Transcript Imprints of Relic Gravitational waves on Cosmic Microwave

CMB Polarization Generated
by Primordial Gravitational
Waves - Analytical Solutions
Alexander Polnarev
Queen Mary, University of London
MG12, Paris, 13 July 2009
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
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)
At a given spatial position the CMB is characterized by:
1) its frequency spectrum:
black body, with temperature ~ 2.728°K.
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/
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:
Choosing a x,y coordinate frame in the plane orthogonal to the line of
these
related tolet
possible
quadratic
time the
averages
Insight,
order
toare
proceed
us firstly
recap
mainof the
electromagnetic field:
characteristics of the radiation field…
I is the total intensity
of the radiation field
Q and U quantify the
direction and the magnitude
of the linear polarization.
V characterizes the degree
of circular polarization
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!
Physics of polarization creation due to the
Thompson scattering of anisotropic radiation:
The structure of the Thompson scattering is such that it requires a quadrupole
component of anisotropy to produce linear polarization!
(Animation taken from Wayne Hu homepage : http://background.uchicago.edu/~whu/)
Introducing a spherical coordinate system
, as a function of the direction
of observation on the sky, the four Stokes parameters form the components of
the polarization tensor Pab :
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:
Two other invariant quantities characterizing the linear polarization can be
which (as was mentioned before)
constructed by covariant differentiation of the symmetric trace free part of
characterizes the total intensity, and
the polarization tensor:
is a scalar under coordinate
Which is known as the E-mode of
transformations.
polarization, and is a scalar.
Which characterizes the degree of
Which
is known as
B-mode of
circular
polarization,
andthe
behaves
and
is pseudoscalar.
as apolarization,
pseudoscalar
under
coordinate
transformations
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
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 l~100.
•Gravitational waves probe the physics of inflation but will
require a thorough understanding of the foregrounds and
the secondary effects for their detection.
Thompson scattering and Equation of radiative transfer:
Symbolically the radiative transfer equation has the form of Liouville equation
in the photon phase space
Where
is a symbolic 3-vector, the components of which are
expressible through the Stokes parameters
Encodes the information on the scattering mechanism
Couples the metric through
In the cosmological context (for z<<10^6), the
covariant differentiation
dominant mechanism is the Thompson scattering!
Is the Chandrasekhar scattering matrix
Where
is the photon
4
for Thompson
scattering.
is the Thompson
momentum,
is the direction of photon
scattering cross section.
is the density of free
propagation, and
is the photon frequency.
electrons
The solution to the radiative transfer equation is sought in the form:
Where the unperturbed part corresponds to an isotropic and homogeneous
radiation field:
(Corresponds to an overall
redshift with cosmological
expansion.)
The first order equation (restricting only to the linear order) takes the form:
Thompson scattering
metric perturbations
(where
and
)
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 (Basko&Polnarev1980):
determines the is
(photon)
frequency
Here
the angle
between the angle of sight e and the wave vector n.
problem
has
toplane
solving
for
two functions
andto n).
of two
dependency
both in
anisotropy
and polarization
(the
And
isThe
theof
angle
the simplified
azimuthal
(i.e.
plane
perpendicular
variables
. (initially we had variables
, 1+3+1+2=7 variables. )
dependence
is same for both)!
determines anisotropy while determines polarization.
The equations for and
have the form of an integro-differential equation in
two variables (Polnarev1985):
where
(Thompson scattering term)
(Density of free electrons)
Anisotropy is generated by the variable g.w. field
Polarization is generated by scattering of anisotropy
and by scattering!
The usual approach to the problem of solving these equations is to
decompose and in over the variable terms of Legendre polynomials.
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!
The expected power spectra
expressed in terms of
and
of anisotropy and polarization can then be
.
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
Further introducing two functions
role in further considerations:
Optical depth
The formal solution to equations for
and
is given by:
Visibility function
Thus both (anisotropy) and
unknown function
!
(polarization) are expressible through a single
The equation for
can be arrived at by substituting the above formal
solution into the initial system for and
.
The result is a single Voltaire type integral equation in one variable:
Where
is the gravitational wave source term for polarization
and the Kernels
are given by:
Precision control
The integral equation for the function Φ allows for a very
simple precision control. A very simple algorithm lets us
control the precision of the numerical evaluation:
Take an arbitrary Φ
and calculate
analytically the
corresponding Φ0
Solve the integral
equation
numerically with
this Φ0.
Compare the
numerically
obtained
Φ with the original Φ!
OR
Find
numerical
solution Φ
for a given Φ0
Insert this Φ
back into the
integral equation
and calculate Φ0.
Compare the new
Φ0 with the original
Φ0 !
Advantages of the integral equation:
1) 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.
1) Computationally it is quicker to solve. Precision control is simpler.
2) Allows for simpler analytical manipulations, and yields a solution in the form of an
infinite series.
3) Allows for an easier understanding of physics (my subjective impression!).
The solution of the integral equation depends crucially on the
Polarization window function Q (η)=q (η) exp(-τ(η)) .
Polarization window function
Polarization window function for secondary ionization
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 l<1000 is a small number).
where
With each term expressible through a recursive relationship:
The main thing to keep in mind is that in the lowest approximation:
Anisotropy is proportional to g.w. wave amplitude at recombination.
While polarization is proportional to the amplitude of the derivative at recombination.
Where Kernels are dependent only on the recombination history:
The solution to the integral equation for various wavenumbers:
(Solid line shows the exact numerical
solution, while the dashed line shows the
zeroth order analytical approximation.)
The solution is localized around the visibility function (around the epoch of recombination).
Physically this is a consequence of the fact that:
on the one hand due to the enormous optical depth before recombination we
cannot see the polarization generated much before recombination
on the other hand polarization generation requires free electrons, which are
absent after the recombination is complete.
Integral Equation in Operator
Form
Low frequency approximation:
High frequency approximation:
Let us introduce the following elementary
integral operators
Low frequency approximation
In limiting case when k0
We have the following expansion for the operator :
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 .
Resonance
=1 or
=-1
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..