Daniela Paoletti Università degli studi di Ferrara INAF/IASF Bologna INFN sezione di Ferrara

1. The Impact of a Stochastic Background of Primordial Magnetic Fields on Scalar Contribution to Cosmic Microwave Background Anisotropies Daniela Paoletti Università degli studi di Ferrara INAF/IASF Bologna INFN sezione di Ferrara Collaboration with Fabio Finelli and Francesco Paci 43 Recontres de Moriond, La Thuile 21 March 2008

2. OUTLINE We study the impact of a stochastic background of primordial magnetic fields on the scalar contribution to CMB anisotropies and on the matter power spectrum. We give both the correct initial conditions for cosmological perturbations and the exact expressions for the energy density and Lorentz force associated with PMF given a power law for their spectra. • Stochastic background of primordial magnetic fields (SB of PMF) • Scalar perturbations with PMF contribution • Lorentz force • Initial conditions • Fully magnetic mode • Magnetic energy density and Lorentz force power spectra • Results F. Finelli, F. Paci, D. P., arXiv:0803.1246

3. PRIMORDIAL MAGNETIC FIELDS • Homogeneous PMF lives in a Bianchi Universe (Barrow, Ferreira and Silk ,1996, put strong limit, B < few nGauss, on this kind of PMF) • We restrict our attention to a SB of PMF which live on a homogeneous and isotropic universe: this can support scalar, vector (Lewis 2004) and tensor perturbations (Durrer et al. 1999) • This SB can be generated in the early universe by a lot of mechanisms. • The scalar contribution has already been studied by several authors through the years (Giovannini et al., Yamazaki et al., Kahniashvili and Ratra): our work improves on initial conditions and PMF power spectra.

4. STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS Fully inhomogeneous PMF do not carry neither energy density nor pressure at the homogeneous level. The absence of a background is the reason why even if PMF are a relativistic massless and with anisotropic stress component, like neutrinos(we have considered only massless neutrinos) and radiation, their behaviour is completely different. PMF EMT Primordial plasma high conductivity justifies the assumptions of the infinite conductivity limit: Conservation equations for PMF simply reduceto a relation between PMF anisotropic stress, energy density and the Lorentz force The energy density is: And evolves as:

5. EFFECTS ON SCALAR PERTURBATIONS PMF modify scalar perturbation evolution through three different effects • PMF gravitate Influence metric perturbations • PMF anisotropic stress Adds to photon and neutrino ones • Lorentz force on baryons Affects baryon velocity Prior to the decoupling baryons and photons are coupled by the Compton scattering Lorentz force acts indirectly also on photons

6. GRAVITATIONAL AND ANISOTROPIC STRESS EFFECT Einstein equations, that govern the evolution of metric perturbations, with PMF contribution become: In order to implement this work on the CAMB code we worked in the synchronous gauge: In the infinite conductivity limit magnetic fields are stationary

7. LORENTZ FORCE PMF induce a Lorentz force on baryons, the charged particles of the plasma. As it is generally used in literature we used a single fluid treatment: we considered all baryons (protons and electrons) together. Primordial plasma is globally neutral Conservation equations for baryons with electromagnetic source term Energy conservation is not affected Baryon Euler equation: During the tight coupling regime the photon velocity equation is:

8. LORENTZ FORCE II The Lorentz force is a forcing term in baryon Euler equation.Using the single fluid treatment leads to a Lorentz term non-vanishing at all times; although it decays as 1/a a late time effect is still present in the baryon velocity (this effect is important in particular for large wave numbers). At late times, i.e. much later than the decoupling time the solution for the baryon velocity is In the figure we show the results for the time evolution of baryon velocity with PMF(dashed) and without(solid): note how numerics and analytic agree very well at late time

9. INITIAL CONDITIONS We calculated the correct initial conditions (Einstein Boltzmann codes for cosmic microwave background radiation with primordial magnetic fields, Daniela Paoletti, Master Thesis, 2007,unpublished).We checked that these are in agreement with the ones recently reported in Giovannini and Kunze 2008. The magnetic contribution drops from the metric perturbations at leading order .This is due to a compensation which nullifies the sum of the leading contribution in the energy density in the Einstein equations and therefore in metric perturbations. There are similar compensations also for a network of topological defects, which does not carry a background EMT as this kind of PMF. C1 characterize the standard adiabatic mode

10. FULLY MAGNETIC MODE I Note that the presence of PMF induces the creation of a fully magnetic mode in metric and matter perturbations. This new indipendent mode is the particular solution of the inhomogeneous Einstein equations,where the homogeneous solution is simply the standard adiabatic mode (or any other isocurvature mode). This mode can be correlated or uncorrelated with the adiabatic one like happens for isocurvature modes, depending on the physics which has generated the PMF. However, the nature of the fully magnetic mode is completely different from isocurvature perturbations and so are its effects. The fully magnetic mode is the particular solution of the inhomogeneous Einstein system sourced by a fully inhomogeneous component, while isocurvature modes are solution of the homogeneous one where all the species carry both background and perturbations.

11. FULLY MAGNETIC MODE II Fully Magnetic mode with fixed PMF amplitude varying the spectral index (n=2,1,-1,-3/2, red, orange, green and blue) compared with the adiabatic one (black curve) Fully magnetic mode (blue) compared with the CDM and neutrinos density isocurvature (red and green curves respectively)

12. PMF POWER SPECTRUM We considered a power law power spectrum PMF Where k* is a reference scale In order to consider the damping of PMF on small scales due to radiation viscosity we considered a sharp cut off in the power spectrum at a scale kD.With this cut off the two point correlation function of PMF is The amplitude of the spectrum is related to the PMF amplitude Is often used in literature to smooth the PMF with a gaussian filter on a comoving scale ks, in this case the relation between the amplitude of the power spectrum and the one of PMF is See our paper for the corrispondence between the two For the convergence of the integrals we need n>-3

13. MAGNETIC ENERGY DENSITY POWER SPECTRUM Magnetic energy density is quadratic in the magnetic fields therefore its Fourier transform is a convolution where: Many author (e.g. Mack et al. (2002)) said that this convolution is not analytically solvable, but they did not have a mad Ph.D. student who worked on it 12 hours a day 7 days a week .  Typically in literature is used an approximation which leads to (Kahniashvili and Ratra 2006) :

14. LORENTZ FORCE POWER SPECTRUM In order to insert the effects of PMF we need other two objects: PMF anisotropic stress and the Lorentz force that are always convolution. Where This time we are lucky because we can use the relation between energy density anisotropic stress and Lorentz force already found and then calculate only one of these convolutions. Obviously we choose the Lorentz force which is easier. The integration technique is the same as the energy density one.

15. INTEGRATION TECHNIQUE The major problem when solving the convolution are the conditions imposed by the sharp cut off: p<kD and |k-p|<kD.The second ones leads to conditions on the angle between k and p, this splits the integration domain in three parts: For this partk and p are in kD units Unfortunately this is not the end of the story, the angular integral solutions contain terms with |k-p|n that makes necessary a further division of the radial integration domain: So in order to solve the convolution you need to solve three angular integrations and seven radial integrations which is quite an hard work

16. EXAMPLES OF THE RESULTS FOR THE ENERGY DENSITY AND LORENTZ FORCE CONVOLUTION An analytical result valid for every generic spectral index is that our spectrum goes to zero for k=2 kD.

17. Things are not always so good…Just to give you an idea

18. SPECTRA COMPARISON: In green and blue figs: Comparison of our spectrum with literature ones Variation with the spectral index Our n=-3/2 Literature n=-3/2 n=-3/2 n=4 Literature n=2 Our n=2 Comparison of density and Lorentz The spectra are in units of Lorentz force For all n, except for n=-3/2, the spectrum is white noise for k<<kD. We found a relation between energy density and Lorentz force for k<<kD: Energy density

19. RESULTS All these theoretical results have been implemented in the Einstein Boltzmann code CAMB (http:cosmologist.info) where originally the effects of PMF are considered only for vector perturbations. We implemented all the effects mentioned earlier, the correct initial conditions and the PMF EMT and Lorentz force spectra. In the following I am going to show you some of the results of this implementation.

20. TEMPERATURE ANGULAR POWER SPECTRUM WITH PMF Solid:adiabatic mode Triple dot-dashed:fully magnetic mode Dotted. Fully correlated mode Dot-dashed: fully anticorrelated mode Short-dashed :uncorrelated mode Long dashed :correlation

21. VARIATION WITH THE SPECTRAL INDEX : n=2 , 1, -1, -3/2, respectively dashed, short-dashed, dot-dashed,dotted VARIATION WITH THE DAMPING SCALE: respectively dotted, dot-dashed, dashed

22. TEMPERATURE E-MODE CROSS CORRELATION APS Solid:adiabatic mode Triple dot-dashed:fully magnetic mode Dotted. Fully correlated mode Dot-dashed: fully anticorrelated mode Short-dashed :uncorrelated mode Long dashed :correlation E-MODE POLARIZATION APS

23. Effect on the APS of the Lorentz force .We compare the results for vanishing(dotted) and non vanishing(dashed) Lorentz force.Note that the decrement of the intermediate multipoles is due to the Lorentz force We show the large effect of the Lorentz force on the evolution of baryons(dashed) and CDM(solid) density contrasts with time, we compare the adiabatic mode(black), with the ones with vanishing(blue) and non vanishing (red) Lorentz force. The large effect is at a wavenumber which now is fully in the non-linear regime,so it can be necessary a non linear treatment for this part.

24. LINEAR MATTER POWER SPECTRUM Solid:adiabatic mode Dashed: with Lorentz force and fully correlated initial conditions Dot-dashed: with Lorentz force and uncorrelated initial conditions Dotted: with vanishing Lorentz force and fully correlated initial conditions

25. CONCLUSIONS We have considered the effects of a SB of PMF on the scalar contribution to CMB anisotropies. We have treated the SB of PMF in the single fluid MHD approximation: we accounted for all the effects: gravitational, due to anisotropic stress and the effects of the Lorentz force. We computed the correct initial conditions for cosmological perturbations and showed the behaviour of the fully magnetic mode. We computed PMF energy density and Lorentz force power spectra exactly, given a power spectrum for the PMF cut at a damping scale. We showed that there are important effects on CMB temperature and polarization APS. Therefore present and future CMB data can constrain PMF to values for rms less than microGauss (a MCMC exploration is underway).