In this paper we present a simple, toy model of single field inflation in which the standard non-Gaussianity consistency condition is violated. In this model the curvature perturbations on superhorizon scales are not conserved and the decaying modes of perturbations are not negligible in the non-atractor phase. As a result a large local non-Gaussianity can be obtained in the squeezed limit which violates the standard non-Gaussianity consistency condition for the single field models.

Oscillating massive fields in the primordial universe can be used as Standard Clocks. The ticks of these oscillations induce features in the density perturbations, which directly record the time evolution of the scale factor of the primordial universe, thus if detected, provide a direct evidence for the inflation scenario or the alternatives. In this paper, we construct a full inflationary model of primordial Standard Clock and study its predictions on the density perturbations. This model provides a full realization of several key features proposed previously. We compare the theoretical predictions from inflation and alternative scenarios with the Planck 2013 temperature data on Cosmic Microwave Background (CMB), and identify a statistically marginal but interesting candidate. We discuss how future CMB temperature and polarization data, non-Gaussianity analysis and Large Scale Structure data may be used to further test or constrain the Standard Clock signals. IntroductionObservationally distinguishing the inflation scenario from other possible alternative scenarios, as the origin of the Big Bang, remains an outstanding challenge for modern astrophysics and cosmology. The simplest inflation models [1-10] have received strong support from observational results on Cosmic Microwave Background (CMB) and Large Scale Structures (LSS) [11][12][13]. In the meanwhile, there are also various attempts to construct alternative scenarios to explain the same observational results, see [14][15][16][17] for recent discussions. One of the reasons that such attempts are still possible is that there are only two parameters in the Standard Model of cosmology that are related to the primordial scenario, namely the amplitude and spectral index of the density perturbations. Therefore, an important research activity is to explore signatures beyond the Standard Model of cosmology, both in theories and most importantly in experiments.While all observational signatures beyond the Standard Model are extremely valuable, not many of them can be used to model-independently distinguish inflation from the alternatives. Many signatures, if discovered, provide information that are helpful to distinguish different models within one scenario.So far, there are two kinds of observational signals that are known to be capable of model-independently distinguishing the inflation scenario from the alternatives.The first and well-known one is the primordial gravitational wave [18][19][20], namely the tensor mode, which may be detected in terms of the B-mode polarization in the CMB [21][22][23]. The tensor mode records the magnitude of the Hubble rate during the primordial epoch, therefore if detected distinguishes scenarios with fast-evolving scale factors, such as inflation, from scenarios with slowly-evolving scale factors, such as Ekpyrosis [24]. However, phenomenologically, it is difficult to use the tensor mode to model-independently distinguish scenarios that all have fast-evolving scale factors and generate scale-invariant tensor modes. An example is...

A detection of large local form non-Gaussianity is considered to be able to rule out all single field inflation models. This statement is based on a single field consistency condition. Despite the awareness of some implicit assumptions in the derivation of this condition and the demonstration of corresponding examples that illustrate these caveats, to date there is still no explicit and selfconsistent model which can serve as a counterexample to this statement. We present such a model in this Letter.Primordial non-Gaussianity is an important probe of the inflation models. Different properties of non-Gaussian correlation functions can reveal different physics of the underlying models . The non-Gaussianity is usually parameterized by [1-3]where R is the curvature perturbation on comoving slices, P R (k) is its power spectrum per unit logarithmic momentum interval, and f N L is the amplitude while the function F (k 1 , k 2 , k 3 ) is the shape of non-Gaussianity. In this Letter we concentrate on the so-called local type non-Gaussianity with the following shapewhere "perm" stands for the cyclic permutation of three momenta. This shape peaks at the squeezed limit. In contrast, other shapes that we will mention later include the equilateral-like shapewhich peaks at the equilateral limit and typically arises in non-slow-roll models with non-canonical kinetic term; and the folded-shapewhich peaks at the folded limit (k 1 + k 2 = k 3 , and cyclic) and qualitatively describes certain features in models with non-Bunch-Davies vacuum.There is an important statement on how the local non-Gaussianity can be used to distinguish inflation models: 0) A detection of a large local non-Gaussian component in the bispectrum can rule out all single field inflation models [4,5].The size of local non-Gaussianity which can be detected with high confidence level in the near future is f loc N L ≫ 1. By single field inflation models, we include not only the slow-roll single field models with Bunch-Davies (BD) vacuum, but also all other inflation models that have one field responsible for both the inflation and creation of curvature perturbation. The statement 0) is based on Maldacena's consistency condition for the single field models [6,4],in which k 3 ≪ k 1 = k 2 and n s is the spectral index. Since the momentum-dependence on the RHS of (5), ∼ 1/k 3 1 k 3 3 , takes the scale-invariant local form, the size of the local non-Gaussianity in the squeezed limit (k 3 ≪ k 1 = k 2 ) is f loc N L ∼ 1−n s , which is of order the slow-variation parameter O(ǫ) ∼ O(0.01) at the leading and non-oscillatory order. Therefore these models predict very small local non-Gaussianity.The derivation of this condition relies on a very general assumption: for single field, the only effect of a long wavelength mode on short wavelength modes is to provide a constant rescaling of the background scale factor. Nonetheless, despite of its generality, it has been noticed that there are a couple of implicit assumptions underlying the derivation of this condition [1,7,8]:A) T...

Non-attractor inflation is known as the only single field inflationary scenario that can violate non-Gaussianity consistency relation with the Bunch-Davies vacuum state and generate large local non-Gaussianity. However, it is also known that the non-attractor inflation by itself is incomplete and should be followed by a phase of slow-roll attractor. Moreover, there is a transition process between these two phases. In the past literature, this transition was approximated as instant and the evolution of non-Gaussianity in this phase was not fully studied. In this paper, we follow the detailed evolution of the non-Gaussianity through the transition phase into the slow-roll attractor phase, considering different types of transition. We find that the transition process has important effect on the size of the local non-Gaussianity. We first compute the net contribution of the non-Gaussianities at the end of inflation in canonical non-attractor models. If the curvature perturbations keep evolving during the transition -such as in the case of smooth transition or some sharp transition scenarios -the O(1) local non-Gaussianity generated in the non-attractor phase can be completely erased by the subsequent evolution, although the consistency relation remains violated. In extremal cases of sharp transition where the super-horizon modes freeze immediately right after the end of the non-attractor phase, the original non-attractor result can be recovered. We also study models with non-canonical kinetic terms, and find that the transition can typically contribute a suppression factor in the squeezed bispectrum, but the final local non-Gaussianity can still be made parametrically large.

In this paper, we point out and study a generic type of signals existing in the primordial universe models, which can be used to model-independently distinguish the inflation scenario from alternatives. These signals are generated by massive fields that function as standard clocks. The role of massive fields as standard clocks has been realized in previous works. Although the existence of such massive fields is generic, the previous realizations require sharp features to classically excite the oscillations of the massive clock fields. Here, we point out that the quantum fluctuations of massive fields can actually serve the same purpose as the standard clocks. We show that they are also able to directly record the defining property of the scenario type, namely, the scale factor of the primordial universe as a function of time a(t), but through shape-dependent oscillatory features in non-Gaussianities. Since quantum fluctuating massive fields exist in any realistic primordial universe models, these quantum primordial standard clock signals are present in any inflation models, and should exist quite generally in alternative-to-inflation scenarios as well. However, the amplitude of such signals is very model-dependent.

Standard Clocks in the primordial epoch leave a special type of features in the primordial perturbations, which can be used to directly measure the scale factor of the primordial universe as a function of time a(t), thus discriminating between inflation and alternatives. We have started to search for such signals in the Planck 2013 data using the key predictions of the Standard Clock. In this Letter, we summarize the key predictions of the Standard Clock and present an interesting candidate example in Planck 2013 data. Motivated by this candidate, we construct and compute full Standard Clock models and use the more complete prediction to make more extensive comparison with data. Although this candidate is not yet statistically significant, we use it to illustrate how Standard Clocks appear in Cosmic Microwave Background (CMB) and how they can be further tested by future data. We also use it to motivate more detailed theoretical model building.

In this paper we develop a formalism for studying the nonrelativistic limit of relativistic field theories in a systematic way. By introducing a simple, nonlocal field redefinition, we transform a given relativistic theory, describing a real, self-interacting scalar field, into an equivalent theory, describing a complex scalar field that encodes at each time both the original field and its conjugate momentum. Our low-energy effective theory incorporates relativistic corrections to the kinetic energy as well as the backreaction of fast-oscillating terms on the behavior of the dominant, slowly varying component of the field. Possible applications of our new approach include axion dark matter, though the methods developed here should be applicable to the low-energy limits of other field theories as well.

In non-attractor single-field inflation models producing a scale-invariant power spectrum, the curvature perturbation on super-horizon scales grows as R ∝ 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 results agree, giving the local-form bispectrum parameter of f local NL = 5(1 + c 2 s )/(4c 2 s ). This result is valid for arbitrary values of the speed of sound parameter, cs, for a particular non-attractor model we consider in this paper.