We develop a general framework for data analysis and phenomenology of the CMB four-point function or trispectrum. To lowest order in the derivative expansion, the inflationary action admits three quartic operators consistent with symmetry: $\dotσ^4$, $\dotσ^2 (\partialσ^2)$, and $(\partialσ)^4$. In single field inflation, only the first of these operators can be the leading non-Gaussian signal. A Fisher matrix analysis shows that there is one near-degeneracy among the three CMB trispectra, so we parameterize the trispectrum with two coefficients $g_{NL}^{\dotσ^4}$ and $g_{NL}^{(\partialσ)^4}$, in addition to the coefficient $g_{NL}^{\rm loc}$ of $ζ^3$-type local non-Gaussianity. This three-parameter space is analogous to the parameter space $(f_{NL}^{\rm loc}, f_{NL}^{\rm equil}, f_{NL}^{\rm orth})$ commonly used to parameterize the CMB three-point function. We next turn to data analysis and show how to represent these trispectra in a factorizable form which leads to computationally fast operations such as evaluating a CMB estimator or simulating a non-Gaussian CMB. We discuss practical issues in CMB analysis pipelines, and perform an optimal analysis of WMAP data. Our minimum-variance estimates are $g_{NL}^{\rm loc} = (-3.80 \pm 2.19) \times 10^5$, $g_{NL}^{\dotσ^4} = (-3.20 \pm 3.09) \times 10^6$, and $g_{NL}^{(\partialσ)^4} = (-10.8 \pm 6.33) \times 10^5$ after correcting for the effects of CMB lensing. No evidence of a nonzero inflationary four-point function is seen.
We present constraints on the patchy reionization by measuring the trispectrum of the Planck 2015 cosmic microwave background (CMB) temperature anisotropies. The patchy reionization leads to anisotropies in the CMB optical depth, and the statistics of the observed CMB anisotropies is altered. We estimate the trispectrum of the CMB temperature anisotropies to constrain spatial variation of the optical depth. We show that the measured trispectrum is consistent with that from the standard lensed CMB simulation at $2σ$. While no evidence of the patchy reionization is found in the Planck 2015 tempe…
Minimum-variance estimators for the parameter fnl that quantifies local-model non-Gaussianity can be constructed from the cosmic microwave background (CMB) bispectrum (three-point function) and also from the trispectrum (four-point function). Some have suggested that a comparison between the estimates for the values of fnl from the bispectrum and trispectrum allow a consistency test for the model. But others argue that the saturation of the Cramer-Rao bound by the bispectrum estimator implies that no further information on fnl can be obtained from the trispectrum. Here we elaborate the nature …
We present trispectrum estimation methods which can be applied to general non-separable primordial and CMB trispectra. We present a general optimal estimator for the connected part of the trispectrum, for which we derive a quadratic term to incorporate the effects of inhomogeneous noise and masking. We describe a general algorithm for creating simulated maps with given arbitrary (and independent) power spectra, bispectra and trispectra. We propose a universal definition of the trispectrum parameter $T_{NL}$, so that the integrated bispectrum on the observational domain can be consistently comp…
Under the existence of chiral non-Gaussian sources during inflation, the trispectrum of primordial curvature perturbations can break parity. We examine signatures of the induced trispectrum of the cosmic microwave background (CMB) anisotropies. It is confirmed via a harmonic-space analysis that, as a consequence of parity violation, such a CMB trispectrum has nonvanishing signal in the $\ell_1 + \ell_2 + \ell_3 + \ell_4 = \text{odd}$ domain, which is prohibited in the concordance cosmology. When the curvature trispectrum is parametrized with Legendre polynomials, the CMB signal due to the Lege…
We study statistical anisotropies generated in the observed two-point function of the cosmic microwave background (CMB) fluctuations if the primordial statistics are non-Gaussian. Focusing on the dipole modulations of the anisotropies, we find that the hemispherical power asymmetry observed in the CMB temperature fluctuations can be modeled by a local-type trispectrum with amplitude $τ_{\rm NL}(k_p=0.05~{\rm Mpc}^{-1}) \approx 2 \times 10^4$ and a large red tilt $n\approx -0.68$. We numerically evaluate the non-Gaussian covariance of the modulation estimators for both temperature and E-mode po…
Considerable recent attention has focussed on the prospects to use the cosmic microwave background (CMB) trispectrum to probe the physics of the early universe. Here we evaluate the probability distribution function (PDF) for the standard estimator tau_nle for the amplitude tau_nl of the CMB trispectrum both for the null-hypothesis (i.e., for Gaussian maps with tau_nl = 0) and for maps with a non-vanishing trispectrum (|tau_nl|>0). We find these PDFs to be highly non-Gaussian in both cases. We also evaluate the variance with which the trispectrum amplitude can be measured, <tau_nle^2>, as a fu…
Cosmic magnetic fields are observed to be coherent on large scales and could have a primordial origin. Non-Gaussian signals in the cosmic microwave background (CMB) are generated by primordial magnetic fields as the magnetic stresses and temperature anisotropy they induce depend quadratically on the magnetic field. We compute the CMB scalar trispectrum on large angular scales, for nearly scale-invariant magnetic fields, sourced via the Sachs-Wolfe effect. The trispectra induced by magnetic energy density and by magnetic scalar anisotropic stress are found to have typical magnitudes of approxim…
We provide an analytical expression for the trispectrum of the Cosmic Microwave Background (CMB) temperature anisotropies induced by cosmic strings. Our result is derived for the small angular scales under the assumption that the temperature anisotropy is induced by the Gott-Kaiser-Stebbins effect. The trispectrum is predicted to decay with a non-integer power-law exponent l^(-r) with 6<r<7, depending on the string microstructure, and thus on the string model. For Nambu-Goto strings, this exponent is related to the string mean square velocity and the loop distribution function. We then explore…
In this paper we demonstrate an efficient method for including both CMB temperature and polarisation data in optimal non-Gaussian estimators. The method relies on orthogonalising the multipoles of the temperature and polarisation maps and results in a reduction by a factor of over 3 the terms required to calculate the estimator. The method is illustrated with the modal method applied to bispectrum estimation via the CMB with the trispectrum included as an appendix. However, the method is quite general and can be applied to any optimal bispectrum or trispectrum estimator including the KSW, binn…
Primordial magnetic fields will generate non-Gaussian signals in the cosmic microwave background (CMB) as magnetic stresses and the temperature anisotropy they induce depend quadratically on the magnetic field. We compute a new measure of magnetic non-Gaussianity, the CMB trispectrum, on large angular scales, sourced via the Sachs-Wolfe effect. The trispectra induced by magnetic energy density and by magnetic scalar anisotropic stress are found to have typical magnitudes of approximately a few times 10^{-29} and 10^{-19}, respectively. Observational limits on CMB non-Gaussianity from WMAP data…
Squeezed primordial non-Gaussianity can strongly constrain early-universe physics, but it can only be observed on the CMB after it has been gravitationally lensed. We give a new simple non-perturbative prescription for accurately calculating the effect of lensing on any squeezed primordial bispectrum shape, and test it with simulations. We give the generalization to polarization bispectra, and discuss the effect of lensing on the trispectrum. We explain why neglecting the lensing smoothing effect does not significantly bias estimators of local primordial non-Gaussianity, even though the change…
We compute the impact of the running of higher order density correlation functions on the two point functions of CMB spectral distortions (SD). We show that having some levels of running enhances all of the SDs by few orders of magnitude which might make them easier to detect. Taking a reasonable range for $ |n_{f_{NL}} |\lesssim 1.1$ and with $f_{NL} = 5$ we show that for PIXIE like experiment, the signal to noise ratio, $(S/N)_{i}$, enhances to $\lesssim 4000$ and $\lesssim 10$ for $μT$ and $yT$ toward the upper limit of $n_{f_{NL}}$. In addition, assuming $ |n_{τ_{NL}}|< 1$ and $τ_{NL} = 10…
Gravitational lensing provides a significant source of cosmological information in modern CMB parameter analyses. It is measured in both the power spectrum and trispectrum of the temperature fluctuations. These observables are often treated as independent, although as they are both determined from the same map this is impossible. In this paper, we perform a rigorous analysis of the covariance between lensing power spectrum and trispectrum analyses. We find two dominant contributions coming from: (i) correlations between the disconnected noise bias in the trispectrum measurement and sample vari…
The kinetic Sunyaev Zel'dovich effect is a secondary CMB temperature anisotropy that provides a powerful probe of the radial-velocity field of matter distributed across the Universe. This velocity field is reconstructed by combining high-resolution CMB measurements with galaxy survey data, and it provides an unbiased tracer of matter perturbations in the linear regime. In this paper, we show how this measurement can be used to probe primordial non-Gaussianity of the local type, particularly focusing on the trispectrum amplitude $τ_{\rm NL}$, as may arise in a simple two-field inflation model t…
We propose a computationally feasible estimator for the needlet trispectrum, which develops earlier work on the bispectrum by Donzelli et al. (2012). Our proposal seems to enjoy a number of useful properties, in particular a) the construction exploits the localization properties of the needlet system, and hence it automatically handles masked regions; b) the procedure incorporates a quadratic correction term to correct for the presence of instrumental noise and sky-cuts; c) it is possible to provide analytic results on its statistical properties, which can serve as a guidance for simulations. …
We use analytic calculations of the post-recombination gravitational effects of cosmic strings to estimate the resulting CMB power spectrum, bispectrum and trispectrum. We place a particular emphasis on multipole regimes relevant for forthcoming CMB experiments, notably the Planck satellite. These calculations use a flat sky approximation, generalising previous work by integrating string contributions from last scattering to the present day, finding the dominant contributions to the correlators for multipoles l > 50. We find a well-behaved shape for the string bispectrum (without divergences) …
We study the covariance in the angular power spectrum estimates of CMB fluctuations when the primordial fluctuations are non-Gaussian. The non-Gaussian covariance comes from a nonzero connected four-point correlation function -- or the trispectrum in Fourier space -- and can be large when long-wavelength (super-CMB) modes are strongly coupled to short-wavelength modes. The effect of such non-Gaussian covariance can be modeled through additional freedom in the theoretical CMB angular power spectrum and can lead to different inferred values of the standard cosmological parameters relative to tho…
Non-Gaussianity of the primordial density perturbations provides an important measure to constrain models of inflation. At cubic order the non-Gaussianity is captured by two parameters $τ_{\rm NL}$ and $g_{\rm NL}$ that determine the amplitude of the density perturbation trispectrum. Here we report measurements of the kurtosis power spectra of the cosmic microwave background (CMB) temperature as mapped by Planck by making use of correlations between square temperature-square temperature and cubic temperature-temperature anisotropies. In combination with noise simulations, we find the best join…
We study the trispectrum of the gravitational waves (GWs) generated through the dynamics of an axionic spectator field and SU(2) gauge fields during inflation. In non-Abelian gauge theory, the gauge fields have four-point self-interactions, which induce the tree-level GW trispectrum. We formulate this type of the GW trispectrum including the non-dynamical contributions and evaluate it in the equilateral limit as a unique signal of this model. We find that the ratio of the GW trispectrum to the cube of the scalar power spectrum can be as large as $\mathcal{O}(10^6)$ in the viable parameter spac…
We investigate an estimator to measure the primordial trispectrum in equilateral type non-Gaussian models such as k-inflation, single field DBI inflation and multi-field DBI inflation models from Cosmic Microwave Background (CMB) anisotropies. The shape of the trispectrum whose amplitude is not constrained by the bispectrum in the context of effective theory of inflation and k-inflation is known to admit a separable form of the estimator for CMB anisotropies. We show that this shape is $87 \%$ correlated with the full quantum trispectrum in single field DBI inflation, while it is $33 \%$ corre…
By decomposing the CMB temperature fluctuations \emph{squared} into spherical harmonics, we can define the \emph{CMB power multipoles} as the fundamental observables to quantify any generic modulations in the CMB. This allows for simple direct measurements for the CMB modulations, and also indirect measurements for the CMB trispectrum, which fully determines the CMB modulations in a statistically isotropic universe. The CMB power multipoles can reach the theoretical accuracies higher than the current constraints from Planck observations. The theoretical accuracy for the dipole modulation, $A\s…
We outline the expected constraints on non-Gaussianity from the cosmic microwave background (CMB) with current and future experiments, focusing on both the third (f_{NL}) and fourth-order (g_{NL} and τ_{NL}) amplitudes of the local configuration or non-Gaussianity. The experimental focus is the skewness (two-to-one) and kurtosis (two-to-two and three-to-one) power spectra from weighted maps. In adition to a measurement of τ_{NL} and g_{NL} with WMAP 5-year data, our study provides the first forecasts for future constraints on g_{NL}. We describe how these statistics can be corrected for the ma…
We investigate the effect of equilateral-type primordial trispectrum on the halo/galaxy bispectrum. We consider three types of equilateral primordial trispectra which are generated by quartic operators naturally appeared in the effective field theory of inflation and can be characterized by three non-linearity parameters, $g_{\rm NL} ^{\dotσ^4}$, $g_{\rm NL} ^{\dotσ^2 (\partial σ)^2}$, and $g_{\rm NL} ^{(\partial σ)^4}$. Recently, constraints on these parameters have been investigated from CMB observations by using WMAP9 data. In order to consider the halo/galaxy bispectrum with the equilatera…
We consider corrections to the underlying cosmology due to secondary contributions from weak gravitational lensing, the integrated Sachs-Wolfe effect, and the Sunyaev-Zel'dovich effect contained in the trispectrum. We incorporate these additional contributions to the covariance of a binned angular power spectrum of temperature anisotropies in the analysis of current and prospective data sets. Although recent experiments such as ACBAR and CBI are not particularly sensitive to these additional non-Gaussian effects, the interpretation of Planck and CMBPol anisotropy spectra will require an accoun…
We present an implementation of an optimal CMB trispectrum estimator which accounts for anisotropic noise and incomplete sky coverage. We use a general separable mode expansion which can and has been applied to constrain both primordial and late-time models. We validate our methods on large angular scales using known analytic results in the Sachs-Wolfe limit. We present the first near-optimal trispectrum constraints from WMAP data on the cubic term of local model inflation $ g_{\rm NL} = (1.6 \pm 7.0)\times 10^5$, for the equilateral model $t_{\rm NL}^{\rm{equil}}=(-3.11\pm 7.5)\times 10^6 $ a…
Under the presence of anisotropic sources in the inflationary era, the trispectrum of the primordial curvature perturbation has a very specific angular dependence between each wavevector that is distinguishable from the one encountered when only scalar fields are present, characterized by an angular dependence described by Legendre polynomials. We examine the imprints left by curvature trispectra on the $TTμ$ bispectrum, generated by the correlation between temperature anisotropies (T) and chemical potential spectral distortions ($μ$) of the Cosmic Microwave Background (CMB). Due to the angula…
The late-time effect of primordial non-Gaussianity offers a window into the physics of inflation and the very early Universe. In this work we study the consequences of a particular class of primordial non-Gaussianity that is fully characterized by initial density fluctuations drawn from a non-Gaussian probability density function, rather than by construction of a particular form for the primordial bispectrum. We numerically generate multiple realisations of cosmological structure and use the late-time matter power spectrum, bispectrum and trispectrum to determine the effect of these modified i…
The leading candidate for the very early universe is described by a period of rapid expansion known as inflation. While the standard paradigm invokes a single slow-rolling field, many different models may be constructed which fit the current observational evidence. In this work we outline theoretical and observational studies of non-Gaussian fluctuations produced by models of inflation and by cosmic strings - topological defects that may be generated in the very early universe during a phase transition. In particular, we consider the imprint of cosmic strings on the cosmic microwave background…
Cosmic microwave background studies of non-Gaussianity involving higher-order multispectra can distinguish between early universe theories that predict nearly identical power spectra. However, the recovery of higher-order multispectra is difficult from realistic data due to their complex response to inhomogeneous noise and partial sky coverage, which are often difficult to model analytically. A traditional alternative is to use one-point cumulants of various orders, which collapse the information present in a multispectrum to one number. The disadvantage of such a radical compression of the da…
We study the $TTμ$ bispectrum, generated by correlations between Cosmic Microwave Background temperature (T) anisotropies and chemical potential ($μ$) distortions, and we analyze its dependence on primordial local trispectrum parameters $g_{\rm NL}$ and $τ_{\rm NL}$. We cross-check our results by comparing the full bispectrum calculation with the expectations from a general physical argument, based on predicting the shape of $μ$-T correlations from the couplings between short and long perturbation modes induced by primordial non-Gaussianity. We show that $both$ $g_{\rm NL}$ and $τ_{\rm NL}$-pa…
We study reconstruction of the lensing potential power spectrum from CMB temperature data, with an eye to the Planck experiment. We work with the optimal quadratic estimator of Okamoto and Hu, which we characterize thoroughly in application to reconstruction of the lensing power spectrum. We find that at multipoles L<250 our current understanding of this estimator is biased at the 15% level by beyond-gradient terms in the Taylor expansion of lensing effects. We present the full lensed trispectrum to fourth order in the lensing potential to explain this effect. We show that the low-L bias, as w…
Non-linear CMB temperature anisotropies up to the third-order on large scales are calculated. On large scales and in the Sachs-Wolfe limit, we give the explicit expression for the observed temperature anisotropy in terms of the primordial curvature perturbation up to the third-order. We derived the final bispectrum and trispectrum of anisotropies and the corresponding non-linear parameters, in which the contributions to the observed non-Gaussianity from primordial perturbations and from the non-linear mapping from primordial curvature perturbation to the temperature anisotropy are transparentl…
Parity Violation in Galaxy Shapes: Primordial Non-Gaussianity
Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理研究所）；Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）；Kavli Institute for the Physics and Mathematics of the Universe (WPI)（卡弗里宇宙物理学与数学研究所）；The University of Tokyo Institutes for Advanced Study (UTIAS)（东京大学高等研究院）；The University of Tokyo（东京大学）
研究通过银河系内对齐性探测 parity 违反的原始非高斯性，利用有效场论框架分析 parity-odd 内对齐功率谱与原始三阶谱的关系，利用 N-body 模拟确定暗物质晕偏置参数，并预测 IA 的约束能力。
We study the reconstruction of the cosmic rotation power spectrum produced by parity-violating physics, with an eye to ongoing and near future cosmic microwave background (CMB) experiments such as BICEP Array, CMBS4, LiteBIRD and Simons Observatory. In addition to the inflationary gravitational waves and gravitational lensing, measurements of other various effects on CMB polarization open new window into the early Universe. One of these is anisotropies of the cosmic polarization rotation which probes the Chern-Simons term generally predicted by string theory. The anisotropies of the cosmic rot…
First Constraints on the Epoch of Reionization Using the non-Gaussianity of the Kinematic Sunyaev-Zel{'}dovich Effect from the South Pole Telescope and {\it Herschel}-SPIRE Observations
We investigate tomography of 21-cm brightness temperature fluctuations during the Dark Ages as a probe for constraining primordial non-Gaussianity. We expand the 21- cm brightness temperature up to cubic order in perturbation theory and improve previous models of the signal by including the effect of the free electron fraction. Using modified standard perturbation theory methods that include baryonic pressure effects we derive an improved secondary bispectrum and for the first time derive the secondary trispectrum of 21-cm brightness temperature fluctuations. We then forecast the amount of inf…
Cosmic Microwave Background (CMB) Anisotropies is a subject of intensive research in several fields of sciences. In this paper we start a systematic development of basic notions and theory in statistics according to the application for CMB. The main result of this paper is the necessary and sufficient condition for isotropy of a non-Gaussian field in terms of spectra. Clear formulae for bi-, tri- and polyspectra and bi-, tri-, and higher order covariances are also given. Keywords: Bispectrum, Trispectrum, Angular poly-Spectra, Cosmic microwave background radiation; Gaussianity; spherical rando…
We generalize the concept of the ordinary skew-spectrum to probe the effect of non-Gaussianity on the morphology of Cosmic Microwave Background (CMB) maps in several domains: in real-space (where they are commonly known as cumulant-correlators), and in harmonic and needlet bases. The essential aim is to retain more information than normally contained in these statistics, in order to assist in determining the source of any measured non-Gaussianity, in the same spirit as Munshi & Heavens (2010) skew-spectra were used to identify foreground contaminants to the CMB bispectrum in Planck data. Using…
Since the first limit on the (local) primordial non-Gaussianity parameter, fNL, was obtained from COBE data in 2002, observations of the CMB have been playing a central role in constraining the amplitudes of various forms of non-Gaussianity in primordial fluctuations. The current 68% limit from the 7-year WMAP data is fNL=32+/-21, and the Planck satellite is expected to reduce the uncertainty by a factor of four in a few years from now. If fNL>>1 is found by Planck with high statistical significance, all single-field models of inflation would be ruled out. Moreover, if the Planck satellite fin…
We demonstrate how to obtain optimal constraints on a primordial gravitational wave component in lensed Cosmic Microwave Background (CMB) data under ideal conditions. We first derive an estimator of the tensor-to-scalar ratio, $r$, by using an error-controlled close approximation to the exact posterior, under the assumption of Gaussian primordial CMB and lensing deflection potential. This combines fast internal iterative lensing reconstruction with optimal recovery of the unlensed CMB. We evaluate its performance on simulated low-noise polarization data targeted at the recombination peak. We c…
We present a new method for delensing $B$ modes of the cosmic microwave background (CMB) using a lensing potential reconstructed from the same realization of the CMB polarization (CMB internal delensing). The $B$-mode delensing is required to improve sensitivity to primary $B$ modes generated by, e.g., the inflationary gravitational waves, axion-like particles, modified gravity, primordial magnetic fields, and topological defects such as cosmic strings. However, the CMB internal delensing suffers from substantial biases due to correlations between observed CMB maps to be delensed and that used…
We present algorithms for searching for azimuthally symmetric features in CMB data. Our algorithms are fully optimal for masked all-sky data with inhomogeneous noise, computationally fast, simple to implement, and make no approximations. We show how to implement the optimal analysis in both Bayesian and frequentist cases. In the Bayesian case, our algorithm for evaluating the posterior likelihood is so fast that we can do a brute-force search over parameter space, rather than using a Monte Carlo Markov chain. Our motivating example is searching for bubble collisions, a pre-inflationary signal …
In this paper we present the tools to optimally extract the Lensing-Integrated Sachs Wolfe (L-ISW) bispectrum signal from future CMB data. We implement two different methods to simulate the non-Gaussian CMB maps with the L-ISW signal: a non-perturbative method based on the FLINTS lensing code and the separable mode expansion method. We implement the Komatsu, Spergel and Wandelt (KSW) optimal estimator analysis for the Lensing-ISW bispectrum and we test it on the non-Gaussian simulations in the case of a realistic CMB experimental settings with an inhomogeneous sky coverage. We show that the es…
Upcoming ground-based cosmic microwave background experiments will provide CMB maps with high sensitivity and resolution that can be used for high fidelity lensing reconstruction. However, the sky coverage will be incomplete and the noise highly anisotropic, so optimized estimators are required to extract the most information from the maps. We focus on quadratic-estimator based lensing reconstruction methods that are fast to implement, and compare new more-optimally filtered estimators with various estimators that have previously been used in the literature. Input CMB maps can be optimally inv…
We propose optimal estimators for bispectra from excited states. Two common properties of such bispectra are the enhancement in the collinear limit, and the prediction of oscillating features. We review the physics behind excited states and some of the choices made in the literature. We show that the enfolded template is a good template in the collinear limit, but does poorly elsewhere, establishing a strong case for an improved estimator. Although the detailed scale dependence of the bispectra differs depending on various assumptions, generally the predicted bispectra are either effectively 1…
One of the principle efforts in cosmic microwave background (CMB) research is measurement of the parameter fnl that quantifies the departure from Gaussianity in a large class of non-minimal inflationary (and other) models. Estimators for fnl are composed of a sum of products of the temperatures in three different pixels in the CMB map. Since the number ~Npix^2 of terms in this sum exceeds the number Npix of measurements, these ~Npix^2 terms cannot be statistically independent. Therefore, the central-limit theorem does not necessarily apply, and the probability distribution function (PDF) for t…
We analyze non-Gaussianity (NG) due to the primordial bispectrum and trispectrum using CMB temperature maps of WMAP 7-year data. We first apply the perturbative formulae of Minkowski functionals up to second-order NG derived by Matsubara (2010), which enable us to give limits on cubic NG parametrized with tau_NL and g_NL as well as various types of quadratic NG parametrized with f_NL. We find no signature of primordial NG in WMAP 7-year data, but give constraints on the local-type, equilateral-type, orthogonal-type f_NL: f_NL(loc)=20+-42, f_NL(eq)=-121+-208, f_NL(ort)=-129+-171, respectively, …
