On the consistency relation of the 3-point function in single field inflation (original) (raw)
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The consistency condition for the three-point function in dissipative single-clock inflation
Journal of Cosmology and Astroparticle Physics, 2012
We generalize the consistency condition for the three-point function in single field inflation to the case of dissipative, multi-field, single-clock models. We use the recently introduced extension of the effective field theory of inflation that accounts for dissipative effects, to provide an explicit proof to leading (non-trivial) order in the generalized slow roll parameters and mixing with gravity scales. Our results illustrate the conditions necessary for the validity of the consistency relation in situations with many degrees of freedom relevant during inflation, namely that there is a preferred clock. Departures from this condition in forthcoming experiments would rule out not only single field but also a large class of multi-field models.
International Journal of Modern Physics, 2009
Inflation is today a part of the Standard Model of the Universe supported by the cosmic microwave background (CMB) and large scale structure (LSS) datasets. Inflation solves the horizon and flatness problems and naturally generates density fluctuations that seed LSS and CMB anisotropies, and tensor perturbations (primordial gravitational waves). Inflation theory is based on a scalar field ϕ (the inflaton) whose potential is fairly flat leading to a slow-roll evolution. This review focuses on the following new aspects of inflation. We present the effective theory of inflation à la Ginsburg-Landau in which the inflaton potential is a polynomial in the field ϕ and has the universal form , where w = O(1), M ≪ M P l is the scale of inflation and N ∼ 60 is the number of efolds since the cosmologically relevant modes exit the horizon till inflation ends. The slow-roll expansion becomes a systematic 1/N expansion and the inflaton couplings become naturally small as powers of the ratio (M/M P l ) 2 . The spectral index and the ratio of tensor/scalar fluctuations are ns -1 = O(1/N ), r = O(1/N ) while the running index turns to be dns/d ln k = O(1/N 2 ) and therefore can be neglected. The energy scale of inflation M ∼ 0.7 × 10 16 GeV is completely determined by the amplitude of the scalar adiabatic fluctuations. A complete analytic study plus the Monte Carlo Markov Chains (MCMC) analysis of the available CMB+LSS data (including WMAP5) with fourth degree trinomial potentials showed: (a) the spontaneous breaking of the ϕ → -ϕ symmetry of the inflaton potential. (b) a lower bound for r in new inflation: r > 0.023 (95% CL) and r > 0.046 (68% CL). (c) The preferred inflation potential is a double well, even function of the field with a moderate quartic coupling yielding as most probable values: ns ≃ 0.964, r ≃ 0.051. This value for r is within reach of forthcoming CMB observations. The present data in the effective theory of inflation clearly prefer new inflation. Study of higher degree inflaton potentials show that terms of degree higher than four do not affect the fit in a significant way. In addition, horizon exit happens for ϕ/[ √ N M P l ] ∼ 0.9 making higher order terms in the potential w negligible. We summarize the physical effects of generic initial conditions (different from Bunch-Davies) on the scalar and tensor perturbations during slow-roll and introduce the transfer function D(k) which encodes the observable initial conditions effects on the power spectra. These effects are more prominent in the low CMB multipoles: a change in the initial conditions during slow roll can account for the observed CMB quadrupole suppression. Slowroll inflation is generically preceded by a short fast-roll stage. Bunch-Davies initial conditions are the natural initial conditions for the fast-roll perturbations. During fast-roll, the potential in the wave equations of curvature and tensor perturbations is purely attractive and leads to a suppression of the curvature and tensor CMB quadrupoles. A MCMC analysis of the WMAP+SDSS data including fast-roll shows that the quadrupole mode exits the horizon about 0.2 efold before fastroll ends and its amplitude gets suppressed. In addition, fast-roll fixes the initial inflation redshift to be zinit = 0.9 × 10 56 and the total number of efolds of inflation to be Ntot ≃ 64. Fast-roll fits the TT, the TE and the EE modes well reproducing the quadrupole supression. A thorough study of the quantum loop corrections reveals that they are very small and controlled by powers of (H/M P l ) 2 ∼ 10 -9 , a conclusion that validates the reliability of the effective theory of inflation. The present review shows how powerful is the Ginsburg-Landau effective theory of inflation in predicting observables that are being or will soon be contrasted to observations. Contents I. Introduction to the Effective Theory of Inflation A. Overview and present status of inflation B. The Standard Cosmological Model C. The Horizon and Flatness problems in non-inflationary cosmology and their inflationary resolution. 1. The horizon problem 2. The flatness problem 3. The solution to the horizon problem in inflation 4. The solution to the flatness problem in inflation 5. The Entropy of the Universe 6. The Age of the Universe D. Inflationary Dynamics in the Effective Theory of Inflation 1. Inflation and Inflaton field dynamics 2. Slow-roll, the Universal Form of the Inflaton Potential and the Energy Scale of Inflation 3. Inflationary Dynamics: Homogeneous Inflaton 4. Fixing the Total Number of Inflation e-folds from Fast-Roll and the CMB Quadrupole suppression E. Gauge invariant Scalar and Tensor Fluctuations 1. Scalar Curvature Perturbations 2. Tensor Perturbations 3. Initial conditions 4. The power spectrum of adiabatic scalar and tensor perturbations 5. The energy scale of inflation and the quasi-scale invariance during inflation. II. Theoretical predictions, MCMC data analysis, early fast-roll stage and CMB quadrupole suppression. A. Ginsburg-Landau polynomial realizations of the Inflaton Potential 1. Binomial inflaton potentials for chaotic inflation 2. Binomial inflaton potentials for new inflation 3. Contrasting the results of new and chaotic binomial inflation. B. Trinomial Chaotic Inflation: Spectral index, amplitude ratio, running index and limiting cases 1. The small asymmetry regime: -1 < h ≤ 0. 2. The flat potential limit h → -1 + 3. The singular limit z = 1 and then h → -1 + yields the Harrison-Zeldovich spectrum 4. The high asymmetry h < -1 regime. C. Trinomial New Inflation: Spectral index, amplitude ratio, running index and limiting cases 1. The weak coupling limit y → 0 2. The strong coupling limit y → ∞ 3. The extremely asymmetric limit |h| → ∞ 4. Regions of n s and r covered by New Inflation and by Chaotic Inflation. D. The Monte Carlo Markov Chain Method of Data Analysis 1. CMB, LSS and Inflation within a MCMC analysis. 2. MCMC results for Trinomial New Inflation. 3. MCMC results for Chaotic Trinomial Inflation. E. Higher degree terms in inflaton potentials 1. Family of models 2. Broken Symmetry models. 3. Field reconstruction for new inflation 4. Chaotic inflation models. 5. Field reconstruction for chaotic inflation 6. Conclusions
Consistency equation hierarchy in single-field inflation models
Physical Review D, 2006
Inflationary consistency equations relate the scalar and tensor perturbations. We elucidate the infinite hierarchy of consistency equations of single-field inflation, the first of which is the well-known relation A 2 T /A 2 S = −nT/2 between the amplitudes and the tensor spectral index. We write a general expression for all consistency equations both to first and second-order in the slow-roll expansion. We discuss the relation to other consistency equations that have appeared in the literature, in particular demonstrating that the approximate consistency equation recently introduced by Chung and collaborators is equivalent to the second consistency equation of .
Inflationary signatures of single-field models beyond slow-roll
If the expansion of the early Universe was not purely de Sitter, the statistical imprints of the primordial density perturbation on the cosmic microwave background can be quite different from those following slow-roll inflation. In this paper we study the inflationary signatures of all single-field models not plagued by ghost-like instabilities. We assume small deviations from exact scale-invariance, as supported by current cosmological constraints, allow for a rapid change of the Hubble parameter and the phase speed of scalar fluctuations. We obtain the propagator of scalar fluctuations and compute the bispectrum, keeping next-order corrections proportional to the deviation of the spectral index from unity. These theories offer an explicit example where the shape and scale dependences of the bispectrum are highly non-trivial for reasonable breaking of slow-roll.
The importance of slow-roll corrections during multi-field inflation
Journal of Cosmology and Astroparticle Physics, 2012
We re-examine the importance of slow-roll corrections during the evolution of cosmological perturbations in models of multi-field inflation. We find that in many instances the presence of light degrees of freedom leads to situations in which next to leading order slow-roll corrections become significant. Examples where we expect such corrections to be crucial include models in which modes exit the Hubble radius while the inflationary trajectory undergoes an abrupt turn in field space, or during a phase transition. We illustrate this with two examples -hybrid inflation and double quadratic inflation. Utilizing both analytic estimates and full numerical results, we find that corrections can be as large as 20%. Our results have implications for many existing models in the literature, as these corrections must be included to obtain accurate observational predictions -particularly given the level of accuracy expected from CMB experiments such as Planck.
The Effective Field Theory of Inflation
2007
We study the effective field theory of inflation, i.e. the most general theory describing the fluctuations around a quasi de Sitter background, in the case of single field models. The scalar mode can be eaten by the metric by going to unitary gauge. In this gauge, the most general theory is built with the lowest dimension operators invariant under spatial diffeomorphisms, like g 00 and K µν , the extrinsic curvature of constant time surfaces. This approach allows us to characterize all the possible high energy corrections to simple slow-roll inflation, whose sizes are constrained by experiments. Also, it describes in a common language all single field models, including those with a small speed of sound and Ghost Inflation, and it makes explicit the implications of having a quasi de Sitter background. The non-linear realization of time diffeomorphisms forces correlation among different observables, like a reduced speed of sound and an enhanced level of non-Gaussianity.
Consistency relations in multi-field inflation
Journal of Cosmology and Astroparticle Physics
We study the consequences of spatial coordinate transformation in multi-field inflation. Among the spontaneously broken de Sitter isometries, only dilatation in the comoving gauge preserves the form of the metric and thus results in quantum-protected Slavnov-Taylor identities. We derive the corresponding consistency relations between correlation functions of cosmological perturbations in two different ways, by the connected and oneparticle-irreducible Green's functions. The lowest-order consistency relations are explicitly given, and we find that even in multi-field inflation the consistency relations in the soft limit are independent of the detail of the matter sector.
2007
We study the effective field theory of inflation, i.e. the most general theory describing the fluctuations around a quasi de Sitter background, in the case of single field models. The scalar mode can be eaten by the metric by going to unitary gauge. In this gauge, the most general theory is built with the lowest dimension operators invariant under spatial diffeomorphisms, like g 00 and K�ν , the extrinsic curvature of constant time surfaces. This approach allows us to characterize all the possible high energy corrections to simple slow-roll inflation, whose sizes are constrained by experiments. Also, it describes in a common language all single field models, including those with a small speed of sound and Ghost Inflation, and it makes explicit the implications of having a quasi de Sitter background. The non-linear realization of time diffeomorphisms forces correlation among different observables, like a reduced speed of sound and an enhanced level of non-Gaussianity.
Possibly Large Corrections to the Inflationary Observables
Modern Physics Letters A, 2008
We point out that the theoretical predictions for the inflationary observables may be generically altered by the presence of fields which are heavier than the Hubble rate during inflation and whose dynamics is usually neglected. They introduce corrections which may be easily larger than both the second-order contributions in the slow-roll parameters and the accuracy expected in the forthcoming experiments.
CMB and LSS constraints on a single-field model of inflation
EPL (Europhysics Letters), 2008
A new inflationary scenario whose exponential potential V (Φ) has a quadratic dependence on the field Φ in addition to the standard linear term is confronted with the tree-year observations of the Wilkinson-Microwave Anisotropy Probe and the Sloan Digital Sky Survey data. The number of e-folds (N), the ratio of tensor-to-scalar perturbations (r), the spectral scalar index of the primordial power spectrum (ns) and its running (dns/d ln k) depend on the dimensionless parameter α multiplying the quadratic term in the potential. In the limit α → 0 all the results of the standard exponential potential are fully recovered. For values of α = 0, we find that the model predictions are in good agreement with the current observations of the Cosmic Microwave Background (CMB) anisotropies and Large-Scale Structure (LSS) in the Universe.