The scalar, vector and tensor contributions of a stochastic background of magnetic fields to cosmic microwave background anisotropies

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1 Mon Not R Astron Soc (9) doi:11111/j x The scalar vector and tensor contributions of a stochastic background of magnetic fields to cosmic microwave background anisotropies D Paoletti 1 F Finelli 4 and F Paci 54 1 Dipartimento di Fisica Università degli studi di Ferrara and INFN via Saragat 1 I-441 Ferrara Italy INAF IASF Bologna Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna Istituto Nazionale di Astrofisica via Gobetti 11 I-419 Bologna Italy INAF OAB Osservatorio Astronomico di Bologna Istituto Nazionale di Astrofisica via Ranzani 1 I-417 Bologna Italy 4 INFN Sezione di Bologna Via Irnerio 46 I-416 Bologna Italy 5 Dipartimento di Astronomia Università degli Studi di Bologna via Ranzani 1 I-417 Bologna Italy Accepted 9 March Received 9 February 18; in original form 8 December 5 ABSTRACT We study the contribution of a stochastic background (SB) of primordial magnetic fields (PMFs) on the anisotropies in temperature and polarization of the cosmic microwave background (CMB) radiation A SB of PMF modelled as a fully inhomogeneous component induces non-gaussian scalar vector and tensor metric linear perturbations We give the exact expressions for the Fourier spectra of the relevant energy momentum components of such a SB given a power-law dependence parametrized by a spectral index n B for the magnetic field power spectrum cut at a damping scale k D For all the values of n B considered here the contribution to the CMB temperature pattern by such a SB is dominated by the scalar contribution and then by the vector one at higher multipoles We also give an analytic estimate of the scalar contribution to the CMB temperature pattern Key words: magnetic fields cosmic microwave background 1 INTRODUCTION The origin of the large-scale magnetic fields observed is an issue of great importance in astrophysics (see Subramanian 6 for a review) Primordial magnetic fields (PMFs) generated in the early Universe are a possible explanation of large-scale magnetic fields in clusters of galaxies and galaxies and might have left an imprint in the anisotropy pattern of the cosmic microwave background (CMB; see Durrer 7 for a review) A stochastic background (SB) is the simplest way to model these random PMFs in an isotropic and homogeneous background A SB of PMF is modelled as a fully inhomogeneous component and its energy momentum tensor (EMT) quadratic in the magnetic fields is considered at the same footing as linear inhomogeneities in the other components and linear metric fluctuations A SB of PMF generates independent modes for all kinds of linear perturbations: there have been several studies for scalar (Koh & Lee ; Kahniashvili & Ratra 6; Giovannini & Kunze 8 and references therein; Finelli Paci & Paoletti 8 hereafter FPP) vector (Subramanian & Barrow 1998; Seshadri & Subramanian 1; Mack Kahniashvili & Kosowsky ; Lewis 4) and tensor (Durrer Ferreira & Kahniashvili ; Mack et al ; Caprini Durrer & Kahniashvili 4) perturbations in the presence of a SB paoletti@iasfboinafit (DP); finelli@iasfboinafit (FF); paci@iasfboinafit (FP) of PMF Limits on the amplitude and spectral index of the PMF were also obtained by an exploration of a flat cold dark matter ( CDM) model in the presence of such SB (Yamazaki et al 6) We study the problem by a numerical Einstein Boltzmann code extending the results obtained in our previous work (FPP) The study of the impact of a SB of PMF on CMB anisotropies requires a detailed study of the initial conditions for fluctuations and of the power spectra of the EMT of the SB of PMF Our paper improves previous results in these two aspects In this paper we obtain the Fourier spectra of the relevant vector and tensor energy momentum components of the SB of PMF along the procedure used in FPP for the scalar components As shown in FPP by solving exactly the convolution integrals for a sequence of values for the spectral slope n B which parametrizes the PMF power spectrum previous results of Mack et al () may be significantly improved With these improved correlators we then investigate the impact of a SB of PMFs on scalar vector and tensor cosmological perturbations and in particular on CMB temperature and polarization anisotropies Our results show that it is very important to study also vector perturbations since these dominate at high l over the scalar ones for any slope of the spectrum of PMF Our paper is organized as follows In Section we introduce our conventions for a non-helical SB and the scalar vector and tensor decomposition of its EMT: we give our exact results for the relevant objects for a set of n B leaving the details in Appendices A and B and we show how our exact results improve on previous results The set of n B now includes also values n B < / which were C 9 The Authors Journal compilation C 9 RAS Downloaded from on July 18

2 54 D Paoletti F Finelli and F Paci not studied in FPP Section presents the decomposition of metric perturbations and Sections 4 6 present the study of scalar vector and tensor cosmological perturbations in the presence of a SB of PMF respectively In Section 7 we present the results obtained by our modified version of the CAMB code (Lewis Challinor & Lasenbly ) We conclude in Section 8 STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS Since the EMT of PMF at homogeneous level is zero at linear order PMFs evolve like a stiff source and therefore it is possible to discard the back reaction of gravity onto the SB of PMF Before the decoupling epoch the electric conductivity of the primordial plasma is very large therefore it is possible at lowest order to consider the infinite conductivity limit in which the induced electric field is zero Within the infinite conductivity limit the magnetic field amplitude scales simply as B(x τ) = B(x)/a(τ) and 1 the EMT of a SB of PMF is τ =ρ B = B (x) 8πa 4 (τ) (1) τ i = () τ i j = 1 B (x) δ i 4πa 4 j (τ) B j (x)b i (x) () The two-point correlation function in the Fourier space for fully inhomogeneous fields is B i (k)b j (k ) =(π) δ(k k )(δ ij ˆk i ˆk j ) P B(k) (4) where P B (k) is the spectrum of PMF parametrized as ( ) nb k P B (k) = A (5) k and k is a reference scale PMFs are damped on small scales by radiation viscosity We model this damping introducing a sharp cutoff in the PMF power spectrum at a damping scale called k D The relation between the amplitude of PMF power spectrum and the amplitude of the field itself is B = B i (x )B i (x) x =x kd = 1 dkk P π B (k) (6) Solving the integral above we obtain B = A π (n B ) k n B D k n B (7) where for the convergence of the integral above is requested n B > We will use to denote the magnetic field amplitude this 1 We choose the standard convention in which at present time t a(t ) = 1 As Fourier transform and its inverse we use Y (kτ) = dxe ik x Y(xτ) dk Y (xτ) = eik x (π) Y(kτ) where Y is a generic function Note that we have changed our Fourier conventions with respect to FPP quantity instead of smearing the field at an additional scale λasin FPP: B λ = 1 dkk P B (k)e λ k (8) π The two definitions can however related simply by B = B k n B D λ n B λ (n B )Ɣ ( n B ) (9) The EMT of PMF is quadratic in the magnetic field and therefore its Fourier transform is a convolution The two-point correlation function of the spatial part of EMT is dqd p τab (k)τ cd(k ) = 64π δ abδ 5 cd B l (q)b l (k q)b m ( p)b m (k p) dqd p π 5 B a(q)b b (k q)b c ( p)b d (k p) We can then obtain scalar vector and tensor correlation functions: (S) (k) (S) (k ) = δ ab δ cd τ ab (k)τ cd(k ) (V) i (k) (V) j (k ) = k a P ib (k)k c P jd(k ) τ ab (k)τ cd(k ) (T) ij (k) (T) tl (k ) = P ia (k)p jb (k) 1 P ij (k)p ab (k) P tc (k )P ld (k ) 1 P tl(k )P cd (k ) τab (k)τ cd(k ) (1) where P ij = δ ij ˆk i ˆk j Such convolutions can be written in terms of spectra as follows: (S) (k) (S) (k ) = (S) (k) δ(k k ) (S) i (k) (V) j (k ) = 1 (V) (k) P ij (k)δ(k k ) (T) ij (k) (T) tl (k ) = 1 4 (T) (k) M ij tl (k)δ(k k ) where M ij tl = P it P jl P il P jt P ij P tl With this choice the spectra take the form: ρ B (k) 1 = d pp 14π 5 B (p) P B ( k p )(1 μ ) (11) (k) = 1 d pp 51π 5 B (p) P B ( k p ) (1 β )(1 γ ) γβ(μ γβ) (1) (T) (k) = 1 d pp 51π 5 B (p) P B ( k p ) (1 γ γ β ) (1) where μ = ˆp (k p)/ k p γ = ˆk ˆp and β = ˆk (k p)/ k p These equations agree within our Fourier convention with previous results by Mack et al () and Durrer et al () One of the main results of our work is the calculation of the correct ie without any approximation expressions for these convolutions given a power spectrum as in equation (5) with a sharp cut-off at k D In the appendices we explain the integration technique and show the results for various spectral indices n B = 1 1 / We use the convention that latin indices run from 1 to while greek indices run from to C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

3 The contribution of PMFs to CMB anisotropies Figure 1 Comparison of k ρ B (k) (solid line) k L (k) (large-dashed line) k i (k) (small-dashed line) k (T) ij (k) (medium-dashed line) in units of B /(14 π )versusk/k D The left- and right-hand panels are for n B = and5/ respectively 5/: in this paper we add n B =5/ to the previously studied n B / values for scalar quantities studied in FPP As for the scalar energy density and Lorentz force discussed in FPP also the vector and tensor anisotropic stresses have support for Fourier modes with modulus smaller than k D WeshowinFig1thebehaviour of scalar vector and tensor quantities for n B = 5/ respectively For n B =5/ the spectrum for k k D is ρ(k) n B =5/ 17A k 5 8π 4 k (14) whose slope but not the amplitude agrees with Kahniashvili & Ratra (6) For n B =5/ L(k) (55/51) ρ(k) for k k D (for n B / we obtained L(k) (11/) ρ(k) for k k D ; FPP) Note also how the tensor contribution dominates over other ones in amplitude of the Fourier spectra in agreement with previous numerical findings (Brown & Crittenden 5) The vector and tensor anisotropic stresses are shown with varying n B in Fig The vector and tensor contributions have a k-dependence very similar to the energy density: for k k D and n B > /(n B =/) (k) (T ) (k) have a white noise (logarithmic divergent) spectrum with (T) (k) (k) whereas for n B = 5/ both become infrared dominated as (T) (k) (94/5) (k) holds The generic behaviour of (k) for k k D and n B > / is (k) A k n D 56π 4 k n( n B) 8 () The pole for n B =/ in equation () is replaced by a logarithmic divergence in k in the exact result reported in Appendix A; the result reported by Mack et al () has a factor 4 instead of the factor 56/ reported in equation () Note also that the relation between the tensor and vector anisotropic stresses is different from the one reported in Mack et al () who predict (in our conventions): (T) (k) = (k) this relation is obtained neglecting the angular part in equations (1) and (1) and is incorrect EVOLUTION OF PERTURBATIONS WITH PMF The presence of PMF influences the evolution of cosmological perturbations mainly in three ways PMFs carry energy and momentum at the perturbation level and therefore gravitate influencing the metric perturbations As a second point they carry anisotropic stress which adds to the ones already present in the plasma given by neutrinos and photons with the caveat that the photon anisotropic stress is negligible before decoupling epoch Thirdly the presence of PMF induces a Lorentz force on baryons which modifies their velocity Due to the tight coupling between photons and baryons prior to the decoupling epoch the Lorentz force has an indirect effect also on photons during this regime The evolution of metric perturbations is described by the Einstein equations These are modified by the presence of PMFs that represent a source term as follows: G μν = 8π(T μν τ μν ) (16) Figure Plot of k (k) (left-hand panel) and k (T) (k) (right-hand panel) in units of B /(14π )versusk/k D for different n B for fixed B The different lines are for n B =5/ / 1 1 ranging from the solid to the longest dashed C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

4 56 D Paoletti F Finelli and F Paci where as usual τ μν represents the PMF EMT The metric chosen in this work is ds = a (τ)dτ (δ ij h ij )dx i dx j (17) where h ij can be decomposed into a trace part h and a traceless part consisting of its scalar vector and tensor part (Ma & Bertschinger 1995): h ij = h ( δ ij i j δ ) ij μ h V ij ht ij (18) The vector part is constructed in terms of a divergenceless vector h V i h V ij = ih V j j h V i (19) The tensor part is traceless and transverse ( i h Ti j = ) 4 THE SCALAR CONTRIBUTION We will focus now on the magnetic scalar contribution to CMB anisotropies The effect on metric perturbations is described by the Einstein equations with a source term given by the PMF EMT We choose to work in the synchronous gauge where the scalar metric perturbation in the Fourier space is described by two scalar potentials namely h(k τ) andη (k τ) The Einstein equations with the contribution of PMF in the synchronous gauge are k η 1 Hḣ =4πGa ( n ρ n δ n ρ B ) k η = 4πGa n (ρ n P n )θ n ( ḧ Hḣ k η =8πGa n c sn ρ nδ n δρ ) B ḧ 6 η H(ḣ 6 η) k η =4πGa n (ρ n P n )σ n σ B (4) where n represents the various species of the plasma ie baryons cold dark matter (CDM) photons and massless neutrinos The conservation of the PMF EMT μ τ μ ν PMF = implies that ρ B (x τ) = ρ B (x τ )/a(τ) 4 and the following relation between the magnetic anisotropic stress σ B the magnetic energy density ρ B and the Lorentz force L holds: σ B = ρ B L (41) Such Lorentz force modifies the Euler equation for baryons velocity leading to observational signatures (see FPP for the most recent discussion on this effect) 41 Initial conditions The magnetized adiabatic mode initial conditions in the synchronous gauge deep in the radiation era are h = C 1 k τ C 1(5 4R ν ) 6( 4R ν ) k4 τ 4 55L B 6( 4R ν ) (55 8R ν) B k 4 τ 4 18( 4R ν ) η = C 1 5 4R ν 6( 4R ν ) C 1k τ B (55 8R ν ) 55L B 168( 4R ν ) 56( 4R ν ) B δ γ = B C 1k τ 6 L B (1 R ν ) δ ν = B C 1k τ B (1 R ν ) 6R ν L B R ν k τ k τ k τ δ b = 4 B C 1 k τ B 8 L B k τ 8(1 R ν ) δ c = 4 B C 1 k τ θ γ = C 1 18 k4 τ B 4 L B k τ 4 (1 R ν ) B 7 L B k 4 τ 4(1 R ν ) θ b = θ γ θ c = θ ν = ( 4R ν) 18( 4R ν ) C 1k 4 τ B (1 R ν ) L B k τ 4R ν 4 R ν (15 14Rν )L B 84R ν ( 4R ν ) (7 161R ν 8R ν ) B 54R ν ( 4R ν ) 4C 1 σ ν = ( 4R ν ) k τ B L B 4R ν 4 R ν B (55 8R ν ) 56R ν ( 4R ν ) F ν = 6 7 B 4R ν 4 L B R ν k 4 τ 165L B 56R ν ( 4R ν ) k τ kτ (4) where B = ρ B /(ρ ν ρ γ ) L B = L/(ρ ν ρ γ ) R ν = ρ ν /(ρ ν ρ γ ) and C 1 is the constant which characterizes the regular growing adiabatic mode as given in Ma & Bertschinger (1995) The result reported is different from the one reported in FPP because we are adding the first non-trivial terms in O( B L B )forh δ c and because of a different truncation in the neutrino hierarchy The term O(k 4 τ 4 ) in h (which also contains the next-to-leading term of the adiabatic mode) has been obtained by taking self-consistently the required order in all the variables in equations (4) For simplicity we have written the O(k 4 τ 4 ) term only in h (since it is the leading term for the magnetic solution) and omitted these higher order terms in all the other variables Here we also choose to truncate the hierarchy at F 4ν = instead of F ν = as in FPP This change affects the magnetic next-to-leading order terms in the velocity and anisotropic stress of neutrinos and in the metric perturbation η The equation for the evolution of the F ν is Ḟ ν = 6 7 kσ ν (4) while the neutrino anisotropic stress equation becomes σ ν = 8 θ ν 5 kf ν 4 ḣ 8 η (44) 5 Note how the presence of a SB of PMFs induces a new independent mode in matter and metric perturbations ie the fully magnetic mode obtained by setting C 1 = in equation (4) This new independent mode is the particular solution of the inhomogeneous system of the Einstein Boltzmann differential equations: the SB of PMF treated as a stiff source acts indeed as a force term in the system of linear differential equations Whereas the sum of the fully magnetic mode with the curvature one can be with any correlation as for an isocurvature mode the nature of the fully magnetic mode and therefore its effect is different: the isocurvature modes C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

5 The contribution of PMFs to CMB anisotropies 57 are solutions of the homogeneous system (in which all the species have both background and perturbations) whereas the fully magnetic one is the solution of the inhomogeneous system sourced by a fully inhomogeneous component It is interesting to note how the magnetic contribution drops from the metric perturbation at leading order although is larger than the adiabatic solution for photons neutrinos and baryons This is due to a compensation which nullifies the sum of the leading contributions (in the long-wavelength expansion) in the single species energy densities and therefore in the metric perturbations A similar compensation exists for a network of topological defects which does not carry a background EMT as the PMF SB studied here 4 4 Analytic description of the scalar magnetic contribution on large angular scales In this section we give an analytic description for the scalar magnetic contribution to CMB anisotropies on large angular scales given by the initial conditions in equation (4) and computed by our modified version of CAMB The Sachs Wolfe term is δt /T SW δ r /4 ψwhereδ r is the radiation density contrast and ψ is one of the metric potentials in the longitudinal gauge: ds = a (η) (1 ψ)dη (1 φ)γ ij dx i dx j (45) For adiabatic initial conditions is simply δt /T SW ψ/since δ r 8ψ/ in the matter-dominated era on large scales The Sachs Wolfe term for the scalar magnetic mode generated by a SB of PMF can be obtained by using the initial conditions given in the synchronous gauge in equation (4) By making a gauge transformation we obtain at leading order in the radiation era: δ r 4 B 4 B(1 R ν ) L B 4R ν ψ (46) B(55 8R ν ) 165L B 14( 4R ν ) In the radiation era because of compensation the metric potential is just proportional to B and not to B /k as obtained by Kahniashvili & Ratra (6) As we will show in the following the same holds in the matter era Now let us assume that in the matter era δt = α B (47) T SW 4 and we compute the scalar contribution to CMB anisotropies for n>/ by solving the integral: C Smagnetic l α 8π kd dkk B j l (kη ) (48) where we have inserted an upper cut-off k D in order to use the infrared expansion of ρ B (k) obtained in FPP: ρ B (k) A k n D 18π 4 k n( n B) (49) By using the result (Abramowitz & Stegun 1965) y dxxj y l1/ (x) = J l1/ (y) J l1/(y)j l/ (y) 4 Note however that a network of topological defects does not scale with radiation and interacts only gravitationally with the rest of matter ie a Lorentz term is absent we obtain for n>/andy 1: C Smagnetic α (n ) B l (4) 51π(n )ρ REL kdη The scalar magnetic contribution to CMB anisotropies on large angular scales is therefore white noise (C Smagnetic l l )forl<4 and n B > /; the slope in l of equation (4) and of the numerical results obtained with our modified version of CAMB agree very well By using an analogous procedure for n B =5/ we obtain analytically C Smagnetic l 1/l to be compared with the numerical results ( 1/l 4 ) obtained with our modified version of CAMB The α parameter in equation (4) that we can fit from our numerical results inherits a dependence on n B k D and is typically O (1 1) 5 THE VECTOR CONTRIBUTION In this section we will review the evolution of vector perturbations induced by a SB of PMF as treated in Lewis (4) The vector metric perturbation is described through a divergenceless vector: h V ij = ih j j h i (51) where we have i h i = (5) The divergenceless condition assures that vector mode does not support density perturbations The Einstein equations in the presence of PMF for the vector metric perturbations simply reduce to ( )/ ḣ V Hh V =16πGa (V) ν (V) γ (V) B k (5) Conservation equations for PMF also in the vector case reduce to a relation between isotropic and anisotropic pressure and the vector Lorentz force: i p B j (V)B ij = L B i (54) The Lorentz force induced on baryons in analogy with what we found for the scalar case modifies the baryon vector velocity equation: v b Hv b = ρ γ 4 ρ b n eaσ T (v b v γ ) LV (55) ρ γ where we have neglected the baryon homogeneous pressure (p b /ρ b 1) In order to investigate the effect of magnetized vector perturbations it is necessary to calculate the Fourier spectra for the vector projection of the PMF EMT and the Lorentz force Since these two quantities are related by L (V) i = k (V) i (56) we need only one spectrum to compute for the vector part as for the tensor part described in the next section and differently from the scalar part 6 THE TENSOR CONTRIBUTION Inflationary tensor modes namely primordial gravitational waves are a key prediction of the standard inflationary model and therefore their indirect observation through CMB anisotropies is one of the crucial point of modern cosmology However PMFs carrying anisotropic stress are themselves a source of tensor perturbations Therefore the presence of PMF affects inflationary tensor modes and creates a new independent fully magnetic tensor mode in analogy to what we found for scalar perturbations The evolution C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

6 58 D Paoletti F Finelli and F Paci equation for the metric tensor perturbation h ij is ḧ ij Hḣ ij k h ij = 16πGa ρ ν π ν ij (BT) (67) which for each polarization state deep in radiation era reads ḧ T k τ ḣt k k h T k = 6 R τ ν σ (T) ν (1 R ν ) (T) B (68) where (T) B represents the time-independent variable (T) B /ρ γ The large-scales solution to this equation can be found expanding h k in powers of (kτ) In order to keep the leading and the next-toleading terms we need to take into account the neutrino octopole (J ) truncating the propagation of anisotropic stress through higher moments by posing J 4 = Hence the neutrino anisotropic stress evolves according to σ (T) ν = 4 ḣk k J J = (T) kσ ν (69) 7 and the solution is then h k = A 1 5(kτ) ( 4R ν ) (1 R ν) (T) B (kτ) 14( 4R ν ) ij σ ν (T) = (1 R ν) (T) B 1 (kτ) R ν 14( 4R ν ) A (kτ) ( 4R ν ) (64) The presence of magnetic fields is responsible for the new leading term in σ ν (T) otherwise absent This is the so-called compensation between collisionless fluid and magnetic anisotropic stresses due to fact that magnetic fields gravitate only at perturbative level Note that the compensation between anisotropic stresses turns on only after neutrino decoupling an epoch which is much earlier than the usual initial time at which cosmological perturbations are evolved in an Einstein Boltzmann code The detailed study of the evolution of gravitational waves during neutrino decoupling is an interesting issue but clearly beyond the purpose of the present project 7 RESULTS FOR CMB ANISOTROPIES In this section we now present the temperature and polarization CMB spectra including the full contribution of SB of PMF In addition to the C l obtained by the adiabatic mode in absence of PMF we add the three contributions C SVT l described in Sections 4 6 ie scalar vector and tensor respectively computed separately by our modified version of CAMBInFigsand4weshowtheresults Figure CMB anisotropies angular power spectrum for temperature (TT hereafter top-left panel) temperature E polarization cross-correlation (TE hereafter top-right panel) E polarization (EE hereafter bottom-left panel) and B polarization (BB hereafter bottom-right panel) The solid line is the adiabatic scalar contribution in TT TE and EE panels whereas it is the tensor homogeneous contribution in the BB panel (for a tensor-to-scalar ratio r = 1); the dotted dot-dashed and dashed lines are the scalar vector and tensor contributions of a SB of PMF respectively for B =75nGn B = andk D = π Mpc 1 The dotted line in the BB panel is the lensing contribution The cosmological parameters of the flat CDM model are b h = c h = 1 z re = 1 n s = 1 H = 1h km s 1 Mpc 1 = 7 km s 1 Mpc 1 C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

7 The contribution of PMFs to CMB anisotropies 59 Figure 4 CMB angular power spectrum for TT (top-left panel) TE (top-right panel) EE (bottom-left panel) and BB (bottom-right panel) The solid line is the adiabatic scalar contribution in TT TE and EE panels whereas it is the tensor homogeneous contribution in the BB panel (for a tensor-to-scalar ratio r = 1);the dotted dot dashed and dashed lines are the scalar vector and tensor contributions of a SB of PMF respectively for B =75nGn B =5/ andk D = π Mpc 1 The dotted line in the BB panel is the lensing contribution The cosmological parameters of the flat CDM model are the same as in Fig for n B = and5/ respectively For the initial conditions of the scalar magnetic mode we use as initial conditions equation (4) with C 1 = ; for those of the tensor mode the ones in equation (64) with A = For the initial conditions for the vector mode we use the ones already implemented in CAMB described in Lewis (4) All the formulae for magnetic spectra needed B L B V T are given in our appendices; the signs of L B and B are taken as opposite as explained in FPP For all the values of n B considered here the CMB temperature pattern generated by the SB of PMF is dominated by the scalar contribution at low and intermediate multipoles; the vector contribution takes over the scalar one at high multipoles whereas the tensor one is always subleading with respect to scalar and vector It is interesting to note that the B polarization signal due to the vector contribution is always larger than the tensor one The B mode produced by vector perturbations has a power spectrum which can be steeper than the one produced by lensing with a peak around l few 1 ; therefore for suitable values of the magnetic field amplitude the B mode produced by a SB of PMF can be larger than the lensing one for any n B Fig 5 shows how the vector contribution to the B spectrum depends on n B Forn B > / the B spectra from the vector contribution are almost indistinguishable for different n B because (V) B is white noise for k k D ;for <n B / we Figure 5 Vector contributions to the CMB angular power spectrum for BB The solid line is the tensor homogeneous contribution for a tensorto-scalar ratio r = 1 and the dotted line is the lensing contribution with cosmological parameters as in the previous figures; the triple-dotted longdashed dot dashed and dashed lines are the vector spectra obtained with B =75nGk D = π Mpc 1 forn B = respectively Note how the spectra for n B =1andn B = are superimposed since the Fourier spectra of the vector part of the PMF EMT are both white noise for k k D for these spectral indices C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

8 5 D Paoletti F Finelli and F Paci note a dependence of the B spectrum on n B Analogous dependence on n B also holds for the vector contribution to TT 8 CONCLUSION We have obtained the Fourier spectra of the relevant scalar vector and tensor energy momentum components of the SB of PMF extending the method used in FPP only for the scalar As already discussed for the scalar sector in FPP we have shown how the correct evaluation of the convolution integrals leads to differences in the vector and tensor parts of the PMF Fourier spectrum previously found in Mack et al () We have then shown the comparison of the scalar vector and tensor contributions to CMB anisotropies of the new inhomogeneous modes generated by the SB of PMF by using the correct convolutions for the EMT of the PMF SB We have shown that the dominant contributions are from scalar and vector perturbations respectively for low and high l We have given an analytic description of the Sachs Wolfe contribution of the scalar mode which agrees very well with the numerical result of our modified version of CAMB (Lewis et al ) and takes into account the compensation effect on large scales which is generic when a fully inhomogeneous source is present The slope in l of the vector power spectrum we obtain numerically agrees very well with previous analytic (Mack et al ) and numerical (Lewis 4) results As already found in (Lewis 4) the B signal by vector perturbations with a peak slightly dependent on n B around l has a shape different either from the inflationary gravitational waves or from the lensing signal We have characterized its dependence on n B and shown how this is non-trivial for n B /: such signal can be constrained by Planck (Planck Collaboration 6) and future small-scale CMB polarization experiments ACKNOWLEDGMENTS We wish to thank Chiara Caprini Ruth Durrer Antony Lewis and Kandaswamy Subramanian for comments and discussions This work has been done in the framework of the Planck LFI activities and is partially supported by ASI contract Planck LFI Activity of Phase E REFERENCES Abramowitz M Stegun I 1965 Handbook of Mathematical Functions with Formulas Graphs and Mathematical Table Dover Publishing New York Brown I Crittenden R 5 Phys Rev D 7 6 Caprini C Durrer R Kahniashvili T 4 Phys Rev D Durrer R 7 New Astron Rev Durrer R Ferreira P G Kahniashvili T Phys Rev D Finelli F Paci F Paoletti D 8 Phys Rev D (FPP) Giovannini M Kunze K E 8 Phys Rev D 77 6 Kahniashvili T Ratra B 6 Phys Rev D 75 Koh S Lee C H Phys Rev D Lewis A 4 Phys Rev D Lewis A Challinor A Lasenby A ApJ Ma C P Bertschinger E 1995 ApJ Mack A Kahniashvili T Kosowsky A Phys Rev D Planck Collaboration 6 Planck: The Scientific Programme preprint (arxiv:astro-ph/6469) Seshadri T R Subramanian K 1 Phys Rev Lett Subramanian K 6 Magnetizing the Universe arxiv:884 Subramanian K Barrow J D 1998 Phys Rev Lett Yamazaki D G Ichiki K Kajino T Mathews G J 6 ApJ APPENDIX A: EMT FOURIER SPECTRA We use the convolutions for the PMF EMT spectra with the parametrization for the magnetic field power spectrum given in equation (5) Since P B (k) = fork>k D two conditions need to be taken into account: p<k D k p <k D (A1) The second condition introduces a k-dependence on the angular integration domain and the two allow the energy power spectrum to be non-zero only for <k<k D Such conditions split the double integral (over γ and over p) in three parts depending on the γ and p lower and upper limit of integration For simplicity we normalize the Fourier wavenumber to k D and we show the integrals with this convention A sketch of the integration is thus the following: (i) <k<1 k dp (ii) 1 <k< k1 dp 1 k p 1 kp dγ dp dγ k 1k p 1 kp dγ dpi c (p k) k1 k dpi a (p k) dpi b (p k) 1k Particular care must be used in the radial integrals In particular the presence of the term k p n in both integrands needs a further splitting of the integral domain for odd n: k dp with p<k (1k) k<1/ dp (1k) dp with p>k k (1k) k>1/ dp with p<k (A) C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

9 (1k) k<1/ dp k>1/ (1k) k (1k) dp with p>k dp with p<k dp with p>k k The contribution of PMFs to CMB anisotropies 51 (k1) { dp 1 <k< dp with p<k (k1) Following the scheme (A) we can now perform the integration over p Our exact results are given for particular values of n B A1 n B = ρ B (k) n B = = A k 9 D 51π 4 k 6 { 4 9 k k 1 5 k 1 4 k k 6 k k 1 k k 1 17 k 6 4 k k 6 k 9 55 for 1 k B (k) n B = = A k 9 D 56π 4 k 6 { k 1 96 k 75 k 9 9 k 4 75 k k k k k k 6 96 k 75 k 9 9 k 4 75 k k 9 75 for 1 k (T) B (k) n B = = A k 9 D 56π 4 k 6 { 56 7 k 111 k 17 k 96 k 4 14 k 6 9 k k 111 k 4 k 96 k 4 14 k 6 9 k 9 for 1 k k k 55 k A n B = ρ B (k) n B = = A kd π 4 k 4 7 k 8 k k k 7 4 B (k) n B = = A kd π 4 k 4 5 k 1 4 k k 1 7 k 5 96 k 7 19 (T) B (k) n B = = A k 7 D 56π 4 k k 6 16 k 7 k 4 1 k k 7 19 A n B = 1 { ρ B (k) n B =1 = A kd 5 4 k k 4 k 4 k π 4 k 8 4 k k 4 k 4 k 5 for 1 k k 5 4 B (k) n B =1 = A k 5 D 56π 4 k { k 1 4 k 5 8 k 4 k k k 4 k 8 75 k 4 4 k 5 8 k 4 k 5 for 1 k (T) B (k) n B =1 = A k 5 D 56π 4 k { 56 7 k 64 k k 8 k 4 4 k k 64 k k 8 k 4 4 k 5 for 1 k 75 k 5 15 k C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

10 5 D Paoletti F Finelli and F Paci A4 n B = ρ B (k) n B = = A kd 9 51π k 16 7 k 8 5 k 96 π k 4 log 1 k k log 1 k 8 k k log 1 k log k k log k k PolyLog 1 k k 4 B (k) n B = = A kd 56π 4 k log 1 k k k 1 k 5 84 k 9 k 64 5 k k 768 log 1 k k 5 log 1 k 16 k k log 1 k 8 5 k log 1 k 96 log 1 k 4 k (T) B (k) n B = = A kd 9 56π k k k k 97 k k k 6 π k 96 log 1 k log 1 k 64 k 5 1 k 5log 1 k 16 k k log 1 k 4 7 k log 1 k 19 k log 1 k log k 8 k log k 16 k PolyLog 1 k k 8 A5 n B =1 ρ B (k) n B =1 = A k D k 51π 4 B (k) n B =1 = A k D k 56π 4 (T) B (k) n B =1 = A k D k 56π 4 { 4 5 k 4 k k k 4 k k for 1 k k 4 { 8 7 k 4 16 k k 5 4 k 4 k 11 k 1 16 k 15 for 1 k { 56 5 k 8 k 15 k k 5 8 k 16 k k 6 8 k 15 k 16 for 1 k A6 n B =/ ρ B (k) n B =/ = A k 51π 4 B (k) n B =/ = A k 56π k k 1 k k π 4 k k 64 k 45 1 k 45 1 k k 9 8log1 1 k 4log k 45 1 k 88 k 9 4 arctan 1 1 k 88 k 1 k k 4 k k 64 k 45 1 k 45 1 k 4 arctan 1 k k k k k 14π k k 4 15 k k 848 k k k k 1 k k k 4 k 448 k 1 k k k 5 15 k 1 k 56 log1 1 k 8 log k k 51 k k k 4 for 1 k k 848 k k k k 1 k k k 4 k 448 k 8 arctan 1 8 arctan 1 k 1 k 1 k 1 k for 1 k C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

11 (T) B (k) n B =/ = A k 56π k k k 8π 56 k k k k k 176 k k k k 1 k k k 64 k 18 k k 51 1 k 51 1 k k k k 6 11 log1 1 k The contribution of PMFs to CMB anisotropies 5 56 log k k 56 k k k k 176 k k k k 1 k k k 64 k 18 k k 56 arctan 1 56 arctan 1 k 51 1 k 51 1 k 6 1 k for 1 k A7 n B =5/ ρ B (k) n B =5/ = A k 5 51π 4 k D B (k) n B =5/ = A k 5 56π 4 k D (T) B (k) n B =5/ = 75 1 k k k 88 k k k 4 k 5 64 k 75 1 k k k k k k k k 4 15 k 77 1 k k k k 44 k k k k 5 64 k 1 1 k A k π 4 kd k k k k k k k k k k k 8 k k k k k k k 1 APPENDIX B: SCALAR PART OF THE LORENTZ FORCE In order to compute the scalar contribution of a SB of PMFs to the cosmological perturbations the convolution for the scalar part of the Lorentz force power spectrum is also necessary The scalar anisotropic stress can be obtained directly from its relation with the Lorentz force and the magnetic energy density in equation (41) We report here the result for the Lorentz force convolution: L(k) 1 = d pp 14π 5 a 8 B (p) P B ( k p )1 μ 4γβ(γβ μ) (B1) with the magnetic field power spectrum in equation (5) for particular values of n B B1 n B = L(k) n B = = A k D 9 51π 4 k 6 44 k 556 k 4 k 164 k 4 4 k 6 11 k k k k k 556 k 75 4 k k k k 9 75 for 1 k B n B = A k 7 D L(k) n B = = 51π 4 k k 8 k k 6 k k 7 67 C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18

12 54 D Paoletti F Finelli and F Paci B n B = 1 A k 5 D L(k) n=1 = 51π 4 k { k k 15 4 k 4 k k k 8 k k k 15 4 k 4 k 5 75 for 1 k B4 n B = L(k) n B = = A k D 51π k 1 4 k 7 48 k 1 19 k 67 k 96 k k 84 log 1 k log 1 k k log 1 k 6 k 8 k 48 log 1 k 16 k 5 B5 n B =1 L(k) n B =1 = A k D k 51π 4 { 44 k 4 k k 4 k for 1 k 15 k 5 k k 15 B6 n B =/ L(k) n=/ = A k 51π k k k k π k k k k k k k 64 k 1 k k k k 1 k 88 log k k k k k 776 k k k k 1 k k k 64 k 44 arctan 1 44 arctan 1 k k k 1 k 44 log k 1 k for 1 k B7 n B =5/ L(k) n B =5/ = A k 5 51π 4 kd 15 1 k k k k k k k k k k k 88 k k k 64 k 15 1 k This paper has been typeset from a TEX/LATEX file prepared by the author C 9 The Authors Journal compilation C 9 RAS MNRAS Downloaded from on July 18