The Impact of a Stochastic The Impact of a Stochastic Background of Primordial Magnetic Background of Primordial Magnetic

Fields on Scalar Contribution to Fields on Scalar Contribution to Cosmic Microwave Background Cosmic Microwave Background

AnisotropiesAnisotropies

Daniela PaolettiDaniela Paoletti

Università degli studi di FerraraUniversità degli studi di Ferrara

INAF/IASF BolognaINAF/IASF Bologna

INFN sezione di FerraraINFN sezione di Ferrara

43 Recontres de Moriond, La Thuile 21 March 200843 Recontres de Moriond, La Thuile 21 March 2008

Collaboration with Fabio Finelli and Francesco PaciCollaboration with Fabio Finelli and Francesco Paci

We study the impact of a stochastic background of primordial magnetic We study the impact of a stochastic background of primordial magnetic fields on the scalar contribution to CMB anisotropies and on the matter fields on the scalar contribution to CMB anisotropies and on the matter

power spectrum.power spectrum.We give both the correct initial conditions for cosmological perturbations We give both the correct initial conditions for cosmological perturbations

and the exact expressions for the energy density and Lorentz force and the exact expressions for the energy density and Lorentz force associated with PMF given a power law for their spectra. associated with PMF given a power law for their spectra.

OUTLINEOUTLINE

•Stochastic background of primordial magnetic fields (SB of PMF)Stochastic background of primordial magnetic fields (SB of PMF)

•Scalar perturbations with PMF contributionScalar perturbations with PMF contribution

•Lorentz forceLorentz force

•Initial conditionsInitial conditions

•Fully magnetic modeFully magnetic mode

•Magnetic energy density and Lorentz force power spectraMagnetic energy density and Lorentz force power spectra

•ResultsResults

F. Finelli, F. Paci, D. P., arXiv:0803.1246F. Finelli, F. Paci, D. P., arXiv:0803.1246

PRIMORDIAL MAGNETIC FIELDSPRIMORDIAL MAGNETIC FIELDS

• Homogeneous PMF lives in a Bianchi Universe (Barrow, Homogeneous PMF lives in a Bianchi Universe (Barrow, Ferreira and Silk ,1996, put strong limit, B < few nGauss, Ferreira and Silk ,1996, put strong limit, B < few nGauss, on this kind of PMF)on this kind of PMF)

• We restrict our attention to a SB of PMF which live on a We restrict our attention to a SB of PMF which live on a homogeneous and isotropic universe: this can support homogeneous and isotropic universe: this can support scalar, vector (Lewis 2004) and tensor perturbations scalar, vector (Lewis 2004) and tensor perturbations (Durrer et al. 1999)(Durrer et al. 1999)

• This SB can be generated in the early universe by a lot of This SB can be generated in the early universe by a lot of mechanisms.mechanisms.

• The scalar contribution has already been studied by several The scalar contribution has already been studied by several authors through the years (Giovannini et al., Yamazaki et authors through the years (Giovannini et al., Yamazaki et al., Kahniashvili and Ratra): our work improves on initial al., Kahniashvili and Ratra): our work improves on initial conditions and PMF power spectra. conditions and PMF power spectra.

STOCHASTIC BACKGROUND OF PRIMORDIAL STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC FIELDSMAGNETIC FIELDS

Fully inhomogeneous PMF do not carry neither energy density nor pressure at the 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 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 relativistic massless and with anisotropic stress component, like neutrinos(we have considered only massless neutrinos) and radiation, their behaviour is completely considered only massless neutrinos) and radiation, their behaviour is completely

different. different. Primordial plasma high Primordial plasma high conductivity justifies the conductivity justifies the

assumptions of the infinite assumptions of the infinite conductivity limit:conductivity limit:

)(

)(),(

0

2

a

xBxB

E

Lρ

σ BB

3

8

2BB

PMF EMTPMF EMT

Conservation equations for PMF Conservation equations for PMF simply reducesimply reduce to a relation between to a relation between

PMF anisotropic stress, energy PMF anisotropic stress, energy density and the Lorentz forcedensity and the Lorentz force

The energy density is:The energy density is:

)(/)(),( 4 akk BB

And evolves as:And evolves as:

EFFECTS ON SCALAR PERTURBATIONSEFFECTS ON SCALAR PERTURBATIONS

• 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

PMF modify scalar perturbation evolution through three different PMF modify scalar perturbation evolution through three different effectseffects

GRAVITATIONAL AND ANISOTROPIC STRESS EFFECTGRAVITATIONAL AND ANISOTROPIC STRESS EFFECT

Einstein equations, that govern Einstein equations, that govern the evolution of metric the evolution of metric

perturbations, with PMF perturbations, with PMF contribution become:contribution become:

In the infinite conductivity In the infinite conductivity limit magnetic fields are limit magnetic fields are

stationarystationary

In order to implement this work on the CAMB code we worked in the In order to implement this work on the CAMB code we worked in the synchronous gauge:synchronous gauge:

)(8 PMFTG

LORENTZ FORCELORENTZ FORCE

As it is generally used in literature we used a single fluid treatment: we As it is generally used in literature we used a single fluid treatment: we considered all baryons (protons and electrons) together.considered all baryons (protons and electrons) together.

Conservation equations for Conservation equations for baryons with baryons with

electromagnetic source electromagnetic source termterm

00 J

))(2

1)()((

4

1)( 2 xBxBxBxL ijiji

Energy Energy conservatconservation is not ion is not affectedaffected

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:

Primordial plasma is globally neutralPrimordial plasma is globally neutral

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.

)(/)(),( 4 axLxL

LORENTZ FORCE IILORENTZ FORCE IIThe Lorentz force is a forcing term in baryon Euler equation.Using the single 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 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 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). is important in particular for large wave numbers).

At late times, i.e. much later At late times, i.e. much later than the decoupling time the than the decoupling time the

solution for the baryon solution for the baryon velocity isvelocity is

In the figure we show the In the figure we show the results for the time evolution results for the time evolution

of baryon velocity with of baryon velocity with PMF(dashed) and PMF(dashed) and

without(solid): note how without(solid): note how numerics and analytic agree numerics and analytic agree

very well at late time very well at late time

INITIAL CONDITIONS INITIAL CONDITIONS

We calculated the correct initial conditions (We calculated the correct initial conditions (Einstein Boltzmann codes for cosmic Einstein Boltzmann codes for cosmic microwave background radiation with primordial magnetic fieldsmicrowave background radiation with primordial magnetic fields , Daniela Paoletti, Master , Daniela Paoletti, Master Thesis, 2007,unpublished).We checked that these are in agreement with the ones recently Thesis, 2007,unpublished).We checked that these are in agreement with the ones recently

reported in Giovannini and Kunze 2008. reported in Giovannini and Kunze 2008.

The magnetic contribution The magnetic contribution drops from the metric drops from the metric

perturbations at leading perturbations at leading order .This is due to a order .This is due to a

compensation which nullifies compensation which nullifies the sum of the leading the sum of the leading

contribution in the energy contribution in the energy density in the Einstein density in the Einstein

equations and therefore in equations and therefore in metric perturbations. There are metric perturbations. There are similar compensations also similar compensations also for a network of topological for a network of topological

defects, which does not carry a defects, which does not carry a background EMT as this kind background EMT as this kind

of PMF.of PMF.

R

CC1 1 characterize characterize

the standard the standard adiabatic modeadiabatic mode

BB

FULLY MAGNETIC MODE IFULLY MAGNETIC MODE I

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. mode in metric and 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.

FULLY MAGNETIC MODE IIFULLY MAGNETIC MODE II

Fully Magnetic mode with fixed PMF Fully Magnetic mode with fixed PMF amplitude varying the spectral index amplitude varying the spectral index (n=2,1,-1,-3/2, red, orange, green and (n=2,1,-1,-3/2, red, orange, green and blue) compared with the adiabatic one blue) compared with the adiabatic one

(black curve) (black curve)

Fully magnetic mode (blue) compared Fully magnetic mode (blue) compared with the CDM and neutrinos density with the CDM and neutrinos density isocurvature (red and green curves isocurvature (red and green curves

respectively)respectively)

PMF POWER SPECTRUMPMF 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

The amplitude of the spectrum The amplitude of the spectrum is related to the PMF amplitudeis related to the PMF amplitude

Is often used in literature to smooth the Is often used in literature to smooth the PMF with a gaussian filter on a comoving PMF with a gaussian filter on a comoving scale kscale kss, in this case the relation between , in this case the relation between

the amplitude of the power spectrum and the amplitude of the power spectrum and the one of PMF is the one of PMF is

For the convergence of the integrals we need n>-3For the convergence of the integrals we need n>-3

Where kWhere k** is a reference scale is a reference scale

See our paper for the corrispondence between the two See our paper for the corrispondence between the two

MAGNETIC ENERGY DENSITY POWER MAGNETIC ENERGY DENSITY POWER SPECTRUMSPECTRUM

Magnetic energy density is quadratic in the magnetic fields therefore its Magnetic energy density is quadratic in the magnetic fields therefore its Fourier transform is a convolutionFourier transform is a convolution

where:where:

Many author (e.g. Mack et al. (2002)) said that this convolution is not 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 analytically solvable, but they did not have a mad Ph.D. student who worked

on it 12 hours a day 7 days a week . on it 12 hours a day 7 days a week .

Typically in literature is used an approximation which leads to Typically in literature is used an approximation which leads to (Kahniashvili and Ratra 2006) :(Kahniashvili and Ratra 2006) :

LORENTZ FORCE POWER SPECTRUMLORENTZ FORCE POWER SPECTRUM

In order to insert the effects of PMF we need other two objects: PMF In order to insert the effects of PMF we need other two objects: PMF anisotropic stress and the Lorentz force that are always convolution. anisotropic stress and the Lorentz force that are always convolution.

This time we are lucky because we can use the relation between energy This time we are lucky because we can use the relation between energy density anisotropic stress and Lorentz force already found and then calculate density anisotropic stress and Lorentz force already found and then calculate

only one of these convolutions. Obviously we choose the Lorentz force which only one of these convolutions. Obviously we choose the Lorentz force which is easier. is easier.

The integration technique is the same as the energy density one.The integration technique is the same as the energy density one.

WhereWhere

INTEGRATION TECHNIQUEINTEGRATION TECHNIQUE

The major problem when solving the convolution are the conditions imposed by The major problem when solving the convolution are the conditions imposed by the sharp cut off: p<kthe sharp cut off: p<kD D and |k-p|<kand |k-p|<kDD.The second ones leads to conditions on the .The second ones leads to conditions on the

angle between k and p, this splits the integration domain in three parts:angle between k and p, this splits the integration domain in three parts:

Unfortunately this is not the end of the story, the angular integral solutions Unfortunately this is not the end of the story, the angular integral solutions contain terms with |k-p|contain terms with |k-p|nn that makes necessary a further division of the radial that makes necessary a further division of the radial

integration domain:integration domain:

So in order to solve the So in order to solve the convolution you need to solve convolution you need to solve three angular integrations and three angular integrations and

seven radial integrations which seven radial integrations which is quite an hard workis quite an hard work

For this partFor this part k and p are in k and p are in

kkDD units units

EXAMPLES OF THE RESULTS FOR THE ENERGY EXAMPLES OF THE RESULTS FOR THE ENERGY DENSITY AND LORENTZ FORCE CONVOLUTION DENSITY AND LORENTZ FORCE CONVOLUTION

2Bn

1Bn

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. .

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

0Bn

2/3Bn

For all n, except for n=-3/2, the For all n, except for n=-3/2, the spectrum is white noise for k<<kspectrum is white noise for k<<kDD. .

We found a relation between We found a relation between energy density and Lorentz force energy density and Lorentz force

for k<<kfor k<<kDD: : 22 |)(|

15

11|)(| kkL BB

23 |)(| kk BComparison of our spectrum with Comparison of our spectrum with

literature onesliterature ones

n=4n=4

n=-3/2n=-3/2

)1024/( 322 BThe spectra are in The spectra are in units ofunits of

Literature Literature n=2n=2

Literature Literature n=-3/2n=-3/2

Our n=2Our n=2

Our n=-3/2Our n=-3/2

SPECTRA COMPARISON:SPECTRA COMPARISON:

Variation with the spectral index Variation with the spectral index

Energy Energy densitydensity

Lorentz Lorentz forceforce

Comparison of density and LorentzComparison of density and Lorentz

In green and blue figs:In green and blue figs:

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.vector perturbations.

We implemented all the effects mentioned earlier, the We implemented all the effects mentioned earlier, the correct initial conditions and the PMF EMT and Lorentz correct initial conditions and the PMF EMT and Lorentz

force spectra.force spectra.

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 implementation.results of this implementation.

TEMPERATURE ANGULAR POWER SPECTRUM WITH PMFTEMPERATURE ANGULAR POWER SPECTRUM WITH PMF

Solid:adiabatic modeSolid:adiabatic mode

Triple dot-dashed:fully magnetic Triple dot-dashed:fully magnetic modemode

Dotted. Fully correlated modeDotted. Fully correlated mode

Dot-dashed: fully anticorrelated modeDot-dashed: fully anticorrelated mode

Short-dashed :uncorrelated modeShort-dashed :uncorrelated mode

Long dashed :correlationLong dashed :correlation

VARIATION WITH THE VARIATION WITH THE SPECTRAL INDEXSPECTRAL INDEX : n=2 , 1, : n=2 , 1,

-1, -3/2, respectively dashed, -1, -3/2, respectively dashed, short-dashed, dot-dashed,dottedshort-dashed, dot-dashed,dotted

VARIATION WITH VARIATION WITH THE DAMPING THE DAMPING

SCALE:SCALE:

2/,,2 Dk

respectively dotted, dot-respectively dotted, dot-dashed, dasheddashed, dashed

TEMPERATURE E-MODE TEMPERATURE E-MODE CROSS CORRELATION APSCROSS CORRELATION APS

E-MODE E-MODE POLARIZATION APSPOLARIZATION APS

Solid:adiabatic modeSolid:adiabatic mode

Triple dot-dashed:fully magnetic Triple dot-dashed:fully magnetic modemode

Dotted. Fully correlated modeDotted. Fully correlated mode

Dot-dashed: fully anticorrelated modeDot-dashed: fully anticorrelated mode

Short-dashed :uncorrelated modeShort-dashed :uncorrelated mode

Long dashed :correlationLong dashed :correlation

Effect on the APS of the Effect on the APS of the Lorentz force .We compare Lorentz force .We compare

the results for the results for vanishing(dotted) and non vanishing(dotted) and non vanishing(dashed) Lorentz vanishing(dashed) Lorentz

force.Note that the decrement force.Note that the decrement of the intermediate multipoles of the intermediate multipoles

is due to the Lorentz forceis due to the Lorentz force

We show the large effect of the Lorentz We show the large effect of the Lorentz force on the evolution of baryons(dashed) force on the evolution of baryons(dashed)

and CDM(solid) density contrasts with and CDM(solid) density contrasts with time, we compare the adiabatic time, we compare the adiabatic mode(black), with the ones with mode(black), with the ones with

vanishing(blue) and non vanishing (red) vanishing(blue) and non vanishing (red) Lorentz force.Lorentz force.

The large effect is at a wavenumber which The large effect is at a wavenumber which now is fully in the non-linear regime,so it now is fully in the non-linear regime,so it

can be necessary a non linear treatment for can be necessary a non linear treatment for this part.this part.

LINEAR MATTER POWER SPECTRUMLINEAR MATTER POWER SPECTRUM

Solid:adiabatic modeSolid:adiabatic mode

Dashed: with Lorentz force and Dashed: with Lorentz force and fully correlated initial fully correlated initial

conditionsconditions

Dot-dashed: with Lorentz force Dot-dashed: with Lorentz force and uncorrelated initial and uncorrelated initial

conditionsconditions

Dotted: with vanishing Lorentz Dotted: with vanishing Lorentz force and fully correlated force and fully correlated

initial conditionsinitial conditions

CONCLUSIONSCONCLUSIONS

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

We have treated the SB of PMF in the single fluid MHD approximation: We have treated the SB of PMF in the single fluid MHD approximation: we accounted for all the effects: gravitational, due to anisotropic stress we accounted for all the effects: gravitational, due to anisotropic stress

and the effects of the Lorentz force.and the effects of the Lorentz force.

We computed the correct initial conditions for cosmological perturbations We computed the correct initial conditions for cosmological perturbations and showed the behaviour of the fully magnetic mode.and showed the behaviour of the fully magnetic mode.

We computed PMF energy density and Lorentz force power spectra We computed PMF energy density and Lorentz force power spectra exactly, given a power spectrum for the PMF cut at a damping scale.exactly, given a power spectrum for the PMF cut at a damping scale.

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 to values for rms less than microGauss (a MCMC exploration is PMF to values for rms less than microGauss (a MCMC exploration is

underway). underway).