The Scalar,Vector and Tensor Contributions
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The Scalar,Vector and Tensor Contributions 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:

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The Scalar,Vector and Tensor Contributionsof 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

2th June 2009 , Cosmological Magnetic Fields, Monte Verità, Ascona


PRIMORDIAL MAGNETIC FIELDS

If primordial magnetic fields(PMF) 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 (Durrer et al., Giovannini, Giovannini and Kunze, Lewis, Mack et al., Kahniashvili and Ratra, Kahniashvili et al., Koh and Lee, Kojima and Ichiki, Seshadri and Subramanian, Subramanian, Yamazaki et al.): 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.

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 like radiation:


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 BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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


LORENTZ FORCE BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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

Conservation equations for baryons with electromagnetic source term

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 BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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 BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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 BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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 BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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 BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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

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 :

Keep this difference in mind when you look at the amplitudes of PMF in our results

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


PMF EMT POWER SPECTRUM BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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

The scalar, vector and tensor two point correlations are:

They can be expressed through the spectra with:

Where we used:


The scalar, vector and tensor Fourier spectra are then: BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

where:

In order to consider all the effects we need to calculate also the scalar part of the Lorentz force and the scalar anisotropic stress , for the last one we can use the relation between anisotropic stress, energy density and Lorentz force that comes from the magnetic conservation equations

In the vector case for the Lorentz force we can use a simple relation with the vector anisotropic stress


INTEGRATION TECHNIQUE BACKGROUND OF PRIMORDIAL MAGNETIC FIELDS ON CMB

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 part k and p are in kD units

Unfortunately this is not the end of the story, the angular integral solutions contain term 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 two angular integrations and seven radial integrations which is quite an hard work


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


The leading terms in the infrared limit are: k^3. The spectra are in units of

Scalar,vector and tensor spectra for n=-1

We found a different relation between vector and tensor than the one reported in Mack et al (2002). There all the angular integrals are neglected and therefore vector and tensor spectra, with our conventions, are the same. Instead we found:

….vector

____scalar density

-- -- -- Lorentz

- - - - - tensor

Paoletti et al. 2008


n=-2 k^3. The spectra are in units of

n=-2.5

….vector ____scalar density -- -- -- Lorentz - - - - - tensor

From the comparison between the spectra we can see how in the Fourier space the dominant contribution comes from the tensor power spectrum while the vector one remains strongly subdominant.

This situation will be completely reverted on the CMB angular power spectra

n=2


RESULTS k^3. The spectra are in units of

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.


VECTOR MODE k^3. The spectra are in units of

Adiabtic scalar

-2.9

-2

0

2

3

Note the different behaviour for n<-1.5 and n>-1.5 due to the change in the magnetic spectrum behaviour from white noise for spectral indices greater than -1.5 to infrared domination for indices smaller than -1.5


TENSOR MODE k^3. The spectra are in units of

Inflationary tensor mode

-2.5

-1.5

1

2

Note again the different behaviour for n<-1.5 and n>-1.5 . Moreover we can note how there are very little differences for the tensor spectra for blu spectral indices


RESULTS FOR TEMPERATURE APS k^3. The spectra are in units of

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 k^3. The spectra are in units of

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 k^3. The spectra are in units of

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 k^3. The spectra are in units of

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 k^3. The spectra are in units of

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 tensor fully magnetic mode and 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 contrary to what happens in the Fourier space for the spectra of the EMT. In particular the scalar mode dominates on large scales while the vector mode is the dominant contribution on small scales.


WORK IN PROGRESS k^3. The spectra are in units of

We are using the implementation that we have done on the CAMB code of the PMF contributions to compare the theoretical prdiction with the present data by WMAP, ACBAR and CBI to give constraints on the PMF amplitude and spectral index.


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