Cosmological Parameters
and the WMAP data
Antony Lewis
CfA, Harvard / CITA, Toronto
http://cosmologist.info
• Standard assumptions – what are the parameters?
• Unexpected features, validity of assumptions?
• Low quadrupole, cut-off/running/dark energy
• Asymmetries
• ‘features’ of WMAP analysis
MCMC sampling for parameter estimation
• MCMC sample points in cosmological parameter space drawn
from the posterior distribution given the data P(parameters|data)
• Each sample gives an equally likely set of parameters given the
data. “possible universes”
• Number density of samples proportional to probability density
• Just requires a function to compute likelihood for each set of
parameters
• CosmoMC code at http://cosmologist.info/cosmomc
uses CAMB (http://camb.info) to generate Cl
• Lewis, Bridle: astro-ph/0205436
Cosmological Parameters:
combining CMB+Weak Lensing
WMAP+ACBAR+CBI+VSA with RCS + weak BBN prior
Contaldi, Hoekstra, Lewis: astro-ph/0302435
Vanilla Universe
marginalized parameter constraints
flat, massless neutrinos, cosmological constant, power law power spectrum, …
 m h 2 ~ 0.144  0.006
large compared to
WMAPext+2dF (0.134±0.006)
Good agreement with
more conservative
independent
CMB+2dF analysis
WMAP TT power spectrum at low l
compared to theoretical power law model (mean over realizations)
Pseudo-Cl data points from http://lambda.gsfc.nasa.gov/
Low quadrupole?
Foreground uncertainty:
WMAP Pseudo-Cl: C2 = 123
Tegmark cleaned map: C2 = 184
(kp2 cut, Pseudo-Cl estimator on map from astro-ph/0302496)
Likelihood modelling:
Likelihood of
theoretical
value given
observed
value
Observed
Standard models
Running ns?
WMAP+CMB+2dF, with and without l =2,3,4 multipoles
Low quadrupole and octopole drive ~1 sigma evidence for running
Need small scale data more reliable than Lyman-α
Cut-off in initial power spectrum?
Bridle, Lewis, Weller, Efstathiou: astro-ph/0302306
P(k)=0 for k<kc ~ 3 x 10-4 Mpc-1
Slightly favoured by the data
Does not give very low quadrupole
because of ISW contribution from
larger k>kc
Contributions to the quadrupole
C 2 ~  P ( k ) |  k |2
Total
ISW
Δk
Last scattering
k MPc
Changing ISW is tricky…
E.g. Dark energy with w > -1,cs2 <1 or
w<-1, cs2≥1 give less ISW than
cosmological constant
Weller, Lewis: astro-ph/0307104
Bean, Doré: astro-ph/0307100
No simple theoretical model
gives a very low quadrupole
The low value is not that unlikely
in a realisation of a standard model
P(k) on smaller scales
Bridle, Lewis, Weller, Efstathiou:
astro-ph/0302306
Asymmetry of low multipoles?
•
after Eriksen et al
astro-ph/0307507:
l <~31 shows unlikely asymmetry:
evaluate binned Cl on half sky as
a function of axis: the lowest ratio
of power on opposite two halves is
small compared to simulations.
Low power in N ecliptic
hemisphere.
•
Also astro-ph/0307282 find
quadrupole and octopole
alignment is unlikely at 1/60 level
WMAP is great, but…
•
Foreground uncertainties significant at low l
– e.g. different analyses of TE power
spectrum. Foreground uncertainties not
included in likelihoods
•
Pseudo-Cl estimators combined with
maximum likelihood error bars not strictly
correct
•
Noise not included in TT likelihood at l<100,
even though larger at l~<100 than l>~100
•
Significant correlation between TT and TE
power spectra neglected – bias on e.g. 
•
Likelihood approximation not valid for outlier
points
•
Is it valid to do parameter estimation with
usual assumptions when Cl not consistent
with Gaussian expectations? Do outliers
bias results? ...
Versions of TE
power spectrum
Conclusions
• Standard ΛCDM cosmology fits the overall shape of the WMAP
power spectrum and is consistent with other data
• Low quadrupole is not that unlikely in standard models, but favours
models predicting low values by factor <~ 10
• Outlier points/asymmetries – quite strong evidence for analysis
problems, foregrounds, or new physics
• Parameter constraints from naïve analysis may be misleading –
should really understand unexpected features first.
In two bins…
1<l<18
17<l<31
No power in northern hemisphere
3-point function?
Measured and
marginalized errors
from simulations:
Eigenmodes: