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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/IASFBologna. 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/IASFBologna
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
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
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
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
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
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.
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 socalled 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
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 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:
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
The major problem when solving the convolution are the conditions imposed by the sharp cut off: p<kD and kp<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 kpn 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:
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=2.5
….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
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.
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
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
Solid: regula adiabatic mode
Dotted: scalar mode
Dashes: tensor mode
DotDashes:vector mode
N=2
N=2.5
All the figures are taken from Paoletti et al. 2008
RESULTS FOR TE CROSSCORRELATION APS
Solid: regula adiabatic mode
Dotted: scalar mode
Dashes: tensor mode
DotDashes:vector mode
N=2
N=2.5
All the figures are taken from Paoletti et al. 2008
Solid: regula adiabatic mode
Dotted: scalar mode
Dashes: tensor mode
DotDashes:vector mode
N=2
N=2.5
All the figures are taken from Paoletti et al. 2008
Solid: regula adiabatic mode
Dotted: lensing
Dashes: tensor mode
DotDashes:vector mode
N=2
N=2.5
All the figures are taken from Paoletti et al. 2008
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.
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.