2727thth May 2009, School of Astrophysics, Bertinoro May 2009, School of Astrophysics, Bertinoro

Daniela PaolettiDaniela Paoletti University and INFN of FerraraUniversity and INFN of Ferrara

INAF/IASF-BolognaINAF/IASF-Bologna

Work in collaboration with Fabio Finelli and Francesco PaciWork in collaboration with Fabio Finelli and Francesco Paci

For more details:For more details:““The Impact of a Stochastic Background of Primordial Magnetic Fields on the Scalar The Impact of a Stochastic Background of Primordial Magnetic Fields on the Scalar

Contribution to Cosmic Microwave Background Anisotropies”Contribution to Cosmic Microwave Background Anisotropies”Finelli, Paci, Paoletti Phys. Rev. D78 (2008) 023510Finelli, Paci, Paoletti Phys. Rev. D78 (2008) 023510

““The Full Contribution of a Stochastic Background of Magnetic Fields to CMB Anisotropies ”The Full Contribution of a Stochastic Background of Magnetic Fields to CMB Anisotropies ”Paoletti, Finelli, Paci Paoletti, Finelli, Paci ArXiv:0811.0230 to appear in MNRASArXiv:0811.0230 to appear in MNRAS

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

to CMB Anisotropiesto CMB Anisotropies

PRIMORDIAL MAGNETIC FIELDSPRIMORDIAL MAGNETIC FIELDS

Magnetic fields are observed everywhere in the universe from galaxies to galaxy Magnetic fields are observed everywhere in the universe from galaxies to galaxy clusters. These observational evidences led to the hypothesis of the existence of clusters. These observational evidences led to the hypothesis of the existence of

primordial magnetic fields (PMF) generated in the early universe.primordial magnetic fields (PMF) generated in the early universe.

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

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

The simplest model of PMF supported in a Robertson Walker The simplest model of PMF supported in a Robertson Walker universe is a stochastic background of primordial magnetic universe is a stochastic background of primordial magnetic

fields (SB of PMF). fields (SB of PMF).

There have been several studies on the effects of PMF on CMB There have been several studies on the effects of PMF on CMB anisotropies (Giovannini et al., Kahniashvili and Ratra for scalar anisotropies (Giovannini et al., Kahniashvili and Ratra for scalar

contribution, Subramanian, Lewis, Durrer et al. for vector and tensor contribution, Subramanian, Lewis, Durrer et al. for vector and tensor respectively): our work improves on PMF EMT Fourier power spectra and respectively): our work improves on PMF EMT Fourier power spectra and

on the initial conditions.on the initial conditions.

STOCHASTIC BACKGROUND OF PRIMORDIAL STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC FIELDSMAGNETIC FIELDS

A SB of PMF does not carry neither energy density nor pressure at the homogeneous 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 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 and with anisotropic stress component, like neutrinos(we have considered only massless neutrinos) and radiation, their behaviour is completely different. massless neutrinos) and radiation, their behaviour is completely different.

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

)(

)(),(

0

2

a

xBxB

E

Lρ

σ BB

3

Conservation equations for PMF simply reduceConservation equations for PMF simply reduce to a to a relation between PMF anisotropic stress, energy relation between PMF anisotropic stress, energy

density and the Lorentz force:density and the Lorentz force:

)(/)(),( 4 akk BB

And the magnetic fields and energy density simply scales as:And the magnetic fields and energy density simply scales as:

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

FIELDS ON CMBFIELDS ON CMB

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

different effectsdifferent effects

• PMF gravitate PMF gravitate Influence metric Influence metric perturbationsperturbations

• PMF anisotropic PMF anisotropic stressstress Adds to photon and Adds to photon and

neutrino onesneutrino ones • Lorentz force on Lorentz force on baryonsbaryons Affects baryon velocityAffects baryon velocity

Prior to the decoupling baryons and Prior to the decoupling baryons and photons are coupled by the Compton photons are coupled by the Compton

scatteringscattering

Lorentz force acts Lorentz force acts indirectly also on indirectly also on

photonsphotons

)(8 PMFTG

SCALAR CONTRIBUTIONSCALAR CONTRIBUTION

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

))(2

1)()((

4

1)( 2 xBxBxBxL ijiji

Baryon Euler equation:Baryon Euler equation:

During the tight coupling regime the photon velocity equation is: During the tight coupling regime the photon velocity equation is:

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

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

become:become:

00 J Energy Energy conservatconservation is not ion is not affectedaffected

Primordial plasma is globally neutralPrimordial plasma is globally neutral

INITIAL CONDITIONS FOR SCALAR COSMOLOGICAL PERTURBATIONSINITIAL CONDITIONS FOR SCALAR COSMOLOGICAL PERTURBATIONS

We calculated the correct initial conditions (Paoletti et al. 2008) truncating the neutrino 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). 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 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 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, 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 PMFwhich does not carry a background EMT as this kind of PMF ..

R

BB

CC1 1 characterize characterize

the standard the standard adiabatic modeadiabatic mode

Paoletti et al. 2008 APaoletti et al. 2008 ArXiv:0811.0230rXiv:0811.0230

SCALAR FULLY MAGNETIC MODESCALAR FULLY MAGNETIC MODE

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

radiation era for matter perturbations.)radiation era for matter perturbations.)

This new indipendent mode is the particular solution of the This new indipendent mode is the particular solution of the inhomogeneous Einstein equations,where the homogeneous solution is inhomogeneous Einstein equations,where the homogeneous solution is simply the standard adiabatic mode (or any other isocurvature mode).simply the standard adiabatic mode (or any other isocurvature mode).

This mode can be correlated or uncorrelated with the adiabatic one like This mode can be correlated or uncorrelated with the adiabatic one like happens for isocurvature modes, depending on the physics which has happens for isocurvature modes, depending on the physics which has

generated the PMF. However, the nature of the fully magnetic mode is generated the PMF. However, the nature of the fully magnetic mode is completely different from isocurvature perturbations and so are its effects.completely different from isocurvature perturbations and so are its effects.

The fully magnetic mode is the particular solution of the The fully magnetic mode is the particular solution of the inhomogeneous Einstein system sourced by a fully inhomogeneous inhomogeneous Einstein system sourced by a fully inhomogeneous

component, while isocurvature modes are solution of the component, while isocurvature modes are solution of the homogeneous one where all the species carry both background and homogeneous one where all the species carry both background and

perturbations.perturbations.

MAGNETICALLY DRIVEN VECTOR MODEMAGNETICALLY DRIVEN VECTOR MODE

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

mode as a consequence rapidly disappears.mode as a consequence rapidly disappears.

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

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

anisotropic stress PMF source vector perturbation. anisotropic stress PMF source vector perturbation.

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

part of the baryon velocity.part of the baryon velocity.

TENSOR CONTRIBUTIONTENSOR CONTRIBUTIONThe tensor primordial perturbations, namely primordial gravitational waves, represent The tensor primordial perturbations, namely primordial gravitational waves, represent

one of the key predictions of the standard inflationary model.one of the key predictions of the standard inflationary model.

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

The tensor initial conditions in the presence of PMF are: The tensor initial conditions in the presence of PMF are:

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

/~

BB

PMF are responsable for the new leading term in the neutrino anisotropic PMF are responsable for the new leading term in the neutrino anisotropic stress otherwise absent. This is the so-called compensation between stress otherwise absent. This is the so-called compensation between

collisionless components and PMF which strongly modifies the effect of collisionless components and PMF which strongly modifies the effect of PMF on tensor modesPMF on tensor modes

Paoletti et al. 2008Paoletti et al. 2008

MAGNETIC FIELDS POWER SPECTRUMMAGNETIC FIELDS POWER SPECTRUM

We considered a power law power spectrum PMFWe considered a power law power spectrum PMF

In order to consider the damping of PMF on small scales due to radiation viscosity 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 kwe considered a sharp cut off in the power spectrum at a scale kD.D.With this cut off With this cut off

the two point correlation function of PMF isthe two point correlation function of PMF is

PMF EMT POWER SPECTRUMPMF EMT POWER SPECTRUM

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

2)8(

2)4(

EXAMPLES OF THE RESULTS FOR THE EMT AND EXAMPLES OF THE RESULTS FOR THE EMT AND LORENTZ FORCE CONVOLUTION LORENTZ FORCE CONVOLUTION

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

2Bn 2Bn

Paoletti et al. 2008Paoletti et al. 2008

)1024/( 322 BIn all the figures the spectra are multiplies for (n+3)^2 k^3. The spectra are in units ofIn all the figures the spectra are multiplies for (n+3)^2 k^3. The spectra are in units of

VectorVectorScalarScalar

Tensor spectra Tensor spectra

Solid n=-2.5Solid n=-2.5

Longest dashed n=3Longest dashed n=3

Scalar Lorentz forceScalar Lorentz force

Solid n=-2.5Solid n=-2.5

Longest dashed n=3Longest dashed n=3

Solid n=-2.5Solid n=-2.5

Longest dashed n=3Longest dashed n=3Solid n=-2.5Solid n=-2.5

Longest dashed n=3Longest dashed n=3

All the figures are taken from Paoletti et al. 2008All the figures are taken from Paoletti et al. 2008

RESULTSRESULTS

All these theoretical results have been implemented in the All these theoretical results have been implemented in the Einstein Boltzmann code CAMB (http:Einstein Boltzmann code CAMB (http:cosmologist.infocosmologist.info) )

where originally the effects of PMF are considered only for where originally the effects of PMF are considered only for vector perturbations, anyway also this part of the code has vector perturbations, anyway also this part of the code has been improved by implementing the correct EMT power been improved by implementing the correct EMT power

spectrum.spectrum.

We implemented all the effects mentioned above.We implemented all the effects mentioned above.

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

Solid: regula adiabatic modeSolid: regula adiabatic mode

Dotted: scalar modeDotted: scalar mode

Dashes: tensor modeDashes: tensor mode

Dot-Dashes:vector modeDot-Dashes:vector mode

RESULTS FOR RESULTS FOR TEMPERATURE APSTEMPERATURE APS

N=-2.5N=-2.5

N=2N=2

nGB 5.72

All the figures are taken from Paoletti et al. 2008All the figures are taken from Paoletti et al. 2008

Solid: regula adiabatic modeSolid: regula adiabatic mode

Dotted: scalar modeDotted: scalar mode

Dashes: tensor modeDashes: tensor mode

Dot-Dashes:vector modeDot-Dashes:vector mode

RESULTS FOR TE RESULTS FOR TE CROSS-CORRELATION CROSS-CORRELATION

APSAPS

N=-2.5N=-2.5

N=2N=2

nGB 5.72

All the figures are taken from Paoletti et al. 2008All the figures are taken from Paoletti et al. 2008

Solid: regula adiabatic modeSolid: regula adiabatic mode

Dotted: scalar modeDotted: scalar mode

Dashes: tensor modeDashes: tensor mode

Dot-Dashes:vector modeDot-Dashes:vector mode

RESULTS FOR E-MODE RESULTS FOR E-MODE APSAPS

N=-2.5N=-2.5

N=2N=2

nGB 5.72

All the figures are taken from Paoletti et al. 2008All the figures are taken from Paoletti et al. 2008

Solid: regula adiabatic modeSolid: regula adiabatic mode

Dotted: lensingDotted: lensing

Dashes: tensor modeDashes: tensor mode

Dot-Dashes:vector modeDot-Dashes:vector mode

RESULTS FOR B-MODE RESULTS FOR B-MODE APSAPS

N=-2.5N=-2.5

N=2N=2

nGB 5.72

All the figures are taken from Paoletti et al. 2008All the figures are taken from Paoletti et al. 2008

CONCLUSIONSCONCLUSIONS

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

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

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

PMF.PMF.

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

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

contribution on small scales.contribution on small scales.

We showed that there are important effects on CMB temperature and We showed that there are important effects on CMB temperature and polarization APS. Therefore present and future CMB data can constrain polarization APS. Therefore present and future CMB data can constrain

PMF up to values for rms of few nGauss.PMF up to values for rms of few nGauss.