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27 th May 2009, School of Astrophysics, Bertinoro

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The Scalar, Vector and Tensor Contribution of a Stochastic Background of Primordial Magnetic Fields to CMB Anisotropies

Daniela Paoletti

University and INFN of Ferrara

INAF/IASF-Bologna

Work in collaboration with Fabio Finelli and Francesco Paci

For more details:

“The Impact of a Stochastic Background of Primordial Magnetic Fields on the Scalar Contribution to Cosmic Microwave Background Anisotropies”

Finelli, Paci, Paoletti Phys. Rev. D78 (2008) 023510

“The Full Contribution of a Stochastic Background of Magnetic Fields to CMB Anisotropies ”

Paoletti, Finelli, Paci ArXiv:0811.0230 to appear in MNRAS

27th May 2009, School of Astrophysics, Bertinoro

PRIMORDIAL MAGNETIC FIELDS

Magnetic fields are observed everywhere in the universe from galaxies to galaxy clusters. These observational evidences led to the hypothesis of the existence of primordial magnetic fields (PMF) generated in the early universe.

If such fields exist they may have left an imprint on cosmic microwave background (CMB) anisotropies in temperature and polarization.

With the present CMB data and the ones coming very soon.. is therefore possible to investigate PMF and constrain the parameters which characterize them.

There have been several studies on the effects of PMF on CMB anisotropies (Giovannini et al., Kahniashvili and Ratra for scalar contribution, Subramanian, Lewis, Durrer et al. for vector and tensor respectively): our work improves on PMF EMT Fourier power spectra and on the initial conditions.

The simplest model of PMF supported in a Robertson Walker universe is a stochastic background of primordial magnetic fields (SB of PMF).

STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS

A SB of PMF does 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.

In the infinite conductivity limit (electric fields nullify) PMF EMT becomes

And the magnetic fields and energy density simply scales as:

Conservation equations for PMF simply reduceto a relation between PMF anisotropic stress, energy density and the Lorentz force:

THE SCALAR, VECTOR AND TENSOR CONTRIBUTIONS OF A STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

PMF induce three types of perturbations: SCALAR, VECTOR and TENSOR perturbations. They act on primordial perturbations 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

SCALAR CONTRIBUTION

PMF EMT acts as a source term in the Einstein equations for metric perturbations:

PMF induce a Lorentz force on baryons, the charged particles of the plasma.

Where the conservation equations for baryons with an electromagnetic source become:

Primordial plasma is globally neutral

Energy conservation is not affected

Baryon Euler equation:

During the tight coupling regime the photon velocity equation is:

INITIAL CONDITIONS FOR SCALAR COSMOLOGICAL PERTURBATIONS

We calculated the correct initial conditions (Paoletti et al. 2008) truncating the neutrino hierarchy at F4=0 instead of F3=0 as in our previous work (Finelli et al. 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

Paoletti et al. 2008 ArXiv:0811.0230

SCALAR FULLY MAGNETIC MODE

Note that the presence of PMF induces the creation of a fully magnetic mode in metric and matter perturbations. (This mode is the leading one in radiation era for 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.

MAGNETICALLY DRIVEN VECTOR MODE

Vector perturbations are induced by vorticity in the primordial plasma. Even if a primordial vorticity is considered in RW it decays rapidly and primordial vector mode as a consequence rapidly disappears.

Vector pertubations can be sourced by a dishomogeneous SB of PMF.

Vector perturbations have vanishing energy density; vector metric perturbations are sourced by the anisotropic stress in the plasma. Carrying anisotropic stress PMF source vector perturbation.

As for the scalar mode, also in the vector case is necessary to take into account the Lorentz force induced on baryons. Therefore PMF also modify the vector part of the baryon velocity.

TENSOR CONTRIBUTION

The tensor primordial perturbations, namely primordial gravitational waves, represent one of the key predictions of the standard inflationary model.

PMF carrying anisotropic stress generate an independent mode in addition to the inflationary one.

Tensor metric pertubation are sourced by the anisotropic stress in the plasma.

The tensor initial conditions in the presence of PMF are:

Paoletti et al. 2008

PMF are responsable for the new leading term in the neutrino anisotropic stress otherwise absent. This is the so-called compensation between collisionless components and PMF which strongly modifies the effect of PMF on tensor modes

MAGNETIC FIELDS POWER SPECTRUM

We considered a power law power spectrum PMF

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

PMF EMT POWER SPECTRUM

PMF EMT is quadratic in the magnetic fields therefore its Fourier transform is a convolution

EXAMPLES OF THE RESULTS FOR THE EMT AND LORENTZ FORCE CONVOLUTION

An analytical result valid for every generic spectral index is that our spectrum goes to zero for k=2 kD.

Paoletti et al. 2008

In all the figures the spectra are multiplies for (n+3)^2 k^3. The spectra are in units of

Scalar

Vector

Solid n=-2.5

Longest dashed n=3

Solid n=-2.5

Longest dashed n=3

Tensor spectra

Scalar Lorentz force

Solid n=-2.5

Longest dashed n=3

Solid n=-2.5

Longest dashed n=3

All the figures are taken from Paoletti et al. 2008

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, anyway also this part of the code has been improved by implementing the correct EMT power spectrum.

We implemented all the effects mentioned above.

In the following I am going to show you some of the results of this implementation.

RESULTS FOR TEMPERATURE APS

Solid: regula adiabatic mode

Dotted: scalar mode

Dashes: tensor mode

Dot-Dashes:vector mode

N=2

N=-2.5

All the figures are taken from Paoletti et al. 2008

RESULTS FOR TE CROSS-CORRELATION APS

Solid: regula adiabatic mode

Dotted: scalar mode

Dashes: tensor mode

Dot-Dashes:vector mode

N=2

N=-2.5

All the figures are taken from Paoletti et al. 2008

RESULTS FOR E-MODE APS

Solid: regula adiabatic mode

Dotted: scalar mode

Dashes: tensor mode

Dot-Dashes:vector mode

N=2

N=-2.5

All the figures are taken from Paoletti et al. 2008

RESULTS FOR B-MODE APS

Solid: regula adiabatic mode

Dotted: lensing

Dashes: tensor mode

Dot-Dashes:vector mode

N=2

N=-2.5

All the figures are taken from Paoletti et al. 2008

CONCLUSIONS

We have considered the effects of a SB of PMF on the CMB anisotropies.

We have considered the magnetically induced perturbations of all kind: scalar vector and tensor .

We calculated the correct initial conditions for scalar and tensor cosmological perturbations and showed the behaviour of both the scalar and the tensor fully magnetic mode and also the vector one sourced by PMF.

We calculated the exact PMF EMT power spectra without any approximation.

The results show that the dominant contributions are the scalar and the vector while the tensor one remains subdominant. In particular the scalar mode dominates on large scales while the vector mode is the dominant contribution on small scales.

We showed that there are important effects on CMB temperature and polarization APS. Therefore present and future CMB data can constrain PMF up to values for rms of few nGauss.