Isocurvature perturbations naturally occur in models of inflation consisting of more than one scalar field. In this paper we calculate the spectrum of isocurvature perturbations generated at the end of inflation for three different inflationary models consisting of two canonical scalar fields. The amount of non-adiabatic pressure present at the end of inflation can have observational consequences through the generation of vorticity and subsequently the sourcing of B-mode polarisation. We compare two different definitions of isocurvature perturbations and show how these quantities evolve in different ways during inflation. Our results are calculated using the open source Pyflation numerical package which is available to download.
We look at the question posed by Parker et al. about the effect of UV regularisation on the power spectrum for inflation. Focusing on the slow-roll $k$-inflation, we show that up to second order in the Hubble and sound flow parameters, the adiabatic regularisation of such model leads to no difference in the power spectrum apart from certain cases that violate near scale invariant power spectra. Furthermore, extending to non-minimal $k$-inflation, we establish the equivalence of the subtraction terms in the adiabatic regularisation of the power spectrum in Jordan and Einstein frames.
We investigate the effect of adiabatic regularization on both the tensor- and scalar-perturbation power spectra in \textit{nonminimally} coupled chaotic inflation. Similar to that of the \textit{minimally} coupled general single-field inflation, we find that the subtraction term is suppressed by an exponentially decaying factor involving the number of $ e $-folds. By following the subtraction term long enough beyond horizon crossing, the regularized power spectrum tends to the "bare" power spectrum. This study justifies the use of the unregularized ("bare") power spectrum in standard calculati…
The evolution of the non-adiabatic pressure perturbation during inflation driven by two scalar fields is studied numerically for three different types of models. In the first model, the fields have standard kinetic terms. The other two models considered feature non-canonical kinetic terms; the first containing two fields which are coupled via their kinetic terms, and the second where one field has the standard kinetic term with the other field being a DBI field. We find that the evolution and the final amplitude of the non-adiabatic pressure perturbation depends strongly on the kinetic terms.
We perform adiabatic regularization of power spectrum in nonminimally coupled general single-field inflation with varying speed of sound. The subtraction is performed within the framework of earlier study by Urakawa and Starobinsky dealing with the canonical inflation. Inspired by Fakir and Unruh's model on nonminimally coupled chaotic inflation, we find upon imposing near scale-invariant condition, that the subtraction term exponentially decays with the number of $ e $-folds. As in the result for the canonical inflation, the regularized power spectrum tends to the "bare" power spectrum as the…
The possibility that quantum corrections break the conservation of superhorizon adiabatic perturbations in single field inflation is examined. I consider the lowest order corrections from massless matter fields in the Hamiltonian formalism. Particular emphasis is therefore laid on the renormalization. The counterterms are the same as in the Lagrangian formalism. The renormalized value of the tadpole is zero. I find a possible secular dependence of the power spectrum at one loop due to the trace anomaly, but this result depends on the approximation of the modes and is inconclusive. The symmetry…
We study the dynamics of inflation driven by an adiabatic self-gravitating medium, extending the previous works on fluid and solid inflation. Such a class of media comprises perfect fluids, zero and finite temperature solids. By using an effective field theory description, we compute the power spectrum for the scalar curvature perturbation of constant energy density hypersurface $ζ$ and the comoving scalar curvature perturbation ${\cal R}$ in the case of slow-roll, super slow-roll and $w$-media inflation, an inflationary phase with $w$ constant in the range $-1 <w <-1/3$. A similar computation…
Non-adiabatic pressure perturbations naturally occur in models of inflation consisting of more than one scalar field. The amount of non-adiabatic pressure present at the end of inflation can have observational consequences through changes in the curvature perturbation, the generation of vorticity and subsequently the sourcing of B-mode polarisation. In this work, based on a presentation at the 13th Marcel Grossmann Meeting, we give a very brief overview of non-adiabatic pressure perturbations in multi-field inflationary models and describe our recent calculation of the spectrum of isocurvature…
We express the in-in functional determinant giving the one-loop effective potential for a scalar field propagating in a cosmological spacetime in terms of the mode functions specifying the vacuum of the theory and then apply adiabatic regularization to make this bare potential finite. In this setup, the adiabatic regularization offers a particular renormalization prescription that isolates the effects of the cosmic expansion. We apply our findings to determine the radiative corrections to the classical inflaton potentials in scalar field inflationary models and also we derive an effective pote…
We propose a new approach for calculating the curvature perturbations produced during inflation in the stochastic formalism. In our formalism, the fluctuations of the e-foldings are directly calculated without perturbatively expanding the inflaton field and they are connected to the curvature perturbations by the $δN$ formalism. The result automatically includes the contributions of the higher order perturbations because we solve the equation of motion non-perturbatively. In this paper, we analytically prove that our result (the power spectrum and the nonlinearity parameter) is consistent with…
We show that, in warm inflation, the nearly constant Hubble rate and temperature lead to an adiabatic evolution of the number density of particles interacting with the thermal bath, even if thermal equilibrium cannot be maintained. In this case, the number density is suppressed compared to the equilibrium value but the associated phase-space distribution retains approximately an equilibrium form, with a smaller amplitude and a slightly smaller effective temperature. As an application, we explicitly construct a baryogenesis mechanism during warm inflation based on the out-of-equilibrium decay o…
We explore the superhorizon generation of large fnl of the local form in two field inflation. We calculate the two- and three-point observables in a general class of potentials which allow for an analytic treatment using the delta N formalism. Motivated by the conservation of the curvature perturbation outside the horizon in the adiabatic mode and also by the observed adiabaticity of the power spectrum, we follow the evolution of fnl^{local} until it is driven into the adibatic solution by passing through a phase of effectively single field inflation. We find that although large fnl^{local} ma…
We calculate the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ in a class of recently proposed two-field no-scale inflationary models in supergravity. We show that, in order to obtain correct predictions, it is crucial to take into account the coupling between the curvature and the isocurvature perturbations induced by the noncanonical form of the kinetic terms. This coupling enhances the curvature perturbation and suppresses the resulting tensor-to-scalar ratio to the per mille level even for values of the slow-roll parameter $ε\sim 0.01$. Beyond these particular models, we e…
We discuss the impact of adiabatic renormalization on the power spectrum of scalar and tensor perturbations from inflation. We show that adiabatic regularization is ambiguous as it leads to very different results, for different adiabatic subtraction schemes, both in the range $v\equiv k/(aH) \gsim 0.1$ and in the infrared regime. All these schemes agree in the far ultraviolet, $v\gg 1$. Therefore, we argue that in the far infrared regime, $v\ll 1$, the adiabatic expansion is no longer valid, and the unrenormalized spectra are the physical, measurable quantities. These findings cast some doubt …
In this work we clarify aspects of renormalization on curved backgrounds focussing on the potential ramifications on the amplitude of inflationary perturbations. We provide an alternate view of the often used adiabatic prescription by deriving a correspondence between the adiabatic subtraction terms and traditional renormalization. Specifically, we show how adiabatic subtraction can be expressed as a set of counter terms that are introduced by redefining the bare parameters of the action. Our representation of adiabatic subtraction then allows us to easily find other renormalization prescripti…
In multi-field inflation one or more non-adiabatic modes may become light, potentially inducing large levels of isocurvature perturbations in the cosmic microwave background. If in addition these light modes are coupled to the adiabatic mode, they influence its evolution on super horizon scales. Here we consider the case in which a non-adiabatic mode becomes approximately massless ("ultralight") while still coupled to the adiabatic mode, a typical situation that arises with pseudo-Nambu-Goldstone bosons or moduli. This ultralight mode freezes on super-horizon scales and acts as a constant sour…
We examine the regime of validity of $N$-point spectra predictions of single field inflation models that invoke transient periods of non-adiabatic evolution. Such models generate oscillatory features in these spectra spanning frequencies up to the inverse time scale of the transient feature. To avoid strong coupling of fluctuations in these theories this scale must be at least $\sim 10^{-2}/c_s$ of the Hubble time during inflation, where $c_s$ is the inflaton sound speed. We show that, in such models, the signal-to-noise ratio of the bispectrum is bounded from above by that of the power spectr…
We investigate one-loop quantum corrections to the power spectrum of adiabatic perturbation from entropy modes/adiabatic mode cross-interactions in multiple DBI inflationary models. We find that due to the non-canonical kinetic term in DBI models, the loop corrections are enhanced by slow-varying parameter $ε$ and small sound speed $c_s$. Thus, in general the loop-corrections in multi-DBI models can be large. Moreover, we find that the loop-corrections from adiabatic/entropy cross-interaction vertices are IR finite.
We consider first order perturbation theory for a non-minimally coupled inflaton field without assuming an adiabatic equation of state. In general perturbations in non-minimally coupled theory may be non-adiabatic. However under the slow-roll assumptions the perturbation theory may look like adiabatic one. We show in the frame-work of perturbation theory, that our results of spectral index and bound on no-minimal coupling parameter agree with the results obtained using the adiabatic equation of state by the earlier authors.
In recent publications we proposed one way to calculate gauge-invariant variables in the local observable universe, which is limited to a portion of the whole universe. To provide a theoretical prediction of the observable fluctuations, we need to preserve the gauge-invariance in the local universe, which is referred to the genuine gauge-invariance. The importance of the genuine gauge-invariance is highlighted in the study of primordial fluctuations in the infrared (IR) limit. The stability against IR loop corrections to primordial fluctuations is guaranteed in requesting the genuine gauge-inv…
In this paper, we investigate the primordial perturbations of inflation model induced from the multi-field mimetic gravity, where there are two field during inflation, and thus both adiabatic and isocurvature perturbation modes are generated. We show that although it is true that the original adiabatic perturbation mode loses the kinetic term due to the constraint equation, by applying the curvaton mechanism where one of the field is viewed as curvaton field, the adiabatic perturbation can actually be transferred from the isocurvature one at the end of inflation. Detailed calculations are perf…
We derive all single-field cosmologies with unit sound speed that generate scale invariant curvature perturbations on a dynamical attractor background. We identify three distinct phases: slow-roll inflation; a slowly contracting adiabatic ekpyrotic phase, described by a rapidly-varying equation of state; and a novel adiabatic ekpyrotic phase on a slowly expanding background. All of these yield identical power spectra. The degeneracy is broken at the 3-point level: unlike the nearly gaussian spectrum of slow-roll inflation, adiabatic ekpyrosis predicts large non-gaussianities on small scales.
We initiate the study of multi-field inflation using holography. Bulk light scalar fields correspond to nearly marginal operators in the boundary theory and the dual quantum field theory is a deformation of a CFT by such operators. We compute the power spectra of adiabatic and entropy perturbations in a simple model and find that the adiabatic curvature perturbation is not conserved in the presence of entropy perturbations but becomes conserved when the entropy perturbations are set to zero or the model is effectively a single scalar model, in agreement with expectations from cosmological pert…
Current data are in good agreement with the predictions of single field inflation. However, the hemispherical asymmetry seen in the cosmic microwave background data, may hint at a potential problem. Generalizing to multi-field models may provide one possible explanation. A useful way of modeling perturbations in multi-field inflation is to investigate the projection of the perturbation along and perpendicular to the background fields' trajectory. These correspond to the adiabatic and isocurvature perturbations. However, it is important to note that in general there are no corresponding adiabat…
We develop a cosmological model where primordial inflation is driven by a 'solid', defined as a system of three derivatively coupled scalar fields obeying certain symmetries and spontaneously breaking a certain subgroup of these. The symmetry breaking pattern differs drastically from that of standard inflationary models: time translations are unbroken. This prevents our model from fitting into the standard effective field theory description of adiabatic perturbations, with crucial consequences for the dynamics of cosmological perturbations. Most notably, non-gaussianities in the curvature pert…
We employ the effective field theory approach for multi-field inflation which is a generalization of Weinberg's work. In this method the first correction terms in addition to standard terms in the Lagrangian have been considered. These terms contain up to the fourth derivative of the fields including the scalar field and the metric. The results show the possible shapes of the interaction terms resulting eventually in non-Gaussianity in a general formalism. In addition generally the speed of sound is different but almost unity. Since in this method the adiabatic mode is not discriminated initia…
We develop a theory of self-resonance after inflation. We study a large class of models involving multiple scalar fields with an internal symmetry. For illustration, we often specialize to dimension 4 potentials, but we derive results for general potentials. This is the first part of a two part series of papers. Here in Part 1 we especially focus on the behavior of long wavelengths modes, which are found to govern most of the important physics. Since the inflaton background spontaneously breaks the time translation symmetry and the internal symmetry, we obtain Goldstone modes; these are the ad…
Extending the analysis of [1011.4934] beyond the bispectrum, we explore the superhorizon generation of local non-gaussianities and their subsequent approach to adiabaticity. Working with a class of two field models of inflation with potentials amenable to treatment with the delta N formalism we find that, as is the case for f_{NL}^{local}, the local trispectrum parameters tau_{NL} and g_{NL} are exponentially driven toward values which are slow roll suppressed if the fluctuations are driven into an adiabatic mode by a phase of effectively single field inflation. We argue that general considera…
We show that higher-order actions for cosmological perturbations in the multi-field DBI-inflation model are obtained by a Lorentz boost from the rest frame of the brane to the frame where the brane is moving. We confirm that this simple method provides the same third- and fourth- order actions at leading order in slow-roll and in the small sound speed limit as those obtained by the usual ADM formalism. As an application, we compute the leading order connected four-point function of the primordial curvature perturbation coming from the intrinsic fourth-order contact interaction in the multi-fie…
In the context of single-field inflation, the conservation of the curvature perturbation on comoving slices, $\R_c$, on super-horizon scales is one of the assumptions necessary to derive the consistency condition between the squeezed limit of the bispectrum and the spectrum of the primordial curvature perturbation. However, the conservation of $\R_c$ holds only after the perturbation has reached the adiabatic limit where the constant mode of $\R_c$ dominates over the other (usually decaying) mode. In this case, the non-adiabatic pressure perturbation defined in the thermodynamic sense, $δP_{na…
Instead of Lorentz invariance, gravitational degrees of freedom may obey Lifshitz scaling at high energies, as it happens in Hořava's proposal for quantum gravity. We study consequences of this proposal for the spectra of primordial perturbations generated at inflation. Breaking of 4D diffeomorphism (Diff) invariance down to the foliation-preserving Diff in Hořava-Lifshitz (HL) gravity leads to appearance of a scalar degree of freedom in the gravity sector, khronon, which describes dynamics of the time foliation. One can naively expect that mixing between inflaton and khronon will jeopardize c…
We compute the low energy effective field theory (EFT) expansion for single-field inflationary models that descend from a parent theory containing multiple other scalar fields. By assuming that all other degrees of freedom in the parent theory are sufficiently massive relative to the inflaton, it is possible to derive an EFT valid to arbitrary order in perturbations, provided certain generalized adiabaticity conditions are respected. These conditions permit a consistent low energy EFT description even when the inflaton deviates off its adiabatic minimum along its slowly rolling trajectory. By …
We study the inflation scenarios, in the framework of superstring theory, where the inflaton is an axion producing the adiabatic curvature perturbations while there exists another light axion producing the isocurvature perturbations. We discuss how the non-trivial couplings among string axions can generically arise, and calculate the consequent cross-correlations between the adiabatic and isocurvature modes through concrete examples. Based on the Planck analysis on the generally correlated isocurvature perturbations, we show that there is a preference for the existence of the correlated isocur…
We argue that $isotropic$ scalar fluctuations in solid inflation are adiabatic in the super-horizon limit. During the solid phase this adiabatic mode has peculiar features: constant energy-density slices and comoving slices do not coincide, and their curvatures, parameterized respectively by $ζ$ and $\mathcal R$, both evolve in time. The existence of this adiabatic mode implies that Maldacena's squeezed limit consistency relation holds after angular average over the long mode. The correlation functions of a long-wavelength spherical scalar mode with several short scalar or tensor modes is fixe…
We study the non-adiabatic cosmological production of ultra light dark matter (ULDM) under a minimal set of assumptions: a free ultra light real scalar as a spectator field in its Bunch-Davies vacuum state during inflation and instantaneous reheating into a radiation dominated era. For (ULDM) fields minimally coupled to gravity, non-adiabatic particle production yields a \emph{distribution function} peaked at \emph{low} comoving momentum $\mathcal{N}_k \propto 1/k^3$. The infrared behavior is a remnant of the infrared enhancement of light minimally coupled fields during inflation. We obtain th…
Constant roll inflation is analyzed in the presence of multi scalar fields which are assumed to be described by a constant roll rate each. The different cases are studied and the corresponding potentials are reconstructed. The exact solutions are obtained, which show a similar behaviour to the single scalar field model. For one of the cases analyzed in the paper, the so-called adiabatic field also constant rolls while entropy perturbations become null, while the second case may lead to non-adiabatic perturbations. Both cases can fit well the Planck data by assuming the appropriate values for t…
In this article we investigate the time evolution of the adiabatic(curvature) and isocurvature (entropy) spectral indices after end of inflation for all cosmological scales and two different initial conditions. For this purpose,we first extract an explicit equation for the time evolution of the comoving curvature perturbation (which may be known as the generalized Mukhanov-Sasaki equation). It shall be manifested that the evolution of adibatic spectral index severely depends on the intial conditions and just for the super-Hubble scales and adiabatic initial conditions is constat as be expected…
We discuss and clarify the validity of effective single field theories of inflation obtained by integrating out heavy degrees of freedom in the regime where adiabatic perturbations propagate with a suppressed speed of sound. We show by construction that it is indeed possible to have inflationary backgrounds where the speed of sound remains suppressed and slow-roll persists for long enough. In this class of models, heavy fields influence the evolution of adiabatic modes in a manner that is consistent with decoupling of physical low and high energy degrees of freedom. We emphasize the distinctio…
Adiabatic or Non-Adiabatic? Unraveling the Nature of Initial Conditions in the Cosmological Gravitational Wave Background
Università degli Studi di Padova（帕多瓦大学）；INFN, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）；INAF - Osservatorio Astronomico di Padova（意大利国家天体物理研究所帕多瓦天文台）；Gran Sasso Science Institute（格兰萨索科学研究所）；Università di Pisa（比萨大学）；INFN, Sezione di Pisa（意大利国家核物理研究所比萨分部）
We study inflationary perturbations in multiple-field models, for which zeta typically evolves until all isocurvature modes decay--the "adiabatic limit". We use numerical methods to explore the sensitivity of the nonlinear parameter fNL to the process by which this limit is achieved, finding an appreciable dependence on model-specific data such as the time at which slow-roll breaks down or the timescale of reheating. In models with a sum-separable potential where the isocurvature modes decay before the end of the slow-roll phase we give an analytic criterion for the asymptotic value of fNL to …
Single-field slow-roll inflation with a non-vacuum initial state has an enhanced bispectrum in the local limit. We numerically calculate the local-type fNL signal in the CMB that would be measured for such models (including the full transfer function and 2D projection). The nature of the result depends on several parameters, including the occupation number N_k, the phase angle θ_k between the Bogoliubov parameters, and the slow-roll parameter ε. In the most conservative case, where one takes θ_k \approx η_0 k (justified by physical reasons discussed within) and ε\lesssim 0.01, we find that 0 <…
The celebrated Weinberg theorem in cosmological perturbation theory states that there always exist two adiabatic scalar modes in which the comoving curvature perturbation is conserved on super-horizon scales. In particular, when the perturbations are generated from a single source, such as in single field models of inflation, both of the two allowed independent solutions are adiabatic and conserved on super-horizon scales. There are few known examples in literature which violate this theorem. We revisit the theorem and specify the loopholes in some technical assumptions which violate the theor…
We study slow-roll inflation and the generation of primordial fluctuations in $F(T)$ gravity's rainbow. We obtain the second order action for scalar and tensor perturbations and then calculate the primordial power spectrum for them. Thus, after calculating the inflationary observables up to first order in slow-roll approximation, namely the scalar spectral index $n_{s}$ and the tensor-to scalar ratio $r$, we confront the predictions of the model with the current PLANCK and BICEP/Keck data.
In Higgs-otic inflation a complex neutral scalar combination of the $h^0$ and $H^0$ MSSM Higgs fields plays the role of inflaton in a chaotic fashion. The potential is protected from large trans-Planckian corrections at large inflaton if the system is embedded in string theory so that the Higgs fields parametrize a D-brane position. The inflaton potential is then given by a DBI+CS D-brane action yielding an approximate linear behaviour at large field. The inflaton scalar potential is a 2-field model with specific non-canonical kinetic terms. Previous computations of the cosmological parameters…
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…
There is an apparent power deficit relative to the $Λ$CDM prediction of the CMB spectrum at large scales, which, though not yet statistically significant, persists from WMAP to Planck data. Proposals that invoke some form of initial condition for the inflation have been made to address this apparent power suppression, albeit with conflicting conclusions. By studying the curvature perturbations of a scalar field in the FLRW universe parameterized by the equation of state parameter $w$, we find that the large-scale spectrum at the end of inflation reflects the super-horizon spectrum of the initi…
We consider $P(ϕ^I,X^{IJ})$ theories of multi-field inflation and ask the question of how to define the adiabatic and entropy perturbations, widely used in calculating the curvature and isocurvature power spectra, in this general context. It is found that when the field perturbations propagate with different speeds, these adiabatic and entropy modes are not generally the fundamental (most natural to canonically quantise) degrees of freedom that propagate with a single speed. The alternative fields which do propagate with a single speed are found to be a rotation in field space of the adiabatic…
We examine the covariant properties of generalized models of two-field inflation, with non-canonical kinetic terms and a possibly non-trivial field metric. We demonstrate that kinetic-term derivatives and covariant field derivatives do commute in a proper covariant framework, which was not realized before in the literature. We also define a set of generalized slow-roll parameters, using a unified notation. Within this framework, we study the most general class of models that allows for well-defined adiabatic and entropic sound speeds, which we identify as the models with parallel momentum and …
We analyse the evolution of primordial magnetic fields in spatially flat Friedmann universes and reconsider the belief that, after inflation, these fields decay adiabatically on all scales. Without~abandoning classical electromagnetism or standard cosmology, we demonstrate that this is not necessarily the case for superhorizon-sized magnetic fields. The underlying reason for this is causality, which confines the post-inflationary process of electric-current formation, electric-field elimination and magnetic-flux freezing within the horizon. As a result, the adiabatic magnetic decay is not a~pr…
Improved quantum algorithm for calculating eigenvalues of differential operators and its application to estimating the decay rate of the perturbation distribution tail in stochastic inflation
We present a model of inflation in which the inflaton field is charged under a triplet of $U(1)$ gauge fields. The model enjoys an internal $O(3)$ symmetry supporting the isotropic FRW solution. With an appropriate coupling between the gauge fields and the inflaton field, the system reaches an attractor regime in which the gauge fields furnish a small constant fraction of the total energy density. We decompose the scalar perturbations into the adiabatic and entropy modes and calculate the contributions of the gauge fields into the curvature perturbations power spectrum. We also calculate the e…
We propose a solution to the quantum measurement problem in Inflation. Our model treats Fourier modes of cosmological perturbations as analogous to particles in a weakly-interacting Bose gas. We generalize the idea of a macroscopic wavefunction to cosmological fields, and construct a self-interaction Hamiltonian that focuses that wavefunction. By appropriately setting the coupling between modes, we obtain the standard adiabatic, scale-invariant power spectrum. Because of Central Limit Theorem (CLT), we recover a Gaussian Random Field, consistent with observations.
