We develop a possibility of generating tensor non-Gaussianity in a kind of anisotropic inflation, where a $U(1)$ gauge field is kinetically coupled to a spectator scalar field. Owing to this coupling, the coherent mode of the electric field appears and softly breaks the isotropy of the Universe. We compute the bispectrum of linearly-polarized tensor perturbations sourced by the gauge field and find that it is strongly red-tilted and has distinctive statistical anisotropies including higher-order multipole moments. Interestingly, the tensor bispectra with the specific combinations of linear polarization modes are dominant, and their amplitudes depend on the different sets of multipole moments. This new type of statistically-anisotropic tensor non-Gaussianity can be potentially testable with the upcoming cosmic microwave background B-mode polarization experiments.
The bispectrum of single-field inflationary trajectories in which the speed of sound of the inflationary trajectories $c_s$ is constant but not equal to the speed of light $c=1$ is explored. The trajectories are generated as random realisations of the Hubble Slow-Roll (HSR) hierarchy and the bispectra are calculated using numerical techniques that extends previous work. This method allows for out-of-slow-roll models with non-trivial time dependence and arbitrarily low $c_s$. The ensembles obtained using this method yield distributions for the shape and scale-dependence of the bispectrum and th…
Realistic models of high-energy physics include multiple scalar fields. Renormalization requires that the fields have nonminimal couplings to the spacetime Ricci curvature scalar, and the couplings can be large at the energy scales of early-universe inflation. The nonminimal couplings induce a nontrivial field-space manifold in the Einstein frame, and they also yield an effective potential in the Einstein frame with nontrivial curvature. The ridges or bumps in the Einstein-frame potential can lead to primordial non-Gaussianities of observable magnitude. We develop a covariant formalism to stud…
We develop an integral form for the bispectrum in general single-field inflation whose domain of validity includes models of inflation where the background evolution is not constrained to be slowly varying everywhere. Our integral form preserves the squeezed-limit consistency relation, allows for fast evaluation of the bispectrum for all triangle configurations expediting the efficient comparison of slow-roll violating models with data, and provides complete and compact slow-roll expressions correct to first order in slow-roll parameters. Motivated by the recent Planck results, we consider as …
We investigate the consistency relation relating the squeezed limit of the bispectrum to the scalar spectral index in single field models of inflation. We give a simple integral formula for the bispectrum in the squeezed limit in terms of the free mode mode functions of the primordial curvature perturbation, in any Lorentz invariant single field model of inflation and without resorting to any approximation, generalizing a recent result obtained by Ganc and Komatsu in the case of canonical kinetic terms. We use our result to verify the consistency relation in an exactly solvable class of models…
The theory and methodology is developed to compute the bispectrum in warm inflation, leading to results for the non-linearity parameter and the shape of the bispectrum. Particular attention is paid to the study of the bispectrum in the regime of weak dissipation and how stochastic fluctuations affect the bispectrum. It is shown that, in contrast to the strong dissipative regime, the amplitude of non-Gaussianity is strongly dependent on the parameters governing the microscopic physics in the intermediate and weak dissipation warm inflation regimes. The most important results concern the shape o…
We compute the full cosmic microwave background temperature bispectrum generated by nonlinearities after single-field inflation. By integrating the photon temperature at second order along a perturbed geodesic in Newtonian gauge, we derive an expression for the observed temperature fluctuations that, for the first time, clarifies the separation of the gravitational lensing and time-delay effects from the purely second-order contributions. We then use the second-order Boltzmann code CosmoLib$2^{\rm nd}$ to calculate these contributions and their bispectrum. Including the perturbations in the ph…
We use WMAP 9-year bispectrum data to constrain the free parameters of an 'effective field theory' describing fluctuations in single-field inflation. The Lagrangian of the theory contains a finite number of operators associated with unknown mass scales. Each operator produces a fixed bispectrum shape, which we decompose into partial waves in order to construct a likelihood function. Based on this likelihood we are able to constrain four linearly independent combinations of the mass scales. As an example of our framework we specialize our results to the case of 'Dirac-Born-Infeld' and 'ghost' i…
We present a complete formulation of the scalar bispectrum in the unified effective field theory (EFT) of inflation, which includes the Horndeski and beyond-Horndeski Gleyzes-Langlois-Piazza-Vernizzi classes, in terms of a set of simple one-dimensional integrals. These generalized slow-roll expressions remain valid even when slow-roll is transiently violated and encompass all configurations of the bispectrum. We show analytically that our expressions explicitly preserve the squeezed-limit consistency relation beyond slow-roll. As an example application of our results, we compute the scalar bis…
Galileon fields arise naturally from the decoupling limit of massive gravities, and possess special self-interactions which are protected by a spacetime generalization of Galilean symmetry. We briefly revisit the inflationary phenomenology of Galileon theories. Working from recent computations of the fluctuation Lagrangian to cubic order in the most general model with second-order equations of motion, we show that a distinct shape is present but with suppressed amplitude. A similar shape has been found in other higher-derivative models. It may be visible in a theory tuned to suppress the leadi…
We explain the motivation and main idea of our work in reference arXiv:0907.2476 [hep-th]. We present a simple model of multifield Dirac-Born-Infeld inflation whose bispectrum exhibits a linear combination of the equilateral and local shapes, which are usually considered as separate possibilities. We also point out the presence of a particularly interesting component of the primordial trispectrum.
We evaluate the dimensionless non-Gaussianity parameter $h_{_{\rm NL}}$, that characterizes the amplitude of the tensor bispectrum, numerically for a class of two field inflationary models such as double inflation, hybrid inflation and aligned natural inflation. We compare the numerical results with the slow roll results which can be obtained analytically. In the context of double inflation, we also investigate the effects on $h_{_{\rm NL}}$ due to curved trajectories in the field space. We explicitly examine the validity of the consistency relation governing the tensor bispectrum in the squee…
We investigate influences of the curved field-space metric of multi-field inflationary models on the squeezed bispectrum. The reduced bispectrum in squeezed limit is computed using the δN formalism. The calculation is performed under the slow-roll approximation and assumption that field derivative of the field-space metric is sufficiently small such that the contributions from Riemann tensor of the field-space can be approximately ignored. Based on these approximations, We compute the analytic expressions for the reduced bispectrum in squeezed limit, and find that, for such a nearly flat field…
Nongaussian statistics are a powerful discriminant between inflationary models, particularly those with noncanonical kinetic terms. Focusing on theories where the Lagrangian is an arbitrary Lorentz-invariant function of a scalar field and its first derivatives, we review and extend the calculation of the observable three-point function. We compute the "next-order" slow-roll corrections to the bispectrum in closed form, and obtain quantitative estimates of their magnitude in DBI and power-law k-inflation. In the DBI case our results enable us to estimate corrections from the shape of the potent…
We examine the momentum dependence of the bispectrum of two-field inflationary models within the long-wavelength formalism. We determine the sources of scale dependence in the expression for the parameter of non-Gaussianity fNL and study two types of variation of the momentum triangle: changing its size and changing its shape. We introduce two spectral indices that quantify the possible types of momentum dependence of the local type fNL and illustrate our results with examples.
We employ the formalism of the effective field theory of inflation to study the effects of a sudden change in the speed of sound of the inflationary perturbations. Such an event generates a feature with high frequency oscillations both in the two- and in the three-point functions of the curvature fluctuations. We study, at first order in the magnitude of the change of the speed of sound, the dependence of the power spectrum and of the bispectrum on the duration of the change. In the limit of a very short duration, the oscillations in the power spectrum persist up to very large momenta and the …
We study the primordial non-Gaussinity predicted from simple models of inflation with a linear potential and superimposed oscillations. This generic form of the potential is predicted by the axion monodromy inflation model, that has recently been proposed as a possible realization of chaotic inflation in string theory, where the monodromy from wrapped branes extends the range of the closed string axions to beyond the Planck scale. The superimposed oscillations in the potential can lead to new signatures in the CMB spectrum and bispectrum. In particular the bispectrum will have a new distinct s…
We present a consistent δN formalism for curvature perturbations in anisotropic cosmological backgrounds. We employ our δN formalism to calculate the power spectrum, the bispectrum and the trispectrum in models of anisotropic inflation with the background gauge fields in Bianchi I universe. Our results coincide exactly with the recent results obtained from in-in formalism. To satisfy the observational constraints the anisotropies generated on power spectrum are kept small but large orientation-dependent non-Gaussianities can be generated. We study the Suyama-Yamaguchi inequality for the amplit…
Using the consistency relation in Fourier space, we derive the observed galaxy bispectrum from single-field inflation in the squeezed limit, in which one of the three modes has a wavelength much longer than the other two. This provides a non-trivial check of the full computation of the bispectrum based on second-order cosmological perturbation theory in this limit. We show that gauge modes need to be carefully removed in the second-order cosmological perturbations in order to calculate the observed galaxy bispectrum in the squeezed limit. We then give an estimate of the effective non-Gaussiani…
In non-attractor single-field inflation models producing a scale-invariant power spectrum, the curvature perturbation on super-horizon scales grows as ${\cal R}\propto a^3$. This is so far the only known class of self-consistent single-field models with a Bunch-Davies initial state that can produce a large squeezed-limit bispectrum violating Maldacena's consistency relation. Given the importance of this result, we calculate the bispectrum with three different methods: using quantum field theory calculations in two different gauges, and classical calculations (the $δN$ formalism). All the resul…
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 s…
A scalar field rolling down a potential with a large initial velocity results in inflation of a finite duration. Such a scenario suppresses the scalar power on large scales improving the fit to the cosmological data. We find that the scenario leads to a hitherto unexplored situation wherein the boundary terms dominate the contributions to the scalar bispectrum over the bulk terms. We show that the consistency relation governing the non-Gaussianity parameter $f_{_{\rm NL}}$ is violated on large scales and that the contributions at the initial time can substantially enhance the value of $f_{_{\r…
We calculate the squeezed limit of the bispectrum produced by inflation with multiple light fields. To achieve this we allow for different horizon exit times for each mode and calculate the intrinsic field-space three-point function in the squeezed limit using soft-limit techniques. We then use the $δN$ formalism from the time the last mode exits the horizon to calculate the bispectrum of the primordial curvature perturbation. We apply our results to calculate the spectral index of the halo bias, $n_{δb}$, an important observational probe of the squeezed limit of the primordial bispectrum and …
We compute the covariant three-point function near horizon-crossing for a system of slowly-rolling scalar fields during an inflationary epoch, allowing for an arbitrary field-space metric. We show explicitly how to compute its subsequent evolution using a covariantized version of the separate universe or "delta-N" expansion, which must be augmented by terms measuring curvature of the field-space manifold, and give the nonlinear gauge transformation to the comoving curvature perturbation. Nonlinearities induced by the field-space curvature terms are a new and potentially significant source of n…
We study an inflationary model driven by a single minimally coupled standard kinetic term scalar field with a step in its mass modeled by an Heaviside step function. We present an analytical approximation for the mode function of the curvature perturbation, obtain the power spectrum analytically and compare it with the numerical result. We show that, after the scale set by the step, the spectrum contains damped oscillations that are well described by our analytical approximation. We also compute the dominant contribution to the bispectrum in the equilateral and the squeezed limits and find new…
The methods of effective field theory are used to study generic theories of inflation with a single inflaton field and to perform a general analysis of the associated non-Gaussianities. We investigate the amplitudes and shapes of the various generic three-point correlators, the bispectra, which may be generated by different classes of single-field inflationary models. Besides the well-known results for the DBI-like models and the ghost inflationary theories, we point out that curvature-related interactions may give rise to large non-Gaussianities in the form of bispectra characterized by a fla…
Primordial non-Gaussianity generated in an inflationary model where inflation is preceded by a radiation era is discussed. It is shown that both bispectrum and trispectrum non-Gaussianities are enhanced due to the presence of pre-inflationary radiation era. One distinguishing feature of such a scenario is that the trispectrum non-Gaussianity is larger than the bispectrum one.
A suitable coupling of the inflaton phi to a vector kinetic term F^2 gives frozen and scale invariant vector perturbations. We compute the cosmological perturbations zeta that result from such coupling by taking into account the classical vector field that unavoidably gets generated at large scales during inflation. This generically results in a too anisotropic power spectrum of zeta. Specifically, the anisotropy exceeds the 1% level (10% level) if inflation lasted ~5 e-folds (~50 e-folds) more than the minimal amount required to produce the CMB modes. This conclusion applies, among others, to…
The decay of massive particles during inflation generates characteristic signals in the squeezed limit of the primordial bispectrum. These signals are in particular distinctive in the regime of the quasi-single field inflation, where particles are oscillating with masses comparable to the Hubble scale. We apply the investigation to a class of scalar particles that experience a so-called waterfall phase transition in the isocurvature direction driven by the symmetry-breaking mechanism based on the hybrid inflation scenario. With a time-varying mass, a novel shape of oscillatory bispectrum is pr…
We study the effects of a general type of features of the inflaton potential on the spectrum and bispectrum of primordial curvature perturbations. These features correspond to a discontinuity in the $n$-th order derivative of the potential which are dumped exponentially away from the value of the field where the feature happens. Interestingly we find that different values of the amplitude and of the order of the feature can lead to the same effects on the power spectrum on both large and short scales, and on the bispectrum at small scales. Only taking into account the bispectrum at large scale…
We calculate the bispectrum of primordial curvature perturbations, ζ, generated during "open inflation." Inflation occurs inside a bubble nucleated via quantum tunneling from the background false vacuum state. Our universe lives inside the bubble, which can be described as a Friedman-Lemaître-Robertson-Walker (FLRW) universe with negative spatial curvature, undergoing slow-roll inflation. We pay special attention to the issue of an initial state for quantum fluctuations. A "vacuum state" defined by a positive-frequency mode in de Sitter space charted by open coordinates is different from the E…
In order to gain new insights into the gauge field couplings in the early universe, we consider the constraints on gauge field production during inflation imposed by requiring that their effect on the CMB anisotropies are subdominant. In particular, we calculate systematically the bispectrum of the primordial curvature perturbation induced by the presence of vector gauge fields during inflation. Using a model independent parametrization in terms of magnetic non-linearity parameters, we calculate for the first time the contribution to the bispectrum from the cross correlation between the inflat…
We study single field inflation in noncommutative spacetime and compute two-point and three-point correlation functions for the curvature perturbation. We find that both power spectrum and bispectrum for comoving curvature perturbation are statistically anisotropic and the bispectrum is also modified by a phase factor depending upon the noncommutative parameters. The non-linearity parameter $f_{NL}$ is small for small statistical anisotropic corrections to the bispectrum coming from the noncommutative geometry and is consistent with the recent PLANCK bounds. There is a scale dependence of $f_{…
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.
We investigate the non-Gaussian signatures of ultra slow-roll inflation. The bispectrum and the trispectrum are calculated with general initial conditions. The trispectrum is of local shape, as in the case of the bispectrum. We show that the prediction of local non-Gaussianity is robust again generalizing the Bunch-Davies vacuum. The Suyama-Yamaguchi relation is saturated in this scenario.
We consider a non-inflationary early Universe scenario in which relevant scalar perturbations get frozen out at some point, but then are defrosted and follow a long nearly Minkowskian evolution before the hot era. This intermediate stage leaves specific imprint on the CMB 3-point function, largely independent of details of microscopic physics. In particular, the CMB bispectrum undergoes oscillations in the multipole l space with roughly constant amplitude. The latter is in contrast to the oscillatory bispectrum enhanced in the flattened triangle limit, as predicted by inflation with non-Bunch-…
We compute analytically the dominant contribution to the tree-level bispectrum in the Starobinsky model of inflation. In this model, the potential is vacuum energy dominated but contains a subdominant linear term which changes the slope abruptly at a point. We show that on large scales compared with the transition scale $k_0$ and in the equilateral limit the analogue of the non-linearity parameter scales as $(k/k_0)^2$, that is its amplitude decays for larger and larger scales until it becomes subdominant with respect to the usual slow-roll suppressed corrections. On small scales we show that …
In previous work, two of us have proposed a model of inflationary magnetogenesis based on a rolling auxiliary field able both to account for the magnetic fields inferred by the (non) observation of gamma-rays from blazars, and to start the galactic dynamo, without incurring in any strong coupling or strong backreaction regime. Here we evaluate the correction to the scalar spectrum and bispectrum with respect to single-field slow-roll inflation generated in that scenario. The strongest constraints on the model originate from the non-observation of a scalar bispectrum. Nevertheless, even when th…
Massive spinning particles, if present during inflation, lead to a distinctive bispectrum of primordial perturbations, the shape and amplitude of which depend on the masses and spins of the extra particles. This signal, in turn, leaves an imprint in the statistical distribution of galaxies; in particular, as a non-vanishing galaxy bispectrum, which can be used to probe the masses and spins of these particles. In this paper, we present for the first time a new theoretical template for the bispectrum generated by massive spinning particles, valid for a general triangle configuration. We then pro…
Deviations from the Bunch-Davies vacuum during an inflationary period can leave a testable imprint on the higher-order correlations of the CMB and large scale structures in the Universe. The effect is particularly pronounced if the statistical non-Gaussianity is inherently large, such as in models of inflation with a small speed of sound, e.g. DBI. First reviewing the motivations for a modified vacuum, we calculate the non-Gaussianity for a general action with a small speed of sound. The shape of its bispectrum is found to most resemble the 'orthogonal' or 'local' templates depending on the ph…
Focusing on the local type primordial non-Gaussianities, we study the bispectrum and trispectrum during a non-minimal slow-roll inflation. We use the so-called $δN$ formalism to investigate the super-horizon evolution of the primordial perturbations in this setup. Firstly we obtain the main equations of the model and introduce the framework of the $δN$ formalism for this case. Then we give analytical expressions for the nonlinear parameters describing the non-Gaussianity in the slow-roll approximation. We analyze the bispectrum by its non-linear parameter, $f_{NL}$. Furthermore, we calculate $…
The first year of observations by the Planck satellite mission shows that the cosmic microwave background (CMB) fluctuations are consistent with gaussian statistics in the primordial perturbations, a key prediction of the simplest models of inflation. However, there are hints of anomalies in the CMB power spectrum and bispectrum. We check for the possibility that some of these anomalous features have a common physical origin in a transient reduction of the inflaton speed of sound. We do this by exploiting predicted correlations between the power spectrum and bispectrum. Our results suggest tha…
In 2004, Creminelli and Zaldarriaga proposed a consistency relation for the primordial curvature perturbation of all single-field inflation models; it related the bispectrum in the squeezed limit to the spectral tilt. We have developed a technique, based in part on the Creminelli and Zaldarriaga argument, that can greatly simplify the calculation of the squeezed-limit bispectrum using the in-in formalism; we were able to arrive at a generic formula that does not rely on a slow-roll approximation. Using our formula, we explicitly tested the consistency relation for power-law inflation and for a…
In this review, 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 non-linear dynamics on super-horizon scales.
We study the primordial bispectrum of curvature perturbation in the uniform- density slicing generated by the interaction between the inflaton and isotropic background gauge fields. We derive the action up to cubic order in perturbation and take into account all the relevant effects in the leading order of slow-roll expansion. We first treat the quadratic vertices perturbatively and confirm the results of past studies, while identifying their regime of validity. We then extend the analysis to include the effect of the quadratic vertices to all orders by introducing exact linear mode functions,…
If during inflation the inflaton couples to a "fossil" field, some new scalar, vector, or tensor field, it typically induces a scalar-scalar-fossil bispectrum. Even if the fossil field leaves no direct physical trace after inflation, it gives rise to correlations between different Fourier modes of the curvature or, equivalently, a nonzero curvature trispectrum, but without a curvature bispectrum. Here we quantify the effects of a fossil field on the cosmic microwave background (CMB) temperature fluctuations in terms of bipolar spherical harmonics (BiPoSHs). The effects of vector and tensor fos…
