toproceed we have to evaluate the following integrals over angular variables I m ,m l ,l = Z d Ω ˆk Z d Ω ˆq (1 + µβγ − γ β )( k + q − kqγ ) n/ (cid:16) Y (+1) ⋆l m ( ˆk ) · Y (+1) l m ( ˆk ) (cid:17) . (35)It is easy to see that integrals over d Ω ˆk and d Ω ˆq separate. First we evaluate the integral over d Ω ˆq . For a particular ˆk , choose the polar axis for the d Ω ˆq integral in the z direction. Since γ = cos θ ˆq the integrand does not depend onthe azimuthal angle φ ˆq and the integration over φ ˆq simply gives 2 π . The integration over d cos θ ˆq is simple to beevaluated and is given as R − dγ (1 − γ ) (cid:16) − qγ ( k + qγ ) k + q − kqγ (cid:17) (cid:0) k + q − kqγ (cid:1) n/ , with q = | q | . Thus I m ,m l ,l = 2 πδ l l δ m m Z − dγ (1 − γ ) (cid:18) − qγ ( k + qγ ) k + q − kqγ (cid:19) (cid:0) k + q − kqγ (cid:1) n/ (36)and finally only the diagonal cross correlations are present. Note that, the rotational symmetry (absence of off-diagonalterms in D m m l l ) is due to the ensemble averaging. B. Bispectrum definition and calculation
The standard approach to study the CMB non-gaussianity consists on the CMB temperature fluctuations inves-tigation. Below we present the self-consistent way to describe the bispectrum. As usual the three-point correlationfunction of CMB temperature anisotropy is defined as ξ ( n , n , n ) ≡ h ∆ TT ( n ) ∆ TT ( n ) ∆ TT ( n ) i = X l i m i h a l m a l m a l m i Y l m ( n ) Y l m ( n ) Y l m ( n ) , i = 1 , , , (37)In order to proceed we need to calculate the following form B m m m l l l ≡ h a l m a l m a l m i = Z d k (2 π ) d k (2 π ) d k (2 π ) h a l m ( k ) a l m ( k ) a l m ( k ) i . (38)Using expression for a lm given by Eq. (25) we get h a l m a l m a l m i = Z d k (2 π ) d k (2 π ) d k (2 π ) d n d n d n e i ( k · n + k · n + k · n )∆ η Y ⋆l m ( n ) Y ⋆l m ( n ) Y ⋆l m ( n ) n i n i n i h Ω i ( k , η dec )Ω i ( k , η dec )Ω i ( k , η dec ) i . (39)First we integrate over d n i by using Eq. (117) on p.227 [29]. Proceeding in the way given in Sec. IIIA we arrive at B m m m l l l = i l + l + l − (2 π ) p l ( l + 1) l ( l + 1) l ( l + 1) Z d k d k d k j l ( k η ) j l ( k η ) j l ( k η ) k k k η Y (+1) l m ⋆ (ˆ k ) | i Y (+1) l m ⋆ (ˆ k ) | i Y (+1) l m ⋆ (ˆ k ) | i h Ω i ( k , η dec )Ω i ( k , η dec )Ω i ( k , η dec ) i . (40)Next we have to consider separately case of large scale approximation Eq. (9) and case of small scale approximationEq. (10). Using Eqs. (17, 18, 19), relation Eq. (A8) and the representation for the Dirac delta function Eq. (A9) aftersome computations for the large scale approximation ( L > L S ) we obtain B m m m l l l = i l + l + l − p l ( l + 1) l ( l + 1) l ( l + 1)(2 π ) [(1 + R dec )( ρ γ + p γ )] (cid:18) η dec η (cid:19) Z k dk k dk k dk j l ( k η ) j l ( k η ) j l ( k η ) Z d Ω ˆ k d Ω ˆ k d Ω ˆ k Z d Ω ˆ q Z q dqM ( | q | ) M ( | k − q | ) M ( | k + q | ) X t r X t r X t r i t + t + t Z x dxj t ( k x ) j t ( k x ) j t ( k x ) G t t t r r r h Y ( − t r (ˆ k ) × ˆ q i a h Y (1) l m ⋆ (ˆ k ) × ( ˆ k − q ) i b h Y ( − t r (ˆ k ) × ˆ q i a h Y (1) l m ⋆ (ˆ k ) × ( ˆ k + q ) i c h Y ( − t r (ˆ k ) × ( ˆ k − q ) i b h Y (1) l m ⋆ (ˆ k ) × ( ˆ k + q ) i c , (41)where G t t t r r r is the usual Gaunt integral, G t t t r r r = Z d Ω ˆ n Y t r (ˆ n ) Y t r (ˆ n ) Y t r (ˆ n ) . (42)The indices a, b, c correspond to the vector components and repeated ones reflect the scalar product of the corre-sponding vectors.Similarly for the small scale approximation ( L < L S ) we obtain B m m m l l l = i l + l + l − p l ( l + 1) l ( l + 1) l ( l + 1)(2 π ) [( L γ / ρ γ + p γ )] η Z dk dk dk j l ( k η ) j l ( k η ) j l ( k η ) Z d Ω ˆ k d Ω ˆ k d Ω ˆ k Z d Ω ˆ q Z q dqM ( | q | ) M ( | k − q | ) M ( | k + q | ) X t r X t r X t r i t + t + t Z x dxj t ( k x ) j t ( k x ) j t ( k x ) G t t t r r r h Y ( − t r (ˆ k ) × ˆ q i a h Y (1) l m ⋆ (ˆ k ) × ( ˆ k − q ) i b h Y ( − t r (ˆ k ) × ˆ q i a h Y (1) l m ⋆ (ˆ k ) × ( ˆ k + q ) i c h Y ( − t r (ˆ k ) × ( ˆ k − q ) i b h Y (1) l m ⋆ (ˆ k ) × ( ˆ k + q ) i c . (43)Using the Wigner D functions and proceeding in the way analogous to Sec. IIIA it can be shown that there arenon-zero cross correlations between non-equal l and m . Again the proper answer assumes accounting for the angulardependence in the power spectra M . IV. CONCLUDING REMARKS
We present the study of the magnetized perturbation vector mode induced CMB two- and three-point correlationfunctions in a common framework. We show that already the vorticity two-point correlation functions reflect theanisotropy of the considered perturbations before ensemble averaging. One of the implications of our results is thatCMB bispectrum computation technique presented in Ref. [30] in the case of magnetized perturbations should beapplied with caution.Note that in the present work we have focused on derivation of main analytic results for the CMB two- and three-point correlation functions arising from the vector mode supported by the stochastic cosmological magnetic field. Weplan to present a detailed analysis of obtained equations and phenomenological estimates in a separate publication.
Acknowledgments
We are grateful to A.Kosowsky for fruitful discussions on different points of present investigation (in particular inderiving the ensemble averaging of the two-point correlation function). We also appreciate useful communicationsand comments from I. Brown, R. Durrer, and A. Lewis, and acknowledge discussions with M. Kamionkowski, B.Ratra, L. Samushia, K. Subramanian, and A. Tevzadze. We acknowledge partial support from Swiss National ScienceFoundation SCOPES grant no. 128040, NSF grant AST1109180 and NASA Astrophysics Theory Program grantNNXlOAC85G. T. K. acknowledges the ICTP associate membership program.
Appendix A: Useful Mathematical Formulae
In this Appendix we list various mathematical results we use in the computations.
1. Wigner D Functions
Wigner D functions relate helicity basis vectors e ′± = ∓ ( e Θ ± i e φ ) / √ e ′ = e r to spherical basis vectors e ± = ∓ ( e x ± i e y ) / √ e = e z (see Eq. (53), p. 11, [29]) through e ′ µ = X ν D νµ ( φ, Θ , e ν , ν, µ = − , , . (A1)In both the spherical basis and the helicity basis the following relations hold: e ν e µ = δ νµ , e µ = ( − µ e − µ , e µ = e ⋆µ , e µ × e ν = − iǫ µνλ e λ .
2. Calculation of B For calculation of the two-point correlation function we must determine the magnetic field energy momentum two-point cross correlation h τ ( B ) ab ( k ) τ ( B ) cd ( k ′ ) i given by Eq. A6 of [11] and Eq. 4.1 of [8]. It is easy to show that the partsof the τ ( B ) ab (and τ ( B ) cd proportional to the δ cd ), see Eq. (7), do not contribute to the integral of Eq. (8). Then we needto compute the following object B abcd ( k , k ) = Z d q (2 π ) d q (2 π ) h B ⋆a ( q ) B ⋆b ( k − q ) B c ( q ) B d ( k − q ) i . (A2)Using Eq. (6) and Wick’s theorem, we obtain that the contribution of the two-point correlation of the magnetic fieldenergy momentum into the vorticity perturbation is given by δ ( k − k ′ )(4 π ) Z d qM ( | q | ) M ( | k − q | ) h P ac ( ˆq ) P bd ( ˆk − q ) + P ad ( ˆq ) P bc ( ˆk − q ) i . (A3)For calculation of bispectrum we need to know following object B abcdef ( k , k , k ) = Z d q (2 π ) d q (2 π ) d q (2 π ) h B a ( q ) B b ( k − q ) B c ( q ) B d ( k − q ) B e ( q ) B f ( k − q ) i . (A4)Assuming that the magnetic field obeys Gaussian statistics we can expand the six-point correlation function usingWick’s theorem. Doing this we will get seven terms proportional to either δ ( k ), δ ( k ) or δ ( k ) (which we neglect)and eight terms proportional to δ ( k + k + k ) which we keep. So, for B abcdef one finds [8, 31]: B abcdef ( k , k , k ) = δ ( k + k + k ) Z d q M ( | q | ) M ( | k − q | ) M ( | k + q | ) { P ac ( q ) P be ( k − q ) P df ( k + q ) + P ac ( q ) P bf ( k − q ) P de ( k + q )+ P ad ( q ) P be ( k − q ) P ef ( k + q ) + P ad ( q ) P bf ( k − q ) P ce ( k + q ) } + δ ( k + k + k ) Z d q M ( | q | ) M ( | k − q | ) M ( | k + q | ) { P ae ( q ) P bc ( k − q ) P df ( k + q ) + P ae ( q ) P bd ( k − q ) P df ( k + q )+ P af ( q ) P bc ( k − q ) P de ( k + q ) + P af ( q ) P bd ( k − q ) P ce ( k + q ) } . (A5)Assuming that projector A abcdef acting on this object is symmetric w.r.t. each pair of indexes, i.e. A abcdef = A bacdef = A abdcef = A abcdfe , (A6)quantity B abcdef can be brought to a compact form B abcdef ( k , k , k ) = 8 δ ( k + k + k ) Z d q M ( | q | ) M ( | k − q | ) M ( | k + q | ) P ac ( q ) P be ( k − q ) P df ( k + q ) . (A7)0
3. Useful relations
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