aa r X i v : . [ a s t r o - ph . C O ] M a r Solid Consistency
Lorenzo Bordin, a , b Paolo Creminelli, c Mehrdad Mirbabayi, c , d Jorge Nore˜na e a SISSA, via Bonomea 265, 34136, Trieste, Italy b INFN, National Institute for Nuclear Physics, Via Valerio 2, 34127 Trieste, Italy c Abdus Salam International Centre for Theoretical PhysicsStrada Costiera 11, 34151, Trieste, Italy d Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA e Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4059, Valpara´ıso, Chile
Abstract
We argue that isotropic scalar fluctuations in solid inflation are adiabatic in the super-horizon limit.During the solid phase this adiabatic mode has peculiar features: constant energy-density slices and co-moving slices do not coincide, and their curvatures, parameterized respectively by ζ and R , both evolvein time. The existence of this adiabatic mode implies that Maldacena’s squeezed limit consistency re-lation holds after angular average over the long mode. The correlation functions of a long-wavelengthspherical scalar mode with several short scalar or tensor modes is fixed by the scaling behavior of thecorrelators of short modes, independently of the solid inflation action or dynamics of reheating. Consistency relations (CRs) in single-field inflation are a consequence of adiabaticity: a long mode islocally unobservable and its effect can be removed by a coordinate redefinition [1, 2]. In the presenceof additional fields, long-wavelength relative fluctuations (entropy modes) can be locally observed andCRs are violated. This common lore is challenged when one considers models of inflation with adifferent symmetry structure that cannot be described in the framework of the EFT of inflation [3].In this paper we focus on the case of solid inflation [4, 5] where the “stuff” that drives inflation hasthe same symmetry as an ordinary solid. Here the situation is different from the usual case. In solidsthere is a single scalar excitation: the longitudinal phonon. However this mode is not adiabatic: theperturbation is anisotropic and this anisotropy is locally observable even at very long wavelengths.The absence of adiabaticity suggests at first sight that one cannot derive any CR.This conclusion is too quick. The existence of CRs also with this different symmetry structurecan be seen in this way. If one considers an isotropic perturbation of the solid, i.e. a dilation orcompression, this will be adiabatic since in solids there is a unique relation between the pressure and he energy density, p ( ρ ). Since the solid experiences all states of compression as the universe expands,this perturbation cannot be locally distinguished from the unperturbed evolution. We are going toverify this statement in Section 2 showing that an isotropic superposition of linear scalar modes isindeed adiabatic. This adiabatic mode is not standard: the two variables ζ and R do not coincide andthey are both time dependent. This stems from the fact that the solids do not admit curved FRWsolution, but only flat ones.The existence of adiabatic modes imply CRs for the variable ζ . This does not happen for R since the diffeomorphism which removes the long mode cannot be written in terms of R in a model-independent way. In Section 3 we are going to verify the CRs in various cases, both when the shortmodes are inside the Hubble radius and outside. The conclusion is that, in models with the symmetrypattern of solid inflation, after reheating correlation functions satisfy the usual CRs once an averageover the relative orientation between long and short modes has been done. We discuss the implicationsof this and open questions in Section 4.Before proceeding, let us recall, following [5], that the dynamics of the solid is described in termsof three scalar fields φ I which parametrise the position of the elements of the solid: φ I = x I + π I . Theaction can be written in terms of SO(3)-invariant objects built out of the matrix B IJ ≡ ∂ µ φ I ∂ µ φ J .One can choose these invariants to be ([ . . . ] indicates a trace) X ≡ [ B ] , Y ≡ [ B ][ B ] , Z ≡ [ B ][ B ] . (1)So the action, at lowest order in derivatives and including gravity, is S = Z d x √− g (cid:20) M P l R + F ( X, Y, Z ) (cid:21) . (2) We want to show that, in solid inflation, a scalar perturbation becomes adiabatic once we average overthe solid angle, i.e. we take a superposition of scalar Fourier modes which is isotropic. We have to provethat this kind of perturbation, in the long-wavelength limit, can be brought back to the unperturbedsolution via a suitable diffeomorphism.We start from the so-called Spatially Flat Slicing Gauge (SFSG) which is defined as the gaugewhere the spatial part of the metric is only perturbed by tensor modes. For the rest of this Sectionwe will only be interested in scalar modes: g ij = a δ ij , φ I = x I + π I . (3)The triplet π I consists of a scalar, π L , plus a transverse vector, π IT , which we neglect in the following. he constraint equations give the lapse N = 1 + δN and the longitudinal part of the shift N L [5]: δN = − a ˙ HkH ˙ π L − ˙ Hπ L /H − Ha /k , N L = − a ˙ H ˙ π L /k + ˙ Hπ L /H − Ha /k . (4)In SFSG, the gauge-invariant variable ζ is given by ζ = ∂π O ( π ) , (5)where ∂π ≡ ∂ i π i . Therefore, by performing the following time diff, x → x + ξ ( t, x ) , with ξ ( t, x ) = 13 H ∂π , (6)we go from SFSG to the ζ − gauge , defined by the condition δρ = 0, where ρ is the energy density. Thespatial part of the metric now reads g ij = a ( t ) (1 + 2 ζ ( t, x )) δ ij , while one can write π L as a functionof ζ . Using eq.s (4) and (6), one can verify that the expression of δN in ζ − gauge is δN = − k ddt (cid:16) π L H (cid:17) − H a k . (7)In the limit k/aH → π L is slow-roll suppressed, see eq. (15), therefore δN → φ i . This can be done by a redefinition of the spatial coordinates x i → x i + ξ i ( t, x ) , with ξ i ( t, x ) = − π i ( t, x ) . (8)Since now the scalars are unperturbed, it is natural to call this Unitary Gauge (UG). In UG the shiftvanishes on super-horizon scales N L = − ddt (cid:16) π L H (cid:17) H − H a k . (9)In this gauge, for long wavelength, δN = N L = π i = 0. However the spatial part of the metric is stillperturbed g ij = a ( t ) (1 + 2 ζ ( t, x ) δ ij + ∂ i ∂ j χ ( t, x )) , with χ ( t, x ) = − ∂ − ζ ( t, x ) . (10)The perturbation is purely anisotropic, i.e. the volume is not perturbed because of the gauge condition δρ = 0. Therefore if one considers a spherically symmetric superposition of scalar modes, the metricperturbations in eq. (10) average to zero Z d ˆ k π (2 ζ k δ ij − k i k j χ k ) = Z d ˆ k π (2 ζ k δ ij − k i ˆ k j ζ k ) = 0 . (11) We use the notation: π i = ∂ i √−∇ π L + π iT and analogously for N i . Notice that when we expand around the background φ I = x I one has ∂ i φ I = δ Ii , so that there is nodistinction between capital and lower-case spatial indeces. his shows that a spherically symmetric superposition of scalar modes is adiabatic.Notice that, if one works in ζ − gauge , the transformation that eliminates a long-wavelength modeand goes back to FRW is a rescaling of the spatial coordinates: this is quite similar to the standardcase of single-field inflation. However here the rescaling is time-dependent and adiabaticity requiresan average over directions. In the next Section we will see that the adiabaticity gives rise to CRs: theonly difference with the standard case is that they hold only after the spherical average. (The timedependence of the rescaling is immaterial because one is usually interested in correlation functions atequal time.)Instead of using time-slices with δρ = 0, one could use slices that are orthogonal to the 4-velocityof the solid. The perturbation of the spatial part of the metric is called ζ in the first case and R in the second. Contrary to the usual case, in Solid Inflation the variables ζ and R differ even onsuper-horizon scales. At linear level R = 1 ǫH ˙ ζ + ǫ H ζ k / a H ǫ . (12)Since the two slicings do not coincide, one needs a time-diff to go from one to the other. This is thedifference of the two time diff.s to go from SFSG to ζ − gauge and to R− gauge respectively: δt R→ ζ = δt ζ − δt R = ζH − R H ≃ − ˙ ζǫH , (13)where the last equation holds on super-horizon scales. The property that δN vanishes in ζ − gauge onlarge scales, eq. (7), will not hold in R− gauge . This means that to go from R− gauge to the unperturbedFRW one has to supplement the rescaling of spatial coordinates with the time diff eq. (13). As we willdiscuss in the next Section this implies that the CR for R will contain an extra piece: the time diffinduces a piece involving the time-derivative of the short modes. In taking the long-wavelength limit k → k ≪ aHǫ / . However one expectsthat the adiabaticity arguments above hold whenever k is comfortably outside the Hubble radius andin particular also in the intermediate regime aH ≫ k ≫ aHǫ / . This is indeed the case. For examplein this regime the expression for the lapse in ζ − gauge is | δN k | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:18) ζ k H (cid:19) − H a k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:18) ζ k H (cid:19) k a H
13 ˙ H (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) k a H (cid:19) . (14)(Notice that in the inequality above we are not assuming that the term proportional to ˙ H in thedenominator dominates.) The same argument works for the shift: also in the intermediate regime thephysical difference with the unperturbed solution are suppressed when the mode is superhorizon.Therefore we expect the CR to hold both in the intermediate and in the super-squeezed, k ≪ aHǫ / , regime. However, to get analytical results one is forced to expand the solution of the constraints n different ways in the two regimes and therefore one has to assume one of the two regimes. This alsoapplies to the expression of the wavefunction, which can be expanded in the two limits to give [5] π L ( τ, k ) ≃ ( B k (cid:0) ic L kτ + c L k τ (cid:1) e ic L kτ , if | c L kτ | ≥ ǫ , B k (cid:2) ǫ (1 + c L ) log( − c L kτ ) (cid:3) ( − c L kτ c ) − s/ − η/ − ǫ , if | c L kτ | ≤ ǫ , (15)with B k = − HM P l c / L ǫ / k / . (16) Solid Inflation is an interesting exception to many general theorems on cosmological perturbations.Weinberg [10, 11] showed that, under quite general assumptions, one can always find an adiabaticmode which features identical and time-independent ζ and R on super-horizon scales. In the Solidcase, ζ and R are neither equal (see eq. (12)) nor time-independent (both ζ and R have a slow-rollsuppressed time-dependence on super-horizon scales). Of course by linearity these properties are notchanged by the spherical average. In the original paper on Solid Inflation [5] (see also [9]) the authorsaddressed the issue of why a scalar Fourier mode does not comply with Weinberg analysis. The pointis that the solid supports a large anisotropic stress, so that even in the long-wavelength limit the stressenergy tensor remains anisotropic and thus locally distinguishable from the unperturbed solution: themode is not adiabatic. The theorem [10, 11] assumes the decay of the anisotropic stress for k → i component of Einstein equations isnot regular for k →
0, since δu diverges in that limit. This does not allow to continue a homogeneousperturbation to a physical one at finite momentum.This however looks rather technical. What is the physical reason why the adiabatic mode we areconsidering is different from the standard case? Why doesn’t adiabaticity ensure that ζ is constant?In the standard case, the conservation of ζ and the relation ζ = R can be understood from a linearizedversion of a spatially curved FRW. Neglecting short-wavelength perturbations and working at linearorder, the curvature of a constant ρ slice is given by (3) R = − a ∂ ζ. (17)Since for a curved FRW the spatial curvature κ = a R/ ζ fluctuationsbetter be time-independent. Moreover for a curved FRW the surfaces of constant density are perpen-dicular to the 4-velocity so we need ζ = R . The adiabatic mode we are discussing in Solid Inflationdoes not have these properties and this is related to the fact that one does not have curved FRWsolution in this model. This can be understood in terms of symmetries: the internal symmetries Curved FRW solutions are allowed if we change the internal metric in the Lagrangian, see [12]. However, f the φ I is isomorphic to the symmetries of flat Euclidean space. This allows to write flat FRWsolutions, but not spatially curved solutions, which have a different group of isometries. In ζ − gauge a long mode averaged over the direction can be removed by a rescaling of the spatialcoordinates. The derivation of the CR is very similar to the standard case, apart from the requiredangular average. In the case of the scalar 3-point function we get Z d ˆ q π h ζ q ζ k ζ − k − q i ′ q ≪ k = − d log k P ζ ( k ) d log k P ζ ( q ) P ζ ( k ) , (18)where here and in the following the prime indicates that a momentum-conserving delta function,(2 π ) δ ( P i k i ), is dropped. We stress that this result, in the limit of exact scale-invariance when theRHS of eq. (18) vanishes, was already discussed in [8]. h ζ ζ ζ i Let us check the CR eq. (18). In Solid Inflation the quadratic action is O ( ǫ ) while the cubic actionis O ( ǫ ). Thus the 3-point function is slow-roll enhanced , f NL ∼ (cid:10) ζ (cid:11) / (cid:10) ζ (cid:11) = O ( ǫ − ) [5]. Sincethe tilt of the 2-point function outside the horizon is O ( ǫ ), a non-trivial (in the sense of non-zero)verification of eq. (18) in this regime would require taking into account corrections to the leadingbispectrum at second-order in slow-roll. This is quite challenging. We content ourselves with the firstorder correction: at this order the two sides of eq. (18) should vanish when all the modes are outsidethe horizon. When the short modes are inside the horizon the scale-dependence of the spectrum is notslow-roll suppressed and the LHS of eq. (18) should thus be non-zero. The check will be done in theregime k ≫ aHǫ / for all the modes.To do this we compute the cubic Lagrangian in SFSG up to O ( ǫ ), calculate the bispectrum andthen transform to ζ − gauge . The O ( ǫ ) corrections to the bispectrum of three super-horizon modeswas studied in detail in [7]. Thus, we skip most of the technical steps. However, we identify a missingterm in [7] that is important for the CR to work. The in-in calculation up to this order consists ofthe sum of three pieces. Schematically, h ζζζ i ∼ L (3) O (1) × π ( τ, k ) O (1) + L (3) O (1) × π ( τ, k ) O ( ǫ ) + L (3) O ( ǫ ) × π ( τ, k ) O (1) , (19)where L (3) O ( ǫ n ) and π ( τ, k ) O ( ǫ n ) are respectively the cubic Lagrangian and the wavefunctions evaluatedat n th order in slow-roll. The leading cubic Lagrangian, L (3) O (1) , was calculated in [5] (eq. (D.2)). In our argument for constancy of ζ in the standard case is based on the fact that curvature is a free parameter ofthe background solution. In the models discussed in [12] curvature is uniquely fixed in terms of energy density,so we don’t expect ζ to be conserved. This also implies the usual curvature problem takes a somewhat different flavour in this class of models. For a discussion of CRs in standard inflation when the short modes are inside the horizon see [6]. ourier space and in the squeezed limit it reduces to L (3) O (1) (cid:12)(cid:12)(cid:12) squeezed = − F Y (cid:0) − ( θ ) (cid:1) π L, q π L, k π L, − q − k , (20)where θ is the relative angle between the long and the short modes. The above expression giveszero when one takes the angular average. This means there is no contribution O ( ǫ − ) to the LHS ofeq. (18). Notice that the cancellation after angular average holds independently of the explicit form ofthe wavefunctions. Therefore, the second term of eq. (19) vanishes and only the last term is relevantfor checking the CR. Cubic scalar Lagrangian at O ( ǫ ) . At first look, expanding eq. (D.1) of [5] up to first order inslow-roll seems like a formidable task. Since Y and Z in (1) are defined in such a way that they startfrom second order in perturbations, one has to expand L (3) O ( ǫ ) = F X δX (3) + F XX δX (1) δX (2) + 16 F XXX ( δX (1) ) + F XY δX (1) δY (2) + ( F Y δY (3) ) O ( ǫ ) + ( Y ↔ Z ) , (21)where subscripts on F denote partial derivatives. However, there are several simplifications [7]. Aswe will see, one only needs the SFSG deformation matrix B IJ at zeroth order in slow-roll parameters.This allows neglecting N i and δN (which are slow-roll suppressed in the regime we are considering): B IJ = 1 a (cid:0) δ IJ + ∂ I π J + ∂ J π I + ∂ k π I ∂ k π J (cid:1) − ˙ π I ˙ π J + O ( ǫ ) (22)Therefore, δX terminates at quadratic order, apart from slow-roll suppressed corrections δX = δ [ B ] = 3 a ( 23 ∂π + 13 ∂ i π j ∂ i π j − a π i ) + O ( ǫ ) . (23)Given that derivatives of F with respect to X are slow-roll suppressed, there is no contribution from F X δX to O ( ǫ ) cubic Lagrangian. Another simplification is that if at some order in perturbations thecorrections to [ B n ] involve at most m ≤ n of the B IJ factors, then δ (cid:18) [ B n ][ B ] n (cid:19) = δ (cid:18) [ B m ][ B ] m (cid:19) , for all n ≥ m . (24)This implies that δY (2) = δZ (2) , (25)and since slow-roll corrections to B IJ start at O ( π ) δY (3) O ( ǫ ) = δZ (3) O ( ǫ ) . (26)Therefore, the following combinations appear in (21)( F Y + F Z ) δY (3) O ( ǫ ) , ( F XY + F XZ ) δX (1) δY (2) . (27)However F Y + F Z = O ( ǫ ) and F XY + F XZ = O ( ǫ ), so these terms are negligible. Using F XX = − a ǫF, F XXX = 2 a ǫF, F = − M H , (28) ne gets L (3) O ( ǫ ) = ǫM P l H a (cid:20)
23 ( ∂π ) ∂ j π k ∂ j π k −
827 ( ∂π ) (cid:21) − ǫM P l H a ( ∂π ) ˙ π i + 427 ( F Y + F Z ) a (cid:2) ( ∂π ) ˙ π i − ∂ i π j ˙ π i ˙ π j (cid:3) . (29)The terms with time derivatives in this equation are absent from eq. (35) of [7]. Note that theappearance of the combination F Y + F Z on the second line is a consequence of (24) since time-derivatives appear in B IJ starting from quadratic order. The angular average of this term is zero,hence it does not contribute to CR while the last term on the first line does contribute. Field redefinition.
Since we are interested in the bispectrum of ζ at f NL = O (1), we need to findthe relation between ζ and π at quadratic order and to zeroth order in ǫ . We start from B IJ in SFSGgiven in (22). The last term can be neglected because in the squeezed limit at least one of the two π ’s will be out of the horizon with a slow-roll suppressed time evolution. The assumption of sphericalsymmetry simplifies the expression, B IJ = δ IJ X ( t ) = δ IJ a (cid:18) ∂π ) + 19 ( ∂π ) (cid:19) . (30)Now we perform the time diffeomorphism that leads to the ζ − gauge (the analogue of eq. (6) but nowat second order), where X ( t ) takes its unperturbed value X ( t + ξ ( t, x ); x ) = ¯ X ( t ) = a − . (31)At non-linear order the spatial part of the ζ − gauge metric is defined as g ij = a e ζ δ ij . Therefore upto quadratic order in π , we obtain ζ = Hξ = 13 ∂π −
118 ( ∂π ) + 19 H ( ∂ ˙ π )( ∂π ) + O (( ∂π ) ) . (32)One can now put together the in-in computation, using the Lagrangian in the first line of eq. (29),with the definition of ζ , eq. (32), to get Z d ˆ q π h ζ q ζ k ζ − q − k i ′ q ≪ k = 127 Z d ˆ q π h ( ∂π ) q ( ∂π ) k ( ∂π ) − k − q i ′ q ≪ k − P ζ ( q ) P ζ ( k ) + 1 H P ζ ( q ) ˙ P ζ ( k ) . (33)When all the modes are outside the horizon the last term on the RHS of eq. (33) is slow-roll suppressedand can be neglected. A straightforward calculation shows that the other two terms cancel each otherconfirming that Z d ˆ q π h ζ q ζ k ζ − k − q i ′ q ≪ k = O ( ǫ ) , (34)as implied by eq. (18). When the short modes are inside the horizon the cancellation between thefirst two terms on the RHS of eq. (33) still holds, but now the last term is non-negligible since the It is challenging to perform the calculation at next order in slow-roll, to test the CR. The cancellation afterangular average in eq. (33) can be seen only after the explicit in-in integral (in contrast to what happens atleading order, eq. (20)). This means that, at higher order, we should compute time integrals involving thewavefunctions at O ( ǫ ). ime dependence is not slow-roll suppressed in this regime. One has Z d ˆ q π h ζ q ζ k ζ − k − q i ′ q ≪ k = 1 H P ζ ( q ) ˙ P ζ ( k ) = − d log k P ζ ( k, τ ) d log k P ζ ( q ) P ζ ( k, τ ) , (35)where in the last passage we used that the power spectrum is of the form P ζ = k − f ( kτ ) as dictatedby scale invariance, up to corrections of order slow-roll. The CR eq. (18) is verified. h ζ γγ i One novel feature of our analysis is that the (spherically averaged) adiabatic modes of solid inflationfeature a time-dependent ζ outside the horizon. Since this time dependence arises at O ( ǫ ) it would benice to check the CR at this order. As discussed this is quite challenging for h ζζζ i , but it is doable for h ζγγ i , where the scalar mode is taken to be long. The leading term in h ζγγ i is O (1) [8], so one justneeds to do the calculation including the first-order slow-roll corrections. The leading Lagrangian hasthe form (see eq. (A.7) of [8]) L (3) O ( ǫ ) ∝ −
13 ( ∂π ) γ ij γ ij + γ ij γ jk ∂ k π i . (36)It averages to zero in the squeezed limit π L q → independently of the wavefunctions. Thus we do notneed to consider the slow-roll corrections to the wavefunctions. We go directly to the computation of L (3) O ( ǫ ) . Cubic Lagrangian at O ( ǫ ) . There are two terms which contribute to L (3) O ( ǫ ) . The first arises fromthe expansion of the function F ( X, Y, Z ), while the second is from the Einstein-Hilbert action. Theexpansion of F gives L (3) O ( ǫ ) ⊃ F X δX + F Z δZ + F XX δX + F XY δXδY + F XZ δXδZ (37)where, δX = a − (cid:20) ∂π ) − γ ij ∂ i π j + 12 γ ij + γ ij γ jk ∂ k π i (cid:21) , (38) δY = (cid:20) − γ ij ∂ i π j + 19 γ ij + 23 γ ij γ jk ∂ k π i − γ ij ( ∂π ) (cid:21) , (39) δZ = (cid:20) − γ ij ∂ i π j + 19 γ ij + 89 γ ij γ jk ∂ k π i − γ ij ( ∂π ) (cid:21) . (40)One gets L (3) O ( ǫ ) = − ǫa M P l H (cid:20) γ ij γ jk ∂ k π j − γ ij ( ∂π ) (cid:21) . (41)This gives zero after the angular average. The contribution which arises from the Einstein-Hilbertaction (plus the appropriate boundary terms) is L (3) O ( ǫ ) ⊃ a h N (3) R + N − ( E ij E ij − E ) i O ( ǫ ) πγγ = − a δN h γ ′ ij + ( ∂ l γ ij ) i − a γ ′ ij ∂ k γ ij N k . (42) he in-in calculation can be done separately in the intermediate regime aH ≫ q ≫ aHǫ / and in thesuper-squeezed regime q ≪ aHǫ / . In both regimes the in-in computation of the 3-point functiongives 13 Z d ˆ q π (cid:10) ( ∂π ) q γ s k γ s − k (cid:11) q ≪ k = 2 ǫ P ζ ( q ) P γ ( k ) . (43) Tensor modes at quadratic order in perturbations.
The final expression of the bispectrumis given once one considers the contribution coming from the time diff that has to be performed to gofrom SFSG to ζ − gauge . This changes the tensor perturbations at quadratic level (see eq. (A.8) of[1]). The interesting part (for us) is γ ζ = γ π + 1 H ˙ γ π ( ∂π )3 , (44)where γ ζ and γ π denote tensor perturbations respectively in ζ − gauge and SFSG. This adds a contri-bution to the bispectrum: h ζγ ζ γ ζ i = h ζγ π γ π i + H h ζ∂πγ π ˙ γ π i , which is given to leading order by1 H P ζ ( q ) ddt P γ ( k ) = − ǫ (1 + c L ) P ζ ( q ) P γ ( k ) . (45)Eqs. (43) and (45) give Z d ˆ q π (cid:10) ζ q γ s k γ s − k (cid:11) q ≪ k = − c L ǫP ζ ( q ) P γ ( k ) = − d log k P γ ( k ) d log k P ζ ( q ) P γ ( k ) . (46)This is exactly what is predicted by the CR. In conclusion, this computation confirms that CRs afterspherical average hold even at slow roll order, i.e. when the time dependence of the long mode cannotbe neglected. R ? There are two differences if one wants to use the variable R instead of ζ . First, as discussed above,starting from R− gauge one needs an extra time diff to map a long mode into an unperturbed FRW.This changes the CR and introduces a time derivative of the short mode 2-point function. Moreoveron super-horizon scale, in both squeezed and super-squeezed regimes, the relation between R and ζ depends on c L and therefore is non-universal, R ≃ − c L ζ. (47)Hence, the spatial rescaling (8), which is determined in terms of ζ , will be model-dependent whenwritten in terms of R . This means that in this gauge we are not going to be able to write an explicitCR, i.e. a model independent relation among correlation functions. Notice that eq. (42), evaluated in the super-squeezed limit is O (1). But δN, N L ∝ ˙ π L which has a slow-rollsuppressed time dependence and so the final expression of the bispectrum is O ( ǫ ). Notice also that, naively,the term proportional to the shift seems to cancel when one performs the angular average. However in thesuper-squeezed limit this term would give a bispectrum ∝ /q for q → N i in eq. (4)).This leading behaviour cancels, even before taking the angular average. The subleading contribution has thecorrect 1 /q dependence and does contribute to the angular average. or instance, consider the squeezed limit of hR γγ i . One needs to go from ζ − gauge to R− gauge withthe time diff δ R→ ζ , see eq. (13). This implies Z d ˆ q π (cid:10) R q γ s k γ s − k (cid:11) ′ q ≪ k = n t hR q ζ − q i ′ h γ k γ − k i ′ + 1 ǫ H D R q ˙ ζ − q E ′ ddt h γ k γ − k i ′ == 2 ǫ − ( c L + 1) c L ! P R ( q ) P γ ( k ) ; (48)where we used the fact that, at first order in slow roll, ˙ ζ q = − H (1 + c L ) ǫ ζ q . This form of CR isnot very useful since it contains parameters that make the relation model-dependent. However, afterreheating ζ = R and they are both time-independent. Therefore for observational purposes the CRstake the form of eqs. (18) and (46). It is remarkable that in solid inflation one has evolution outside the horizon, even when the mode isspherically averaged and therefore adiabatic. The evolution is slow-roll suppressed during inflation,but it will become significant during reheating unless some extra assumption about this phase ismade. (For instance the limit of instantaneous reheating was taken in [5].) Therefore one cannoteven relate the normalization of the spectrum to the parameters during inflation, since the changeduring reheating may be significant. From this point of view, it is quite surprising that one can derivemodel-independent (and reheating independent) relations like eq. (18).Physically the angular-averaged consistency relations we found imply, within the symmetry patternof solid inflation, that local (angle-independent) non-Gaussianity cannot be generated. (As in the usualcase, the tiny non-Gaussianity in the squeezed limit, which is implied by the CR should be considered insome sense “unobservable”, see for example [13].) This however does not prevent large non-Gaussianityin the squeezed limit, as long as it vanishes after angular average.The consistency relation we discussed here is the analogue of the original Maldacena’s CR. Anatural question would be to look for other CRs, analogue of the conformal ones in standard inflation[14, 15]. Another natural question is to try and apply these methods to other symmetry breakingpatterns for instance Gauge-flation [16] or supersolid inflation [17, 18]. In all these cases however thereare multiple scalar excitations: similarly to standard multifield inflation, one does not expect any CR,unless further conditions are imposed.Finally, even though our explicit checks were limited to the lowest derivative solid action, the CRsare just based on symmetry considerations, and therefore they are robust when one considers higherderivative operators. cknowledgements We would like to thank Mohammad Akhshik, Alberto Nicolis, and Riccardo Penco for valuable dis-cussions.
References [1] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationarymodels,” JHEP , 013 (2003) [astro-ph/0210603].[2] P. Creminelli and M. Zaldarriaga, “Single field consistency relation for the 3-point function,”JCAP , 006 (2004) [astro-ph/0407059].[3] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, “The Effective FieldTheory of Inflation,” JHEP , 014 (2008) [arXiv:0709.0293 [hep-th]].[4] A. Gruzinov, “Elastic inflation,” Phys. Rev. D , 063518 (2004) [astro-ph/0404548].[5] S. Endlich, A. Nicolis and J. Wang, “Solid Inflation,” JCAP , 011 (2013) [arXiv:1210.0569[hep-th]].[6] L. Senatore and M. Zaldarriaga, “A Note on the Consistency Condition of Primordial Fluctua-tions,” JCAP , 001 (2012) [arXiv:1203.6884 [astro-ph.CO]].[7] M. Akhshik, “Clustering Fossils in Solid Inflation,” JCAP , no. 05, 043 (2015)[arXiv:1409.3004 [astro-ph.CO]].[8] S. Endlich, B. Horn, A. Nicolis and J. Wang, “Squeezed limit of the solid inflation three-pointfunction,” Phys. Rev. D , no. 6, 063506 (2014) [arXiv:1307.8114 [hep-th]].[9] M. Akhshik, H. Firouzjahi and S. Jazayeri, “Cosmological Perturbations and the Weinberg The-orem,” JCAP , no. 12, 027 (2015) [arXiv:1508.03293 [hep-th]].[10] S. Weinberg, “Adiabatic modes in cosmology,” Phys. Rev. D , 123504 (2003) [astro-ph/0302326].[11] S. Weinberg, “Cosmology,” Oxford, UK: Oxford Univ. Pr. (2008) 593 p[12] C. Lin and L. Z. Labun, JHEP , 128 (2016) doi:10.1007/JHEP03(2016)128 [arXiv:1501.07160[hep-th]].[13] E. Pajer, F. Schmidt and M. Zaldarriaga, “The Observed Squeezed Limit of Cosmological Three-Point Functions,” Phys. Rev. D , no. 8, 083502 (2013) [arXiv:1305.0824 [astro-ph.CO]].[14] P. Creminelli, J. Nore˜na and M. Simonovi´c, “Conformal consistency relations for single-fieldinflation,” JCAP , 052 (2012) [arXiv:1203.4595 [hep-th]].[15] K. Hinterbichler, L. Hui and J. Khoury, “An Infinite Set of Ward Identities for Adiabatic Modesin Cosmology,” JCAP , 039 (2014) [arXiv:1304.5527 [hep-th]].
16] A. Maleknejad and M. M. Sheikh-Jabbari, “Gauge-flation: Inflation From Non-Abelian GaugeFields,” Phys. Lett. B , 224 (2013) [arXiv:1102.1513 [hep-ph]].[17] N. Bartolo, D. Cannone, A. Ricciardone and G. Tasinato, “Distinctive signatures of space-timediffeomorphism breaking in EFT of inflation,” JCAP , no. 03, 044 (2016) [arXiv:1511.07414[astro-ph.CO]].[18] A. Ricciardone and G. Tasinato, “Primordial gravitational waves in supersolid inflation,”arXiv:1611.04516 [astro-ph.CO]., no. 03, 044 (2016) [arXiv:1511.07414[astro-ph.CO]].[18] A. Ricciardone and G. Tasinato, “Primordial gravitational waves in supersolid inflation,”arXiv:1611.04516 [astro-ph.CO].