1. Testing Primordial non-Gaussianities in CMB Anisotropies
Michele Liguori
University of Cambridge, DAMTP

2. 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

3. 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:

4. From primordial perturbations to CMB 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

5. From primordial perturbations to CMB 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)
radiative
transfer
second order
correction
F is the primordial gravitational potential at the end of inflation. This generally
has a primordial NG contribution:

6. 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

7. 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

8. Non-Gaussian CMB maps Problems
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

9. Non-Gaussian CMB maps From primordial potential to CMB multipoles
Transfer functions
in real space
“Potential
Multipoles”
SW effect
Late ISW
effect
Acoustic
oscillations

10. From Gaussian potential multipoles to non-Gaussian
From white noise coefficients to Gaussian potential multipoles

11. Liguori, Matarrese and Moscardini (2003)

13. Applications fNL = - 5 ± 85 (1s c.l.)
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)

14. 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

15. Komatsu and Spergel (2001)

16. 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
Experiment
lmax
fNL
COBE 30
600
WMAP
500 20
Planck
3000 5
Standard single-field inflation: fNL ~ 1
Ideal experiment : fNL = 3

17. Polarization BTTT + BTTE + BTEE + BEEE
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)

18. Expexcted signal-to-noise ratio
for NG for WMAP, Planck and
an ideal experiment, including
polarization in the analysis.
Babich and Zaldarriaga (2004)

19. 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

20. Scale dependent fNL
In the full second order treatment the averaged bispectrum
becomes:
Combination
of 3j and 6j symbols
New l.o.s.
integral
Now the line of sight
Integral has a different
expression

21. Previous terms (KS) +
5 corrections
Liguori et al. (2005)

22. 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 > 500 Planck could detect
non-Gaussianity arising from standard single-field inflation

23. Liguori et al. 2005 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

24. 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

25. 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