non-Gaussianity Of Local Type Research Articles

Mapping the large-angle deviation from Gaussianity in simulated CMB maps

[Abridged] In recent works we have proposed two new large-angle non-Gaussianity indicators based on skewness and kurtosis of patches of CMB sky-sphere, and used them to find out significant deviation from Gaussianity in frequency bands and foreground-reduced CMB maps. Simulated CMB maps with assigned type and amplitude of primordial non-Gaussianity are important tools to determine the strength, sensitivity and limitations of non-Gaussian estimators. Here we investigate whether and to what extent our non-Gaussian indicators have sensitivity to detect non-Gaussianity of local type, particularly with amplitude within the seven-year WMAP bounds. We make a systematic study by employing our statistical tools to generate maps of skewness and kurtosis from several thousands of simulated maps equipped with non-Gaussianity of local type of various amplitudes. We show that our indicators can be used to detect large-angle local-type non-Gaussianity only for relatively large values of the non-linear parameter $f_{\rm NL}^{\rm local}$. Thus, our indicators have not enough sensitivity to detect deviation from Gaussianity with the non-linear parameter within the seven-year WMAP bounds. This result along with the outcomes of frequency bands and foreground-reduced analyses suggest that non-Gaussianity captured in the previous works by our indicators is not of primordial origin, although it might have a primordial component. We have also made a comparative study of non-Gaussianity of simulated maps and of the full-sky WMAP foreground-reduced seven-year ILC-7yr map. An outcome of this analysis is that the level of non-Gaussianity of ILC-7yr map is higher than that of the simulated maps for $f_{\rm NL}^{\rm local}$ within WMAP bounds. This provides quantitative indications on the suitability of the ILC-7yr map as a Gaussian reconstruction of the full-sky CMB.

ANALYSIS OF PRIMORDIAL NON-GAUSSIANITIES IN CMB DATA

In less than twenty years cosmology has evolved from being a theoretical endeavor to a science where observational data is becoming so precise that can rule out, or confirm, models and theories. This spectacular advance in observational cosmology was mainly due to the COBE and WMAP satellites which mapped the cosmic microwave background (CMB) radiation. In particular, the recent high quality measurements from WMAP-7yr constitute at present the most precise data to understand physics of the very early universe. In the standard model of cosmology, inflation is considered the dominant paradigm for the generation of the primordial density perturbations regarded as the seeds for structure formation. A powerful probe to investigate the nature of these primordial seeds is performed through the analysis of Gaussian deviations in the temperature fluctuations of the CMB. In fact, present and forthcoming CMB data may be sensitive enough to detect tiny Gaussian deviations induced by small non-linear primordial perturbations at the inflation era. Here we investigate the use of wavelets transform in the analyses of small non-Gaussianities of local type in CMB data from WMAP-7yr.

On classification of models of large local-type non-Gaussianity

We classify models generating large local-type non-Gaussianity into some categories by using some ``consistency relations'' among the non-linearity parameters fNLlocal,τNLlocal and gNLlocal, which characterize the size of bispectrum for the former and trispectrum for the latter two. Then we discuss how one can discriminate models of large local-type non-Gaussianity with such relations. We first classify the models by using the ratio of τNLlocal/(6fNLlocal/5)2, which is unity for ``single-source'' models and deviates from unity for ``multi-source'' ones. We can make a further classification of models in each category by utilizing the relation between fNLlocal and gNLlocal. Our classification suggests that observations of trispectrum would be very helpful to distinguish models of large non-Gaussianity and may reveal the generation mechanism of primordial fluctuations.

LOCAL NON-GAUSSIANITY IN THE COSMIC MICROWAVE BACKGROUND THE BAYESIAN WAY

We introduce an exact Bayesian approach to search for non-Gaussianity of local type in Cosmic Microwave Background (CMB) radiation data. Using simulated CMB temperature maps, the newly developed technique is compared against the conventional frequentist bispectrum estimator. Starting from the joint probability distribution, we obtain analytic expressions for the conditional probabilities of the primordial perturbations given the data, and for the level of non-Gaussianity, f_nl, given the data and the perturbations. We propose Hamiltonian Monte Carlo sampling as a means to derive realizations of the primordial fluctuations from which we in turn sample f_nl. Although being computationally expensive, this approach allows us to exactly construct the full target posterior probability distribution. When compared to the frequentist estimator, applying the Bayesian method to Gaussian CMB maps provides consistent results. For the analysis of non-Gaussian maps, however, the error bars on f_nl do not show excess variance within the Bayesian framework. This finding is of particular relevance in the light of upcoming high precision CMB measurements obtained by the Planck satellite mission.

The primordial non-Gaussianity of local type (flocalNL) in the WMAP 5-year data: the length distribution of CMB skeleton

We present skeleton studies of non-Gaussianity in the CMB temperature anisotropy observed in the WMAP5 data. The local skeleton is traced on the 2D sphere by cubic spline interpolation which leads to more accurate estimation of the intersection positions between the skeleton and the secondary pixels than conventional linear interpolation. We demonstrate that the skeleton-based estimator of non-Gaussianity of the local type (f_NL) - the departure of the length distribution from the corresponding Gaussian expectation - yields an unbiased and sufficiently converged f_NL-likelihood. We analyse the skeleton statistics in the WMAP5 combined V- and W-band data outside the Galactic base-mask determined from the KQ75 sky-coverage. The results are consistent with Gaussian simulations of the the best-fitting cosmological model, but deviate from the previous results determined using the WMAP1 data. We show that it is unlikely that the improved skeleton tracing method, the omission of Q-band data, the modification of the foreground-template fitting method or the absence of 6 extended regions in the new mask contribute to such a deviation. However, the application of the Kp0 base-mask in data processing does improve the consistency with the WMAP1 results. The f_NL-likelihoods of the data are estimated at 9 different smoothing levels. It is unexpected that the best-fit values show positive correlation with the smoothing scales. Further investigation argues against a point-source or goodness-of-fit explanation but finds that about 30% of either Gaussian or f_NL samples having better goodness-of-fit than the WMAP5 show a similar correlation. We present the estimate f_NL=47.3+/-34.9 (1sigma error) determined from the first four smoothing angles and f_NL=76.8+/-43.1 for the combination of all nine. The former result may be overestimated at the 0.21sigma-level because of point sources.

Constraining primordial non-Gaussianity with high-redshift probes

We present an analysis of the constraints on the amplitude ofprimordial non-Gaussianity of local type described by thedimensionless parameter fNL. These constraints are set bythe auto-correlation functions (ACFs) of two large scale structureprobes, the radio sources from NRAO VLA Sky Survey (NVSS) and theQSO catalogue of Sloan Digital Sky Survey Release Six (SDSS DR6QSOs), as well as by their cross-correlation functions (CCFs) withthe cosmic microwave background (CMB) temperature map (IntegratedSachs-Wolfe effect). Several systematic effects that may affect theobservational estimates of the ACFs and of the CCFs areinvestigated and conservatively accounted for.Our approach exploits the large-scale scale-dependence of thenon-Gaussian halo bias. The derived constraints on {fNL}coming from the NVSS CCF and from the QSO ACF and CCF are weakerthan those previously obtained from the NVSS ACF, but stillconsistent with them. Finally, we obtain the constraints on fNL = 53±25 (1 σ) and fNL = 58±24 (1 σ)from NVSS data and SDSS DR6 QSO data, respectively.

Scale dependence of halo bispectrum from non-Gaussian initial conditions in cosmological N-body simulations

We study the halo bispectrum from non-Gaussian initial conditions. Based on a set of large N-body simulations starting from initial density fields with local type non-Gaussianity, we find that the halo bispectrum exhibits a strong dependence on the shape and scale of Fourier space triangles near squeezed configurations at large scales. The amplitude of the halo bispectrum roughly scales as fNL2. The resultant scaling on the triangular shape is consistent with that predicted by Jeong & Komatsu based on perturbation theory. We systematically investigate this dependence with varying redshifts and halo mass thresholds. It is shown that the fNL dependence of the halo bispectrum is stronger for more massive haloes at higher redshifts. This feature can be a useful discriminator of inflation scenarios in future deep and wide galaxy redshift surveys.

PRIMORDIAL NON-GAUSSIANITY AND THE NRAO VLA SKY SURVEY

The NRAO VLA Sky Survey (NVSS) is the only dataset that allows an accurate determination of the auto-correlation function (ACF) on angular scales of several degrees for Active Galactic Nuclei (AGNs) at typical redshifts $z \simeq 1$. Surprisingly, the ACF is found to be positive on such large scales while, in the framework of the standard hierarchical clustering scenario with Gaussian primordial perturbations it should be negative for a redshift-independent effective halo mass of order of that found for optically-selected quasars. We show that a small primordial non-Gaussianity can add sufficient power on very large scales to account for the observed NVSS ACF. The best-fit value of the parameter $f_{\rm NL}$, quantifying the amplitude of primordial non-Gaussianity of local type is $f_{\rm NL}=62 \pm 27$ ($1\,\sigma$ error bar) and $25<f_{\rm NL}<117$ ($2\,\sigma$ confidence level), corresponding to a detection of non-Gaussianity significant at the $\sim 3\,\sigma$ confidence level. The minimal halo mass of NVSS sources is found to be $M_{\rm min}=10^{12.47\pm0.26}h^{-1}M_{\odot}$ ($1\,\sigma$) strikingly close to that found for optically selected quasars. We discuss caveats and possible physical and systematic effects that can impact on the results.

Non-Gaussianity of quantum fields during inflation

In this paper, we discuss how non-Gaussianity of cosmological perturbations arises from inflation. After introducing the in–in formalism to calculate the n-point correlation function of quantum fields, we present the computation of the bispectrum of the curvature perturbation generated in general single-field inflation models. The shapes of the bispectrum are compared with the local-type non-Gaussianity that arises from nonlinear dynamics on super-horizon scales.

Use of δN formalism—difficulties in generating large local-type non-Gaussianity during inflation

We discuss the generation of non-Gaussianity in density perturbation through the super-horizon evolution during inflation by using the so-called δN formalism. We first provide a general formula for the nonlinearity parameter generated during inflation. We find that it is proportional to the slow-roll parameters, multiplied by the model-dependent factors that may enhance non-Gaussianity to the observable ranges. Then we discuss three typical examples to illustrate how difficult it is to generate sizable non-Gaussianity through the super-horizon evolution during inflation. The first example is the double inflation model, which shows that temporal violation of slow-roll conditions is not enough for the generation of non-Gaussianity. The second example is the ordinary hybrid inflation model, which illustrates the importance of taking into account perturbations on small scales. Finally, we discuss the Kadota–Stewart model. This model gives an example in which we have to choose rather unnatural initial conditions even if large non-Gaussianity can be generated.

Rotating a curvaton brane in a warped throat

In this paper we study a curvaton model obtained by considering a probeanti-D3-brane with angular motion at the bottom of a KS throatwith approximate isometries. We calculate the spectrum ofcurvature perturbations and the non-Gaussianities of this model.Specifically, we consider the limit of relativistic rotation ofthe curvaton brane which leads to a small sound speed, and thus itcan be viewed as an implementation of the DBI-curvaton mechanism. We findthat the primordial power spectrum is nearly scale-invariant whilethe non-Gaussianity of local type is sizable and that ofequilateral type is usually large and negative. Moreover, we studyboth the theoretical and observational constraints on this model,and find that there exists a sizable allowed region for the phasespace of this model.

Analytic Minkowski functionals of the cosmic microwave background: Second-order non-Gaussianity with bispectrum and trispectrum

Analytic formulas of Minkowski functionals in two-dimensional random fields are derived, including effects of second-order non-Gaussianity in the presence of both the bispectrum and trispectrum. The set of formulas provides a promising method to constrain the primordial non-Gaussianity of the universe by temperature fluctuations in the cosmic microwave background radiation. In a case of local-type non-Gaussianity, the Minkowski functionals are analytically given by powers of quadratic and cubic parameters, ${f}_{\mathrm{NL}}$ and ${g}_{\mathrm{NL}}$ (and ${\ensuremath{\tau}}_{\mathrm{NL}}$). Our formulas are not restricted to this particular model, and applicable to a wide class of non-Gaussian models. The analytic formulas are compared to numerical evaluations from non-Gaussian realizations of temperature maps, showing very good agreement.

Probing local non-Gaussianities within a Bayesian framework

Aims: We outline the Bayesian approach to inferring f_NL, the level of non-Gaussianity of local type. Phrasing f_NL inference in a Bayesian framework takes advantage of existing techniques to account for instrumental effects and foreground contamination in CMB data and takes into account uncertainties in the cosmological parameters in an unambiguous way. Methods: We derive closed form expressions for the joint posterior of f_NL and the reconstructed underlying curvature perturbation, Phi, and deduce the conditional probability densities for f_NL and Phi. Completing the inference problem amounts to finding the marginal density for f_NL. For realistic data sets the necessary integrations are intractable. We propose an exact Hamiltonian sampling algorithm to generate correlated samples from the f_NL posterior. For sufficiently high signal-to-noise ratios, we can exploit the assumption of weak non-Gaussianity to find a direct Monte Carlo technique to generate independent samples from the posterior distribution for f_NL. We illustrate our approach using a simplified toy model of CMB data for the simple case of a 1-D sky. Results: When applied to our toy problem, we find that, in the limit of high signal-to-noise, the sampling efficiency of the approximate algorithm outperforms that of Hamiltonian sampling by two orders of magnitude. When f_NL is not significantly constrained by the data, the more efficient, approximate algorithm biases the posterior density towards f_NL = 0.

PRIMORDIAL NON-GAUSSIANITY, SCALE-DEPENDENT BIAS, AND THE BISPECTRUM OF GALAXIES

The three-point correlation function of cosmological fluctuations is a sensitive probe of the physics of inflation. We calculate the bispectrum, Bg(k1, k2, k3), Fourier transform of the three-point function of density peaks (e.g., galaxies), using two different methods: the Matarrese–Lucchin–Bonometto formula and the locality of galaxy bias. The bispectrum of peaks is not only sensitive to that of the underlying matter density fluctuations, but also to the four-point function. For a physically motivated, local form of primordial non-Gaussianity in the curvature perturbation, Φ = ϕ + fNLϕ2 + gNLϕ3, where ϕ is a Gaussian field, we show that the galaxy bispectrum contains five physically distinct pieces: (1) non-linear gravitational evolution, (2) non-linear galaxy bias, (3) fNL, (4) f2NL, and (5) gNL. While (1), (2), and a part of (3) have been derived in the literature, (4) and (5) are derived in this paper for the first time. We also find that, in the high-density peak limit, (3) receives an enhancement of a factor of ∼15 relative to the previous calculation for the squeezed triangles (k1 ≈ k2 ≫ k3). Our finding suggests that the galaxy bispectrum is more sensitive to fNL than previously recognized, and is also sensitive to a new term, gNL. For a more general form of local-type non-Gaussianity, the coefficient f2NL can be interpreted as τNL, which allows us to test multi-field inflation models using the relation between the three- and four-point functions. The usual terms from Gaussian initial conditions, (1) and (2), have the smallest signals in the squeezed configurations, while the others have the largest signals; thus, we can distinguish them easily. We cannot interpret the effects of fNL on Bg(k1, k2, k3) as a scale-dependent bias, and thus replacing the linear bias in the galaxy bispectrum with the scale-dependent bias known for the power spectrum results in an incorrect prediction. As the importance of primordial non-Gaussianity relative to the non-linear gravity evolution and galaxy bias increases toward higher redshifts, galaxy surveys probing a high-redshift universe are particularly useful for probing the primordial non-Gaussianity.

Inflation with multiple sound speeds: A model of multiple DBI type actions and non-Gaussianities

In this Letter we study adiabatic and isocurvature perturbations in the frame of inflation with multiple sound speeds involved. We suggest this scenario can be realized by a number of generalized scalar fields with arbitrary kinetic forms. These scalars have their own sound speeds respectively, so the propagations of field fluctuations are individual. Specifically, we study a model constructed by two DBI type actions. We find that the critical length scale for the freezing of perturbations corresponds to the maximum sound horizon. Moreover, if the mass term of one field is much lighter than that of the other, the entropy perturbation could be quite large and so may give rise to a growth outside sound horizon. At cubic order, we find that the non-Gaussianity of local type is possibly large when entropy perturbations are able to convert into curvature perturbations. We also calculate the non-Gaussianity of equilateral type approximately.

Extracting Primordial Non-Gaussianity without Cosmic Variance

Recent work has emphasized the possibility to probe non-Gaussianity of local type by measuring the power spectrum of highly biased tracers of large scale structure on very large scales. This method is limited by the cosmic variance, by the finite number of structures on the largest scales, and by the partial degeneracy with other cosmological parameters that can mimic the same effect. We propose an alternative method based on the fact that on large scales, halos are linearly biased, but not stochastic, tracers of dark matter: by correlating a highly biased tracer of large scale structure against an unbiased tracer, one eliminates the cosmic variance error, which can lead to a significant increase in signal to noise. For an ideal survey out to z approximately 2, the error reduction can be as large as a factor of 7, which should guarantee a detection of non-Gaussianity from an all-sky survey of this type.

Crinkles in the last scattering surface: Non-Gaussianity from inhomogeneous recombination

The perturbations in the electron number density during recombination contributes to the Cosmic Microwave Background bispectrum through second order terms. Perturbations in the electron density can be a factor of ~5 larger than the baryon density fluctuations on large scales as shown in the calculations by Novosyadlyj. This raises the possibility that the contribution to bispectrum arising from perturbations in the optical depth may be non-negligible. We calculate this bispectrum and find it to peak for squeezed triangles and of peak amplitude of the order of primordial non-Gaussianity of local type with fNL of 0.05 ~ -1 depending on the l-modes being considered. This is because the shape of the bispectrum is different from the primordial one although it peaks for squeezed configurations, similar to the local type primordial non-Gaussianity.

Fast Estimator of Primordial Non‐Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background. II. Partial Sky Coverage and Inhomogeneous Noise

In our recent paper (Yadav et al. 2007) we described a fast cubic (bispectrum) estimator of the amplitude of primordial non-Gaussianity of local type, f_{NL}, from a combined analysis of the Cosmic Microwave Background (CMB) temperature and E-polarization observations. In this paper we generalize the estimator to deal with a partial sky coverage as well as inhomogeneous noise. Our generalized estimator is still computationally efficient, scaling as O(N^3/2) compared to the O(N^5/2) scaling of the brute force bispectrum calculation for sky maps with N pixels. Upcoming CMB experiments are expected to yield high-sensitivity temperature and E-polarization data. Our generalized estimator will allow us to optimally utilize the combined CMB temperature and E-polarization information from these realistic experiments, and to constrain primordial non-Gaussianity.