CMB Polarization Generated by Primordial Gravitational Waves Analytical Solutions

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CMB Polarization Generated by Primordial
Gravitational Waves - Analytical Solutions

• Alexander Polnarev
• Queen Mary, University of London
• MG12, Paris, 13 July 2009

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• The Beginning of Polarization Theory in
  Cosmology
• Originally proposed by Rees (1968) as an
  observational signature of an Anisotropic
  Universe Caderni et al (1978), Basko, Polnarev
  (1979), Lubin et al (1979), Nanos et al (1979)
• The polarization of the cosmic microwave
  background (CMB) was unobserved for 34 years

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• This presentation is using the results
  of the following papers
• I. The polarization of the Cosmic Microwave
  Background due to Primordial Gravitational
  Waves,
• B.G.Keating, A.G.Polnarev, N.J.Miller and
  D.Baskaran.
• International Journal of Modern Physics A,
  Vol. 21, No. 12, pp. 2459-2479
• (2006)
• II. Imprints of Relic Gravitational Waves on
  Cosmic Microwave Background Radiation,
• D.Baskaran, L.P.Grishchuk and
  A.G.Polnarev,
• Physical Review D 74, 083008
• (2006)

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• At a given spatial position the CMB is
  characterized by

1) its frequency spectrum black body, with
temperature 2.728K.
2) angular anisotropy (i.e. variations in CMB
intensity in different directions) 1 part in
105 (excluding the dipole).

• 3) polarization of CMB.
• 1 part in 106

Pictures taken from LAMBDA archive
http//lambda.gsfc.nasa.gov/
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At a given observation point, for a particular
direction of observation the radiation field can
be characterized by four Stokes parameters
conventionally labeled as I,Q,U,V
In order to proceed let us firstly recap the main
characteristics of the radiation field
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The metric perturbations directly couple only to
anisotropies, i.e. they only directly create an
unpolarized temperature anisotropy.
Polarization is created by the Thompson
scattering of this anisotropic radiation from
free electrons!
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The components of the polarization tensor are not
invariant under rotations, and transform through
each other under a coordinate transformation. For
this reason it is convenient to construct
rotationally invariant quantities out of Pab
Two of the obvious quantities are
which (as was mentioned before) characterizes the
total intensity, and is a scalar under coordinate
transformations.
Which characterizes the degree of circular
polarization, and behaves as a pseudoscalar under
coordinate transformations
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The most important thing is that the E mode is
different from the B mode
The mathematical formulae from the previous slide
allow to separate the two types
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Gravitational Waves

• Gravitational waves show  a power spectrum with
  both the E and the B mode contributions
• Gravitational waves contribution to the B-modes
  is a few tenths of a µK at l100.
• Gravitational waves probe the physics of
  inflation but will require a thorough
  understanding of the foregrounds and the
  secondary effects for their detection.

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Thompson scattering and Equation of radiative
transfer
Symbolically the radiative transfer equation has
the form of Liouville equation in the photon
phase space
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The solution to the radiative transfer equation
is sought in the form
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Due to the linear nature of the problem and in
order to simplify the equations, we can Fourier
(spatial) decompose the solution, and consider
each individual Fourier mode separately
For each individual Fourier mode, without the
loss of generality the solution can be sought in
the form (BaskoPolnarev1980)
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where
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This procedure leads to an infinite system of
coupled ordinary differential equations for each
l !
The standard numerical codes like CMBFast and
CAMB are based on solving an (appropriately cut)
version of these equations!
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A n alternative approach to the problem is to
reduce above equations to a single integral
equation.
In order to do this let us first introduce two
quantities which will play an important role in
further considerations
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Precision control
The integral equation for the function F allows
for a very simple precision control. A very
simple algorithm lets us control the precision of
the numerical evaluation
OR
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Advantages of the integral equation

• The integral equation allows to recast the
  problem in a mathematically closed form.
• It is a single equation instead of an
  infinite system of coupled differential
  equations.
• Computationally it is quicker to solve. Precision
  control is simpler.
• Allows for simpler analytical manipulations, and
  yields a solution in the form of an infinite
  series.
• Allows for an easier understanding of physics (my
  subjective impression!).

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The solution of the integral equation depends
crucially on the Polarization window function Q
(?)q (?) exp(-t(?)) .
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The integral equation can be either solved
numerically, or the solution can be presented in
the form of a series in over (which
for wavelengths of our interest llt1000 is a
small number).
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The solution to the integral equation for various
wavenumbers
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Integral Equation in Operator Form
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• Low frequency approximation

High frequency approximation
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Let us introduce the following elementary
integral operators
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Low frequency approximation
In limiting case when k?0
We have the following expansion for the
operator
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High frequency approximation
In opposite limiting case when k??
We have the following asymptotic expansion for
the operator
where C plays the role of constant of integration
over k and can be obtained from asymptotic k?? .

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Resonance 1 or -1
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Summary and Conclusions
We have conducted a semi-analytical study of
anisotropy and polarization of CMB due to
primordial gravitational waves. Mathematically
the problem has been formulated in terms of a
single Voltairre type integral equation (instead
of a infinite system of coupled differential
equations). This method allows for a simpler
numerical evaluation, as well as a clearer
understanding of the underlying physics. The
main features in the anisotropy and polarization
spectra due to primordial g.w. have been
understood and explained. With the currently
running and future planned CMB experiments there
seems to be a good chance to observe primordial
g.w.s . CMB promises to be our clearest window to
observe primordial g.w..