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.

We point out the existence of a consistency relation involving the 3-point function of scalar perturbations which is valid in any inflationary model, independently of the inflaton Lagrangian under the assumption that the inflaton is the only dynamical field. The 3-point function in the limit in which one of the momenta is much smaller than the other two is fixed in terms of the power spectrum and its tilt. This relation, although very hard to verify experimentally, could be easily proved wrong by forecoming data, thus ruling out any scenario with a single dynamical field in a model independent way. PACS numbers: 98.80.Cq In the last few years various modifications of the single field slow-roll inflation scenario have been proposed. The basic mechanism for solving the standard cosmological problems remains unaltered: a rapid expansion with an approximately constant Hubble parameter. What is different in these recent models are the characteristics of the produced density perturbations. These are modified if we change the inflaton dynamics with respect to the minimal slow-roll case, for example with higher derivative terms in the inflaton Lagrangian [1,2,3,4] or adding sharp features in the inflaton potential [5]. Another possibility is to assume that density perturbations are created by another field, different from the inflaton, whose quantum fluctuations are finally converted into adiabatic perturbations [6,7].These alternative models generically give distinctive predictions for the shape of the scalar spectrum, the gravitational wave contribution and for the scalar 3-point function, which should allow to distinguish them from a minimal slow-roll model. Without much theoretical guidance about which kind of modification is more likely, it would be extremely useful to make model independent statements about the observable quantities.The purpose of this letter is to indicate the existence of a consistency relation involving the 3-point function of scalar perturbations, which is valid in any inflationary model irrespectively of the inflaton dynamics ( 1 ), under the assumption that the inflaton is the only dynamical field during inflation. It is not based on any slow-roll approximation and it is valid for any inflaton Lagrangian. We will show that the consistency relation is in some sense kinematical, being just a consequence of the assumption that there are no other fields which evolve (classically or quantum mechanically) during inflation. The inflaton is the only "clock of the Universe" and it fixes the Hubble parameter: fluctuations of the inflaton are 1 A different kind of consistency relation valid in a particular set of models has been recently pointed out in [8].therefore equivalent to a relative rescaling of the scale factor in different parts of the Universe. The consistency relation relates a particular geometrical limit of the 3-point function of density perturbations to the spectrum and tilt of the 2-point function:On the left-hand side we have the 3-point function of the ζ variable, which is the non-...

We present a consistent effective theory that violates the null energy condition (NEC) without developing any instabilities or other pathological features. The model is the ghost condensate with the global shift symmetry softly broken by a potential. We show that this system can drive a cosmological expansion withḢ > 0. Demanding the absence of instabilities in this model requireṡ H < ∼ H 2 . We then construct a general low-energy effective theory that describes scalar fluctuations about an arbitrary FRW background, and argue that the qualitative features found in our model are very general for stable systems that violate the NEC. Violating the NEC allows dramatically non-standard cosmological histories. To illustrate this, we construct an explicit model in which the expansion of our universe originates from an asymptotically flat state in the past, smoothing out the big-bang singularity within control of a low-energy effective theory. This gives an interesting alternative to standard inflation for solving the horizon problem. We also construct models in which the present acceleration has w < −1; a periodic ever-expanding universe; and a model with a smooth "bounce" connecting a contracting and expanding phase.

We study the dependence on configuration in momentum space of the primordial 3-point function of density perturbations in several different scenarios: standard slow-roll inflation, curvaton and variable decay models, ghost inflation, models with higher derivative operators and the DBI model of inflation. We define a cosine between the distributions using a measure based on the ability of experiments to distinguish between them. We find that models fall into two broad categories with fairly orthogonal distributions. Models where non-Gaussianity is created at horizon-crossing during inflation and models in which the evolution outside the horizon dominates. In the first case the 3-point function is largest for equilateral triangles, while in the second the dominant contribution to the signal comes from the influence of long wavelength modes on small wavelength ones. We show that, because the distributions in these two cases are so different, translating constraints on parameters of one model to those of another based on the normalization of the 3-point function for equilateral triangles can be very misleading. rather robust. Although the precise number is model-dependent, in most models |n − 1| is of order 1/N e , where N e is the number of e-folds to the end of inflation when relevant scales exit the horizon. Present limits are of order |n − 1| 0.05 (e.g. [1,2,3]), so that we are entering in the interesting region. A deviation from a flat spectrum would strongly support the slow-roll inflation picture and it would allow to distinguish it from 'ghost inflation' [4] for example, where |n − 1| is expected to be negligible. However, if no tilt is detected slow-roll inflation cannot be safely ruled out: it is easy to build models with a tilt as small as we like.Gravity wave (GW) contribution. The contribution of GWs is directly related to the value of the Hubble constant H during inflation. The detection of a GW signal would therefore point towards models with big vacuum energy (V 1/4 10 16 GeV). Inflationary models fall into two broad categories. Models with small vacuum energy (which is equivalent to a very small ǫ, ǫ ≪ 1/N e , as H/(M P √ ǫ) is fixed by the spectrum normalization) with totally negligible productions of GWs and models with big vacuum energy (usually with ǫ ∼ η ∼ 1/N e ), where the GW contribution should be close to the present experimental limit, r 0.5 (e.g. [1,2]) . The distinction is quite sharp because the two categories can also be distinguished by the variation of the inflaton field during inflation: much smaller than the Planck scale in the first case, comparable to the Planck scale in the second. A possible criticism against models with a sensible production of GWs is that a variation of the inflaton field much bigger than M P seems out of control of the effective field theory [5]. Extra dimensional UV completions provide examples in which this is not true [6]. On the other hand models with very small ǫ have been considered unnatural as they require a hierarchy between the two slow-roll paramet...

The observation of GW170817 and its electromagnetic counterpart implies that gravitational waves travel at the speed of light, with deviations smaller than a few×10^{-15}. We discuss the consequences of this experimental result for models of dark energy and modified gravity characterized by a single scalar degree of freedom. To avoid tuning, the speed of gravitational waves must be unaffected not only for our particular cosmological solution but also for nearby solutions obtained by slightly changing the matter abundance. For this to happen, the coefficients of various operators must satisfy precise relations that we discuss both in the language of the effective field theory of dark energy and in the covariant one, for Horndeski, beyond Horndeski, and degenerate higher-order theories. The simplification is dramatic: of the three functions describing quartic and quintic beyond Horndeski theories, only one remains and reduces to a standard conformal coupling to the Ricci scalar for Horndeski theories. We show that the deduced relations among operators do not introduce further tuning of the models, since they are stable under quantum corrections.