Michele Liguori

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Testing Primordial non-Gaussianities in CMB
Anisotropies

• Michele Liguori
• University of Cambridge, DAMTP

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Outline

• CMB non-Gaussianity primordial and
  post-inflationary 2nd order contributions.
• Tests of Gaussianity NG estimators and NG CMB
  maps
• The CMB angular bispectrum
• Predictions for the detectability of a NG signal
  in the CMB with WMAP and Planck

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CMB non-Gaussianity from inflation
Primordial non-Gaussianity is usually
parametrized as
Chi-squared non-Gaussian term
Gaussian random field

• fNL measures the expected level of
  non-Gaussianity
• It is model dependent, e.g.

Standard single-field slow roll inflation fNL
1 Multi-field inflation fNL 100

• In Fourier space

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From primordial perturbations toCMB anisotropies
The CMB temperature fluctuations are related to
the primordial potential through the radiation
transfer functions
primordial potential
Radiation transfer function
At linear level F Gaussian DT/T
Gaussian
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From primordial perturbations toCMB anisotropies
A rigorous treatment of CMB non-Gaussianity
cannot use linear perturbation theory
Non-Gaussianity in the CMB is the combination of
two effects primordial NG and second-order post
inflationary evolution of perturbations.
Pyne and Carroll (1992)
second order correction
radiative transfer
F is the primordial gravitational potential at
the end of inflation. This generally has a
primordial NG contribution
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The observed fNL is then given by
In single-field inflation fp is predicted to be
very small
Acquaviva et al. (2004) Maldacena (2004)
The dominant contribution to CMB non-Gaussianity,
in this case, is then given by fF . For this
reason we say that single field inflation gives
fNL 1
fF 1 is only a rough estimate. A full second
order treatment and second order transfer
functions are required (Bartolo et al. 2006)
In an accurate treatment fNL is no longer a
constant. There is a momentum dependence
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Tests of Gaussianity
Goal putting quantitative constraints on fNL
using CMB datasets
1. Apply the NG estimator to the measured CMB
map 2. Compute the expected value of the
estimator for several different fNL 3. Build a
c2 statistic

• Analytical predictions for the expected value
  of some estimators have been
• obtained (Komatsu and Spergel 2001 Kogo and
  Komatsu 2006
• Chiaki, Komatsu and Matsubara 2006)
• For most estimators such analytical predictions
  are impossible to get
• Also when an analytical approach is possible,
  it is in general very difficult
• to keep into account all the realistic
  experimental effects (noise, beam etc.)
• Monte Carlo simulations of NG maps are
  necessary

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Non-Gaussian CMB maps

• How to generate a non-Gaussian (NG)
  inflation-motivated CMB map ?
• 1. Generate the Gaussian part of the
  primordial potential
• 2. Square it in real space to get the non
  Gaussian part
• 3. Convolve with the radiation transfer
  functions

Problems
Non-Gaussianity has a simple form in real space
but we want to start from uncorrelated Gaussian
r.v. We need a big simulation box (side
present cosmic horizon) We need a fine sampling
to accurately resolve the thickness of the last
scattering surface
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Non-Gaussian CMB maps
From primordial potential to CMB multipoles
Transfer functions in real space
Potential Multipoles
SW effect
Late ISW effect
Acoustic oscillations
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From Gaussian potential multipoles to
non-Gaussian potential multipoles
From white noise coefficients to Gaussian
potential multipoles
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Liguori, Matarrese and Moscardini (2003)
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(No Transcript)
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Applications

• Tests of Gaussianity on WMAP data using a set of
  300 simulated NG maps.
• Wavelets
  Curvature test

fNL - 5 85 (1s c.l.)
Cabella, M.L., et al. (2004)
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Primordial Bispectrum
Standard fNL parametrization all the relevant
information is contained in the reduced
bispectrum
Angular bispectrum
Averaged bispectrum
Reduced bispectrum
The reduced bispectrum can be obtained through a
line of sight integral
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Komatsu and Spergel (2001)
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Primordial Bispectrum
After computing the Bispectrum we can perform a
Fisher analysis to estimate the expected
signal-to-noise-ratio for a given experiment
Higher angular resolution More bispectrum
modes Higher S/N

The signal-to-noise ratio roughly grows as l
Standard single-field inflation fNL 1 Ideal
experiment fNL 3
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Polarization
Using a Bispectrum-based statistics and Minkowski
functionals the WMAP team obtained the following
constraint from the WMAP 3 years dataset
The analysis so far has only included temperature
data. Polarization E-mode can be used to
increase S/N by adding new Bispectrum modes
BTTT BTTE BTEE
BEEE
Babich and Zaldarriaga (2004) estimated an
improvement of a factor 2 from polarization
measurements
NG polarization CMB maps are needed for the
analysis. A preliminary set of 100 polarization
maps at WMAP angular resolutions will be ready
soon and allow a full temperature
polarization analysis (Yadav, M.L. et al. 2006,
in preparation)
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Expexcted signal-to-noise ratio for NG for WMAP,
Planck and an ideal experiment,
including polarization in the analysis. Babich
and Zaldarriaga (2004)
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Scale dependent fNL

• Usual parametrization for primordial
  non-Gaussianity fNL is constant

• A full second order perturbative approach for
  single-field yields a
• momentum-dependent fNL

The momentum-dependent part accounts for the
growth of non-Gaussianity due to
post-inflationary non-linear evolution
Model dependent (intrinsic NG)
Model independent Post-inflationary evolution
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Scale dependent fNL
In the full second order treatment the averaged
bispectrum becomes
New l.o.s. integral
Combination of 3j and 6j symbols
Now the line of sight Integral has a different
expression
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Previous terms (KS)

5 corrections
Liguori et al. (2005)
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Scale dependent fNL

• We perform a Fisher analysis at WMAP angular
  resolution
• Three scenarios
• Standard single-field inflation
• Inhomogenous reheating
• Curvaton

• Standard single-field
  inflation
• The signal is still undetectable at WMAP angular
  resolution
• S/N grows faster than in the standard
  parametrization
• If S/N keeps growing at lmax gt 500 Planck could
  detect
• non-Gaussianity arising from standard
  single-field inflation

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Planck detection treshold as calculated in
Komatsu and Spergel (2001)
fNL required in the standard parametrization in
order to reproduce S/N obtained at a given lmax
from the momentum-dependent parametrization
Liguori et al. 2005
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Conclusions

• CMB primordial non-Gaussianity is a powerful
  tool to make consistency tests
• of inflationary models
• The predicted non-Gaussianity from inflation is
  generally small high angular
• resolution experiments (WMAP, Planck) and
  optimized statistical tools are needed
• We presented an algorithm able to produce NG
  maps at Planck angular
• resolution. NG maps are a fundamental tool to
  make tests of Gaussianity
• Only temperature NG maps have been produced so
  far. We are extending the
• present algorithms to include polarization in
  the analysis (present fNL bounds from
• WMAP can be improved using polarization)
• Using a bispectrum-based approach we
  complemented previous analyses for the
• detectability of a primordial NG signal with
  WMAP and Planck to include
• a scale-dependent fNL
• Our results show that post-inflationary
  evolution of perturbations due to non-linear
• gravitational effects has a significant
  impact on the observed level of NG

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CMB non-Gaussianity from inflation

• CMB fluctuations are obtained by linear
  filtering the
• primordial potential with the radiation
  transfer functions
• If the primordial potential is Gaussian also
  the CMB
• temperature anisotropies are Gaussian
• Different inflationary models predict different
  levels of
• non-Gaussianity in the primordial potential
• Detection of non-Gaussianity in the CMB can be
  used as a test
• to discriminate between different models of
  inflation !

Caveats i) the filtering is linear only at
first order in the
perturbations ii) foregrounds
and experimental noise introduce
NG in the data