Poster - damtp

Weak Gravitational Lensing of The CMB
and Cosmological Parameter Constraints
Laurence Perreault Levasseur and Gilbert Holder
Departement of Physics, McGill University, 3600 University, Montréal, QC, Canada
Abstract: We compute the lensing potential traversed by the CMB photons as they journey along the line of sight from last scattering surface to us for Ωm from 0.002 to 0.502, for Ωk from -0.10 to 0.10, for wΛ from -1.1 to -0.1, and for
n from 0.95 to 1.05. The amplitude of the lensing potential turns out to be very sensitive to Ωk and Ωm, and shows little sensitivity to wΛ and n, while its shape is most influenced by this last parameter. We compute the B polarization power
spectrum generated by the lensing of the primordial E modes. Assuming Gaussian covariance in the B modes power spectrum, we obtain, through a Fisher matrix analysis, a forecast of the error on the measure of each of the four studied
parameters, for experiment covering various fractions of the sky (from 1/1000 to 1) with successive instrumental noises of 0, 10, 30 and 100 µK·(arcmin). Finally, including an estimate of effects on our results of the non-Gaussian
covariance of the B field, we correct the predicted errors.
4. Lensing of the primordial E modes into B modes
1. Introduction
3. Lensing potential calculation
Cosmic microwave background (CMB) discovery and study has and will continue
to revolutionize our understanding of cosmology by many of its aspects, one of
which is its conjectured polarization field and the subsequent lensing of such a
field. Under the working cosmological model paradigm, density perturbations
produced during the inflationary era caused the build up of acoustic oscillations in
the primordial photon-baryon fluid, which in turn induced a complete spectrum of
temperature fluctuations and became anisotropies of the CMB at last scattering
surface. Thomson scattering of temperature anisotropies gave rise to a linear
polarization field in the CMB photons, which were then gravitationally lensed along
the line of sight as these photons travelled from last scattering surface to us.
The focus of this project is this so called weak lensing phenomena of the
polarization field, and its utilization to understand the role and to constrain the
value of various cosmological parameters. The selected parameters of study are the
matter density of the Universe (Ωm), its curvature (Ωk ), the equation of state of dark
energy (wΛ), and the scalar spectral index (n). We accomplish this task through the
following main steps:
• Definition of the studied parameters;
• Computation of the lensing potential;
• Lensing of the primordial E modes into B modes;
• Gaussian and estimated non-Gaussian errors prediction.
•The deflection angle α of the CMB photons due to gravitational lensing is
determined by the integral of the comoving distance to redshift z, D(z), of each
local deflection angle, projected into an observed angle, along the line of sight
from last scattering surface to us:
2. Parameters Definition
! = "2 #
D(zrec )
D(0)
dD(z)
DA (D(zrec ) " D(z))
$ n̂ %(D(z)n̂; &rec " &z ),
DA (D(zrec ))DA (D(z))
where the subscript rec stands for recombination, DA(z) is the comoving angular
diameter distance to redshift z, n is the direction in the sky, ηrec-ηz is the
conformal time at which the photon was at position D(z)n, and ▽ n Φ represents the
angular derivative of the gravitational potential.
•The lensing potential power spectrum is define by:
! = " n̂ #.
•The definition of its power spectrum Cl ϕϕ’ is obtained by expanding this lensing
potential in spherical harmonics:
!( n̂) = " ! lmYlm ( n̂);
! lm ,! 'l ' m ' = # ll '# mm 'C
*
!! '
l
.
lm
•We employ a fiducial model with h=0.72, Ω mh2=0.142, Ω k =0, wΛ=-1 and n=1.
•Figures 1 and 2 show the resulting power spectra when allowing freedom in the
studied parameters.
where G is Newton’s constant.
• The curvature, Ωk, is defined by:
!" = w" p" ,
X( n̂) =
d2l
" (2!)
2
X(l)* X '(l ') = (2!)2 #(l $ l ')ClXX ' .
X(l)eil i n̂ ;
• The lensed polarization power spectra, C l XX’, where X is again either E or B, are defined
similarly.
• Cl BB’ lensed from a purely E modes unlensed field is given by:
C! lBB ' =
d2l '
$ (2!)
2
'
[l '" (l # l ')]2C %%
C EE ' sin 2 (2&')
l#l ' l
• By approximating the B polarization field function as Gaussian, we construct the Fisher
matrix for an hypothetical experiment. The inverse of this matrix provides an estimate of
the error in each parameter
• We approximate the effects of the non-Gaussianities of the polarization field by a factor
dividing the number of samples of each measurement by 10.
• Figure 5 shows the error forcast on each studied parameter for various experimental setups,
for the Gaussian and non-Gaussian cases.
• σnoise=10 to 30 µK·(arcmin) is the precision we hope to reach in the next generation of
experiments.
3H 02
# m a $3 (t)
8"G
H2
k = ! 20 " k ,
c
setting k=0 in the case of a flat Universe, k=1 for a closed Universe, and k=-1
for an open Universe.
• The equation of state of dark energy, wΛ, relates its density ρΛ to its pressure p Λ
as:
• Assuming only scalar modes perturbations to the primordial fluid, namely only
compressional density fluctuations in the photon-baryon fluid before recombination, only E
modes will be produced at last scattering surface. It follows that the detection of B modes
today in the CMB polarization field comes from secondary effects, among which we count
the weak lensing of the CMB photons.
• Treating the sky as flat, we decompose the polarization fields of the CMB into their Fourier
moments, which determines the unlensed power spectra Cl XX’, where X is either E or B:
5. Application to future datasets
The studied parameters are formally defined as follows:
• The mean matter density, Ω m , is the density of non-relativistic (cold) matter
(baryons and cold dark matter) today, in units of the critical density (the precise
density required for the Universe to be flat). It is related to the matter density
ρm by:
!m (t) =
• The polarization field of the CMB is parametrized by the scalar E and pseudo-scalar B,
instead of the usual Stokes parameters U and Q (assuming only linear polarization). They
are defined as in Figure 3.
k
–E modes: parallel to the wavevector k
(positive E) and orthogonal to k
(negative E)
E modes
B modes
–B modes: ±B modes make a ±45o angle
Figure 3: Representation of a Fourier mode k as a plane
with k
wave and classification of its polarization into E and B modes.
Figure 1: Lensingϕϕ’potential power
spectra for variation of a) Ωm and b) wΛ from the fiducial model. a) top and
2
bottom both show Cl for Ωm h going from 0.002 (red) to 0.502 (yellow) by steps of 0.020, with the reference
spectrum dashed. b) top and bottom both show Cl ϕϕ’ for wΛ going from -0.1 (red) to -1.1 (yellow) by steps of 0.1;
again the reference spectrum is dashed.
• The scalar spectral index, n, is defined by:
n )1
# k &
! 2" (k, z) = A(z) %
T 2 (k),
$ H 0 ('
where Δ Φ2(k, z) is the power spectrum of fluctuations in the gravitational
potential Φ in Fourier space, T(k) is the transfer function of Φ (the ratio of Φ at
late times to its initial value at recombination), and A(z) is a normalization
factor depending on the redshift z.
In the above definitions, a(t) is the scale factor and H 0 is the Hubble constant
today.
fsky
fsky
Figure 5: Comparison of the forecast of the Gaussian and estimated non-Gaussian errors for an experiment probing Ωm, Ωk,
wΛ , and n as a function of the fraction of the sky covered (f sky); for an instrumental noise of σnoise=0µK·(arcmin) (Top left)
(the theoretical precision limit that can be achieved through weak lensing), 10µK·(arcmin) (Top right), 30µK·(arcmin)
(Bottom left), and 100µK·(arcmin) (Bottom right).
Table 1: Gaussian Parameters’ Error
for fsky minimizing the error given σnoise
Figure 3:
Lensing potential power spectra for variation of a) Ωk and b) n from the fiducial model. a) top and
bottom both show Clϕϕ’ for Ωk going from -0.10 (yellow) to 0.10 (red) by steps of 0.01, with the reference
spectrum dashed b) top and bottom both show Cl ϕϕ’ for n going from 0.95 (red) to 1.05 (yellow) by steps of 0.01;
again the reference spectrum is dashed.
Table 2: Non-Gaussian Parameters’ Error
for fsky minimizing the error given σnoise