Abstract
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, nongaussianities in the curvature perturbations are unusually large, with , and have a novel shape: peaked in the squeezed limit, with anisotropic dependence on how the limit is approached. Other unusual features include the absence of adiabatic fluctuation modes during inflation—which does not impair their presence and near scaleinvariance after inflation—and a slightly blue tilt for the tensor modes.
Solid inflation
[1cm] Solomon Endlich, Alberto Nicolis, and Junpu Wang
[0.4cm]
Department of Physics and ISCAP,
Columbia University, New York, NY 10027, USA
1 Introduction
There is certainly no shortage of models for primordial inflation. We regret to inform the reader that—as anticipated in the Abstract—we are going to add our own to this list. However, we feel that the inflationary model we introduce here presents conceptually novel features that make it stand out as a radical alternative to the standard inflationary scenario. The main reason for this is that our model does not conform to the standard symmetry breaking pattern of inflationary models, and this has farreaching implications.
In the usual cases, the matter fields feature timedependent cosmological background solutions , which spontaneously break time translations. As a result, there is one fluctuation mode that can be identified with the associated Goldstone excitation. Roughly speaking, it can be thought of as an insync perturbation of all the matter fields, of the form
(1.1) 
When coupling to gravity is taken into account, such a mode describes adiabatic perturbations. As usual for Goldstone bosons, the spontaneously broken symmetry puts completely general, nontrivial constraints on these perturbations’ dynamics [1]. This property is at the basis of the modelindepedent approach that goes under the name of “effective field theory of inflation” [2], whose tenets are particularly compelling since they encompass—at first glance—all cosmological models: cosmology is about timedependent, homogeneous, and isotropic field configurations.
However, as we will see, there are other possibilities. In our case, we will be dealing with matter fields featuring timeindependent, dependent background solutions. Apparently, this contradicts two facts about inflationary cosmology:

The universe is homogeneous and isotropic;

In an expanding universe physical quantities depend on time and, more to the point, that one needs a physical ‘clock’—a timedependent observable—to tell the universe when to stop inflating.
As for item 1: dependent solutions can be compatible with the homogeneity and isotropy we want for cosmological solutions and for the dynamics of their perturbations, provided extra symmetries are imposed. For instance, to get an FRW solution for the gravitational field, we need an homogeneous and isotropic background stressenergy tensor. This can arise from matter fields that are not homogeneous nor isotropic, provided there are internal symmetries acting on the fields that can reabsorb the variations one gets by performing translations and rotations. The simplest example is that of a scalar field with a vacuum expectation value
(1.2) 
Such a configuration breaks translations along . However, if one postulates an internal shift symmetry , then the configuration above is invariant under a combined spacial translation/internal shift transformation. As we will see, this is enough to make the stressenergy tensor and the action for small perturbations invariant under translations. To recover isotropy as well, one needs more fields, and more symmetries. For instance—in fact, this is the case that we will consider in this paper—one can use three scalar fields (), with internal shift and rotational symmetries
(1.3)  
(1.4) 
so that the background configurations
(1.5) 
are invariant under combined spacial translation/internal shift transformations, and under combined spacial/internal rotations. As we will review in sect. 2, such a system has the same dynamics as those of the mechanical deformations of a solid—the phonons. In this sense, the cosmological model that we are putting forward corresponds to having a solid driving inflation. If this interpretation causes the reader discomfort—in particular, if having a solid that can be stretched by a factor of without breaking sounds implausible—one should think of our model just as a certain scalar field theory. As we will see, the structure of such a theory is the most general one compatible with the postulated symmetries—and the impressive stretchability we need can also be motivated by an approximate symmetry—so that from an effective field theory standpoint, ours is a perfectly sensible inflationary model. From this viewpoint, the fact that the solids we are used to in everyday life behave quite differently—quantitively, not qualitatively—seems to be an accident: they lack the ‘stretchability symmetry’.
As for apparent contradiction number 2: In our model the role of the physical clock will be played by the metric. More precisely, it will be played by (gauge invariant) observables, made up of our scalars and of the metric, like for instance the energy density or the pressure. These can depend on time even for purely spacedependent scalar backgrounds, because in the presence of a nontrivial stressenergy tensor, the metric will depend on time, in a standard FRW fashion. Doesn’t this correspond to a spontaneous breakdown of time translations too? At some level it is a matter of definition, but we will argue in sect. 4 that the operationally useful answer is ‘no’, in the sense that there is no associated Goldstone boson, and that one cannot apply to our case the standard construction of the effective field theory of inflation as given in [2].
Formal considerations aside, our peculiar symmetrybreaking pattern has concrete physical implications, with striking observational consequences. For instance, it predicts a threepoint function for adiabatic perturbations with a ‘shape’ that is not encountered in any other model we are aware of. Without going into details here, we display it in fig. 1 for the benefit of the nongaussianity aficionados: it diverges in the squeezed limit, but in a way that depends on the direction in which one approaches the limit, with a quadrupolar angular dependence. Its overall amplitude is also unusually large, corresponding to .
Before proceeding to spelling out our model in detail, we close this Introduction with two qualifications. The first is that we have been using (and will be using) a somewhat misleading, but fairly standard, language: when spontaneously broken symmetries are gauged, the associated wouldbe Goldsone bosons are not in the physical spectrum—rather, they are ‘eaten’ by the longitudinal polarizations of the gauge fields. In fact, there is a gauge, the socalled unitary gauge, in which the Goldstone fields are set to zero. In our case we will be dealing with spontaneously broken translations and rotations, and when gravity is dynamical, these are gauged. In unitary gauge one can set the scalars to their vevs (1.5), and have the corresponding excitations show up in the metric. The Goldstone language is still useful though, in that it captures the correct high energy/short distance dynamics of these excitations. For massive gauge theories, this statement goes under the name of “the equivalence theorem”. For cosmological models, it is just the statement that at subcosmological distances and timescales, in first approximation one can neglect the mixing between matter perturbations and gravitational ones. We hope the Goldstone boson nomenclature will be more useful than misleading.
The second qualification is that our model is not entirely new. A different formulation of essentially the same inflationary model was put forward and briefly analyzed by Gruzinov in [3]. Our emphasis here will be on the peculiar symmetry breaking pattern, on the effective field theory viewpoint, on a systematic analysis of cosmological perturbations—including their nongaussian features—and in general on the conceptual and technical differences with more standard inflationary models. Cosmological solids have also been used as an exotic model for dark matter [5], and more recently as dynamical media with negative pressure but wellbehaved excitations [6].
As a general guide for the reader, the bulk of our (long) paper can be schematically divided into three parts:
A brief summary of our results is contained in sect. 11, along with a number of possible generalizations of our scenario. Much technical material is contained in the Appendix.
2 Effective field theory for solids (and fluids)
To begin with, let us review how one can describe the mechanical degrees of freedom of a solid in modern effective field theory terms. The first systematic approach to this question has probably been that of ref. [7]. Here instead we will review the equivalent construction of [8, 9] (see also [10]), whose notation we will follow.
Consider a medium, filling space. Let’s neglect for the moment gravitational effects and let’s take the metric to be flat. The medium’s configuration space can be parameterized by its volume elements’ individual positions. We can attach a (threedimensional) comoving label (with ) to each volume element, and follow the volume element’s trajectory in physical space, as a function of its comoving label and of time, . We can also do the opposite, which for our purposes turns out to be more convenient: since at any given time the mapping between and is invertible, one can parameterize the system via , that is, by keeping track of the comoving labels of the volume elements going through physical position , as a function of time. This is a completely equivalent description, yet it has the advantage of implementing the spacetime symmetries in the usual way: are just the usual Minkowski coordinates, and the ’s are three Poincaréscalars. Our problem then is reduced to constructing a relativistic lowenergy effective field theory for three scalar fields in Minkoswki space obeying certain internal symmetries, which we will discuss in a moment. To avoid future confusion with standard cosmological nomenclature, in the following we will call the ’s ‘internal’ coordinates, and reserve ‘comoving’ for the standard FRW coordinates for when we introduce gravity.
Now, apart from faithfully following the volume elements they are associated with, the internal coordinates are completely arbitrary. There is a coordinate system however that makes discussing internal symmetries—the symmetries that act on the fields—particularly easy: it is the choice that reduces to (1.5) for a medium at rest, in equilibrium, in an homogeneous state at given external pressure. The parameter measures the compression level, which can be dialed physically by changing the external pressure. Then, since we are supposed to be describing an homogeneous medium, but the expectation values (1.5) break translations, we are forced to postulate the existence of internal shift symmetries
(2.1) 
so that there is an unbroken linear combination of spacial and internal translations that mimics the physical homogeneity we are trying to reproduce. In other words, thanks to the symmetry (2.1), the internal space of the medium is always homogeneous. Whether this homogeneity is realized in physical space depends on the medium’s configuration, or state. For a state like (1.5), it is. This logic is, of course, the same as we were employing in the Introduction to motivate our cosmological model.
We may want to impose rotational symmetries. The background configuration (1.5) breaks spacial rotations. For a crystalline solid, like e.g. one with a cubic lattice, a discrete subgroup of these should be preserved. This can be achieved by demanding an internal symmetry
(2.2) 
for a generic matrix belonging to the desired subgroup of , so that the background (1.5) is invariant under combined spacial and internal rotations. One can also impose full invariance, in which case one would be describing an isotropic solid with no preferred axes—a ‘jelly’. For simplicity, in this paper we will consider this invariant case only. We will come back to comment briefly on less symmetric cases in sect. 11.
So, we are led to consider the most general lowenergy theory for three scalar fields obeying Poincaré invariance and the internal symmetries (1.3) and (1.4). The shift (1.3) forces each field to appear with at least one derivative. Lorentz invariance forces these derivatives to contract among themselves. At lowest order in the derivative expansion, the only Lorentzscalar, shiftinvariant quantity is the matrix
(2.3) 
We then have to construct invariants out of this matrix. For a matrix, there are only three independent ones, which we can take for instance to be the traces
(2.4) 
where the brackets are shorthand for the trace of the matrix within. Alternatively, one could take the determinant, and two of the traces above. In the following, we will find it convenient to use one invariant—say —to keep track of the overall ‘size’ of the matrix , and to choose the other two such that they are insensitive to an overall rescaling of , e.g.
(2.5) 
The most general solid action therefore is
(2.6) 
where is a generic function that depends on the physical properties of the solid—e.g. its equation of state—and the dots stand for higherderivative terms, which are negligible at low energies and momenta.
As a selfconsistency check, notice that the background configuration (1.5) solves the equations of motion for any value of , which can then be thought of as being determined by the boundary conditions (the external pressure we alluded to above.) The eom for the action above are
(2.7) 
For a linear configuration like (1.5), all terms in parentheses are constant, since they depend on the s’ first derivatives, which are constant. The eom are thus trivially obeyed.
As a side note, a perfect fluid fits into the same general description. Only, it features infinitely many more internal symmetries, which express the physical fact that for a fluid one can move volume elements around in an adiabatic manner without paying any energy price—only their compression matters. By contrast, if one tries to displace a solid’s volume elements, even without compressing or dilating them, one encounters stresses that want to bring the volume elements back to their rest positions. Mathematically, in our language this property of a fluid is expressed as the invariance of the dynamics under internal volumepreserving diffeomorphisms:
(2.8) 
Of the invariants above, only one particular combination survives: the determinant of ,
(2.9) 
which is of course insensitive to multiplication of by unitdeterminant Jacobians. At lowest order in the derivative expansion, a fluid’s action is thus
(2.10) 
So, in this formalism, a fluid is just a very symmetric solid.
Back to the solid. The background configurations (1.5) spontaneously break some of our symmetries. There are associated Goldsone bosons, which are nothing but fluctuations of the ’s about such a background,
(2.11) 
We get these fluctuations’ free action by expanding our action (2.6) to second order in . Using
(2.12) 
after integrating by parts and neglecting boundary terms we get
(2.13) 
where the subscripts stand for partial derivatives, which are to be evaluated at the background values
(2.14) 
These Goldstone excitations are the solid’s phonons. For what follows it will be convenient to split the phonon field into a longitudinal part and a transverse one,
(2.15) 
It is straightforward to extract the longitudinal and transverse propagation speeds from the phonon’s action:
(2.16) 
in terms of which the quadratic action is simply
(2.17) 
If we expand eq. (2.6) to higher orders, we get the interactions among the phonons. The expansion is straightforward, but already at cubic order the result is quite messy, and not particularly illuminating. Below, we will display explicitly the cubic interactions in a particular limit, which yields some simplifications, and which is the physically relevant limit for an inflationary background. For the moment, it suffices to say that, by construction, the th order interaction terms will be schematically of the form , where the derivatives can be spacial or temporal, and the indices are contracted in all possible ways. The coefficients of these interactions terms—the coupling constants—will be given by suitable derivatives of , evaluated on the background solution. Like for all derivativelycoupled theories, our interactions become strong in the UV, at some energy scale . For our theory to be predictive for cosmological observables, we will need this strongcoupling scale to be above the Hubble rate , for the whole duration of inflation.
3 Inflation
We can now allow for a cosmological spacetime metric and for dynamical gravity, which, operationally, is trivial: the indexcontraction in (2.3) should be done via rather than , and the measure in (2.6) should carry a . As usual, ‘minimal coupling’ corresponds to the most general coupling one can have between a matter system and gravity at lowest order in the derivative expansion. Then our solid’s stressenergy tensor is
(3.1) 
As to the scalar fields’ background configuration, the in (1.5) should now be interpreted as comoving FRW coordinates. The reason is that the FRW metric is invariant under translations and rotations acting on the comoving coordinates, and we want the l.h.s. and the r.h.s. in (1.5) to transform in the same way under the symmetries we are trying to preserve. We can also choose the normalization of the comoving coordinates to set the parameter to one, so that from now on the background configuration is simply
(3.2) 
When computed on the background, the stressenergy tensor reduces to the standard , with
(3.3) 
where the subscript stands for partial derivative, and and are evaluated at the background values for our invariants:
(3.4) 
Notice that—by construction— is the only invariant that depends on the scale factor; and were designed to be insensitive to an overall rescaling of . This is the reason why only appears in the pressure: for an FRW solution, the pressure is related to the response of the system to changing the scale factor, i.e., the volume. For a more general configuration, the stressenergy tensor (3.1) has a more complicated structure, which depends on and as well, which we report here for later use:
(3.5) 
Now, in order to have near exponential inflation, we need
(3.6) 
Via the Friedmann equations ^{1}^{1}1We are defining the Planck scale as ,
(3.7) 
and eq. (3.3), we can express directly in terms of our Lagrangian :
(3.8) 
where we used eq. (3.4) for the background value of . We thus see that if we want our solid to drive near exponential inflation, we need a very weak dependence for . Which is not surprising, since is the only invariant that is sensitive to the volume of the universe: for inflation to happen, the solid’s energy should not change much if we dilate the solid by ; this is only possible if the solid’s dynamics do not depend much on .
This also suggests how to enforce the smallness of via an approximate symmetry. Consider the scale transformation
(3.9) 
The matrix changes by an overall factor, which affects but not nor . Therefore, the smallness of can be interpreted as an approximate invariance under (3.9): If only depended on and , it would be exactly invariant under (3.9), and this would prevent quantum corrections from generating some dependence. If we start with a small at tree level, the symmetry is only approximate, yet all further dependence generated at quantum level will be suppressed by the small symmetrybreaking coupling constant— itself.
Notice that in general, ordinary scale invariance—that acting on the spacetime coordinates as —cannot be readily used as an approximate symmetry to enforce the smallness of symmetrybreaking parameters. For instance, one cannot solve the Higgsmass hierarchy problem this way [11]. Moreover, it is generically anomalous, that is, broken by quantum effects, as clearly displayed by the running of coupling constants for interactions that are scaleinvariant at tree level. Here instead, we are dealing with a purely internal symmetry—eq. (3.9)—which commutes with all spacetime symmetries. It has nothing to do with spacetime scaleinvariance. It is on an equal footing with our other internal symmetries (1.3), (1.4), and, like those, is nonanomalous and can be used to constrain the structure of the Lagrangian. To avoid confusion, in the following we will refer to the symmetry (3.9) as ‘internal scale invariance’.
Notice in passing that at lowest order in the derivative expansion, imposing internal scale invariance gives us full (internal) conformal invariance as a byproduct. The reason is the following. The invariants we are using in our lowestorder action are combinations of traces of the form . Under a generic internal diff
(3.10) 
these traces transform as
(3.11) 
where is the diff’s Jacobian matrix, and we have used the cyclicity of the trace. Now, by definition, conformal transformations are the subgroup of diffs that change the flat metric only by an overall scalar factor, that is, they have a proportional to the identity. So, under a general conformal transformation, our traces only change by an overall (dependent) factor. Our and combinations are insensitive to such a change, and as long as the Lagrangian only depends on those, it is invariant under all internal conformal transformations. It is interesting that even though our internal scale invariance has in principle nothing to do with ordinary spacetime scale invariance, it shares with it the unavoidable company of special conformal transformations. In both cases, special conformal transformations are not needed to close the symmetry group, yet they are respected by generic scaleinvariant dynamics. In the spacetime symmetry case, there are fundamental reasons why that happens [12, 13]. In our case, it appears to be an accidental feature of the lowest order truncation of the derivative expansion.
Back to physics. The smallness of in our model does not come for free. To see this, notice that by dialing we could go continuously through and end up with negative , that is, positive , which violates the null energy condition (NEC). But our EFT conforms to the hypotheses of the general theorem of [8], which links NEC violations to such pathologies as ghost or gradientinstabilities and superluminality. For small but positive , we could be dangerously close to these pathologies ^{2}^{2}2These considerations are irrelevant for standard slowroll inflation, where the smallness of is achieved via an approximately flat potential. One cannot play with the potential’s slope and end up with positive . Positive would require flipping the sign of the inflaton’s kinetic energy, which would of course entail ghostinstabilities. On the other hand, in our case the sign of is the same as that of , which is the Lagrangian parameter we are playing with to make small.. Our quadratic action for the phonons—eq. (2.13)—is quite explicit about this: the phonon’s kinetic energy is suppressed by ,
(3.12) 
For very small , this can in principle lead to two problems for our theory:

Superluminality: the gradient energies in (2.13) are not explicitly suppressed by , and as a consequence the propagation speeds (2.16) are formally of order , unless the numerators are also small. In an effective field theory like ours, with spontaneously broken Lorentz invariance, superluminal signal propagation is not necessarily an inconsistency. However, it prevents the theory from admitting a standard Lorentzinvariant UVcompletion [14]. We therefore feel that it should be avoided.

Strong coupling: unless interactions are also suppressed by suitable powers of —and it turns that they are not—a smaller kinetic energy means stronger interactions. This is obvious if one goes to canonical normalization for , by absorbing the prefactor in (3.12) into a redefined phonon field. Then inverse powers of will show up in the interaction terms, thus signaling that the strong coupling scale of the theory is suppressed by some (positive) power of . We have to make sure that this strong coupling scale is above , for the whole duration of inflation.
As for the former issue, notice first of all that the term proportional to in the expression for is forced to be close to . The reason is that not only do we need the ‘slowroll’ condition (3.6) for inflation to happen, we also need
(3.13) 
for inflation to last many folds.^{3}^{3}3In the computations that follow, we will assume all the slowroll parameters to be of the same order of magnitude. This forces the second derivative also to be small. In particular, given eq. (3.8), and
(3.14) 
we get
(3.15) 
So that, at lowest order in and , the propagation speeds (2.16) reduce to
(3.16) 
It is quite interesting that in this limit they depend on exactly the same combination; we see no obvious reason why this should be the case. As a result, the two speeds are not independent: they are related by ^{4}^{4}4To all orders in and , the exact relation is
(3.18) 
We thus see that for both speeds to be subluminal, we need
(3.19) 
that is, positive (recall that is negative, because is). We do not want to be too positive though—otherwise, we end up with negative squared speeds, that is, exponentially growing modes. We thus need to fit into a small window,
(3.20) 
We were able to motivate the smallness of ,
(3.21) 
via an approximate symmetry. It would be desirable to do the same for this new combination of derivatives. One possibility is of course to say that all derivatives of are small, that is
(3.22) 
This formally corresponds to an approximate invariance under all internal diffs
(3.23) 
that is, to the statement that the value of the Lagrangian does not depend much on the fields. Less formally, and more physically, it corresponds to saying that the bulk of the solid’s energy density and pressure are dominated by a cosmological constant, which does not depend on the fields. Although this is of course a technically natural choice—having a large cosmological constant was never a problem—it is not particularly interesting. It would be more interesting to find a symmetry that allows large derivatives of ,
(3.24) 
but that—in the limit of exact symmetry—forces the combination to vanish. Another possibility would be a symmetry that makes saturate the upper bound in (3.20), that is, that makes vanish. This is not as unlikely as it sounds: for instance the perfect fluid action (2.10)—whose structure is protected by the volumepreserving diff symmetry—features vanishing propagation speed for the transverse phonons [8, 9], as can be checked explicitly in (2.16), by using expression (2.9) for the determinant. We have not been able to find a symmetry that enforces the condition (3.20) while preserving (3.24). We have to take such a condition as an assumption, which might involve some fine tuning, but which is nonetheless consistent and necessary for the consistency of our inflationary solution.
As for the strong coupling issue, we have to estimate the strong coupling scale in our small limit, and make sure that cosmological perturbations are weakly coupled at horizon crossing, that is, at frequencies of order . Expanding the action (2.6) to all orders in we get interactions of the form
(3.25) 
where is some typical derivative of . In our case some combinations of derivatives are small,
(3.26) 
but we do not expect this to yield a substantial weakening of interactions. For instance, we will see below that in our approximation the coefficient weighing cubic interactions is , which, as we argued above, can be as large as the background energy density, . Assuming that is a good estimate for the coefficients encountered in interaction terms, and assuming for the moment that both and are of order of the speed of light—so that there is no parametric difference between time and spacederivatives—we can estimate very easily the strong coupling scale: We can go to canonical normalization for the kinetic term
(3.27) 
so that the th order interaction becomes
(3.28) 
This is a dimension interaction, weighed by a scale
(3.29) 
(recall that and have massdimension four.) If is of the same order as , this is simply
(3.30) 
which, for and , is an increasing function of . The lowest of all such scales—which defines the strong coupling scale of the theory—is thus that associated with :
(3.31) 
If on the other hand is much smaller that , say of order , we get that all interactions are weighed by the same scale, which then defines strong coupling:
(3.32) 
Either way, the strong coupling scale is a fractional power of smaller than the scale associated with the solid’s energy density.
If the propagation speeds are nonrelativistic, the estimate of the strong coupling scale depends on the specific structure of the interaction terms, that is, on how many timederivatives there are. In general, one may expect stronger interactions, i.e., lower strongcoupling scales for nonrelativistic excitations (see e.g. [9] for a systematic analysis of this phenomenon in a different limit of our solid action.) Notice first of all that, because of (3.18), the transverse phonon speed is always relativistic,
(3.33) 
So, our estimates above always work for the transverse phonons’ selfinteractions. For longitudinal phonons with , we can repeat the estimate using the cubic interaction, assuming this is still a good indicator of the strong coupling scale of the theory. Expanding (2.6) up to cubic order, and neglecting terms that are proportional to derivatives of or to , we find
(3.34) 
where is the matrix of spacial derivatives of , is its transpose, and the brackets stand for the trace. Notice in particular that there are no timederivatives. For , one can estimate the strong coupling scale via the following trick [9]. We can redefine the time variable as . Now in the kinetic energy term there is no hierarchy between time and spacederivatives,
(3.35) 
and we can apply the usual orderofmagnitude estimates as for relativistic theories. The cubic interaction becomes schematically
(3.36) 
To get the lowest possible value for the strongcoupling scale—that is the most dangerous one—we can take to be as large as possible, . ^{5}^{5}5There is a third possibility where . While this may not be a problem when analyzing the strong coupling scale of the theory, it is still very unnatural to have the background energy scale much smaller than the other scales in the theory. We already have one cosmological constant problem in cosmology, it would maybe be better if we tried not to introduce a second one. Going to canonical normalization and estimating the strongcoupling scale as above we get ^{6}^{6}6If one were to repeat the same analysis for a more generic th order interaction, also weighed by like eq. (3.34), and also involving spatial derivatives only, one would get
(3.38) 
This is the strong coupling momentum scale, or equivalently, the strongcoupling energy scale in units that are appropriate for our new variable. To convert to the original units of energy, we have to multiply by an extra :
(3.39) 
As we mentioned above, cosmological perturbation theory is under control only if
(3.40) 
Relating and via the Friedmann equation, , we get a lower bound on the combination ,
(3.41) 
In principle our can be several orders of magnitudes smaller than the Planck scale, in which case this bound is not particularly restrictive. Still, it is a nontrivial condition for the selfconsistency of the perturbative computations we will perform.
Once the strongcoupling danger is exorcised, large interactions are demoted (or promoted) from a problem to an exciting feature of the model: they imply huge nongaussianities for our cosmological perturbations. As we will see in sect. 5, our nongaussian signal is peaked on squeezed triangles, with the same sizedependence as the socalled local forms of nongaussianity, but with a different angular dependence. The corresponding parameter is of order , which is a factor of bigger than what one finds—at the same value of —in singlefield models with nonrelativistic sound speed.
A clarification is in order: we have been analyzing the viability of our model focusing on the phonons’ dynamics, neglecting the background spacetime curvarture and the phonons’ mixing with gravitational perturbations. Of course this is not entirely correct. As we mentioned in the Introduction however, at energies much bigger than , or equivalently, for timescales much shorter than , curvature and mixing have negligible effects, and in first approximation they can be neglected. Our conditions above, (3.20) and (3.41), should then be thought of as necessary and sufficient for our system to be wellbehaved in the UV, at very short distances and time scales. Our detailed analysis of cosmological perturbations in sect. 5 will confirm these results.
We should also point out that although we will be using standard ‘slowroll’ nomenclature for the conditions (3.6), (3.13) and for the associated perturbative expansion, nothing is ‘rolling’ in our system, slowly or otherwise: our scalars are exactly constant in time. As usual however, the socalled slowroll expansion really relies on the slowness of certain timedependent observables like , , etc., which are well defined regardless of the presence of a rolling scalar. We will still use ‘slowroll’ to refer to such a weak timedependence, hoping that this will not cause confusion. As we emphasized, in our case the physical origin of this slowness is the near independence of the dynamics on , which is, among our invariants, the only one that depends on time. Besides and , in the following we will need one more slowroll parameter,
(3.42) 
which is small, because depends on time only via the Lagrangian’s dependence.
Finally, we should comment on why we are focusing on a solid rather than on a perfect fluid. First, since eventually we will be interested in quantum mechanical effects—as usual, quantum fluctuations will be the ‘seed’ for cosmological perturbations—we focus on a solid because we do not know yet how to consistently treat the perfect fluid effective theory as a quantum theory [9]. The problem has to do with the transverse excitations, which appear to be strongly coupled at all scales. Second, even forgetting about the transverse excitations and focusing on the longitudinal ones, we would not be able to keep those weakly coupled for many folds. As clear from (3.17), to have vanishing (which is one of the defining features of a fluid) and small , we need . But, by definition, , so that we need to decrease by an order one factor over one Hubble time, i.e. to decrease like some orderone power of . has to be nearly constant over many folds, which means that it is actually the numerator that is tracking . But it is precisely combinations like that control the strongcoupling scale for longitudinal excitations in a fluid [9], which means that we cannot have decrease by exponentially large factors without making the system strongly coupled at frequencies of order at some point during inflation.
4 Physical ‘clocks’ and reheating
Eventually we want our inflation to end and to be followed by a standard hot BigBang phase, that is, we want the universe to reheat and to become radiation dominated ^{7}^{7}7It should be noted that [3] avoided going through a specific model of reheating by simply evolving the solid until , and demanding that the solid turn into a perfect fluid at that point. Given our analysis above, if we want to keep all the slow roll parameters small, as is reflected in (3.33) cannot become much smaller than . Via our EFT approach we thus see that the reheating model of [3] necessarily entails a breakdown of the slowroll expansion before reheating, which is, of course, a consistent possibility.. In our case, this process can be thought of as a phasetransition from a solid state to a relativistic fluid state. The advantage of our language in dealing with such a transition is that, as we emphasized in sect. 2, it describes both solids and fluids in terms of the same longdistance degrees of freedom, our scalars . Only, the fluid action (2.10) enjoys (many) more symmetries. So, regardless of the microscopic dynamics that are actually responsible for the phase transition, at long distances and time scales reheating corresponds to some sort of symmetry enhancement of our action. We will be more specific about this in a moment.
In terms of our infrared degrees of freedom, what triggers reheating? In standard slowroll inflation, it is the inflaton itself, when its timedependent background field reaches a critical value. On the other hand, in the absence of perturbations, our ’s are exactly timeindependent: . However the metric is not, and there are gaugeinvariant combinations like our
(4.1) 
or the energy density and pressure in eq. (3.3), that do depend on time. Usually we are used to solids turning into liquids—that is, melting—when the temperature exceeds a critical value. But we can also envisage a solid that ‘melts’ at zero temperature, when one of the physical quantities above goes past a critical value. Helium offers an example of such a phenomenon: at zero temperature one can turn liquid helium into a solid by raising the pressure beyond atm, and melt it back again by lowering the pressure below that value. In our case, we need this zerotemperature melting to be associated with a substantial release of latent heat, so that the fluid we end up with is (very) hot ^{8}^{8}8In the fluid phase, which is described by the action (2.10), the temperature is given by [22].. As far as we can tell, this does not violate any sacred principles of thermodynamics.
Notice that as far as the background solution is concerned, to lowest order in the derivative expansion all choices for what observable triggers our solid’s melting are physically equivalent: this is evident in our parameterization of the action (2.6), where all timedependent observables depend on time only through , or, equivalently, through . However, when we include fluctuations, we break this equivalence. For instance, the three invariants , , and are independent combinations of the fields’ derivatives. In the presence of fluctuations, the hypersurface defined by reaching its critical reheating value is different from that defined by or reaching their critical reheating values. As a result, some of our predictions for cosmological correlation functions might depend on the physical variable chosen to trigger reheating. Notice that after reheating, in the hot fluid phase, there is no ambiguity: the lowestorder action (2.10) only depends on one variable, the determinant of . It is thus natural, although not obviously necessary, to postulate that reheating is triggered by the value of .
So, in terms of our action, our assumption is that for large the action has the general structure (2.6), whereas for below a critical value, the action has the more restricted form (2.10). In the space of our , , invariants, this corresponds to dividing up the space into two regions, where the action has different symmetries: eqs. (1.3), (1.4) for the former, eq. (2.8) for the latter. Moreover, the slowroll condition during inflation is protected by another (approximate) symmetry of the solid phase, eq (3.9), which we want to be maximally violated in the postreheating fluid phase, which has . Since renormalization is local in field space, the existence of different regions with different symmetries is protected by precisely those symmetries, and is thus a consistent and natural assumption ^{9}^{9}9A similar mechanism is at work in ghost inflation [23], or in the EFT description of finitetemperature superfluids [24].. For illustrative purposes, consider for instance the following action:
(4.2) 
where is a generic function that evaluates to one for the background values , , and we suppressed an overall common factor, which defines the density at reheating. The ‘gluing’ at can be smoothed at will. In the first regime, describes a solid driving an inflationary phase with constant (for fixed and , scales like —hence the power). In the second regime, describes an ultrarelativistic fluid, with [9, 22]. The two regimes have different internal symmetries, as discussed above, but they share the same degrees of freedom. The classical evolution of the background solutions and of perturbations can then be followed smoothly through the transition region, as the long wavelength degrees of freedom are the same all along, and the equations of motion are regular ^{10}^{10}10As mentioned above, one should refrain from performing quantum computations in the fluid phase [9]. So, eq. (4.2) should not be thought of as a quantum effective theory in the second regime. Still, since at reheating all relevant modes are well outside the horizon and, thanks to the usual reasons, can be treated as classical, we only need (4.2) to be a consistent classical field theory, which it is..
Notice that we have been implicitly assuming that reheating is instantaneous, that is, that our solid/fluid phase transition happens in a time interval that is much shorter than the Hubble scale, which is reasonable in principle, but not necessary. One can also consider much slower transitions, which in field space would correspond to replacing the sharp critical values we have been talking about for our observables, with much more continuous transition regions. All our physical considerations above apply unaltered. Since, as we will discuss, some of our predictions are potentially modeldependent, for what follows we need to assume a specific model for reheating. So, we will assume that reheating is fast, much faster than , and that it is controlled by the value of .
4.1 Why not the EFT of inflation?
Before turning to a detailed analysis of cosmological perturbations, we close this section by discussing why our model does not conform to the standard EFT of inflation. As we just saw, we do have physical ‘clocks’, that is timedependent background observables, so why can’t we use the standard results for spontaneously broken timetranslations? The reason is that these timedependent observables depend on time only because the metric does.
To see why this subtlety is important, consider first the dynamics of our system at very short distances—in the socalled decoupling limit—where the matter fluctuations decouple from the gravitational ones and the Goldstone boson language is appropriate. In first approximation, this limit corresponds formally to setting to zero. But without gravity, our background solution has no timedependence whatsoever! All observables like density, pressure, etc. are now exactly constant in time, and only spacial translations and rotations are broken by the background configuration . As a result, the Goldstone bosons, whose existence and properties have to be assessed in the decoupling limit—because as recalled in the Introduction, only in this limit does it make sense to talk about them—are those associated with this spontaneous symmetry breaking pattern, not with timetranslations.
It is not surprising then, that once we reintroduce gravity, the dynamics of cosmological perturbations at all scales are quite different than for the EFT of inflation. This is manifest in the socalled unitary gauge, where one chooses the timevariable according to a physical clock. In the standard case, that clock would be the inflaton, and choosing the equaltime surfaces to be the equalinflaton surfaces automatically sets to zero the inflaton perturbations and makes the metric the only fluctuating field. In our case, if we choose one of our timedependent observables—say or —to define unitary gauge, we are still left with matter perturbations, because the background timedependence that we are using to set the gauge is carried by the metric, not by matter fields. We can either include the matter perturbations explicitly in the Lagrangian terms that we write down in this gauge, or we can set them to zero, by choosing the spacial coordinates now so that . This is a complete gaugefixing—all spacetime coordinates have been unambiguously defined—and is of course quite a different gauge choice than the standard unitary gauge. In particular, it is inequivalent to choosing the socalled gauge for spacial diffs. Either way, the Lagrangian terms one would write down are quite different than for the standard EFT of inflation. We will go into the details of this new unitary gauge in sect. B.
The presence of matter fluctuations in the ‘naive’ unitary gauge cannot be taken as a sign that we are dealing with what would be called a multifield model in the standard classification. First, because there is a gauge—our ‘improved’ unitary gauge—in which all matter fluctuations are set to zero. Second, because our spectrum of cosmological perturbations only includes one scalar mode, as clear from the Goldstone quadratic action (2.17). Furthemore, as we will see, this scalar mode is not adiabatic. In other words, in our system there are no adiabatic modes of fluctuation. This is yet another manifestation that we are dealing with a truly unconventional cosmological system.
Notice that at the classical level, a subtle, isolated exception to all of the above is offered by a perfect fluid. On the one hand, in our language a perfect fluid is just a very symmetric solid. In particular, it features the same symmetry breaking pattern. On the other hand however, because of powerful conservation laws for vorticity, classically one can consistently set to zero the transverse excitations—the vortices—and be left with an EFT for the compressional modes only [8]. This admits a description in terms of a single scalar which spontaneously breaks timetranslations—a theory—to which the standard EFTofinflation construction is applicable ^{11}^{11}11Here we are using the standard notation of the community: is a generic function and here does not refer to the we have been considering throughout the text but rather , where is some scalar field whose background vev is .. In particular, for a perfect fluid scalar cosmological perturbations are adiabatic. Once quantum effects are taken into account however, transverse excitations cannot be neglected any longer. In fact, for a fluid they are not particularly well behaved quantum mechanically [9], which is one of the reasons why we have been considering a more generic solid rather than the special perfectfluid case.
As a technical aside, we should also emphasize that in a gauge where the matter fields are unperturbed, , our matrix reduces simply to , and our Lagrangian thus becomes the sum of the EinsteinHilbert action and of a particular function of , that is, it reduces to a Lorentzviolating theory of massive gravity. Theories like this have been studied in broad generality in [25]. The reader familiar with the EFT of inflation might wonder why we are not writing down directly the action for the perturbations in this gauge—the analog of unitary gauge in that case—, instead of going through the (apparently) unnecessary burden of writing an action for the full fields, solving for the background solution, and then expanding the action in small perturbations. The technical reason is that, unlike or for the EFT of inflation, our does not transform covariantly under the residual diffs, which are just time diffs for us. The reason is that does, but its background value, does not. It is then technically more convenient to write an action for the full , which just amounts to writing an action for , like we have done.
5 Cosmological perturbations
The three sections that follow contain a technical analysis of cosmological perturbations. Before skipping directly to sect. 8, the reader uninterested in the details of the derivations should be aware of our results: the scalar tilt (6.34), the tensortoscalar ratio (6.35), the tensor tilt (6.19), and the threepoint function of scalar perturbations (7.15) (which is analyzed in some detail in sect. 8).
As the background stress tensor takes the usual homogeneous and isotropic form represented by , all the interesting repercussions of our peculiar symmetry breaking pattern lie in the dynamics of perturbations around the slow roll background. In order to best isolate the dynamical degrees of freedom of the gravitational field it is most convenient to work in the ADM parametrization of the metric:
(5.1) 
It is straightforward to check that the inverse metric is given by
(5.2) 
where is the inverse spatial metric: . For the background FRW metric , , and .
Following [18] we can write the action as
(5.3) 
where is the 3dimensional Ricci scalar constructed out of and , with denoting the extrinsic curvature of equaltime hypersurfaces. The constraint equations given by varying (5.3) with respect to and are:
(5.4)  
(5.5) 
The derivatives of with respect to and can be calculated easily by noting that our (and hence ) can be expressed in ADM variables as
(5.6) 
It is usually convenient to work in a gauge where scalar perturbations are removed from the matter fields and appear only in the metric, as , see e.g. [18]. This possibility is not available to us: By using up the three spatial diffs, we can set the matter field perturbations to zero, , but then the spatial metric has an extra scalar mode, proportional to , which we now cannot remove in the usual manner. However, we are still free to use time diffs, but these at best can set the scalar in front of , not , to zero. A more useful choice is to use the time diff to set a physical “clock”—like those we discussed in the last section—to its unperturbed value. If this clock controls reheating, then reheating will happen at the same time for all observers in this gauge. We review this gauge choice, which we call ‘unitary’, in Appendix B.
For the moment we find it more convenient to work in spatially flat slicing gauge (SFSG)—defined in Appendix B—where we can write the fluctuations about the FRW background as
(5.7) 
where is transverse and traceless, i.e.
(5.8) 
We can also further split the and fields in terms of their longitudinal scalar and transverse vector components. We therefore write:
(5.9) 
where . From now on we will stop differentiating between internal indices and spacial ones. The reason is that of the full original symmetry, only the diagonal combination is preserved by the background . and both transform as vectors under this unbroken , and therefore they carry the same kind of index.
For our purposes here we are interested only in the leading nongaussian behavior. Barring accidental cancellations, this can be captured by keeping terms that are cubic in the fluctuations. In order to reproduce these terms it turns out to be necessary to only know and to first order in the fluctuations ^{12}^{12}12This lucky fact is because the higher order terms in and will be multiplying the constraint equations. In particular: the third order term of and multiplies the zeroth order constraint equations, and the second order the first order constraint equations [18]. If we where, however, to try and generate the fourth order terms we would need and to second order.. From now on, we find it easier to work in spacial Fourier space, with our convention defined for any field by:
(5.10) 
For convenience, however, we will drop the twiddle as which field variable we intend will be obvious from the arguments. And so, solving the constraint equations (5.4) and (5.5) to first order in fluctuations we have
(5.11)  
(5.12)  
(5.13) 
where the dot denotes a timederivative.
Now, plugging these solutions back into (5.3) will give us the correct action for the fluctuations up to cubic order. For instance, the trilinear solid action after mixing with gravity is contained in Appendix D while the quadratic actions for the tensor, vector and scalar modes are contained in the next section.
Now that we have the correct action for the perturbations in the presence of an inflating background we can compute correlation functions. In the end, we are interested in the postreheating correlation functions of curvature perturbations, parameterized by either of the gauge invariant (at linearorder) combinations
(5.14) 
where we have followed the notation of [19] ^{13}^{13}13The general (i.e. before gauge fixing) perturbed metric (to the linearorder) is parametrized by
(5.19) 
where the nonlocal piece of comes from solving the constraint equation for .
Two peculiarities concerning the behavior of these variables during solid inflation are worth mentioning at this point. First, and do not coincide on superhorizon scales. Second, neither of them is conserved. These properties are in sharp contrast with what happens for adiabatic perturbations in standard cosmological models, and stem from the fact that during solid inflation, there are no adiabatic modes of fluctuation! We will clarify why this is the case in sect. 9.
6 Twopoint functions
Upon plugging the expressions (5.11)–(5.13) back into the action, the quadratic action for tensor, vector, and scalar fluctuations reads:
(6.1)  
(6.2)  
(6.3)  
(6.4) 
Notice the quite nontrivial dependence for and in Fourier space, which would translate into a (spacially) nonlocal structure in real space.
6.1 Tensor perturbations
Using (6.2) we can calculate the twopoint function of the tensor perturbations. As usual, it is a simpler calculation than the scalar case and will serve as a warmup. We decompose the tensor modes into their polarizations
(6.5) 
with . The transverse, traceless conditions on now simply become . We further decompose each as
(6.6) 
where and