CMB Bispectrum Research Articles

High-resolution CMB bispectrum estimator with flexible modal bases

High-resolution CMB bispectrum estimator with flexible modal bases

The CMB bispectrum from bouncing cosmologies

In this paper we compute the CMB bispectrum for bouncing models motivated by Loop Quantum Cosmology. Despite the fact that the primordial bispectrum of these models is decaying exponentially above a large pivot scale, we find that the cumulative signal-to-noise ratio of the bispectrum induced in the CMB from scales ℓ<30 is larger than 10 in all cases of interest and therefore can, in principle, be detected in the Planck data.

Isocurvature modes: joint analysis of the CMB power spectrum and bispectrum

We perform a joint analysis of the power spectrum and the bispectrum of the CMB temperature and polarization anisotropies to improve the constraints on isocurvature modes. We construct joint likelihoods, both for the existing Planck data, and to make forecasts for the future LiteBIRD and CMB-S4 experiments. We assume a general two-field inflation model with five free parameters, leading to one isocurvature mode (which can be CDM density, neutrino density or neutrino velocity) arbitrarily correlated with the adiabatic mode. We theoretically assess in which cases (of detecting and/or fixing parameters) improvements can be expected, to guide our subsequent numerical analyses. We find that for Planck, which detected neither isocurvature modes nor primordial non-Gaussianity, the joint analysis does not improve the constraints in the general case. However, if we fix additional parameters in the model, the improvements can be highly significant depending on the chosen parameter values. For LiteBIRD+CMB-S4 we study in which regions of parameter space compatible with the Planck results the joint analysis will improve the constraints or the significance of a detection. We find that, while for CDM isocurvature this region is very small, for the neutrino isocurvature modes it is much larger. In particular for neutrino velocity it can be about half of the Planck-allowed region, where the joint analysis reduces the isocurvature error bars by up to 70%. In addition the joint analysis can also improve the error bars of some of the standard cosmological parameters, by up to 30% for θMC for example, by breaking the degeneracies with the correlation parameter between adiabatic and isocurvature modes.

CMB-S4 forecast on the primordial non-Gaussianity parameter of feature models

We present forecasts on the primordial non-Gaussianity parameter $f_\text{NL}$ of feature models for the future Cosmic Microwave Background Stage-4 (CMB-S4) experiments. The Fisher matrix of the bispectrum estimator was computed using noise covariances expected for preliminary CMB-S4 specifications including ones for the Simons Observatory. We introduce a novel method that improves the computation by orthonormalising the covariance matrix. The most sensitive CMB-S4 experiment with 1' beam and 1$\mu K$-arcmin noise would yield a factor of 1.7-2.2 times more stringent constraints compared to Planck. Under the Simons Observatory baseline conditions the improvement would be about 1.3-1.6 times to Planck. We also thoroughly studied the effects of various model and experimental parameters on the forecast. Detailed analysis on the constraints coming from temperature and E-mode polarisation, in particular, provided some insight into detecting oscillatory features in the CMB bispectrum.

Detecting higher spin fields through statistical anisotropy in the CMB bispectrum

Inflation may provide a suitable collider to probe physics at very high energies. In this paper we investigate the impact on the CMB bispectrum of higher spin fields which are long-lived on super-Hubble scales, e.g. partially massless higher spin fields. We show that distinctive statistical anisotropic signals on the CMB three-point correlator are induced and we investigate their detectability.

Efficient evaluation of angular power spectra and bispectra

Angular statistics of cosmological observables are hard to compute. The main difficulty is due to the presence of highly-oscillatory Bessel functions which need to be integrated over. In this paper, we provide a simple and fast method to compute the angular power spectrum and bispectrum of any observable. The method is based on using an FFTlog algorithm to decompose the momentum-space statistics onto a basis of power-law functions. For each power law, the integrals over Bessel functions have a simple analytical solution. This allows us to efficiently evaluate these integrals, independently of the value of the multipole ℓ. In particular, this method significantly speeds up the evaluation of the angular bispectrum compared to existing methods. To illustrate our algorithm, we compute the galaxy, lensing and CMB temperature angular power spectrum and bispectrum.

Using inpainting to construct accurate cut-sky CMB estimators

The direct evaluation of manifestly optimal, cut-sky CMB power spectrum and bispectrum estimators is numerically very costly, due to the presence of inverse-covariance filtering operations. This justifies the investigation of alternative approaches. In this work, we mostly focus on an inpainting algorithm that was introduced in recent CMB analyses to cure cut-sky suboptimalities of bispectrum estimators. First, we show that inpainting can equally be applied to the problem of unbiased estimation of power spectra. We then compare the performance of a novel inpainted CMB temperature power spectrum estimator to the popular apodised pseudo-$C_l$ (PCL) method and demonstrate, both numerically and with analytic arguments, that inpainted power spectrum estimates significantly outperform PCL estimates. Finally, we study the case of cut-sky bispectrum estimators, comparing the performance of three different approaches: inpainting, apodisation and a novel low-l leaning scheme. Providing an analytic argument why the local shape is typically most affected we mainly focus on local type non-Gaussianity. Our results show that inpainting allows to achieve optimality also for bispectrum estimation, but interestingly also demonstrate that appropriate apodisation, in conjunction with low-l cleaning, can lead to comparable accuracy.

Optimal estimator for resonance bispectra in the CMB

We propose an (optimal) estimator for a CMB bispectrum containing logarithmically spaced oscillations. There is tremendous theoretical interest in such bispectra, and they are predicted by a plethora of models, including axion monodromy models of inflation and initial state modifications. The number of resolved logarithmical oscillations in the bispectrum is limited due to the discrete resolution of the multipole bispectrum. We derive a simple relation between the maximum number of resolved oscillations and the frequency. We investigate several ways to factorize the primordial bispectrum, and conclude that a one dimensional expansion in the sum of the momenta $\sum k_i = k_t$ is the most efficient and flexible approach. We compare the expansion to the exact result in multipole space and show for $\omega_{\rm eff}=100$ that $\mathcal{O}(10^3)$ modes are sufficient for an accurate reconstruction. We compute the expected $\sigma_{f_{\rm NL}}$ and find that within an effective field theory (EFT) the overall signal to noise scales as $S/N\propto \omega^{3/2}$. Using only the temperature data we find $S/N\sim\mathcal{O}(1-10^2)$ for the frequency domain set by the EFT.

Constraining Galileon inflation

In this short paper, we present constraints on the Galileon inflationary model from the CMB bispectrum. We employ a principal-component analysis of the independent degrees of freedom constrained by data and apply this to the WMAP 9-year data to constrain the free parameters of the model. A simple Bayesian comparison establishes that support for the Galileon model from bispectrum data is at best weak.

The Komatsu Spergel Wandelt estimator for oscillations in the cosmic microwave background bispectrum

Oscillating shapes of the primordial bispectrum are present in many inflationary models. The Planck experiment has recently published measurements of oscillating shapes, which were however limited to the efficient frequency range of the used analysis method. Here we study the KSW estimator for oscillations in the CMB bispectrum, that allows to examine arbitrary oscillation frequencies for separable oscillating bispectrum shapes. We study the precision with which amplitude, phase and frequency can be determined with our estimator. An examination of the 3-point function in real space gives further insight into the estimator.

Cosmic microwave background bispectrum from nonlinear effects during recombination

We study cosmological perturbations by solving the governing Boltzmann and Einstein field equations up to second order and calculate the corresponding CMB bispectrum during recombination. We include all the second-order Liouville and collision terms, truncating the multipole hierarchy at $l=10$, consistently including all $m\ensuremath{\ne}0$ terms when calculating the bispectrum in the flat-sky limit. At this stage, we focus on contributions at and around recombination, and we neglect second-order vector and tensor perturbations, lensing effects, and late-time nonlinear ISW effects. We find that the signal-to-noise for the bispectrum is 0.69 for ${l}_{\mathrm{max}}=2000$, yielding an overall signal ${F}_{\mathrm{NL}}=3.19$ (normalized relative to the local model). We find that the effective ${f}_{\mathrm{NL}}$'s of the equilateral and local type are 5.11 and 0.88, respectively. This recombination signal will have to be taken into account in a quantitative analysis of the Planck data.

Effect of intermediate Minkowskian evolution on CMB bispectrum

We consider a non-inflationaryearly Universe scenario in which relevant scalar perturbations get frozen outat some point, but then are defrosted and follow a long nearly Minkowskian evolution beforethe hot era. This intermediate stage leaves specific imprinton the CMB 3-point function, largely independent ofdetails of microscopic physics.In particular, the CMB bispectrum undergoes oscillationsin the multipole l space withroughly constant amplitude. The latter is in contrastto the oscillatory bispectrum enhancedin the flattened triangle limit, as predicted by inflation withnon-Bunch-Davies vacuum.Given this and other peculiar features of the bispectrum,stringent constraints imposed by the Planck data may not apply.The CMB 3-point function is suppressed bythe inverse duration squared of the Minkowskian evolution, but can be ofobservable size for relatively short intermediate Minkowskian stage.

The Observed squeezed limit of cosmological three-point functions

The squeezed limit of the three-point function of cosmological perturbations is a powerful discriminant of different models of the early Universe. We present a conceptually simple and complete framework to relate any primordial bispectrum in this limit to late time observables, such as the CMB temperature bispectrum and the scale-dependent halo bias. We employ a series of convenient coordinate transformations to capture the leading non-linear effects of cosmological perturbation theory on these observables. This makes crucial use of Fermi Normal Coordinates and their conformal generalization, which we introduce here and discuss in detail. As an example, we apply our formalism to standard slow-roll single-field inflation. We show explicitly that Maldacena's results for the squeezed limits of the scalar bispectrum [proportional to (ns-1) in comoving gauge] and the tensor-scalar-scalar bispectrum lead to no deviations from a Gaussian universe, except for projection effects. In particular, the primordial contributions to the squeezed CMB bispectrum and scale dependent halo bias vanish, and there are no primordial "fossil" correlations between long-wavelength tensor perturbations and small-scale perturbations. The contributions to observed correlations are then only due to projection effects such as gravitational lensing and redshift perturbations.

Dark radiation and the CMB bispectrum

Non-Gaussianities in the cosmic microwave background maps arising from correlations between lensing and time variations of the gravitational potential (the so-called integrated Sachs-Wolfe effect) are one of the most important contaminants to the determination of the primordial inflationary bispectrum and may bias its determination. The presence of an extra dark radiation component, as suggested by some recent osmic microwave background measurements from the South Pole Telescope, could bias the expected value of the local bispectrum. In this paper we investigate the impact of dark radiation on the local bispectrum. As a by-product we also quantify the additional information on the dark radiation component that could come from a future precise measurement of the lensing-integrated-Sachs-Wolfe bispectrum.

Perturbed recombination from dark matter annihilation

We show that dark matter annihilation around the time of recombination can lead to growing ionization fraction perturbations, that track the linear collapse of matter over-densities. This amplifies small scale cosmological perturbations to the free electron density by a significant amount compared to the usual acoustic oscillations. Electron density perturbations distort the CMB, inducing secondary non-gaussianity. We calculate the CMB bispectrum from recombination, that is marginally observable by Planck. Even though electron perturbations can be markedly boosted compared with the Standard Model prediction, the dark matter effect in the CMB bispectrum turns out to be small and will be difficult to disentangle from the standard model in the foreseeable future.

The CMB bispectrum

We use a separable mode expansion estimator with WMAP7 data to estimate the bispectrum for all the primary families of non-Gaussian models, including non-scaling feature (periodic) models,the flat (trans-Planckian) model, DBI and ghost inflation, as well as previously constrainedsimple cases. We review the late-time mode expansion estimator methodology which can be applied to any non-separable primordial and CMB bispectrum model, and we demonstrate how the method can be used to reconstruct the CMB bispectrum from an observational map. We extend the previous validation of the general estimatorusing local map simulations. We apply the estimator to the coadded WMAP 7-year V and W channel maps, reconstructing the WMAP bispectrum using l < 500 multipoles and n = 50 orthonormal 3D eigenmodes;both the mode expansion parameters and the reconstructed 3D WMAP bispectrum are plotted. We constrain all popular nearly scale-invariant models, ensuring that the theoretical bispectrum is well-described by a convergent mode expansion. Constraints from the local model fNL = 20.31±27.64 and the equilateral model fNL = 10.19±127.38(FNL = 1.90±23.79) are consistent with previously published results. (Here, we use a nonlinearity parameter FNL normalised to the local case, to allow more direct comparison between different models.) Notable new constraints from our method include those for the constant model FNL = 7.82±24.57, the flat model FNL = 7.31±26.22, and warm inflation FNL = 2.10±25.83. We investigate feature models,which break scale invariance, surveying a wide parameter range for both the scale and phase (scanning for feature models with an effective period l* > 150). We find no significant evidence of non-Gaussianity for all cases well-described by the given eigenmodes. In the overall non-Gaussian analysis, we find one anomalous mode n = 33 with a 3.39σ amplitude which could give rise to an oscillatory model signal with l* ⩽ 150. We propose a measure F̄NL for the total integrated bispectrum and find that the measured value is consistent with the null hypothesis that CMB anisotropies obey Gaussian statistics. We argue that this general bispectrum survey with the WMAP data represents the best test of Gaussianity to date and we discuss future prospects with higher precision and resolution, notably from the Planck satellite.

Conformal invariance, dark energy, and CMB non-gaussianity

In addition to simple scale invariance, a universe dominated by dark energy naturally gives riseto correlation functions possessing full conformal invariance. This is due to the mathematicalisomorphism between the conformal group of certain three dimensional slices of de Sitter spaceand the de Sitter isometry group SO(4,1). In the standard homogeneous, isotropic cosmologicalmodel in which primordial density perturbations are generated during a long vacuum energydominated de Sitter phase, the embedding of flat spatial ℝ3 sections in de Sitterspace induces a conformal invariant perturbation spectrum and definite prediction for the shapeof the non-Gaussian CMB bispectrum. In the case in which the density fluctuations are generatedinstead on the de Sitter horizon, conformal invariance of the \U0001d54a2 horizon embeddingimplies a different but also quite definite prediction for the angular correlations of CMB non-Gaussianityon the sky. Each of these forms for the bispectrum is intrinsic to the symmetries of de Sitter space,and in that sense, independent of specific model assumptions. Each is different from the predictionsof single field slow roll inflation models, which rely on the breaking of de Sitter invariance. Wepropose a quantum origin for the CMB fluctuations in the scalar gravitational sector from theconformal anomaly that could give rise to these non-Gaussianities without a slow roll inflaton field,and argue that conformal invariance also leads to the expectation for the relation nS−1 = nTbetween the spectral indices of the scalar and tensor power spectrum. Confirmation of this predictionor detection of non-Gaussian correlations in the CMB of one of the bispectral shape functions predictedby conformal invariance can be used both to establish the physical origins of primordial density fluctuations,and distinguish between different dynamical models of cosmological vacuum dark energy.

Isocurvature modes in the CMB bispectrum

We study the angular bispectrum of local type arising from the (possibly correlated) combination of a primordial adiabatic mode with an isocurvature one. Generically, this bispectrum can be decomposed into six elementary bispectra. We estimate how precisely CMB data, including polarization, can enable us to measure or constrain the six corresponding amplitudes, considering separately the four types of isocurvature modes (CDM, baryon, neutrino density, neutrino velocity). Finally, we discuss how the model-independent constraints on the bispectrum can be combined to get constraints on the parameters of multiple-field inflation models.

The full squeezed CMB bispectrum from inflation

The small-scale CMB temperature we observe on the sky is modulated by perturbations that were super-horizon at recombination, giving differential focussing and lensing that generate a non-zero bispectrum even for single-field inflation where local physics is identical. Understanding this signal is important for primordial non-Gaussianity studies and also parameter constraints from the CMB lensing bispectrum signal. Because of cancellations individual effects can appear larger or smaller than they are in total, so a full analysis may be required to avoid biases. I relate angular scales on the sky to physical scales at recombination using the optical equations, and give full-sky results for the large-scale adiabatic temperature bispectrum from Ricci focussing (expansion of the ray bundle), Weyl lensing (convergence and shear), and temperature redshift modulations of small-scale power. The δN expansion of the beam is described by the constant temperature 3-curvature, and gives a nearly-observable version of the consistency relation prediction from single-field inflation. I give approximate arguments to quantify the likely importance of dynamical effects, and argue that they can be neglected for modulation scales l≲100, which is sufficient for lensing studies and also allows robust tests of local primordial non-Gaussianity using only the large-scale modulation modes. For accurate numerical results early and late-time ISW effects must be accounted for, though I confirm that the late-time non-linear Rees-Sciama contribution is negligible compared to other more important complications. The total corresponds to fNL ∼ 7 for Planck-like temperature constraints and fNL ∼ 11 for cosmic-variance limited data to lmax = 2000. Temperature lensing bispectrum estimates are affected at the 0.2σ level by Ricci focussing, and up to 0.5σ with polarization.

The CMB bispectrum in the squeezed limit

The CMB bispectrum generated by second-order effects at recombination can be calculated analytically when one of the three modes has a wavelength much longer than the other two and is outside the horizon at recombination. This was pointed out in [1] and here we correct their results. We derive a simple formula for the bispectrum, fNLloc = −(1/6+cos 2θ)⋅(1−1/2⋅dln (lS2CS)/dln lS), where CS is the short scale spectrum and θ the relative orientation between the long and the short modes. This formula is exact and takes into account all effects at recombination, including recombination-lensing, but neglects all late-time effects such as ISW-lensing. The induced bispectrum in the squeezed limit is small and will negligibly contaminate the Planck search for a local primordial signal: this will be biased only by fNLloc ≈ −0.4. The above analytic formula includes the primordial non-Gaussianity of any single-field model. It also represents a consistency check for second-order Boltzmann codes: we find substantial agreement with the current version of the CMBquick code.