Second part of the talk

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1 Second part of the talk

2 Homogeneous vs Inhomogeneous No matter where they come from, PMFs could have left an imprint on the CMB anisotropy pattern in temperature and polarization. In case of an homogeneous PMF, our Universe would be described by the Bianchi metric. Barrow, Ferreira & Silk 1997 Inhomogenous PMFs (respecting homogeneity and isotropy globally) can source scalar, vector and tensor non-gaussian perturbations. CMB predictions in presence of a SB of PMFs has drawn a lot of interest as a non-standard cosmological model beyond the simplest ΛCDM rich of observational signatures (B-mode polarization from the vector contribution, Faraday rotation, non-gaussianities, effects due to non-zero helicity): most of these effects are larger at high multipoles, which have been characterized better and better by several experiments in the last years (ACBAR, SPT, ACT, Planck...). Affecting cosmological perturbations, PMF could leave an imprint also on the LSS.

3 Inhomogeneous Magnetic Field A SB of PMFs is modeled as a fully inhomogeneous component (i.e. does not affect the evolution of the Hubble parameter). The infinite conductivity limit is assumed (E is zero and the SB is comoving with the expansion of the universe). On considering B 2 at the same level of metric and density fluctuations, PMFs affects matter and metric perturbations in three ways: a. PMFs carry energy density and pressure and therefore gravitate at the level of perturbations. b. PMFs have anisotropic stress - differently from perfect fluids - which adds to the photon and neutrino ones (the photon anisotropic stress is negligible before the decoupling epoch). This anisotropic stress has scalar, vector and tensor component. c. PMFs induce a Lorentz force on baryons, which indirectly affects also photons during the tight coupling regime. The Lorentz force has a scalar and vector component. PMFs therefore lead to inhomogeneous solutions to scalar, vector and tensor metric and matter perturbations. These inhomogeneous solutions inherit the statistics of the source terms in B 2 which is non- Gaussian (more precisely a chi-square distribution) and this is an important point for the imprint on CMB non-gaussianities.

4 The relevant GR equations Einstein equations: G µ =8 G(T µ + T PMF µ ) Due to the infinite conductivity limit a. the induced electric field vanishes b. the magnetic field is frozen in: B(x, )= B(x) a 2 ( ) T 0PMF 0 = B = B(x) 2 8 a 4 T 0PMF i =0 T i PMF j = 1 4 a 4 B(x) 2 2 i j = i j B 3 + i j i i =0 B j (x)b i (x) As already stated, is a stiff source of the Einstein equations acting only at the level of perturbations (same treatment of a network of topological defects, a similar component which also does not respect the symmetries of isotropy and homogeneity locally). The presence of PMFs in a globally neutral plasma induces a Lorentz force on baryons: T PMF µ L(x, )= L(x) a 4 L i (x) = 1 4 apple B j (x)r j B i (x) 1 2 r i(b j (x)b j (x))

5 Metric Perturbations Inhomogeneous perturbations around homogeneous background: g µ (x, )=g (0) µ ( )+ g µ (x, ) T µ (x, )=T (0) µ ( )+ T µ (x, ) For simplicity flat spatial sections: g (0) µ ( ) =a 2 ( ) diag (, +, +, +) All except PMF have homogeneous EMT: non relativistic components (baryons, CDM), relativistic components (radiation, massless neutrinos), dark energy (only homogeneous if ). Metric perturbations have ten degrees of freedom (dofs) which can be divided in scalar, vector and tensor as for any symmetric tensor: A ij = A ij j ij 3 r2 Ā + A V ij + A T ij Trace Scalar potential for the traceless part Vector part Tensor part: traceless and transverse A V ij i A V j j A V i A Ti i i A Ti j =0 A V i Vector i A Vi =0

6 The ten metric perturbations (4 scalar, 4 vector and 2 tensor) are redundant and is convenient to keep separate the physical perturbations from the infinitesimal changes in the coordinate system (called gauge) transformations: x µ = x µ + µ (x) g µ = g µ r µ r µ The four dofs of the infinitesimal coordinate transformation corresponds to 2 scalar and 2 vector functions. The tensor part is not affected by gauge transformations and is directly gauge invariant. This split into scalar, vector and tensor components holds as well for the PMF anisotropic stress i (with the difference that is traceless). j

7 A stochastic background of PMF We assume the magnetic field satisfy in Fourier space: h i hb i (k)bj (k 0 )i = (2 )3 (k k 0 ) ( ij ˆkiˆkj )P S (k)+i ijlˆkl P A (k) 2 The power spectra are related to the symmetric part and to the helical part: hb i (k)b i (k 0 )i =(2 ) 3 (k k 0 )P S (k) h(r B) i (k)b i (k 0 )i =(2 ) 3 (k k 0 )P A (k) Above definition of helicity is gauge-invariant, similar to kinetic helicity. Other definitions of helicity are available as ha i (k)b i (k 0 )i with B = r A which is not gauge invariant unless the normal direction of the B field perpendicular to the surface decays sufficiently faster. With this latter definition, the helicity would be constant due to the large conductivity.

8 Power spectrum smoothed on a scale B 2 = B 2 = Z Z d 3 k (2 ) 3 P S(k)exp( d 3 k (2 ) 3 k P A(k) exp( (tipically chosen as 1 Mpc). 2 k 2 )= A S (2 ) 2 1 n S +3 2 k 2 )= A A (2 ) 2 1 n A +3 ns +3 2 na +4 2 Also used the mean square of B: hb 2 i = Z d 3 k (2 ) 3 P S(k) = A S 2 2 (n S + 3) The helical term must satisfies a Schwarz inequality: P A (k) applep S (k) k n S+3 D k n S = B 2 k n S+3 D (n S + 3) n S +3 ns +3 2 Example of a coupling to a pseudo-scalar field in which the helicity basis is convenient: P A / A + 2 A 2 P S / A A 2 Analysis including helicity in preparation, selected comparison with non-helical results for completeness.

9 Power-spectra of the PMF EMT We assume the magnetic field satisfy (no helicity is considered, note the change in notation): B i (k)b j (k ) = (2 ) 3 (k k )( ij ˆkiˆkj ) P B(k) 2 Power spectrum damped at small scales larger than kd by radiation viscosity: P B (k) = A k k nb = 0 only for k < kd, where k is a pivot scale and k D is a damping scale. The spectra of the energy momentum components of the PMF are 4-point functions of the magnetic field. Example of energy density: h B (k) B(k 0 )i =(2 ) 3 (k k 0 ) B (k) 2 h B (k) B(k 0 )i/ Wick theorem = Z Z Z dp dqhb i (k)b i (k p)b j (k 0 )B j (k 0 q)i Z dp dqhb i (k)b j (k 0 )ihb i (k p)b j (k 0 q)i+ hb i (k)b j (k 0 q)ihb i (k p)b j (q)i

10 Wick theorem (example for four point correlation function): hx 1 X 2 X 3 X 4 i = hx 1 X 2 ihx 3 X 4 i + hx 1 X 3 ihx 2 X 4 i + hx 1 X 3 ihx 2 X 4 i Spatial components of the EMT: h ab(k) cd (k 0 )i = Z dqdp 64 5 ab cd hb l (q)b l (k q)b m (p)b m (k 0 p)i Z dqdp 32 5 hb a(q)b b (k q)b c (p)b d (k 0 p)i Scalar, vector and tensor of the spatial anisotropic stress: h (S) (k) (S) (k 0 )i =(2 ) 3 (S) (k) 2 (k k 0 )= ab cd h ab(k) cd (k 0 )i h (V ) i (k) (V ) j (k 0 )i = (2 )3 (V ) (k) 2 P ij (k) (k k 0 )=k a P ib (k)k 0 2 cp jd (k 0 )h ab(k) cd (k 0 ) h (T ) (k 0 )i = (2 )3 (T ) (k) 2 M ijtl (k) (k k 0 ) ij (k) (T ) tl = 4 apple P ia (k)p jb (k) 1 2 P ij(k)p ab (k) apple P tc (k 0 )P ld (k 0 ) 1 2 P tl(k 0 )P cd (k 0 ) h ab(k) cd (k 0 )i where M ijtl = P it P jl + P il P jt P ij P tl

11 With these inputs: B (k) 2 = L(k) 2 = Z Z dpp B (p) P B ( k p )(1 + µ 2 ) h B (k)l B (k 0 )i = (k k0 ) d 3 pp B (p) P B ( k p )[1 + µ 2 +4 ( µ)] Z dpp B (p) P B ( k p )(1 1( )+2 µ µ 2 ) Z (V ) (k) 2 1 = dpp B (p) P B ( k p )[(1 + 2 )(1 2 )+ (µ )] (T ) (k) 2 = 1 Z dpp B (p) P B ( k p )( ), where µ = p (k p) p k p, = k p k p, = k (k p) k k p Durrer, Ferreira & Kahniashvili 2000 Mack, Kahniashvili & Kosowsky 2002 Caprini, Durrer & Kahniashvili 2004 FF, Paci & Paoletti 2008

12 Integration scheme The region of integration is defined where P(k) is non-zero as: p<k D k p <k D 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 0 < k < 2k 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. A sketch of the integration is thus the following: 1) 0 <k<k D : Z kd 0 k Z 1 Z kd Z 1 dp d + dp k 1 k D k 2 +p 2 k D 2 2kp d Z kd 0 k dpi a (p, k)+ Z kd k D k dpi b (p, k) 2) k D <k<2k D : Z kd k k D dp Z 1 k 2 +p 2 k D 2 2kp d Z kd k k D dpi c (p, k) k 2.5 FF, Paci & Paoletti c b a p Careful computation of angular integrals

13 B (k) 2 n B =2 = A2 kd k 4 " L(k) 2 n B =2 = A2 k 7 D k 4 " # k k 2 k k 7 2 k k 2 15 k 3 6 # k k k = k k D k 3 B (k) 2 < B 2 > 2 /( ) 5 4 k 3 L(k) 2 < B 2 > 2 /( ) FF, Paci & Paoletti 2008 nb=-5/2,-3/2,-1,0,1,2, from the solid to large dashed k/kd For k << kd and nb>-3/2 we obtain k/kd B (k) 2 A 2 k 2n+3 D k 2n (3 + 2n B ) For nb=-3/2 we have a logarithmic dependence and for -3 < nb < -3/2 Differences with respect to previous results in the literature in the infrared part due to the exact computation of the angular part There is non-zero correlation between density and Lorentz B (k) 2 k 2n+3 Kahniashvili & Ratra 2007

14 (V ) B (k) 2 n B =2 = A2 k 7 D k 4 " k k 2 15 k k k # (T ) B (k) 2 n B =2 = A2 k 7 D k 4 " k k k 3 24 # 13 k k Similarly we obtain for the vector and tensor part of the anisotropic stress k 3 (V ) (k) 2 B 2 2 /( ) k 3 (T ) (k) 2 B 2 2 /( ) Paoletti, FF & Paci k/kd k/kd nb=-5/2,-3/2,-1,0,1,2,3 from the solid to large dashed Differences with respect to previous results in the literature in the infrared part due to the exact computation of the angular part Mack, Kahniashvili & Kosowsky 2002 Caprini, Durrer & Kahniashvili 2004

15 nb=2 7 6 nb=-5/ k/kd solid - scalar energy density large dashed - scalar Lorentz short dashed - vector stress medium dashed - tensor stress k/kd All contributions (scalar energy density and Lorentz force, vector and tensor anisotropic stress) are comparable The various contributions differ because of the transfer functions to Cl.

16 k D = ( ) The damping scale The damping of the magnetic field is determined by the Alfven velocity and the Silk damping scale: 1 n B +5 B 2 n B +5 ng 2 nb +3 n B +5 h 1 n B +5 Mpc This allows us to fix the damping scale by other cosmological and magnetic parameters and vary only B and nb Subramanian & Barrow 1998 Jedamzik, Katalinic & Olinto 1998 Kahniashvili & Ratra 2006 Paoletti & FF, 2012 kd [Mpc -1 ] Posterior probability for kd [Mpc -1 ] obtained from a Monte Carlo Markov Chain on the six cosmological parameters, B and nb plus nuisance foreground parameters. This posterior probability is marginalized over the other parameters varied in the Monte Carlo.

17 Wavelengths relevant for CMB anisotropies are almost not affected by the quantitative details with which magnetic fields are damped by viscosity. B (k) 2 nb = Damping modeled as a sharp cut-off Damping modeled as a Gaussian smoothing k/kd Analytical integration feasible for generic spectral index only by a damping with a sharp cut-off. The methodology of computing the correlators analitically is therefore quite powerful for the investigation of probes for PMF on scales much smaller than those relevant for CMB.

18 Predictions of inhomogeneous PMFs on CMB anisotropies Vector Contribution Tensor Contribution Scalar contribution Subramanian & Barrow 1998 Seshadri & Subramanian 2001 Mack, Kahniashvili & Kosowsky 2002 Lewis 2004 Yamazaki et al Paoletti, FF & Paci Durrer, Ferreira & Kahniashvili 2000 Mack, Kahniashvili & Kosowsky 2002 Caprini, Durrer & Kahniashvili 2004 Lewis 2004 Paoletti, FF & Paci Adams et al., 1997 Koh & Lee 2002 Giovannini 2004, 2005, 2006 (2) Kahniashvili & Ratra 2006 Yamazaki et al Giovannini 2007 FF, Paci & Paoletti 2008 Paoletti, FF & Paci 2008 Giovannini & Kunze 2008 (6) Giovannini 2009 (2)...

19 Predictions Although several pioneering works were analytical, our approach is numerical. Several public codes which compute the evolution of linear fluctuations in the Einstein- Boltzmann system and compute the spectrum of CMB anisotropies and matter: Code for Anisotropies in the Microwave Background Lewis & Challinor, Doran, Robbers, Mueller the Cosmic Linear Anisotropy Solving System Lesgourgues 2011 The progenitors of the CAMB and CMBEASY, COSMICS (Ma & Bertschinger 1995) and CMBFAST (Seljak & Zaldarriaga ) are no longer supported. To predict the effects of PMF on cosmological perturbations and CMB anisotropies: Modifications of the Einstein-Boltzmann system of equations to include PMF contribution Inclusion of power spectra for the EMT components of PMF Computation of initial conditions for cosmological perturbations deep in the radiation era The results are from a modified version of the CAMB code created and mantained by Daniela Paoletti.

20 Scalar fluctuations Synchronous gauge (or longitudinal gauge, quite common choices): ds 2 = a 2 ( ) d 2 +( ij + h ij ) dx i dx j h 0µ =0 = a 2 ( ) d 2 (1 + 2 )+(1 2 ) ij dx i dx j By the decomposition of symmetric tensor, the two scalar potentials are trace) and (being 6 the potential for the traceless part): k Hḣ =4 Ga2 ( n n n + B ), k 2 =4 Ga 2 n ( n + P n ) n, ḧ +2Hḣ 2k2 = 8 Ga 2 ( n c 2 sn n n + B 3 ), ḧ +6 +2H(ḣ +6 ) 2k2 = 24 Ga 2 [ n ( n + P n ) n + B ] h (the n = 4 components (baryons, CDM, radiation, neutrinos) Scalar part of the Lorentz force: r 2 L (S) = 1 4 r 2 L (S) r i L i h (r i B j (x))r j B i (x) From the conservation of the PMF EMT: 1 i 2 r2 (B i (x)b i (x)) B = B 3 + L B

21 Baryons velocity Lorentz force acting on baryons velocity scalar potential: b = H b + k 2 c 2 sb b k 2 L(S) b Eq. for photons velocity during tight coupling prior to recombination: = k 2 + an e T ( b ) 4 = R 1 b + H b c 2 sk 2 b + k 2 L + k 2 b 4

22 Initial conditions: scalar perturbations h(k, ) = (k, ) = 1 8 B 3 4 B B 2 2 (k, ) = B + B 2 (k, ) = B + B 2 3 B 2 2 Initial conditions for the radiation era 64 b(k, ) = 3 B B 8 3 B c(k, ) = + 3 B (k, ) = 4 3L B k 2 4( 1 + R ) 8 with matter corrections + ( 165L B 55 B + 28R B ) k (15 + 4R ) 3 B B k2 2 B 9 64 B 2 2 B 4 k2 + (3L B + B R B ) k 2 2 6( 1 + R ) (3L B + B R B ) 6R k B L B k 2 2 8( 1 + R ) k 2 2 9L B ( 1 + R c ) 16( 1 + R ) 2 + ( 4 + R + 3R c ) B 16( 1 + R ) (k, ) = 3L Bk 2 4R k 2 ( 1 + R ) B 4R k2 2 B b (k, ) = 3L B k 2 4( 1 + R ) 1 4 B k 2 + k 2 2 9L B ( 1 + R c ) 16( 1 + R ) 2 + ( 4 + R + 3R c ) ] 16( 1 + R ) c (k, ) = 0 (k, ) = F 3 (k, ) = 3L B + B + B k 2 (55 28R ) L Bk 2 2 4R 56R (15 + 4R ) 56R (15 + 4R ) 3k (3L B + B ) + 165L B + 55 B 28R B 14R 7(430R + 112R ) 2 Terms proportional to PMF energy and Lorentz density is the regular inhomogeneous solution a. with FF, Paci, Paoletti 2008 Paoletti, FF, Paci 2009 Paoletti, FF 2010 i 1 + w i = b. exhibits compensation (the same for pressures, velocities) c. B, L B O(k 2 2 ), i.e. this inhomogeneous mode does not carry curvature on large scales and is regular at early times d. subsequently in the matter era this inhomogeneous mode source curvature perturbation on large scales. e. neutrino hierarchy needs to be closed to higher order wtr to the adiabatic homogeneous solution f. other initial conditions for the inhomogeneous solution can be chosen (isocurvature modes can also be studied in presence of PMFs, Giovannini & Kunze 2008) g. agrees with Shaw and Lewis 2009 B R = j 1 + w j + + B 0 B L L B = mh 0 + R c = c b + c

23 Vector fluctuations In absence of inhomogeneous sources or free streaming particles, vector fluctuations are usually neglected since corresponds to decaying modes. In presence of PMF vector modes play a relevant role. ḣ V i +2Hh V i = 16 Ga 2 h (V ) i + (V ) i + (V ) Bi i /k As for the scalar sector the anisotropic stress in related to the Lorentz force. L (V ) i = k (V ) i The vector part of the velocity field of baryons is driven by the (vector part of the) Lorentz force v bi + Hv bi = b 4 3 n ea T (v bi v i ) L V i

24 Tensor fluctuations Tensor perturbations: ḧ ij +2Hḣij + k 2 h ij = 16 Ga 2 ( ij + (B,T) ij ) The neutrino tensor anisotropic stress satisfy a differential equation of the Boltzmann hierarchy. Deep in the radiation era: Initial conditions: ḧ T k + 2 ḣt k + k 2 h T k = 6 2 [R (T ) (T ) +(1 R ) B ] (T ) = 4 15ḣk J 3 = 3 7 k (T ) J 4 =0 The absence of a constant mode on large scales is the consequence of compensation - as for the scalar mode - for the regular tensor inhomogeneous solution in presence of PMF and free streaming neutrinos. k 3 J 3 (T ) B (k )2 h k = 15(1 R ) 56(15 + 4R ) (T ) = (1 R ) R (T ) B h 1 15(k ) 2 i 14(15 + 4R )

25 CMB anisotropies Photons travelling from the surface of last scattering surface when matter and radiation decoupled. Anisotropies of the cosmic black-body radiation with T= K T T (, )=X`m a`m Y`m (, ) C` = X` a`m 2 2` +1 ha`0m 0 a`m i = C` ``0 mm 0 The anisotropy spectrum is a convolution of the 3-D Fourier spectrum Z apple C` =4 d ln k k 3 R(k) ( e,k)j`(k ) v b ( e,k)j 0`(k )+ Z o e ( )d 2 Sachs-Wolfe term Doppler term Analogous formula for vector and tensor contribution to CMB anisotropies Integrated Sachs-Wolfe term

26 Status of CMB temperature measurements Planck 2013 results XXV: CMB power spectra and likelihood 10 4 Planck WMAP9 ACT SPT Planck 2013 results XXVI: Cosmological parameters D`[µK 2 ] `

27 CMB polarization Thomson scattering generates partial linear polarization from density perturbations close to the last scattering surface Two additional observables: Stokes parameters Q, U at the map level E,B decomposition at the power spectrum level B modes generated only by vector and tensor fluctuations as primary anisotropies or by lensing of scalar fluctuations as a secondary anisotropy Only cross-correlation between T and E for parity symmetry Courtesy from A. Challinor

28 Status of CMB polarization measurements

29 PMF scalar contribution Scalar mode driven by PMF

30 PMF vector contribution Lewis 2003 Vector mode driven by neutrino hierarchy Vector mode driven by PMF

31 PMF tensor contribution Tensor inflationary mode Tensor mode driven by PMF

32 Dependence on the spectral index: scalar Black Primary CMB Blue n=2 Cyan n=1 Green n=0 Yellow n=-1.5 Orange n=-2.5 Red n=-2.9 For k << kd and nb>-3/2 the Fourier spectra of the scalar part of the PMF EMT have a white noise spectrum and the same occurs for the scalar contribution to CMB anisotropies.

33 Dependence on the spectral index: vector Black Primary CMB Blue n=2 Cyan n=1 Green n=0 Yellow n=-1.5 Orange n=-2.5 Red n=-2.9 For k << kd and nb>-3/2 the Fourier spectra of the vector part of the PMF EMT have a white noise spectrum and the same occurs for the vector contribution to CMB anisotropies.

34 Tutti insieme: TT

35 Tutti insieme: EE

36 Tutti insieme: TE

37 Tutti insieme: BB

38 Comparison of CMB generated by compensated and passive modes So far we have shown results based on initial conditions which satisfy the Einstein-Boltzmann system after neutrino decoupling and correspond to cosmological perturbations which are regular (these perturbation carry an infinitesimal curvature on long wavelength as for the so-called isocurvature modes) It has been argued that passive scalar and tensor fluctuations are generated at neutrino decoupling by the matching of pre-existing scalar and tensor fluctuations (which are characterized by modes which are singular and not regular before neutrino decoupling) which were not compensated by the free streaming neutrinos. These passive modes correspond to adiabatic scalar and tensor homogeneous solution of the Einstein-Boltzmann system with an amplitude fixed by the spectrum of the PMF R'R( B ) h ' (T ) B 3 2 B apple 5 ln + 1 B 8R apple 5 ln + 1 B 8R Shaw and Lewis 2010

39 Shaw and Lewis 2010 Interesting results which deserve further attention (including a careful matching through the neutrino decoupling process)

40 The impact of helicity Ballardini, FF, Paoletti 2013 An helical component of a SB of PMF would open a new window on the physics of early Universe or in our understanding of turbulence. hb i (k)b j (k 0 )i = (2 )3 2 h i (k k 0 ) ( ij ˆkiˆkj )P S (k)+i ijlˆkl P A (k) PA contributes to all the PMF EMT Fourier correlations shown so far (parity even correlations). As an example consider the tensor anisotropic stress (T ) (k) 2 = 2 (4 ) 5 Z h d 3 p P S (p)p S ( k p )(1 + 2 )(1 + 2 )+4P A (p)p A ( k p ) k 3 (T ) (k) 2 ns=na=-3/2, -1,0,1,2,3 from the solid to large dashed Z / k/kd dp P S (k)p S ( k p ) / Z k/kd dp P A (k)p A ( k p ) Caprini, Durrer, Kahnia Plot which shows the relative magnitude in the case of a maximally helical background: AS=AA, ns=na

41 A non-zero helical component generates also correlations which do not respect parity symmetry (parity odd). As an example consider the tensor part: A (T ) (k) = 4 (4 ) 5 Z apple d 3 p P S (p)p A ( k p )(1 + 2 ) + P A (p)p S ( k p ) (1 + 2 ) k 3 A (T ) (k) k/kd nb=-3/2,-1,0,1,2,3 from the solid to large dashed The role of exact computation of the angular part is emphasized in comparison to previous works. Caprini, Durrer, Kahniashvili 2004