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

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

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:

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

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:

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

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

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

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

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

Liguori, Matarrese and Moscardini (2003)

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)

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

Komatsu and Spergel (2001)

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

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)

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

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

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

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

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

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

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

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