Presentazione di PowerPoint

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The Impact of a Stochastic Background of
Primordial Magnetic Fields on Scalar Contribution
to Cosmic Microwave Background Anisotropies
Daniela Paoletti Università degli studi di
Ferrara INAF/IASF Bologna INFN sezione di Ferrara
Collaboration with Fabio Finelli and Francesco
Paci
43 Recontres de Moriond, La Thuile 21 March 2008
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OUTLINE
We study the impact of a stochastic background of
primordial magnetic fields on the scalar
contribution to CMB anisotropies and on the
matter power spectrum. We give both the correct
initial conditions for cosmological perturbations
and the exact expressions for the energy density
and Lorentz force associated with PMF given a
power law for their spectra.

• Stochastic background of primordial magnetic
  fields (SB of PMF)
• Scalar perturbations with PMF contribution
• Lorentz force
• Initial conditions
• Fully magnetic mode
• Magnetic energy density and Lorentz force power
  spectra
• Results

F. Finelli, F. Paci, D. P., arXiv0803.1246
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PRIMORDIAL MAGNETIC FIELDS

• Homogeneous PMF lives in a Bianchi Universe
  (Barrow, Ferreira and Silk ,1996, put strong
  limit, B lt few nGauss, on this kind of PMF)
• We restrict our attention to a SB of PMF which
  live on a homogeneous and isotropic universe
  this can support scalar, vector (Lewis 2004) and
  tensor perturbations (Durrer et al. 1999)
• This SB can be generated in the early universe by
  a lot of mechanisms.
• The scalar contribution has already been studied
  by several authors through the years (Giovannini
  et al., Yamazaki et al., Kahniashvili and Ratra)
  our work improves on initial conditions and PMF
  power spectra.

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STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC
FIELDS
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 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 reduce to a
relation between PMF anisotropic stress, energy
density and the Lorentz force
The energy density is
And evolves as
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EFFECTS ON SCALAR PERTURBATIONS
PMF modify scalar perturbation evolution 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
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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
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LORENTZ FORCE
PMF induce a Lorentz force on baryons, the
charged particles of the plasma.
As it is generally used in literature we used a
single fluid treatment we considered all baryons
(protons and electrons) together.
Primordial plasma is globally neutral
Conservation equations for baryons with
electromagnetic source term
Energy conservation is not affected
Baryon Euler equation
During the tight coupling regime the photon
velocity equation is
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LORENTZ FORCE II
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 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).
At late times, i.e. much later than the
decoupling time the solution for the baryon
velocity is
In the figure we show the results for the time
evolution of baryon velocity with PMF(dashed) and
without(solid) note how numerics and analytic
agree very well at late time
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INITIAL CONDITIONS
We calculated the correct initial conditions
(Einstein Boltzmann codes for cosmic microwave
background radiation with primordial magnetic
fields, Daniela Paoletti, Master Thesis,
2007,unpublished).We checked that these are in
agreement with the ones recently reported in
Giovannini and Kunze 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
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FULLY MAGNETIC MODE I
Note that the presence of PMF induces the
creation of a fully magnetic mode in metric and
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.
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FULLY MAGNETIC MODE II
Fully Magnetic mode with fixed PMF amplitude
varying the spectral index (n2,1,-1,-3/2, red,
orange, green and blue) compared with the
adiabatic one (black curve)
Fully magnetic mode (blue) compared with the CDM
and neutrinos density isocurvature (red and green
curves respectively)
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PMF POWER SPECTRUM
We considered a power law power spectrum PMF
Where k is a reference scale
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
See our paper for the corrispondence between the
two
For the convergence of the integrals we need ngt-3
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MAGNETIC ENERGY DENSITY POWER SPECTRUM
Magnetic energy density is quadratic in the
magnetic fields therefore its Fourier transform
is a convolution
where
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 on it 12 hours a day 7 days a week . ?
Typically in literature is used an approximation
which leads to (Kahniashvili and Ratra 2006)
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LORENTZ FORCE POWER SPECTRUM
In order to insert the effects of PMF we need
other two objects PMF anisotropic stress and the
Lorentz force that are always convolution.
Where
This time we are lucky because we can use the
relation between energy density anisotropic
stress and Lorentz force already found and then
calculate only one of these convolutions.
Obviously we choose the Lorentz force which is
easier. The integration technique is the same
as the energy density one.
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INTEGRATION TECHNIQUE
The major problem when solving the convolution
are the conditions imposed by the sharp cut off
pltkD and k-pltkD.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 terms with
k-pn that makes necessary a further division of
the radial integration domain
So in order to solve the convolution you need to
solve three angular integrations and seven radial
integrations which is quite an hard work
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EXAMPLES OF THE RESULTS FOR THE ENERGY DENSITY
AND LORENTZ FORCE CONVOLUTION
An analytical result valid for every generic
spectral index is that our spectrum goes to zero
for k2 kD.
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Things are not always so goodJust to give you an
idea
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SPECTRA COMPARISON
In green and blue figs
Comparison of our spectrum with literature ones
Variation with the spectral index
Literature n-3/2
Our n-3/2
n-3/2
n4
Literature n2
Our n2
Comparison of density and Lorentz
The spectra are in units of
Lorentz force
For all n, except for n-3/2, the spectrum is
white noise for kltltkD. We found a relation
between energy density and Lorentz force for
kltltkD
Energy density
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RESULTS
All these theoretical results have been
implemented in the Einstein Boltzmann code CAMB
(httpcosmologist.info) where originally the
effects of PMF are considered only for vector
perturbations. We implemented all the effects
mentioned earlier, the correct initial conditions
and the PMF EMT and Lorentz force spectra. In the
following I am going to show you some of the
results of this implementation.
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TEMPERATURE ANGULAR POWER SPECTRUM WITH PMF
Solidadiabatic mode Triple dot-dashedfully
magnetic mode Dotted. Fully correlated
mode Dot-dashed fully anticorrelated
mode Short-dashed uncorrelated mode Long dashed
correlation
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VARIATION WITH THE SPECTRAL INDEX n2 , 1, -1,
-3/2, respectively dashed, short-dashed,
dot-dashed,dotted
VARIATION WITH THE DAMPING SCALE
respectively dotted, dot-dashed, dashed
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TEMPERATURE E-MODE CROSS CORRELATION APS
Solidadiabatic mode Triple dot-dashedfully
magnetic mode Dotted. Fully correlated
mode Dot-dashed fully anticorrelated
mode Short-dashed uncorrelated mode Long dashed
correlation
E-MODE POLARIZATION APS
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Effect on the APS of the Lorentz force .We
compare the results for vanishing(dotted) and non
vanishing(dashed) Lorentz force.Note that the
decrement of the intermediate multipoles is due
to the Lorentz force
We show the large effect of the Lorentz force on
the evolution of baryons(dashed) and CDM(solid)
density contrasts with time, we compare the
adiabatic mode(black), with the ones with
vanishing(blue) and non vanishing (red) Lorentz
force.
The large effect is at a wavenumber which now is
fully in the non-linear regime,so it can be
necessary a non linear treatment for this part.
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LINEAR MATTER POWER SPECTRUM
Solidadiabatic mode Dashed with Lorentz force
and fully correlated initial conditions Dot-dashed
with Lorentz force and uncorrelated initial
conditions Dotted with vanishing Lorentz force
and fully correlated initial conditions
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CONCLUSIONS
We have considered the effects of a SB of PMF on
the scalar contribution to CMB anisotropies. We
have treated the SB of PMF in the single fluid
MHD approximation we accounted for all the
effects gravitational, due to anisotropic stress
and the effects of the Lorentz force. We computed
the correct initial conditions for cosmological
perturbations and showed the behaviour of the
fully magnetic mode. We computed PMF energy
density and Lorentz force power spectra exactly,
given a power spectrum for the PMF cut at a
damping scale. We showed that there are important
effects on CMB temperature and polarization APS.
Therefore present and future CMB data can
constrain PMF to values for rms less than
microGauss (a MCMC exploration is underway).