Global adiabaticity and non-Gaussianity consistency condition

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, $\delta P_{nad}\equiv\delta P-c_w^2\delta\rho$ where $c_w^2=\dot P/\dot\rho$, usually becomes also negligible on superhorizon scales. Therefore one might think that the adiabatic limit is the same as thermodynamic adiabaticity. This is in fact not true. In other words, thermodynamic adiabaticity is not a sufficient condition for the conservation of $\R_c$ on super-horizon scales. In this paper, we consider models that satisfy $\delta P_{nad}=0$ on all scales, which we call global adiabaticity (GA), which is guaranteed if $c_w^2=c_s^2$, where $c_s$ is the phase velocity of the propagation of the perturbation. A known example is the case of ultra-slow-roll(USR) inflation in which $c_w^2=c_s^2=1$. In order to generalize USR we develop a method to find the Lagrangian of GA K-inflation models from the behavior of background quantities as functions of the scale factor. Applying this method we show that there indeed exists a wide class of GA models with $c_w^2=c_s^2$, which allows $\R_c$ to grow on superhorizon scales, and hence violates the non-Gaussianity consistency condition.

In the context of single-field inflation, the conservation of the curvature perturbation on comoving slices, Rc, 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 Rc holds only after the perturbation has reached the adiabatic limit where the constant mode of Rc dominates over the other (usually decaying) mode. In this case, the non-adiabatic pressure perturbation defined in the thermodynamic sense, δP nad ≡ δP − c 2 w δρ where c 2 w =Ṗ /ρ, usually becomes also negligible on superhorizon scales. Therefore one might think that the adiabatic limit is the same as thermodynamic adiabaticity. This is in fact not true. In other words, thermodynamic adiabaticity is not a sufficient condition for the conservation of Rc on super-horizon scales. In this paper, we consider models that satisfy δP nad = 0 on all scales, which we call global adiabaticity (GA), which is guaranteed if c 2 w = c 2 s , where cs is the phase velocity of the propagation of the perturbation. A known example is the case of ultra-slow-roll(USR) inflation in which c 2 w = c 2 s = 1. In order to generalize USR we develop a method to find the Lagrangian of GA K-inflation models from the behavior of background quantities as functions of the scale factor. Applying this method we show that there indeed exists a wide class of GA models with c 2 w = c 2 s , which allows Rc to grow on superhorizon scales, and hence violates the non-Gaussianity consistency condition.

I. INTRODUCTION
A period of accelerated expansion during the early stages of the evolution of the Universe, called inflation [1][2][3], is able to account for several otherwise difficult to explain features of the observed Universe such the high level of isotropy of the cosmic microwave background (CMB) [4] radiation and the small value of the curvature. Some of the simplest inflationary models are based on a single slowly-rolling scalar field, and they are in good agreement with observations. It is commonly assumed in slow-roll models that adiabaticity in the thermodynamic sense, δP nad ≡ δP − c 2 w δρ = 0 where c 2 w =Ṗ /ρ, implies the conservation of the curvature perturbation on uniform density slices ζ, and hence the conservation of the curvature perturbation on comoving slices R c , on superhorizon scales.
In [5] it was shown that there can be important exceptions, i.e. in some cases thermodynamic adiabaticity does not necessarily imply the super-horizon conservation of R c and ζ, and that they can differ from each other. This can happen even for models in which c 2 w = c 2 s . An example is ultra-slow-roll (USR) inflation [6,7], which has exact adiabaticity δP nad = 0 on all scales. In USR inflation, both R c and ζ exhibit super-horizon growth but their behavior is very different from each other. As has been stressed in [8], the non-freezing of R c has important phenomenological consequences. Since the freezing of R c on superhorizon scales is a necessary ingredient [9] for Maldacena's consistency relation [10] to hold, models that do not conserve R c can actually violate that consistency condition.
In this paper focusing on K-inflation, i.e., Einsteinscalar models with a general kinetic term, we explore in a general way other single field models which have c 2 w = c 2 s , hence satisfy δP nad = 0 on all scales which we call globally adiabatic (GA), but which may not conserve R c . We find a generalization of the USR model. A different generalization without imposing the condition c 2 w = c 2 s was discussed in [11,12]. The method we adopt is based on establishing a general condition for the non-conservation of R c in terms of the dependence of the background quantities, in particular the slow-roll parameter ǫ ≡ −Ḣ/H 2 and the sound velocity c s , on the scale factor a.
We first derive the necessary condition for the comoving curvature perturbation R c to grow on superhorizon scales. Next we determine ρ(a) and P (a) by solving the continuity equation. Then using the equivalence between barotropic fluids and K-inflationary models which satisfy the condition c 2 w = c 2 s [13,14], we determine the corresponding Lagrangian for the equivalent scalar field model. Using this method we obtain a new class of GA scalar field models which do not conserve R c .
Throughout the paper we denote the proper-time derivative by a dot (˙= d/dt), the conformal-time derivative by a prime (′ = d/dη = a d/dt) and the Hubble expansion rates in proper and conformal times by H =ȧ/a and H = a ′ /a, respectively. We also use the terminology "adiabaticity" for thermodynamic adiabaticity δP nad = 0 throughout the paper.

II. CONSERVATION OF Rc AND GLOBAL ADIABATICITY
We set the perturbed metric as In [5] it was shown that independently of the gravity theory and for generic matter the energy-momentum conservation equations imply where the subscript c means a quantity evaluated on comoving slices defined by δT i 0 = 0 (or equivalently slices on which the scalar field is homogeneous). In the case of general relativity, the additional relation A c =Ṙ c /H gives an important relation for the time derivative of R c The non-adiabatic pressure perturbation is given according to its thermodynamics definition This definition of δP nad is important because it is gauge invariant and δP nad = δP ud , where δP ud is the pressure perturbation on uniform density slices. It appears in the equation for the curvature perturbation on uniform density slices ζ ≡ R ud obtained from the energy conservation law [15], where v is the 3-velocity potential (v = δφ/φ ′ for a scalar field). In general, the curvature perturbations on uniform density and comoving slices are related as A common interpretation of these equations (see for example [16,17]) is that when δP nad = 0 with c 2 w = c 2 s , ζ and R c are equal because of eq. (6), and they are both conserved on super-horizon scales because of eq. (3). The equation (3) is the key relation to understand how R c depends on the non-adiabatic pressure δP nad . First of all let us note that this equation is valid on any scale. The advantage of it with respect to eq. (5) is that it does not involve gradient terms, so it allows us to directly relate δP nad toṘ c if c 2 w = c 2 s , while in eq. (5)ζ depends on spatial gradients, which in the case of USR are not negligible on super-horizon scales [5]. This explains while in USR in which c 2 w = c 2 s = 1, both R c and ζ are not conserved despite δP nad = 0.
It should be noted here that for slow-roll attractor models c 2 w = c 2 s in general, and R c is time-varying on sub-horizon scales. This implies that the non-adiabatic pressure perturbation δP nad on sub-horizon scales is not zero. In other words, the attractor models are adiabatic only on super-horizon scales, and we call these models super-horizon adiabatic (SHA).
From eq. (3) we can immediately deduce that in general relativity there are two possible scenarios for the non-conservation of R c , The second case was studied in [11,12]. Here we focus on the first case. It is trivial to see that because of the gauge invariance of δP nad the condition c 2 w = c 2 s automatically implies δP nad = 0. The models satisfying the condition c 2 s − c 2 w = δP nad = 0 are adiabatic on any scale, and because of this we call them globally adiabatic (GA). In GA models an explicit calculation can reveal the superhorizon behavior of R c , and ζ, as was shown in [5] in the case of USR. Below, we develop an inversion method to find a new class of models that violate the conservation of R c without solving the perturbations equations.

III. GLOBALLY ADIABATIC K-ESSENCE MODELS
The condition c 2 w = c 2 s has been studied in the context of K-inflation [13] described by the action and it was shown that it is satisfied by scalar field models with the Lagrangian of the form, These models are equivalent to a barotropic perfect fluid, i.e. a fluid with equation of state P (ρ). See also [18][19][20][21]. We note again that these models are adiabatic on any scale (GA), contrary to the slow-roll attractor models, which are adiabatic only on super-horizon scales (SHA). The fact that they are mutually exclusive can be readily seen by considering the hypothetical case of δP nad = 0 and c 2 w = c 2 s . In this case Eq. (3) which is valid on any scale would mean R c should be frozen on all scales. In contrast, the condition c 2 w = c 2 s allows for the curvature perturbation to evolve both on sub-horizon and superhorizon scales.
In [13] it was shown that is possible to associate any barotropic perfect fluid with an equivalent K-inflation model according to where F (P ) = ρ(P ) + P and Y = g(φ)X with X = −g µν ∂ µ φ∂ ν φ/2. These models are the ones which could violate the conservation of R c for adiabatic perturbations, since they satisfy c 2 w = c 2 s . It is noted of course that the global adiabaticity is not the sufficient condition for the non-conservation of R c . Not all GA models violate the conservation of R c on super-horizon scales.

IV. GENERAL CONDITIONS FOR SUPER-HORIZON GROWTH OF Rc
From the equation for the curvature perturbation on comoving slices, we can deduce, after re-expressing the time derivative in terms of the derivative respect to the scale factor a, that on superhorizon scales there is (apart from a constant solution) a solution of the form, where we have introduced the function f (a) for later convenience. In conventional slow-roll inflation c 2 s and ǫ are both slowly varying, hence the integral rapidly approaches a constant, rendering R c conserved. The time dependent part of the above solution corresponds to the decaying mode.
The necessary and sufficient condition for superhorizon freezing is that there exists some δ > 0 for which lim a→∞ a δ f (a) = 0.
By definition of inflation, H must be sufficiently slowly varying; ǫ = −Ḣ/H 2 ≪ 1. So we may neglect the time dependence of H in (12) at leading order, while ǫ and c 2 s may vary rapidly in time. For models for which ǫ ≈ a −n and c 2 s ≈ a q we get hence the condition for freezing is If this condition is violated, i.e. q + n − 3 ≥ 0, then the solution (12) will grow on super-horizon scales. This happens for example in USR., which corresponds to c 2 s = 1 and ǫ ∝ a −6 , i.e. q = 0, and n = 6. (The superhorizon growth of R c in USR can also be understood as a direct consequence of the non-attractor nature of USR [22].) In general, we expect that q would not become very large. This implies ǫ should decrease sufficiently rapidly. Conversely, if ǫ decreases sufficiently rapidly, then the growth of R c on superhorizon scales will follow.

V. BAROTROPIC MODEL
We have shown that GA models could violate the super-horizon conservation of R c , so now we will look for GA K-essence models which do indeed violate it, based on the freezing condition in eq. (13). Inspired by the equivalence between barotropic fluids and GA K-essence models [13] we will first look for barotropic fluids that can give the growing curvature perturbation on superhorizon scales. From the very beginning we will set c 2 w = c 2 s . Using the Friedmann equation we can write the slowroll parameter ǫ as In terms of the scale factor and ǫ the energy conservation equation reads We may now define the quantity b(a) = 2ǫρ. It appears naturally in the continuity equation and plays a crucial role in regards to the super-horizon behavior of curvature perturbations because the function f (a) can be re-written in terms of it as Integrating the energy conservation equation we get Using eq. (16), we then obtain The sound velocity is given by We now consider the behavior of f (a) introduced in (12). As mentioned before, we consider the case when ǫ decreases sufficiently rapidly. In this case, ρ = 3H 2 M 2 P approaches a constant rapidly. Hence the time dependence of ρ may be neglected compared to that of other quantities that vary far more rapidly. With this approximation, assuming ǫ ∝ a −n , we find which means q ≈ 0, and which satisfies the condition for the growth if n > 3, in accordance with the original anticipation. In passing, it is interesting to note that the condition n > 3 implies c 2 s > 0, a necessary condition to avoid the gradient instability of the perturbation. Thus virtually all GA models that are free from the gradient instability exhibit superhorizon growth of the comoving curvature perturbation R c .

VI. SCALAR FIELD MODEL
Let us now find a scalar field model that corresponds to the barotropic model discussed in the previous section. As a warm-up, let us consider the USR case, whose fluid interpretation has already been studied in [23]. In this case, we exactly have c 2 s = 1. From eq. (21), this implies b/2 = ǫρ = 3(ρ + P )/2 ∝ a −6 . Also c 2 s = 1 implies ρ = P + const. Inserting this into eq. (10) gives Thus up to a constant term P and Y are the same, Absorbing g(φ) in Y into the definition of the scalar field by g 1/2 dφ → dφ, this is indeed the Lagrangian for a minimally coupled massless scalar with a cosmological constant: This is consistent with ρ + P = 2X ∝ ǫρ ∝ a −6 . Let us generalize the USR case. As in the previous section, we consider models that have the behavior of ǫρ as where b(a) should decrease faster than a −3 asymptotically at a → ∞ but otherwise is an arbitrary function. Then we have which gives For dP , using the energy conservation law, we may rewrite it as This is consistent with the USR case in which b(a) ∝ a −6 and Y = X ∝ a −6 . This relation is quite useful since it allows to rewrite the freezing function f (a) as from which we can deduce that Y (a) determines the super-horizon behavior of R c . In particular, for the models we are considering in which c s is constant, we infer that super-horizon growth can happen in the limit Y → 0. For a given choice of b(a), eq. (32) can be inverted to give the scale factor as a function of Y , a = a(Y ). Also eq. (30) can be integrated to give P = P (a). Combining these two, one can obtain the Lagrangian for the scalar field, L = P = P (Y ).
Note that in GA models there is a one-to-one correspondence between the scale factor and state variables such as P (a) and ρ(a), which is the reason why we can also write a barotropic equation of state P (ρ) = P (a(ρ)). Once any of the functions P (a), ρ(a), b(a), ǫ(a), Y (a) is specified, all the others are specified too, as well as the equation of state P (ρ) or its scalar field equivalent Lagrangian P (Y ), which is in fact the basis of the inversion method that we are developing in this paper.

VII. EXAMPLES
Here we give a couple of specific K-inflation models that are globally adiabatic and violate the convervation of R c . Given the parametric behaviour of b ≡ 2ǫρ, our inversion method allows us to deduce the Lagrangian.
A. Ex 1: Generalized USR Let us consider a specific case where b(a) is a power-law function, where c is a constant. We assume n > 3 in order to have the growth on superhorizon scales. From eq. (32) we have Now eq. (30) gives Plugging eq. (35) into this, we finally obtain Since this may be regarded as a natural generalization of the USR case, which corresponds to the case n = 6, we call it the generalized USR (GUSR) model. Lagrangians involving Y α terms have already been studied in [11,24,25], but those models are either not exactly globally adiabatic because of the presence of a not constant potential or they satisfy the relation ǫ ∝ a −n only approximately and during a limited time range, while for GUSR ǫ ∝ a −n is an exact relation and is valid at any time. As the Lagrangian is of the type described in eqs. (9) and (26) (remember that after a field transformation Y can be made equal to X), we understand that this scalar field model is indeed equivalent to a barotropic fluid. Hence we have c 2 w = c 2 s , and therefore δP nad = 0. Indeed the second condition for super-horizon growth of R c given in eq. (7) is satisfied. More precisely, we note that for the GUSR model, the sound velocity is exactly constant, The power spectrum of the comoving curvature perturbation can be explicitly computed for this model. One finds [26] that the spectral index is a function of n: n s − 1 = 6 − n, in agreement with the scale invariant spectrum of the original ultra slow-roll inflation in which one has n = 6. Hence, the model can be constrained by the observational value. Note as well, from eq. (38), that to have a slightly red-tilted spectrum, we need a slightly superluminal speed of sound.

B. Ex 2: Lambert Inflation
As another example, let us consider the case when ǫ is a power-law function, ǫ(a) = ǫ 0 a −n . (39) As before, we assume n > 3. In this case, since d log ρ/d log a = −2ǫ ∝ a −n , we find It is clear that ρ approaches a constant ρ 0 asymptotically at a → ∞. Inserting eq. (39) and eq. (40) into eq. (21), the sound velocity is given by Thus c 2 s is time dependent, but it rapidly approaches a constant as ǫ decays out. Also from eq. (39) and eq. (40), we find b(a) = 2ǫρ = 2ǫρ 0 exp 2ǫ n .
Thus we have which implies To find the Lagrangian, we manipulate eq. (30) as Therefore, integrating this we obtain One can invert eq. (44) to find ǫ as a function of Y , and then insert it into the above to obtain the Lagrangian. Specifically, we introduce the Lambert function W (x) defined by the inverse function of X(z) = ze z , Setting we have 4ǫ 2n − 6 = W (y) ; y ≡ Y n/(2n−6) .
Inserting this into eq. (46), we finally obtain where y = y(Y ) is given in eq. (49). Note that this model has been derived without making any approximation, and it gives exactly ǫ ∝ a −n . However, as we mentioned before, in the late time limit, there is no difference between ǫ ∝ a −n and ρǫ ∝ a −n . Thus the two models discussed above are essentially the same at late times. This can be easily checked by expanding W (y) around y = 0, At leading order in y = Y n/(2n−6) , this gives By absorbing the constant coefficient into g(φ) in the definition of Y , Y = g(φ)X, and absorbing ρ 0 into the constant V 0 , eq. (50) reduces to which indeed coincides with the GUSR model, see eq. (37). Higher order terms in the expansion give an infinite class of models of the type where β i are appropriate coefficients. Finally, note that in USR and as well in the two examples considered here, the shift symmetry in the potential (V (φ) = V 0 ) is a direct consequence of the demand c 2 w = c 2 s , which in turn follows from the global adiabaticity of the model. That is in line with the general statement [27,28] that for a k-essence theory to describe a fluid, one needs a shift symmetry (i.e., there is no physical clock, the model is of the non-attractor type).

VIII. CONCLUSIONS
By introducing the notion of global adiabaticity, namely, c 2 w = c 2 s and δP nad = δP − c 2 w δρ = 0, where c 2 w =Ṗ /ρ and c s is the propagation (phase) speed of the perturbation, we have determined the general conditions for the non-conservation of the curvature perturbations on comoving slices R c on super-horizon scales. We have found that globally adiabatic K-essence models can exhibit this behavior.
We have then developed a method to construct the Lagrangian of a K-essence globally adiabatic (GA) model by specifying the behavior of background quantities such as ǫρ where ǫ is the slow-roll parameter, using the equivalence between barotropic fluids and GA K-essence models. We have applied the method to find the equations of state of the fluids and derive the Lagrangian of the equivalent single scalar field models. Interestingly, we have found that the requirement to avoid the gradient instability, ie, c 2 s > 0 is almost identical to the condition for the non-conservation on superhorizon scales.
The advantage of our approach is that we have not solved any perturbation equation explicitly, since we have proceeded in the opposite way solving the inversion problem consisting of requiring certain properties to the behavior of the perturbation equation we are interested in. In other words, instead of starting from a Lagrangian and then solve the perturbations equations we have determined the equation of state or equivalently the Lagrangian which admits a solution of the perturbation equation with the particular behavior we are interested in.
We have shown that the main difference between attractor models and GA models is that the latter are adiabatic on all scales, while attractor models are approximately adiabatic in the sense of δP nad = 0 only on superhorizon scales and c 2 w = c 2 s . The detailed study of the new models found in this paper will be done in a separate upcoming work [26] but we can already predict that they can be compatible with observational constraints on the spectral index thanks to the extra parameter n which is not present in USR. Furthermore they can violate the Maldacena's consistency condition and consequently produce large local shape non-Gaussianity.
In the future it will be interesting to apply the inversion method we have developed to other problems related to primordial curvature perturbations, or to develop a similar method for the adiabatic sound speed as function of the scale factor.