partial uv completion of p x from a curved field space3tsung-dao lee institute, shanghai jiao tong...
TRANSCRIPT
-
YITP-20-130, IPMU20-0108
Partial UV Completion of P (X) from a Curved Field Space
Shinji Mukohyama1, 2, ∗ and Ryo Namba3, †
1Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan2Kavli Institute for the Physics and Mathematics of the Universe (WPI),
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan3Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China
The k-essence theory is a prototypical class of scalar-field models that already gives rich phe-nomenology and has been a target of extensive studies in cosmology. General forms of shift-symmetric k-essence are known to suffer from formation of caustics in a planar-symmetric con-figuration, with the only exceptions of canonical and DBI-/cuscuton-type kinetic terms. With thisin mind, we seek for multi-field caustic-free completions of a general class of shift-symmetric k-essence models in this paper. The field space in UV theories is naturally curved, and we introducethe scale of the curvature as the parameter that controls the mass of the heavy field(s) that wouldbe integrated out in the process of EFT reduction. By numerical methods, we demonstrate thatthe introduction of a heavy field indeed resolves the caustic problem by invoking its motion nearthe would-be caustic formation. We further study the cosmological application of the model. Byexpanding the equations with respect to the curvature scale of the field space, we prove that theEFT reduction is successfully done by taking the limit of infinite curvature, both for the backgroundand perturbation, with gravity included. The next leading-order computation is consistently con-ducted and shows that the EFT reduction breaks down in the limit of vanishing sound speed of theperturbation.
I. INTRODUCTION
Scalar fields play important roles in modern cosmology, in both early and late epochs. In the inflationary scenario ofthe early universe the graceful exit from a quasi-de Sitter expansion requires breaking of the temporal diffeomorphisminvariance and correspondingly the introduction of an inflaton, i.e. a field recording the time remaining before theend of the quasi-de Sitter expansion. Usually, the inflaton is chosen to be a scalar field or a combination of scalarfields. On the other hand, while the late-time acceleration does not necessarily require the same type of symmetrybreaking pattern, it is usually considered necessary to introduce extra degrees of freedom if one seeks the origin of theacceleration other than the cosmological constant/vacuum energy. In this case the simplest choice is to introduce anextra degree of freedom via a scalar field. In either epoch, from the viewpoint of the effective field theory (EFT), if apart of the diffeomorphism invariance is broken at the cosmological scale, then there is a priori no reason why the speedlimit of a scalar field that plays cosmological roles should agree with the speed of light. For this reason, the kineticterm of a scalar field considered in modern cosmology is often non-canonical. The leading operators in the contextof EFT of single-field inflation/dark energy [1–3] are captured by the action of the form
∫d4x√−gP (ϕ,X), provided
that the background is sufficiently away from PX(ϕ,X) = 0,1 where ϕ is a scalar field, X = −gµν∂µϕ∂νϕ/2 and a
subscript X denotes derivative with respect to X. In this sense the P (ϕ,X) model, often called a k-essence [8–11], isa prototype of a scalar field theory in the context of cosmology.
It is known that the P (ϕ,X) model invariant under an arbitrary constant shift of ϕ, i.e. the P (X) model oftencalled a shift-symmetric k-essence, is equivalent to a vorticity-free perfect fluid with the following parametric form ofa barotropic equation of state,
ρ = 2PX(X)X − P (X) , p = P (X) , (1)
where ρ and p are the energy density and the pressure. In the context of fluid dynamics it has been known that afluid with a generic equation of state tends to form caustics. By employing techniques developed in the research of
∗Electronic address: [email protected]†Electronic address: ryo˙[email protected] On the other hand, if the system enjoys the shift symmetry (or an approximate shift symmetry) and if PX admits a positive root, thenPX = 0 (or PX ≈ 0) is an attractor. When the background is sufficiently close to the attractor PX = 0, the sound speed becomes sosmall that a higher dimensional operator dominates what is usually the dominant gradient term and thus the fluctuations are describedby the EFT of ghost condensate [4, 5] or the scordatura theory [6, 7].
arX
iv:2
010.
0918
4v1
[he
p-th
] 1
9 O
ct 2
020
-
2
partial differential equations and fluid dynamics [12–15], it was shown in [16] that the P (X) model generically formscaustics, where the second and higher derivatives of the scalar field diverge. Initially in [16], only the canonical scalarfield with P (X) = AX, where A is constant, was identified as a model in which the so called simple waves do notlead to caustics. Later in [17], it was found that the Dirac-Born-Infeld (DBI) model with P (X) =
√AX +B, where
A and B are constants, is also free from caustics as far as simple waves are concerned (see also [18–20]).2 This lattercase in fact contains the so-called cuscuton model [23–25] as one of the limits, B → 0 [18].
For a generic P (X) model in which simple waves form caustics, before the formation of caustics the system exitsthe regime of validity of the EFT and should be taken over by a more fundamental description, i.e. a (partial) UVcompletion. For example, as suggested in [16] and elaborated in [26, 27], the P (X) model may emerge as a low-energy effective description of a two-field model with canonical kinetic terms when one of the fields is integrated out.Obviously, in this situation the P (X) description is valid only when second and higher derivatives of the field aresufficiently small in the unit of the mass of the extra field that is integrated out. However, for the two-field modelsstudied in [26, 27], once the field space metric is required to be regular at the origin (so that there is no conicalsingularity) and the range of ϕ is kept non-vanishing (e.g. 2π), the mass of the extra field is determined by the formof P (X) itself. In a way the mass is solely controlled by the dynamics of the low-energy physics, and it is thereforesomewhat contrived to manage the regime of validity of the P (X) description for general P (X).
One of the purposes of the present paper is to extend the two-field model of [26, 27] so that the mass of the extrafield can be made arbitrarily heavy for a given form of P (X). This goal is achieved by promoting the two-dimensionalfield space, which was taken to be flat in the previous works, to a curved one and by considering the curvature scaleof the field space as a parameter that controls the mass of the extra field. In particular, the hyperbolic field space ismaximally symmetric and inferred by the so called distance conjecture [28], which is one of the most conservative casesamong all swampland conjectures proposed so far, and thus may be ideal as an ingredient of a possible (partial) UVcompletion of the P (X) model. We shall also discuss relations to the two-field models studied in the literature [29–32].As noted in [17], adding higher-order Horndeski terms to P (X) does not ameliorate the caustic problem, and thuswe here focus on resolving the issue in the models described by the Lagrangian scalar P (X) without the higher-orderterms.
Another purpose of the present paper is to see how the extra field behaves as the system approaches the incidentof a caustic formation. For this purpose we study a planar symmetric configuration of the two-field model in theMinkowski spacetime. We employ numerical methods to integrate the full two-field system of nonlinear coupledequations as well as the corresponding equations in a single-field P (X) model, which is essentially obtained by theLegendre transformation from the two-field model with one field being infinitely massive. The observation of causticsin the original single-field model and of its resolution in the corresponding two-field completion evidently shows thevalidity of our approach, which is the first explicit numerical demonstration to our knowledge. The result also givesa clear illustration of the necessity of the second field in order to avoid the caustic formation.
In view of further applications in realistic setups, we also study cosmology in the obtained two-field model minimallycoupled to General Relativity. We expand the equations of motion by the curvature scale of the field space so thatthe deviation from the P (X) description can be systematically studied. It is then shown that both at the level ofthe Friedmann–Lemâıtre–Robertson–Walker (FLRW) background and at the level of linear perturbations, the P (X)description is valid at energies and momenta sufficiently lower than the mass of the extra field that is controlled by thecurvature of the field space. We then proceed to the next order in the expansion and illustrate how the higher-orderequations can be systematically obtained by iteratively integrating out the heavy field. It is found that the cutoffscale of the single-field EFT is related to the sound speed of the perturbation and, in particular, the EFT expansionwith respect to the curvature scale would break down in the limit of vanishing sound speed, the finding consistentwith a generic expectation [2].
The rest of the paper is organized as follows. In Sec. II, we introduce the class of P (X) models that we studyand then promote it to those of two fields in a curved field space. We consider both linear-kinetic and DBI-typecompletions. We then conduct numerical computation of the obtained model on a planar-symmetric configurationin Sec. III. This provides an explicit demonstration of the avoidance of caustic formations in the two-field model,observing that the would-be divergence of second derivatives of the light field is smoothed out by the onset of themotion of the heavy field. In Sec. IV, we consider the cosmology of the two-field models, both the FLRW backgroundand the perturbations around it, showing that a consistent expansion in terms of the curvature scale of the fieldspace can be done. Sec. V summarizes our results and discusses their implications. In Appendix A we collect some
2 If the shift symmetry on ϕ is abandoned and P (X) is multiplied by a function of ϕ, the resultant non-shift-symmetric DBI would ingeneral form caustics [21], which can be interpreted in the stringy setups as the relative difference in the light cone structure betweenthe effective metric for open strings and that for closed ones [22].
-
3
technicalities in changing variables with derivatives involved and obtaining the action of the new variable.
II. TWO-FIELD MODEL
It was demonstrated in [16–18] that models of the (shift-symmetric) k-essence and Horndeski types are in generalvulnerable to formations of caustic singularities in the flat spacetime. In [17], it was shown that the classes of modelsimmune to such pathological behaviors consist not only of canonical scalar fields, which had already been shown in [16],but also of the Dirac-Born-Infeld (DBI) scalars, and that this exhausts the list.3 Caustic formations do not necessarilyimply a breakdown of the evolution of a considered system but rather hint a departure from the validity regime ofthe k-essence/Horndeski model used as an effective theory. Before caustics form, operators in a more fundamentaltheory that are integrated out in the effective description start being in action. In this section, we provide a class ofcaustic-free completion of k-essence models by introducing an additional scalar field.
We aim to complete shift-symmetric k-essence by a two-field system with a curved field space that has a lineelement,
γIJ dΦIdΦJ = dχ2 + f(βχ) dϕ2 , (2)
where a non-negative function f determines the shape of the field space spanned by ΦI = (χ, ϕ), and for a fixedfunction f a constant β controls the curvature of the field space, the mass of the extra field and thus the cutoff scaleof the corresponding single-field effective field theory (EFT). The curvature associated with the field-space metricγIJ = diag(1, f) is quantified by the Ricci tensor, given by RIJ = diag
(−β2f−1/2(f1/2)′′, −β2f1/2(f1/2)′′
), where
prime denotes derivative with respect to the argument βχ. Some classifications of the field space are
√f(βχ) =
βχ , flat (with β = 1) ,
exp(βχ) , hyperboloidal ,
sin(βχ)
β, spheroidal ,
(3)
where for the flat case β = 1 is required by the avoidance of a conical singularity at χ = 0 while for the spheroidal caseβ−1 is introduced for the same reason. Our primary interest is the case of nontrivial field space geometry RIJ 6= 0,that is (f1/2)′′ 6= 0. In the following subsections, we therefore consider a linear and DBI-type kinetic terms of twofields whose space geometry is curved according to (2).
A. Two-field model with linear kinetic terms
In this subsection we develop a two-field system with a linear kinetic terms that serves as a (partial) UV completionof the general P (ϕ,X) models. We first consider a simpler case with shift symmetry, i.e. P (X) models, and thenextend it to more general P (ϕ,X) models.
1. Equivalent description of P (X)
As a preparation for the construction of a two-field completion of P (X) models, we first rewrite the Lagrangianscalar P (X) as
Llin-EFT = f(βχ)X − V (βχ) , (4)
where V is a function to be determined so that Llin-EFT reduces to P (X) after integrating out the auxiliary field χ. Thevalue of χ is determined by its equation of motion that is obtained by taking variation of the action
∫d4x√−gLlin-EFT
3 In fact, the result in [17] already included the so-called cuscuton model L ∝√X, where X ≡ −(∂ϕ)2/2, as one of the limits, whose
caustic-free nature was explicitly stated in [18]. The latter reference [18] extended the analysis to an SO(p) symmetry in an arbitrarynumber of space dimensions. However, the most restrictive, that is the most conservative, caustic-free condition is found to emerge fromthe one with a planar-symmetric configuration, as studied in [17].
-
4
with respect to χ,
dv
df= X . (5)
Here, we have considered V (βχ) as a function of f and denoted it as v(f), assuming that f ′(βχ) 6= 0 in the range ofβχ that is of our interest. By algebraically solving (5) with respect to f , we obtain f = f(X). Plugging this solutioninto (4), we can rewrite Llin-EFT as a function of X and then demand that this function of X coincides with P (X),that is,
fX − v(f) = P (X) . (6)
In fact, the relation (6) is interpreted as v(f) being the Legendre transformation of P (X), or P (X) being the Legendretransformation of v(f). This fact immediately results in the following relations: dv/df = X, which is (5), PX = f ,and d2v/df2 = 1/PXX . The invertiblity of the Legendre transformation requires that P (X) be a convex (or concave)function, i.e. PXX > 0 (or PXX < 0). It then follows that v(f) is also a convex (or concave) function, i.e. d
2v/df2 > 0(or d2v/df2 < 0). In other words, the class of the models given by (4) can in principle cover all the shift-symmetrick-essence models P (X) that respect PXX 6= 0.
By varying∫
d4x√−gLlin-EFT, where Llin-EFT is given by (4), with respect to ϕ, we find the equation of motion
for ϕ as
∇µ (f∇µϕ) = 0 , (7)
where ∇µ is the covariant derivative associated with the spacetime metric. By taking derivative of P (X) in (6) withrespect to X with the use of (5), we find the correspondence
PX = f , (8)
which was already inferred from the fact that v(f) is a Legendre transformation of P (X), and thus one can easilyidentify (7) with the equation of motion for the k-essence models. The energy-momentum tensor corresponding to(4) is found as
T lin-EFTµν = f ∂µϕ∂νϕ+ (fX − V ) gµν , (9)
and the corresponding energy density and pressure can be found by ρlin-EFT = nµnνT lin-EFTµν and Plin-EFT =
hµνT lin-EFTµν /3, respectively, where nµ = ∂µϕ/√
2X is the unit vector normal to the constant-ϕ hypersurface withthe inverse induced metric hµν = gµν + nµnν . We find
ρlin-EFT = 2fX − (fX − V ) , Plin-EFT = fX − V . (10)
which reproduce (1) upon using (6) and (8).In this equivalent description of the single-field P (X) model, the parameter β has no physical meaning since it can
be absorbed by rescaling χ in (4) or (6) . However, in completing the single-field theory to a two-field UV theorybelow, it plays a fundamental role by controlling the energy scale ∝ β−1 for which the dynamics of χ becomes relevantto resolve the caustic singularities, as demonstrated below and in Sec. III.
2. Two-field completion by adding kinetic term for extra scalar
For a multi-field completion of the effective theory (4) by the curved field space (2), we assume that the UV theoryabove the cutoff scale of the single-field EFT consists of multi scalar fields in a curved field space. Hence our completeLagrangian scalar takes the form
Llin = −1
2γIJ∇µΦI∇µΦJ − V (ΦI) , (11)
where the curvature of γIJ is negative. To capture the mechanism of our interest, it is sufficient to identify a flatdirection in the field space with ϕ and to denote by χ a representative massive direction. The Lagrangian scalar (11)then reduces to a 2-field one with γIJ identified with the one in (2),
Llin = −1
2(∂χ)
2 − f(βχ)2
(∂ϕ)2 − V (βχ) , (12)
-
5
where V is independent of a flat direction ϕ.4 Now it is evident that β is a parameter of mass dimension −1 thatcontrols the “mass” of the massive mode χ.
The equations of motion for χ and ϕ are, respectively,
−∇2χ+ β[f ′
2(∂ϕ)
2+ V ′
]= 0 , (13)
−∇µ (f∇µϕ) = 0 . (14)
In the limit β →∞, (13) reduces toV ′
f ′= −1
2(∂ϕ)
2, (15)
which is identical to the constraint (5). Since the equation of motion for ϕ in (14) is exactly the same as the effectivemodel (4), it is clear that the dynamics of this system is identical to that of the effective theory (12) as long as β →∞is a justified limit to give (15). The energy momentum tensor of the full theory (12) is
T linµν = ∂µχ∂νχ+ f ∂µϕ∂νϕ+ gµν
[−1
2(∂χ)
2 − f2
(∂ϕ)2 − V
]. (16)
Notice that, in the limit β →∞, the field χ is a massive mode and thus ∂χ→ 0, in which case the energy momentumtensor in (16) also becomes identical to the EFT one (9). This proves the equivalence between the effective theory(4) and its two-field completion (12) in the limit β →∞.
3. Reconstruction of v(f) ≡ V (βχ)
For a given f and V , the corresponding P (X) theory can be obtained by (6), together with βχ as a function of Xthat is the solution of (5). In order to express the potential V for a given P (X) and the form of f that is determinedfrom the field space curvature as in (2), let us consider a generic expansion of P (X),
P (X) =∑n
cnXn , (17)
where cn are the coefficients of X polynomials. Using (6) with the relation (5), one obtains an equation
v − f2 dvdf
+∑n
cn
[dv
df
]n= 0 , (18)
where V ′/f ′ = dv/df has been used with the identification V (βχ) = v(f). By solving the differential equation (18),the form of v, and consequently V , can be found. Eq. (18) is a necessary condition for v to satisfy in order to representa given P (X) model. This is however not sufficient, and as discussed below (6), the Legendre transform can be doneif and only if PXX 6= 0. The first nontrivial example, which includes up to X2 terms in P (X) = c0 + c1X + c2X2,i.e. cn = 0 for all n except for c0, c1 and c2, yields a solution
5
v = −c0 +1
4 c2(f − c1)2 . (20)
From a theoretical/model-building point of view, if a UV theory that takes the form (11), or (12), induces the dynamicssimilar to a P (X) theory truncated at X2, then the corresponding potential should be identified with the one of theform (20). In Secs. III and IV, we focus on the the two-field theory (12) for detailed analyses, and in Sec. III weexplicitly utilize the form (20) for numerical demonstration of caustic formation and resolution.
4 In principle, there can be a mixing kinetic term of the form ∇µϕ∇µχ, but it can be removed by a field redefinition, at the price ofadditional terms in the potential. Imposing shift symmetry on the resultant ϕ, V is left independent of ϕ. Then the kinetic term of χcan be canonically normalized without loss of generality.
5 Eq. (18) gives another set of solutionsv = −c0 + C (f − c1)− C2c2 , (19)
where C is an integration constant. However this does not meet the condition of the Legendre transformation, since the second derivativevanishes, i.e. d2v/df2 = 0. Thus these solutions are not taken as appropriate forms of v(f).
-
6
4. Extension to P (ϕ,X)
The extension of the two-field completion of P (X) models to more general P (ϕ,X) models is straightforward.Assume that P (ϕ,X) is convex (or concave) with respect to X, i.e. PXX > 0 (or PXX < 0), so that the Legendretransformation of P (ϕ,X) with respect to X exists and is unique. Let v(ϕ, f) be the Legendre transformation ofP (ϕ,X) with respect to X (and thus P (ϕ,X) be the Legendre transformation of v(ϕ, f) with respect to f). Thetwo-field completion with the field space metric (2) is then given by the action
∫d4x√−gLlin, where
Llin = −1
2(∂χ)
2 − f(βχ)2
(∂ϕ)2 − V (ϕ, βχ) , V (ϕ, βχ) ≡ v(ϕ, f(βχ)) . (21)
The single-field P (ϕ,X) model is recovered in the limit β →∞.
B. Two-field model with DBI-type kinetic terms
In the previous subsection we have developed a (partial) UV completion of the class of P (X) and P (ϕ,X) modelsby a two-field system that has a linear kinetic terms with the curved field space (2). In the next sections we shalluse it to resolve the problem of caustic singularities and study cosmology. Before that, in this subsection we consideranother possible (partial) UV completion of a class of P (X) models by the curved field space (2) but with Dirac-Born-Infeld (DBI)-type kinetic terms. Extension to P (ϕ,X) is straightforward and discussed in Sec. II B 4. Readerswho are interested in the resolution of the caustic singularities and cosmological application of the completion withlinear kinetic terms may skip this subsection and directly go to the next sections.
As shown in [17], not only the canonical scalar field but also the DBI model with P (X) =√AX +B, where A and
B are constants, is also free from caustics as far as simple waves are concerned. Also, string theory allows not onlyscalar fields with linear kinetic terms but also those with DBI-type kinetic terms that stem from D-branes moving inextra dimensions [33]. For these reasons, it is reasonable to ask whether we can extend the (partial) UV completion ofP (X) models to a two-field system with DBI-type kinetic terms. In this subsection we answer this question positivelyby explicitly constructing such a two-field system.
1. Equivalent description of P (X)
In order to construct a two-field system with DBI-type kinetic terms that can (partially) UV-complete a class ofP (X) models, as the first step we now consider an single-field EFT of the form
LDBI-EFT = −√
1− 2f(βχ)X − V (βχ) , (22)
where again X = −(∂ϕ)2/2, χ is an auxiliary field, f (> 0) and V are functions of βχ, and β is a constant. In thissingle-field description, the value of χ is determined by the constraint equation
dv
df=
X√1− 2fX , (23)
where again we have regarded V (βχ) as a function of f and denoted it as v(f), assuming that f ′(βχ) 6= 0 in the rangeof βχ that is of our interest. Solving this for f in terms of X and plugging it back into (22) results in a class of P (X)theories. We determine v(f) such that
−√
1− 2fX − v(f) = P (X) . (24)
As stated below (6), the parameter β carries no physical meaning at this stage, as it can be absorbed in the redefinitionof χ. Its role as the controlling cutoff scale of the EFT will soon be clear once its two-field completion is introducedbelow. The equation of motion and the energy-momentum tensor associated with (22) are, respectively,
∇µ(
f∇µϕ√1− 2fX
)= 0 , TDBI-EFTµν =
f ∂µϕ∂νϕ√1− 2fX + gµν
(−√
1− 2fX − V). (25)
By observing PX = f/√
1− 2fX by the use of (23), we identify the system governed by the above E.o.M. andenergy-momentum tensor with those of the corresponding P (X) theory.
-
7
2. Two-field completion by adding kinetic term for extra scalar
Our two-field completion of (22) by the curved field space (2) is done by the inclusion of the kinetic term of χ inthe following manner:
LDBI = −√
1 + γIJ∇µΦI∇µΦJ − V (ΦI)
= −√
1 + (∂χ)2
+ f(βχ) (∂ϕ)2 − V (βχ) , (26)
keeping the ϕ direction shift-symmetric. The dynamics of this two-field system is governed by the equations of motion
−∇µ
∇µχ√1 + (∂χ)
2+ f (∂ϕ)
2
+ β1
2
f ′ (∂ϕ)2√
1 + (∂χ)2
+ f (∂ϕ)2
+ V ′
= 0 , (27)−∇µ
f∇µϕ√1 + (∂χ)
2+ f (∂ϕ)
2
= 0 . (28)In the limit β → ∞, (27) gives χ ∝ β−1, and thus (27) and (28) exactly reduce to the constraint (23) and theE.o.M. (25), respectively, in the single-field EFT. The energy-momentum tensor associated with the Lagrangianscalar (26) is
TDBIµν =∂µχ∂νχ+ f ∂µϕ∂νϕ√
1 + (∂χ)2
+ f (∂ϕ)2
+ gµν
[−√
1 + (∂χ)2
+ f (∂ϕ)2 − V
], (29)
and reduces to TDBI-EFTµν in (25) in the limit β →∞. Therefore, the two-field theory (26) derives the single-field one(22) as its single-field EFT limit, even with the inclusion of (minimally coupled) gravity.
3. Reconstruction of v(f) ≡ V (βχ)
The form of the potential V for a given f , i.e. v(f) = V (βχ), with the corresponding single-field EFT of the form(17), can be obtained by solving a differential equation similar to (18), that is
v − f dvdf±√
1 + f
(dv
df
)2+∑n
cn
−f (dvdf
)2±√
1 + f2(
dv
df
)4n = 0 . (30)This apparently does not admit a closed analytical solution for a general P (X), but numerical reconstruction of V isstraightforward.
4. Extension to P (ϕ,X)
We discuss the extension of the two-field completion with DBI-type kinetic terms in order to accommodate P (ϕ,X)models in this subsubsection. While, unlike the case with linear kinetic terms, the relation (24) does not host animmediate interpretation as a Legendre transformation due to the square root structure of DBI, the extension itself isstraightforward. We promote the function V in (22) to the one on both βχ and ϕ, i.e. V (ϕ, βχ). Then the constraintequation (23) is modified to be
∂v
∂f=
X√1− 2fX , (31)
where v(ϕ, f) is V regarded as a function of ϕ and f , provided f ′(βχ) 6= 0. This equation can be algebraically solvedto give the relation of f to ϕ and X. Then we can rewrite the action as a function of ϕ and X and then require thatit coincide with P (ϕ,X), i.e.
−√
1− 2fX − v(ϕ, f) = P (ϕ,X) . (32)
-
8
Utilizing the two equations above, one can determine the form of v, or V , to reconstruct the corresponding P (ϕ,X)theory. Then for our two-field completion, using the obtained V (ϕ, βχ), we promote the Lagrangian scalar to theform
LDBI = −√
1 + (∂χ)2
+ f(βχ) (∂ϕ)2 − V (ϕ, βχ) , V (ϕ, βχ) ≡ v(ϕ, f(βχ)) , (33)
acquiring the corresponding (partial) UV theory. The limit β →∞ properly recovers the single-field P (ϕ,X) model.
C. Comparison with other proposals
In this subsection we discuss relations to the two-field models studied in the literature as (partial) UV completionof P (ϕ,X) models [26, 27, 29–32]. In the prescription of [26, 27], the field space metric is flat. Therefore, afterrequiring the absence of conical singularities in the field space, there is no parameter that describes the properties ofthe field space, see eq. (2). As a result, the mass of the extra field is determined by the form of P (X). For a givenform of P (X), there is no parameter controlling the mass of extra field and thus the scale at which the correspondingsingle-field EFT breaks down. This means that the two-field model of [26, 27] does not allow for a limit in which thesingle-field P (X) model is recovered with an arbitrarily high precision.
More general prescription called gelaton scenario was originally proposed in [29] and then further developed in[30]. There is a parameter that controls the mass of the extra field and thus the scale at which the correspondingsingle-field EFT breaks down. By taking a particular limit, one can therefore recover the single-field P (ϕ,X) modelwith an arbitrarily high precision. However, in this scenario the field space metric and the potential in the two-fieldmodel are simultaneously determined by the form of P (ϕ,X), meaning that the field space metric is less controllablethan the prescriptions developed in the present paper.
In [31], specialized to a single-field DBI action with or without shift symmetry, an extended version of the gelatonscenario was developed, in which the field space metric is specified independently from the form of the single-fieldaction 6. In particular the field space is specified to the hyperbolic one. The curvature of the hyperbolic field spacemetric controls the mass of extra field and thus the scale at which the corresponding single-field EFT breaks down. Inthe limit where the curvature of the field space is infinite, the single-field DBI action with or without shift symmetryis recovered with an arbitrarily high precision.
In the present paper we have developed two different prescriptions of two-field completion: one with linear kineticterms in subsection II A and the other with the DBI-type kinetic terms in subsection II B. The first prescription can beconsidered as a direct generalization of the extended gelaton scenario developed in [31] to the general P (ϕ,X) models.It can be applied to any P (ϕ,X) that is convex (or concave) with respect to X, i.e. PXX > 0 (or PXX < 0). The fieldspace metric can be specified to the form (2) with an arbitrary positive and non-constant function f , independentlyfrom the form of P (ϕ,X). The parameter β parameterizing the curvature of the field space then controls the mass ofthe extra field and thus the scale at which the single-field EFT breaks down. In the limit where the curvature of thefield space is infinite, the single-field P (ϕ,X) model is recovered with an arbitrarily high precision.
The second prescription developed in the present paper utilizes a two-field system with DBI-type kinetic terms.This is motivated by the two facts: DBI scalars are free from simple wave caustics [17]; and DBI scalars exist in stringtheory [33]. The field space metric (corresponding to the metric of extra dimensions in which the D-brane moves) canbe specified to the form (2) with an arbitrary positive and non-constant function f , independently from the form ofP (ϕ,X). The parameter β parameterizing the curvature of the field space then controls the mass of the extra fieldand thus the scale at which the single-field EFT breaks down. In the limit where the curvature of the field space isinfinite, the single-field P (ϕ,X) model is recovered with an arbitrarily high precision.
III. CAUSTIC AVOIDANCE
As shown in [16, 17], models of k-essence theory in general run into caustics formation within a finite time in theMinkowski spacetime, with a planar-symmetric configuration of the scalar field ϕ. The implication of this nature isthat such a k-essence model should be interpreted as an effective theory, and that a more complete theory needs totake over before the formation of caustics. This section is devoted to the demonstration of the resolution of a caustics
6 General P (ϕ,X) models were also studied in [31] but with the original gelaton scenario that we have explained above. On the otherhand, [32] adopted the original gelaton scenario for all single-field models including the DBI model.
-
9
singularity that exists in the single-field effective theory of the type (4), by promoting it to a two-field theory of thetype (12) that is valid beyond the would-be caustics.
In [16, 17], the existence of caustics in k-essence is shown analytically. For our two-field completion in this paper,we employ numerical integrations, due to the nature of highly nonlinear, coupled partial differential equations in theconsidered model. For a concrete demonstration, we make use of the same example single-field model as in [16, 17],namely a k-essence with
P (X) = X +c
2X2 , (34)
where c is a constant. This choice corresponds to c0 = 0, c1 = 1, c2 = c/2 and all other cn = 0 in (17). Using (20),this model can be mapped to the choice of V (βχ) in (12) as
V =1
2c[f(χ̃)− 1]2 , χ̃ = βχ , (35)
for a given f(χ̃). Indeed, eq. (5) together with this V (βχ) = v(f) gives the relations f = 1 + cX and V = cX2/2, andeq. (4) recovers (34) as a single-field effective model.
A. Planar symmetric configuration
Throughout this section we take a planar-symmetric configuration in a Minkowski spacetime, i.e. without loss ofgenerality,
ϕ = ϕ(t, x) , χ = χ(t, x) , (36)
where t and x denote the temporal and one spatial directions, respectively. With the flat metric, the equations ofmotion (13) and (14) respectively reduce to
∂2t χ− ∂2xχ+ β[V ′ − 1
2
(τ2 − ζ2
)f ′]
= 0 , (37)
∂tτ − ∂xζ + β (τ ∂tχ− ζ ∂xχ)f ′
f= 0 , (38)
where we have defined τ ≡ ∂tϕ and ζ ≡ ∂xϕ, which suffice to represent the degrees of freedom of the shift-symmetricfield ϕ. These two equations, together with the integrability condition
∂tζ = ∂xτ , (39)
closes the system of equations. Note that X = (τ2 − ζ2)/2 in the planar-symmetric configuration.The one-field reduction of the above model in the limit β →∞ is the k-essence (34). The constraint equation (5)
in this case reads
f = 1 + cX . (40)
Then the only propagating degree of freedom is ϕ, and its equation of motion yields
∂tτ −4 c τζ
2 + c (3τ2 − ζ2) ∂xτ −2 + c
(τ2 − 3ζ2
)2 + c (3τ2 − ζ2) ∂xζ = 0 , ∂tζ − ∂xτ = 0 , (41)
where χ has been integrated out thanks to the use of (40). These equations (41) for τ and ζ indeed exactly matchthe ones obtained starting from the P (X) model in (34).
B. Setup of numerical calculation
We now describe the setup of numerical integrations both for the single-field EFT and for the two-field completion.From here on, c = 1 is taken for a computational purpose. Also, as a representative (partial) UV completion, weconsider a hyperboloidal field space by taking
f(χ̃) = e2χ̃ , χ̃ ≡ βχ . (42)
-
10
We first solve the system of equations (41) for τ and ζ to observe the formation of caustics in the single-field system,and then compare that to the solutions of the two-field system (37), (38) and (39) to show the singularity resolutionby invoking the motion of the second field χ.
We choose to have our demonstration go along the example given in [17]. To this end, we set the initial conditionat t = 0 as follows: the reduced single-field case (41) admits a class of solutions that obey the equation of differentialsin terms of the variables X = (τ2 − ζ2)/2 and v ≡ −ζ/τ [17],7
dX
Xcs(X)− 2 dv
1− v2 = 0 , at t = 0 , (43)
where cs is defined by and given as
cs ≡√
PX2XPXX + PX
=
√1 +X
1 + 3X. (44)
At time t = 0, we set v = 0.8 exp(−x2) and solve the nonlinear ordinary differential equation (43) for X in one spatialdimension, with the boundary condition fixing X = 2 at the boundaries. The exact locations of the spatial boundaries,which we take x = −10 and x = 20 for the numerical computation, are not important as long as they are sufficientlyaway from the origin x = 0. We take the τ > 0 branch of this solution as the initial condition for the time evolution.For all the solutions shown in this section, we set the size of each spatial increment to be ∆x = 4 × 10−4 and timestep to be ∆t = 2 × 10−4 and take the periodic boundary condition for each variable. The numerical methods wereimplemented in two independent codes, one in C and and the other in Python (with FEniCS package [34, 35]), andthe results have been cross-confirmed.
C. Results
In Fig. 1, the numerical solution of the single-field EFT (41) is shown on the right panels, while the case of thestandard canonical scalar field L = X is on the left panels as a reference point. The height of the wave at the initialtime differs between the two cases, because the initial condition is obtained using (43) for each case, i.e. cs = 1for the canonical scalar and cs given in (44) for the k-essence, with the same v and the same boundary conditionfor X. In the case of the canonical scalar, the wave simply travels without any change indefinitely. On the otherhand, the k-essence wave gets distorted while traveling. Its shape changes as if it would fall over in the direction ofthe propagation. Consequently, the derivative of τ , i.e. second derivative of ϕ, increases over time, and it becomesdivergent around t = 4.5, as observed in the right bottom panel of Fig. 1. The numerical evolution is stopped atthis point, beyond which the result could not be trusted. This divergence in the second derivative of the field ϕ isinterpreted as the formation of caustics in the considered k-essence model.
Fig. 2 shows the avoidance of the caustics formation in the two-field model (12) with f and V given in (42) and(35), respectively. Here we use {τ, ζ, χ̃} as the variables and solve (37), (38) and (39) for the numerics. The sameinitial conditions are taken for τ and ζ as in the single-field case. For the initial conditions for χ and ∂tχ, we assumethat χ is stabilized such that the EFT constraint equation (40) is respected both for its value and for its derivative.In particular, the current case with c = 1 yields to satisfy at the initial time,
χ̃ =1
2ln (1 +X) , ∂tχ̃ =
X[(
1− v2)
(1 +X) + 2vcsX]∂xv
(1− v2) (1 +X) [(1 +X) v2 − 1− 3X] , at t = 0 , (45)
where the time derivatives of τ and ζ have been replaced by using the equations of motion on the constrainedhypersurface, i.e. (41), and the spatial derivative of X is replaced by (43). Using (44) for cs, taking v = 0.8 exp(−x2)and fixing X at the initial time as explained around (43), the above equations uniquely determine the initial conditionsfor χ̃ and ∂tχ̃. We again take the periodic boundary condition.
Comparing Figs. 1 and 2, it is evident that the two-field case is free from the divergence in ∂τ/∂x, which is a secondderivative of the field ϕ, appearing around t = 4.5, implying that the two-field completion indeed removes the causticsingularity that appears present in its low-energy single-field EFT. The single-field EFT well describes the evolutionof the more fundamental, underlying system until it is about to evolve into the caustics. It is the parameter β that
7 In terms of the notations used in [17], the solutions of this class are along the C− characteristics, together with a constant Riemanninvariant Γ−, defined in eq. (20) of [17].
-
11
−5 0 5 10x
−4
−2
0
2
4
6
8
10τ
canonical scalar
−5 0 5 10x
−2
0
2
4
6
τ
single field
−5 0 5 10x
−15
−10
−5
0
5
10
15
∂τ/∂x
canonical scalar
−5 0 5 10x
−400
−300
−200
−100
0
∂τ/∂x
single field
FIG. 1: Numerical solutions for single-scalar field models. The right panels show the result for the P (X) = X + X2/2 model(34), as compared to the standard canonical case L = X on the left panels. The top panels depict the propagation of τ in eachmodel, and the bottom ones that of ∂τ/∂x. In each panel, the wave travels to the right, and the snapshots of the wave aretaken, from left to right, at t = 0, 1.5, 3, 4.5, 6, 7.5 for the left panels, and at t = 0, 1.5, 3, 4.5 for the right panels. Clearly,the wave propagates trivially for the standard canonical scalar, and on the other hand, formation of caustics can be seen neart = 4.5 in the case of the P (X) model.
controls the scale at which the effect of the second field χ starts playing a role. As is seen from (37), β corresponds tothe mass scale of χ, and the larger the value of β, the larger the mass. Thus, for a smaller value of β, the single-fieldEFT breaks down at a lower energy scale, i.e. at an earlier stage of a caustic formation. This expectation is verified byobserving that the β = 0.5 case starts deviating from the single-field EFT dynamics already around t = 3, while thedeviation starts occurring only around t = 4.5 for β = 2. As a result, the former goes through a smoother evolutionthan the latter, which carries sharper peaks both in τ and ∂τ/∂x (and other variables as well).
In the EFT limit β → ∞, the constraint equation (5), or (40), should be fully respected. In other words, thedeviation of χ̃ from the value of ln(1 + X)/2 is an indicator of the departure from the EFT. Fig. 3 compares χ̃ inthe two-field model with β = 0.5 and ln(1 + X)/2 computed from the numerical result of the single-field case. Thecaustics would form around t = 4.5 for the latter, and indeed the deviation increases toward this moment. While theshape of the wave appears to fall over in the direction of the propagation in the single-field case, it is smoothed out inthe two-field case. The presence of this second field χ is crucial for the resolution of the caustic singularity. Finally,Fig. 4 shows the time evolution of the minimum values (i.e. the largest amplitudes) of ∂τ/∂x to compare the cases ofsingle field, two fields with β = 0.5 and two fields with β = 2. This clearly depicts the divergence in the single-fieldEFT, while the values of ∂τ/∂x are well under control in the two-field completed model. It is again shown that theβ = 0.5 case renders the caustics harmless more efficiently than β = 2. To our knowledge, this is the first numericalpresentation of the caustic formation in a k-essence model and of its resolution in its UV completed version.
To summarize, the two-field completion indeed removes the caustics, and such a partial UV completion is necessaryto describe the evolution of a physical system close to and beyond the (would-be) singularity. The parameter β controlthe mass/energy scale of the EFT breakdown, and essentially a measure of the onset of the UV physics. Hence itis a natural question to ask and is of interest to investigate what and how much influence the UV effects produce
-
12
−5 0 5 10x
−2
−1
0
1
2
3
4
5
6
7
τ
β = 0.5
−5 0 5 10x
−2
−1
0
1
2
3
4
5
6
7
τ
β = 2
−5 0 5 10x
−300
−200
−100
0
100
200
300
∂τ/∂x
β = 0.5
−5 0 5 10x
−300
−200
−100
0
100
200
300
∂τ/∂x
β = 2
FIG. 2: Numerical solutions for the two-field completion model (12) with f and V given by (42) and (35), respectively, withc = 1, for two different values of β. The left panels show the solutions for, from top to bottom, τ and ∂τ/∂x in the case ofβ = 0.5, while the right panels are those for β = 2. The initial conditions are taken to correspond to Fig. 1. In each panel, thewave travels to the right, and the snapshots are taken, from left to right, at t = 0, 1.5, 3, 4.5, 6, 7.5. The caustics singularityappearing in the right panels of Fig. 1 is resolved, more smoothly for a smaller mass scale β = 0.5 than for a larger one β = 2.
on physical systems. In the next section, we therefore explore and analyze the two-field system in the cosmologicalsettings.
IV. COSMOLOGICAL APPLICATIONS
In this section, we analyze the two-field completion model of k-essence theory proposed in Sec. II, aiming forcosmological applications. For the purpose of computational ease and of intuitive illustration, we focus our detailedanalysis on the case of linear kinetic terms presented in subsection II A. Using the two-field model with linear kineticterms that is minimally coupled to gravity, the full action of our interest is
S =
∫d4x√−g
[M2Pl
2R− 1
2(∂ϕ)
2 − f(βχ)2
(∂ϕ)2 − V (βχ)
], (46)
where MPl is the reduced Planck mass and R is the Ricci scalar associated with the spacetime metric. In the followingsubsections, we first consider the flat Friedmann-Lemâıtre-Robertson-Walker (FLRW) background and then proceedto the perturbations around it. In view of the cosmological application, we keep including gravity throughout ouranalysis, which is a secure improvement compared to the one in [27].
-
13
1 2 3 4 5 6
x
0.5
0.6
0.7
0.8
0.9
1.0
1.1
χ̃
FIG. 3: Comparison between χ̃ = βχ in the two-field system with β = 0.5 (blue solid curve) and the value of ln(1 + X)/2in the single-field system (black dotted) around the time of caustic formation. These two quantities coincide in the limitβ →∞ through the constraint equation (40) with f given in (42), and in this sense, this plot indicates the deviation from thesingle-field EFT. The snapshots of the waves are taken at t = 3, 3.5, 4, 4.5, 5, 5.5, although the evolution of the single-fieldmodel is stopped at t = 4.5.
0 1 2 3 4 5 6 7
t
−350
−300
−250
−200
−150
−100
−50
0
min
of∂τ/∂x
β = 0.5
β = 2
single field
FIG. 4: The time evolution of the minimum values of ∂τ/∂x (correspondingly to the largest values of |∂τ/∂x|), for the single-field and the two-field cases. Caustics forms in the single-field k-essence (black solid curve) around t = 4.5, and it is amelioratedin the two-field completion (blue dotted and orange dashed curves respectively for β = 0.5 and β = 2).
A. FLRW background
For the cosmological background, we take the flat FLRW metric
ds2 = −N2(t) dt2 + a2(t) δij dxidxj , (47)and the homogeneous modes for the fields
〈ϕ〉 = φ(t) , 〈χ〉 = χ(t) , (48)with some abuse of notation. Here a(t) is the scale factor and N(t) is the lapse function, which we will later set tounity. Then the background action of (46) reads
S(0) = V∫Ndt a3
[−3M2Pl
(∂ta)2
a2N2+
1
2
(∂tχ)2
N2+f
2
(∂tφ)2
N2− V
], (49)
-
14
where V is the comoving spatial 3-volume, and hereafter f and V denote the background values of the correspondingfunctions, i.e. f = f(βχ(t)) and V = V (βχ(t)), respectively. The background dynamics is governed by the equationsof motion
∂2t χ
N2+
(3H − ∂tN
N2
)∂tχ
N+ β
[V ′ − f
′
2
(∂tφ)2
N2
]= 0 , (50)
∂2t φ
N2+
(3H − ∂tN
N2+ β
f ′
f
∂tχ
N
)∂tφ
N= 0 , (51)
where prime on f and V denotes derivative with respective their argument, together with the Friedmann equation
3M2PlH2 =
f
2
(∂tφ)2
N2+
1
2
(∂tχ)2
N2+ V , (52)
where H ≡ ∂ta/(aN) is the Hubble expansion rate. The above three equations close the system. The backgroundenergy density ρ̄ and the pressure p̄ read
ρ̄ =f
2
(∂tφ)2
N2+
1
2
(∂tχ)2
N2+ V , p̄ =
f
2
(∂tφ)2
N2+
1
2
(∂tχ)2
N2− V . (53)
So far the above equations are for the full two-field system.We now proceed to the EFT reduction of the full system to a low-energy single-field one and then to the leading-
order correction for β � 1. From here on, we choose to set N = 1. For a consistent expansion for large β, we expandas
χ = �χ1 + �2χ2 + �
3χ3 + . . . , φ = φ0 + �φ1 + �2φ2 + . . . , H = H0 + �H1 + �
2H2 + . . . , (54)
where � is the expansion parameter, with β = O(�−1), and subscripts 0, 1, 2, . . . keep track of the expansion order.Note that φ and χ start from the 0th and 1st orders in �, respectively. We expand the equations of motion, (50)and (51), and the Friedmann equation (52) for small �. We first notice that the equation of motion for χ starts withO(�−1), and the leading order for the rest of the equations is O(�0). Picking up the leading order of each equation of(50), (51) and (52), we find
V ′0 = f′0X0 , (55)
∂2t φ0 + 3H0 c2s,0 ∂tφ0 = 0 , (56)
3M2PlH20 = f0X0 + V0 , (57)
where V0 ≡ V (βχ1), f0 ≡ f(βχ1), and X0 ≡ (∂tφ0)2 /2, and the 0th-order sound speed cs,0 in this model takes theform
c2s,0 ≡∂tp0∂tρ0
=ff ′0V
′′0 − ff ′′0 V ′0
ff ′0V′′0 + (2f
′02 − ff ′′0 )V ′0
, (58)
where ρ0 ≡ f0X0 + V0 and p0 ≡ f0X0 − V0 are the 0th-order energy density and pressure, respectively. In deriving(56), the time derivative of (55) was also imposed. Recalling the correspondence with the P (X) theory, P ↔ fX −Vand PX ↔ f , eqs. (56) and (57) exactly reproduce the equations for the effective single-field P (X) theory.
To compute the higher orders in small β−1, let us expand the energy density as
ρ̄ = ρ0 + ρ1 + ρ2 + . . . , (59)
where ρ0 is given below (58) and
ρ1 ≡ 6M2PlH0H1 =f0c2s,0
∂tφ0 ∂tφ1 , (60)
ρ2 ≡ 3M2Pl(2H0H2 +H
21
)=
f0c2s,0
∂tφ0 ∂tφ2 +f0
6c4s,0
(3 +
∂tc2s,0
c2s,0H0
)(∂tφ1)
2
− f20
β2f ′02c2s,0
[3(1− c2s,0
)2f0X0
M2Pl+ 3
(1− c2s,0
)H0 ∂tc
2s,0 −
9(1− c2s,0
)22
(3 c2s,0 + 2
(1− c2s,0
) f0f ′′0f ′0
2
)H20
], (61)
-
15
where the second equality in each equation above comes from the Friedmann equation (52). The higher orders of χ,i.e. χ2, χ3, . . . , can be solved iteratively from (50) with respect to other variables of lower orders,
χ2 =1− c2s,02βf ′0X0
ρ1 , (62)
χ3 =1− c2s,02βf ′0X0
ρ2 +
[2 ∂tc
2s,0 − 3
(1− c2s,0
)(4 c2s,0 +
(1− c2s,0
) f0f ′′0f ′0
2
)H0
]ρ21
48βH0f0f ′0X20
−[2 ∂tc
2s,0 − 3
(1− c2s,0
)(3 c2s,0 − 1 +
(1− c2s,0
) f0f ′′0f ′0
2
)H0
]3(1− c2s,0
)H0f
20
4β3f ′03X0
− 3(1− c2s,0
)2f30
2β3M2Plf′03
. (63)
Combining with the above equations, the φ’s E.o.M. (51) translates to the equations for ρ1 and ρ2, giving
∂tρ1 +
[3H0
(1 + c2s,0
)+
f0X0M2PlH0
]ρ1 = 0 , (64)
∂tρ2 +
[3H0
(1 + c2s,0
)+
f0X0M2PlH0
]ρ2 =
(∂tc
2s,0
4f0X0− 1 + c
2s,0
2M2PlH0+
f0X012M4PlH
30
)ρ21
− 9(1− c2s,0
)f20H
20
β2f ′02
[∂tc
2s,0 +
(1− c2s,0
) (3(1− 3 c2s,0
)M2PlH
20 + 2f0X0
)2M2PlH0
− 3(1− c2s,0
)2 f0f ′′0f ′0
2H0
]. (65)
We have obtained up to the first 3 orders of equations. We note that, at the first order, only the dispersion relationfor ρ1 is modified as seen in (64), and then (65) indicates that the lower-order terms act as source for the second andhigher orders. Higher-order equations can be obtained by a straightforward extension of the above methodology.
In this subsection, we have shown that the single-field reduction from the two-field UV theory correctly reproducesthe expected k-essence as EFT in the limit of large mass of χ field, β →∞. The corrections to the leading-order EFTcan be unambiguously calculated by the method of perturbative expansion, up to an arbitrary order in β−1. We haveso far demonstrated this for the flat FLRW background, and, in the following subsection, we extend the analysis tothe cosmological perturbations.
B. Cosmological perturbations
In this subsection, we proceed to the perturbations around the FLRW background. It can be trivially seen that thetensor and vector sectors are as standard as in any models of scalar fields minimally coupled to gravity, and thus welook into the details of the scalar sector below. For the scalar sector, we expand the variables as
ϕ(t,x) = φ(t) + δϕ(t,x) , χ(t,x) = χ(t) + δχ(t,x) , (66)
for the scalar fields and
g00(t,x) = −1− 2Φ(t,x) , g0i(t,x) = a(t) ∂iB(t,x) , gij(t,x) = a2(t) [(1 + 2 Ψ(t,x)) δij + 2 ∂i∂jE(t,x)] , (67)
for the metric. The linear-order action vanishes after using the background equations. In deriving the quadraticaction, we take the spatially flat gauge, namely Ψ = E = 0. It is then clear that Φ and B are non-dynamicalvariables and can be eliminated by replacing them in terms of the dynamical degrees of freedom. By employing theFaddeev-Jackiw method [36], we obtain the quadratic action in terms only of the dynamical variables δi ≡ (δϕ, δχ)as, in the Fourier space,
S(2)scalar =
1
2
∫dtd3k
(∂tδ†i Tij ∂tδj + ∂tδ
†i Xij δj − δ†i Xij ∂tδj − δ†i Ω2ij δj
), (68)
-
16
up to total derivatives. Here, T , X and Ω2 are real, 2 × 2 matrices constructed by the background quantities andhave the properties TT = T , XT = −X and (Ω2)T = Ω2. Their explicit expressions are
T = a3(f 00 1
), X = a3
0 β2 f ′ ∂tφ−β
2f ′ ∂tφ 0
, (69)Ω211 = a
3f
[k2
a2+
3f (∂tφ)2
M2Pl− f (∂tφ)
2
2M4PlH2
(f (∂tφ)
2+ (∂tχ)
2)]
, (70)
Ω212 = Ω221 = a
3
[−β
2
2
(f ′2
f− f ′′
)∂tφ∂tχ+
βf ∂tφ
M2PlHV ′ +
3f ∂tφ∂tχ
M2Pl− f ∂tφ∂tχ
2M4PlH2
(f (∂tφ)
2+ (∂tχ)
2)]
, (71)
Ω222 = a3
[k2
a2+ β2
(V ′′ − f
′′
2(∂tφ)
2
)+
2β ∂tχV′
M2PlH+
3 (∂tχ)2
M2Pl− (∂tχ)
2
2M4PlH2
(f (∂tφ)
2+ (∂tχ)
2)]
. (72)
Variation of (68) with respect to δi provides the equations of motion,
∂2t δ + T−1 (2X + ∂tT ) ∂tδ + T
−1 (Ω2 + ∂tX) δ = 0 , (73)in the matrix form. This is so far the genuine two-field system. In what follows, we show that reduction to theone-field EFT is successfully done for large β and that higher-order corrections can be iteratively computed with noambiguity.
In order to expand the system in terms of small β−1, we decompose the background quantities as in (54) and theperturbation variables as
δϕ = δϕ0 + � δϕ1 + . . . , δχ = � δχ1 + �2δχ2 + . . . , (74)
where � = O(β−1). Note that, similarly to the background, the massive field δχ in the EFT reduction starts at theorder of O(�), while the propagating field δϕ in the EFT starts at O(�0), as seen below. Collecting the leading orderof each term, the equation of motion for δχ from (73) reduces to
� ∂2t δχ1︸ ︷︷ ︸O(�)
+ � 3H20 ∂tδχ1︸ ︷︷ ︸O(�)
+ �β2[V ′′0 −
f ′′02
(∂tφ)2
]δχ1︸ ︷︷ ︸
O(�−1)
= βf ′0 ∂tφ0 ∂tδϕ0︸ ︷︷ ︸O(�−1)
− βf0 ∂tφ0V′0
M2PlH0δϕ0︸ ︷︷ ︸
O(�−1)
, (75)
where again f0 ≡ f(βχ1) and V0 ≡ V (βχ1). We observe that the leading order is O(�−1), and all the time derivativesof δχ1 drop out. This implies that δχ1 is algebraically given by the other variables without its own dynamics,i.e. “integrated out,” and that the above equation plays a role of a constraint rather than E.o.M.. Picking up theO(�−1) terms, we find, from (75),
δχ1 =c−2s,0 − 1β ∂tφ0
(f0f ′0∂tδϕ0 −
f20f ′0
2
V ′0M2PlH0
δϕ0
), (76)
where in the last equality � is absorbed into δχ1, and c2s,0 is defined in (58). From (76) it manifests that the leading
order of δχ is already at the β−1 order. It is worth noting that this EFT reduction would be impossible if φ is notmoving, i.e. ∂tφ0 = 0, or X0 = 0, or if there is no coupling between the two scalar fields, i.e. f = const., as is clearfrom (76). Now we plug (76) and its time derivative back into the equation of motion for δϕ in (73) to obtain theEFT equation. The leading order for δϕ is O(�0), and, using the background equations, its equation of motion isfound as
∂2t δϕ0 +
(3 c2s,0 −
∂tc2s,0
c2s,0H0
)H0 ∂tδϕ0 +
[c2s,0 p
20 +
3 (1 + w0)
2
(3(1 + c2s,0
)+
∂tc2s,0
c2s,0H0
)− 9 (1 + w0)
2
2
]H20 δϕ0 = 0 ,
(77)where w0 ≡ p0/ρ0 and p0 ≡ k/(aH0). This exactly coincides with the case of the k-essence model P (X) with thereplacement
f0 ↔ PX , c2s,0 ↔PX
2XPX + PX, (78)
-
17
which is the correct relation as deduced from (8) and (58). This proves that, starting from the partial UV completion(46) with the two fields, the single-field k-essence EFT is correctly induced as the limit of infinite mass β →∞. It isa nontrivial reduction, with an arbitrary coupling function f(βχ), on the non-flat, cosmological background and withthe gravitational interaction included.
To go to one higher order, integrating out δχ2 and collecting the terms of O(�), the equation of motion for δϕ1reads
∂2t δϕ1 + F0 ∂tδϕ1 + Ω20 δϕ1 + F1 ∂tδϕ0 + Ω
21 δϕ0 = 0 , (79)
where
F0 = 3H0 c2s,0 −
∂tc2s,0
c2s,0, (80)
Ω20 = c2s,0
k2
a2+
3 (1 + w0)
2
[3(1 + c2s,0
)+
∂tc2s,0
c2s,0H0
]H20 −
9 (1 + w0)2
2H20 , (81)
F1 =∂tφ1∂tφ0
[3 (1 + w0)
2H0 −
∂tc2s,0
c2s,0−(∂tc
2s,0
)23 c6s,0H0
+∂2t c
2s,0
3 c4s,0H0
], (82)
Ω21 =∂tφ1∂tφ0
[9 (1 + w0)
(w0 − c2s,0
)22 c2s,0
H20 −(c2s,0
k2
a2− 9
(1− w20
)4
H20
)∂tcs,0
3 c4s,0H0+
1 + w02 c4s,0
((∂tc
2s,0
)2c2s,0N
2− ∂2t c2s,0
)].
(83)
As can be seen explicitly from the above, the coefficients F0 and M0 are the same as those for the 0th order in (77),and F1 and Ω
21 are suppressed by β
−1 because of the overall ∂tφ1. Then, combining (77) and (79), we obtain theequation of motion for δϕ up to this order,
∂2t δϕ+ (F0 + F1) ∂tδϕ+(Ω20 + Ω
21
)δϕ = 0 . (84)
On the other hand, the effective action for δϕ must have the form, in the Fourier space,
S(2)eff =
1
2
∫dtd3k a3 T
(|∂tδϕ|2 − Ω2 |δϕ|2
), (85)
and thus the equation of motion reads
∂2t δϕ+
(3H +
∂tT
T
)∂tδϕ+ Ω
2 δϕ = 0 . (86)
Comparing the mass terms in (84) and (86), we find
Ω2 = Ω20 + Ω21 , (87)
and comparing the friction terms in (84) and (86) order by order with the use of the background equations, we obtain
∂tT0T0
= 3H0(c2s,0 − 1
)− ∂tc
2s,0
c2s,0N=⇒ T0 =
f0c2s,0
, (88)
∂t
(T1T0
)= ∂t
[(1− c2s,0c2s,0
+∂tc
2s,0
3 c4s,0H0
)∂tφ1∂tφ0
]=⇒ T1 =
f0c2s,0
(1− c2s,0c2s,0
+∂tc
2s,0
3 c4s,0NH0
)∂tφ1∂tφ0
. (89)
Therefore the action up to the 1st order in β−1 takes the form (85) with T = T0 +T1 and Ω2 = Ω20 +Ω
21. We have now
derived the action (85) and the E.o.M. (79) for δϕ, which is the only dynamical degree of freedom in this iterativeprocedure of EFT reduction. The above expressions indicate that the current expansion should break down in thelimit c2s,0 → 0 since T1 � T0 and Ω21 � Ω20, i.e. the EFT description would be invalidated in this limit. In orderto ensure that this result is not an artifact of the choice of the variable, in the following subsection we shall employanother variable that has a more physically transparent meaning in itself.
-
18
C. EFT expansion in terms of gauge-invariant energy density perturbation
In deriving the higher-order equations in �, it is instructive to proceed with the gauge-invariant perturbation ofthe energy density instead of δϕ, for a more transparent physical interpretation. We define the energy density byρ ≡ nµnνTµν , where nµ = (1/N , N i/N ) is the normal vector with respect to the 3-D spatial hypersurface, and Nand N i are the full-order lapse and shift functions, respectively. Note that N = N = 1 and N i = 0 at the backgroundlevel, and δN = Φ and δN i = ∂iB/a at the linear perturbation with the decomposition given in (67). Then the linearperturbation of ρ takes the form
δρ = f ∂tφ∂tδϕ+ ∂tχ∂tδχ+ β
[f ′
2(∂tφ)
2+ V ′
]δχ−
[f (∂tφ)
2+ (∂tχ)
2]
Φ . (90)
The gauge-invariant combination of δρ we choose to use in this work is the one on a slice comoving with the ϕdirection, defined by
δρGI = δρ−∂tρ̄
∂tφδϕ , (91)
where the background energy density ρ̄ is defined in (53). This choice is natural in the regime of the single-fieldEFT; on the other hand, once the system recovers to the genuine two-field dynamics, choosing other gauge-invariantquantities may be more appropriate, e.g. those in [37, 38]. We stick to the variable (91) in this work, however, sinceour primary goal is to demonstrate successful reduction to the EFT and the consistent procedure to compute thecorrections to it for large β. Using the background equations and (90) and the Hamiltonian constraint equation inthe spatially flat gauge Ψ = E = 0, i.e.,
Φ =f ∂tφ δϕ+ ∂tχ δχ
2M2PlH, (92)
we obtain the expression for the gauge-invariant energy density contrast, given by
δGI ≡ρ̃GIρ̄
=1
ρ̄
(f ∂tφ∂tδϕ−
f ∂tφ
2M2PlH
[f (∂tφ)
2+ (∂tχ)
2]δϕ+
3H
∂tφ
[f (∂tφ)
2+ (∂tχ)
2]δϕ
+ ∂tχ∂tδχ+ β
[f ′
2(∂tφ)
2+ V ′
]δχ− ∂tχ
2M2PlH
[f (∂tφ)
2+ (∂tχ)
2]δχ
).
(93)
We use this variable as an independent variable in the following analysis, and in fact, since δχ can be integrated outiteratively order by order, it is the only dynamical variable in the EFT expansion.
Let us now perform the expansion in terms of small β−1. Expanding as in (54) for the background and (74) for theperturbations, the gauge-invariant density contrast (93) at the leading order reduces to
δGI,0 =1
ρ0
2f0X0c2s,0
(∂tδϕ0∂tφ0
− f02M2PlH0
∂tφ0 δϕ0
)+
3H0ρ0
f0 ∂tφ0 δϕ0 (94)
=1 + w0c2s,0
[∂tδϕ0∂tφ0
− 3 (1 + w0)2
H0∂tφ0
δϕ0
]+ 3 (1 + w0)
H0∂tφ0
δϕ0 , (95)
after using the constraint equations (55) for χ1 and (76) for δχ1 and replacing f0 in favor of w0 in the second equality.Note that this expression is exactly the same as the equivalent variable in the single-field k-essence model. Followingthe procedure summarized in Appendix A, we obtain the quadratic action for δGI,0 in the Fourier space, given by
S(2)0 =
1
2
∫dtd3k a3T̃0
(|∂tδGI,0|2 − Ω̃20 |δGI,0|2
), (96)
where again p0 = k/(aH0), and
T̃0 =3M2Pl
(1 + w0) p20, Ω̃20 =
3M2Pl(1 + w0) p20
(c2s,0 p
20 + 15 + 9 c
2s,0 − 21 (1 + w0) +
9 (1 + w0)2
2
)H20 . (97)
-
19
One can easily confirm that this expression can be exactly recovered by starting from the corresponding k-essencetheory, with the correct identification (78). This concludes the successful reduction from the two-field theory to thesingle-field EFT as the leading order in the expansion β →∞.
Proceeding to the first-order in the expansion of small β−1, the effective action is given by (85) together with thecoefficients (81), (83), (88) and (89). In this computation, we use the same variable as in (95) (with the replacementδGI,0 → δGI), only aiming for the calculations of the corrections to the EFT dynamics from the higher order, insteadof making observable predictions. Including up to the first sub-leading order in � = O(β−1), we find the action of theform
S(2)0&1 =
1
2
∫dtd3k a3
(T̃0 + T̃1
) [|∂tδGI|2 −
(Ω̃20 + Ω̃
21
)|δGI|2
], (98)
where T̃0 and Ω̃20 are the same as given above, and
T̃1 =3M2Pl
(1 + w0) p40
∂tφ1∂tφ0
[− p20 +
1
c22,0
(p20 − 9 (1 + w0) +
27 (1 + w0)2
4
)− 9 (1 + w0)w
20
2 c4s,0
− 3(
1− 2 p20
9 c2s,0− 1 + w0
2 c2s,0+
1− w204 c4s,0
)∂tc
2s,0
c2s,0H0−(∂tc
2s,0
)2c6s,0H
20
+∂2t c
2s,0
c4s,0H20
]. (99)
The full expression of Ω̃21 is rather lengthy and is not important for our purpose, and thus we only write the expressionwith constant c2s,0 here, giving
Ω̃21 =3H20 (1 + w0)
16 p2∂tφ1∂tφ0
[− 108 (1− w0) (1 + 3w0)− 8 (1 + 9w0) p2
+45 (1 + 3w0)
2(1− w0) + 8
(5− 7 (1 + w0) + 3 (1 + w0)2
)p2
c2s,0− 18w
20 (1− w0) (7 + 9w0)
c4s,0
]. (100)
From this expression, it is evident that the EFT expansion breaks down for c2s,0 → 0, as it drives T̃1 � T̃0 andΩ̃21 � Ω̃20, which is expected by a general argument of EFT [2].
In this section, therefore, we have explicitly shown that the EFT reduction is successfully done as the leading orderin the limit β →∞, given in the 0th-order (85) for δϕ and (96) for the gauge-invariant density contrast δGI, and thatthe sub-leading corrections can be unambiguously derived by iteratively expanding the orders of small β−1, found inthe 1st-order (85) for δϕ and (98) for δGI. In passing, we also observe that the c
2s,0 → 0 limit triggers the departure
from the EFT description.
V. SUMMARY AND DISCUSSION
The class of k-essence models are widely used in the context of cosmological applications, for both early- andlate-time accelerated expansion. The dynamics of the scalar field(s) in these models drive the expansion and lead tothe predictions of inflationary observables as well as the fate of the universe. While the k-essence has attracted muchattention in this respect, it has been pointed out that models of its shift-symmetric version generically form causticsingularities in the spacetime regions where a planar-symmetric configuration is well respected [16–18]. Two classesof shift-symmetric k-essence are known to be free from the caustics, namely the standard canonical scalar [16] andthe scalar field with the DBI-type kinetic term [17]. In this paper, with this knowledge in mind, we have studiedtwo-field completions of some general classes of shift-symmetric single-field k-essence models for those two cases. Tothis end, we have introduced a parameter β that controls the mass scale of the second field χ, so that the single-fieldEFT description should be recovered in the limit β →∞, equivalently mχ →∞, by integrating out the second field.
In Sec. II, we have introduced the class of k-essence we consider as an EFT and then its (partially) UV-completedmodel by promoting a second field to a dynamical degree of freedom on a curved field space. We have exemplifiedthe flat, hyperboloidal and spheroidal geometry of the field space. The completion has been done both for the linearkinetic terms and for the DBI-type kinetic terms, and in each case, we have shown that the two-field model is formallyreduced to the expected single-field k-essence EFT in the β →∞ limit.
Sec. III has been devoted to the explicit demonstration of the caustic formation in the single-field EFT and ofits resolution by the two-field hyperboloidal field space, by performing numerical integrations. To our knowledge,
-
20
this is the first numerical illustration of the formation and resolution of caustics in a k-essence model and its UV-completed theory. From the numerical result, it is evident that the dynamics of the EFT evolves into the formationof caustics as the second derivative of the scalar field, ∂2ϕ, diverges. This singularity is resolved in the two-fieldcase by transferring the energy to the second field χ prior to the caustic formation, and consequently the secondderivative ∂2ϕ is smoothed out. This is the moment when the EFT description breaks down and the system turnsinto a full two-field dynamics. For a smaller value of the controlling parameter β, the system deviates from the EFTat a lower energy scale, i.e. at an earlier time during the evolution. This expectation has indeed been confirmed inthe numerical calculation, and consequently the shape of the wave stays smoother for a smaller β than for a largerone. This completes the demonstration of the partial UV completion of the shift-symmetric k-essence, with the useof a curved field space in the UV sector.
In Sec. IV, we have then considered the above-verified UV model in view of cosmological applications. We havefirst derived the background equations on the flat FLRW metric. Expanding for small β−1 and collecting the leading-order terms in each equation, we have observed that the leading order of the heavy field χ is in fact O(β−1) toderive the O(β0) equations for the light field ϕ. The resulting leading EFT equations have been shown to exactlyreproduce those obtained starting from the corresponding k-essence model P (X). The sub-leading corrections canalso be deduced iteratively in a straightforward manner in the small β−1 expansion. Turning to the cosmologicalperturbations around the background, we have conducted a detailed study of the scalar sector, as the vector andtensor perturbations are unchanged from the standard canonical single-field model. As in the background calculation,we have first derived the equations of the genuine two-field system and then expanded them for small β−1. In thisexpansion, δχ can be iteratively integrated out order by order, and the system is effectively reduced to a single-fieldone at each order. This master equation of the linear perturbation indeed reproduces the corresponding k-essenceequation as the leading order in β−1 → 0. The higher-order corrections have again been computed iteratively withoutambiguity. For a transparent physical interpretation, we have converted the single variable to the gauge-invariantdensity contrast δGI = δρGI/ρ̄ and obtained the quadratic action in terms of δGI using the procedure summarized inAppendix A. Looking at the leading and first-order contributions to the action, we have observed that this expansionbreaks down in the limit of vanishing sound speed c2s → 0, which is consistent with the discussion in the language ofEFT seen in e.g. [2]. Therefore, in Sec. IV, we have provided the explicit demonstration that the correct reductionfrom the two-field model to the single-field EFT as the β−1 → 0 limit, with the gravity taken into account, that thesub-leading terms can be iteratively computed, and that the cutoff scale of the EFT description decreases arbitrarilyin the limit c2s → 0.
Our detailed analysis is focused primarily on the completion by the linear kinetic terms (with a curved field space).It can be extended to the case of the DBI-type kinetic terms in a straightforward, but perhaps more tedious, manner.We expect the main qualitative conclusions in Secs. III and IV to be unchanged. Also, as shown in [17], the avoidance ofcaustics in a planar-symmetric configuration only requires an appropriate choice of the k-essence part in the Horndeskitheory [39–41]. Thus the UV completion introduced in Sec. II of this work should be applicable in the presence ofthe higher-order (shift-symmetric) Horndeski terms. Extending the computation done in Sec. IV to such Horndeskimodels is also of interest for further investigation. Finally, our computation in Sec. IV concentrates on the EFTreduction from the UV theory. It would be exciting to see how the β−1 suppressed contributions, i.e. the effects fromthe UV, modify the observables such as inflationary predictions that are computed only from the single-field EFT.We leave these considerations to upcoming studies and would like to come back to these issues in the near future.
acknowledgement
R.N. is grateful to Elisa G.M. Ferreira and Motoo Suzuki for casual discussions on the topic. The work of S.M. wassupported in part by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H02890,No. 17H06359, and by World Premier International Research Center Initiative, MEXT, Japan.
AppendicesAppendix A: Change of variables with derivatives and derivation of its action
In this appendix, we formulate the derivation of quadratic action/Lagrangian in terms of the variable that consistsof a linear combination of the original variable and its first time derivative. This technique is introduced in AppendixB of [42] (see also [43]), and here we keep track of the time dependence of all the coefficients. For our purpose,i.e. derivation in the Fourier space and on an isotropic and homogeneous background, it suffices to consider a one-
-
21
variable classical-mechanical system of a quadratic Lagrangian
L =T
2q̇2 − M
2q2 , (A1)
where q is the physical variable, dot denotes derivative with respect to time, and T and M are in general functions oftime. We aim to describe the dynamics using another variable, say
Q = C q̇ +D q , (A2)
instead of q. To this end, we rewrite the Lagrangian (A1) as
L =T
2C2(C q̇ +D q)
2 − 12
[M +
TD2
C2− ∂t
(TD
C
)]q2 − ∂t
(TD
2Cq2)
=T
2C2[2Q (C q̇ +D q)−Q2
]− 1
2
[M +
TD2
C2− ∂t
(TD
C
)]q2 − ∂t
(TD
2Cq2).
(A3)
Varying this with respect to Q and plugging the expression for Q back into the Lagrangian, it is clear to that theoriginal action (A1) is restored up to total derivatives. Now, we further manipulate the above expression as, bycompleting the square for q,
L = −12
[M +
TD2
C2− ∂t
(TD
C
)][q −
TDC2 Q− ∂t
(TC Q
)M + TD
2
C2 − ∂t(TDC
)]2
+1
2
[TDC2 Q− ∂t
(TC Q
)]2M + TD
2
C2 − ∂t(TDC
) − T2C2
Q2 + ∂t
(T
CQq − TD
2Cq2).
(A4)
Provided
M +TD2
C2− ∂t
(TD
C
)6= 0 , (A5)
we can vary the action with respect to q and solve an algebraic equation for q. Then the first line of (A4) vanishes,and the Lagrangian becomes
L =1
2
[TC Q̇− TDC2 Q+ ∂t
(TC
)Q]2
M + TD2
C2 − ∂t(TDC
) − T2C2
Q2 + (total derivatives)
=1
2
T 2 Q̇2
MC2 + TD2 − C2∂t(TDC
)− 1
2
{TM + TD ∂t
(TC
)− T 2C Ḋ − C2
[∂t(TC
)]2MC2 + TD2 − C2∂t
(TDC
) − ∂t [ T 2DC − TC ∂t (TC )MC2 + TD2 − C2∂t
(TDC
)]}Q2 + (total derivatives)′ .(A6)
where prime is just a bookmark to note that it is a total derivative different from the previous line. This Lagrangianis fully expressed in terms of Q, while its physical content is completely equivalent to the original action (A1), at leastclassically.
1. Relation to Canonical Transformation
In this subsection, we show that the above transformation of the Lagrangian is indeed a canonical transformation,as a consistency check. From (A1), the conjugate momentum of q is
p ≡ δLδq̇
= T q̇ , (A7)
and the Hamiltonian is
H = pq̇ − L = p2
2T+M
2q2 . (A8)
-
22
The Poisson bracket with respect to the {q, p} canonical pair is defined by
{X, Y } ≡ δXδq
δY
δp− δX
δp
δY
δq, (A9)
and it is obvious that {q, p} = 1.On the other hand, from (A6), the conjugate momentum of Q is
P ≡ δLδQ̇
=T 2 Q̇
MC2 + TD2 − C2∂t(TDC
) , (A10)and the Hamiltonian is
H = PQ̇− L
=MC2 + TD2 − C2∂t
(TDC
)2T 2
P 2
+1
2
{TM + TD ∂t
(TC
)− T 2C Ḋ − C2
[∂t(TC
)]2MC2 + TD2 − C2∂t
(TDC
) − ∂t [ T 2DC − TC ∂t (TC )MC2 + TD2 − C2∂t
(TDC
)]}Q2 .(A11)
From the construction, the transformation (q, p)→ (Q,P ) is a canonical one, and we show this explicitly below.From the original Hamiltonian (A8), the Euler-Lagrange equations are
q̇ = {q,H} = pT, ṗ = {p,H} = −Mq . (A12)
Using this, Q and P can be expressed in terms of q and p as
Q =C
Tp+D q , P =
T
MC2 + TD2 − C2∂t(TDC
) [(D + T ∂t(CT
))p+
(TḊ −MC
)q
]. (A13)
Then the Poisson bracket (A9) of Q and P reads
{Q, P} = D T(D + T∂t
(CT
))MC2 + TD2 − C2∂t
(TDC
) − CT
T(TḊ −MC
)MC2 + TD2 − C2∂t
(TDC
) = 1 . (A14)Therefore, (q, p)→ (Q,P ) is a canonical transformation, and we can treat (Q,P ) as a canonical pair to compute thePoisson bracket,
{X, Y }′ ≡ δXδQ
δY
δP− δXδP
δY
δQ, (A15)
because
{X, Y } = δXδq
δY
δp− δX
δp
δY
δq
=
(δX
δQ
δQ
δq+δX
δP
δP
δq
)(δY
δQ
δQ
δp+δY
δP
δP
δp
)−(δX
δQ
δQ
δp+δX
δP
δP
δp
)(δY
δQ
δQ
δq+δY
δP
δP
δq
)=δX
δQ
δY
δP− δXδP
δY
δQ= {X, Y }′ .
(A16)
Also it is then immediate to see, by taking time derivative of Q and P using the Poisson brackets {•, •} with respect
-
23
to (q, p),
Q̇ = ∂t
(C
T
)p+ Ḋ q +
C
T{p, H}+D {q, H}
=MC2 + TD2 − C2∂t
(TDC
)T 2
P ,
Ṗ = ∂t
[T(D + T ∂t
(CT
))MC2 + TD2 − C2∂t
(TDC
)] p+ ∂t T
(TḊ −MC
)MC2 + TD2 − C2∂t
(TDC
) q
+T(TD + T ∂t
(CT
))MC2 + TD2 − C2∂t
(TDC
) {p, H}+ T(TḊ −MC
)MC2 + TD2 − C2∂t
(TDC
) {q, H}= −
{TM + TD ∂t
(TC
)− T 2C Ḋ − C2
[∂t(TC
)]2MC2 + TD2 − C2∂t
(TDC
) − ∂t [ T 2C (D + T ∂t (CT ))MC2 + TD2 − C2∂t
(TDC
)]}Q .
(A17)
These equations are precisely the Euler-Lagrange equations that can be obtained from the transformed Hamiltonian(A11). Therefore, the dynamics of the (q, p) system is reproduced by that of the (Q,P ) system in the exact manner.This concludes the equivalence of the two systems.
[1] P. Creminelli, M. A. Luty, A. Nicolis, and L. Senatore, JHEP 12, 080 (2006), hep-th/0606090.[2] C. Cheung, P. Creminelli, A. Fitzpatrick, J. Kaplan, and L. Senatore, JHEP 03, 014 (2008), 0709.0293.[3] P. Creminelli, G. D’Amico, J. Norena, and F. Vernizzi, JCAP 02, 018 (2009), 0811.0827.[4] N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, JHEP 05, 074 (2004), hep-th/0312099.[5] N. Arkani-Hamed, P. Creminelli, S. Mukohyama, and M. Zaldarriaga, JCAP 04, 001 (2004), hep-th/0312100.[6] H. Motohashi and S. Mukohyama, JCAP 01, 030 (2020), 1912.00378.[7] M. A. Gorji, H. Motohashi, and S. Mukohyama (2020), 2009.11606.[8] C. Armendariz-Picon, T. Damour, and V. F. Mukhanov, Phys. Lett. B 458, 209 (1999), hep-th/9904075.[9] C. Armendariz-Picon, V. F. Mukhanov, and P. J. Steinhardt, Phys. Rev. D 63, 103510 (2001), astro-ph/0006373.
[10] T. Chiba, T. Okabe, and M. Yamaguchi, Phys. Rev. D 62, 023511 (2000), astro-ph/9912463.[11] C. Armendariz-Picon, V. F. Mukhanov, and P. J. Steinhardt, Phys. Rev. Lett. 85, 4438 (2000), astro-ph/0004134.[12] R. Courant and K. Friedrichs, Supersonic Flow and Shock Waves (Interscience Publ., New York, 1948), ISBN 978-0-387-
90232-6.[13] P. Lax, The Initial Value Problem for Nonlinear Hyperbolic Equations in Two Independent Variables (1954), vol. 33, pp.
211–229.[14] P. Lax, Comm. Pure Appl. Math. 10, 537 (1957).[15] A. Jeffrey and T. Taniuti, Non-Linear Wave Propagation: With Applications to Physics and Magnetohydrodynamics
(Elsevier, 1964), ISBN 978-0-12-374917-8, 978-0-08-095780-7.[16] E. Babichev, JHEP 04, 129 (2016), 1602.00735.[17] S. Mukohyama, R. Namba, and Y. Watanabe, Phys. Rev. D 94, 023514 (2016), 1605.06418.[18] C. de Rham and H. Motohashi, Phys. Rev. D 95, 064008 (2017), 1611.05038.[19] N. Tanahashi and S. Ohashi, Class. Quant. Grav. 34, 215003 (2017), 1704.02757.[20] K. Pasmatsiou, Phys. Rev. D 97, 036008 (2018), 1712.02888.[21] G. N. Felder, L. Kofman, and A. Starobinsky, JHEP 09, 026 (2002), hep-th/0208019.[22] S. Mukohyama, Phys. Rev. D 66, 123512 (2002), hep-th/0208094.[23] N. Afshordi, D. J. Chung, and G. Geshnizjani, Phys. Rev. D 75, 083513 (2007), hep-th/0609150.[24] N. Afshordi, D. J. Chung, M. Doran, and G. Geshnizjani, Phys. Rev. D 75, 123509 (2007), astro-ph/0702002.[25] N. Afshordi, Phys. Rev. D 80, 081502 (2009), 0907.5201.[26] E. Babichev and S. Ramazanov, JHEP 08, 040 (2017), 1704.03367.[27] E. Babichev, S. Ramazanov, and A. Vikman, JCAP 11, 023 (2018), 1807.10281.[28] H. Ooguri and C. Vafa, Nucl. Phys. B 766, 21 (2007), hep-th/0605264.[29] A. J. Tolley and M. Wyman, Phys. Rev. D 81, 043502 (2010), 0910.1853.[30] B. Elder, A. Joyce, J. Khoury, and A. J. Tolley, Phys. Rev. D 91, 064002 (2015), 1405.7696.[31] S. Mizuno, S. Mukohyama, S. Pi, and Y.-L. Zhang, JCAP 09, 072 (2019), 1905.10950.[32] A. R. Solomon and M. Trodden, JCAP 09, 049 (2020), 2004.09526.[33] J. Polchinski, String theory. Vol. 2: Superstring theory and beyond, Cambridge Monographs on Mathematical Physics
(Cambridge University Press, 2007), ISBN 978-0-511-25228-0, 978-0-521-63304-8, 978-0-521-67228-3.
-
24
[34] M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N.Wells, Archive of Numerical Software 3 (2015).
[35] A. Logg, K.-A. Mardal, G. N. Wells, et al., Automated Solution of Differential Equations by the Finite Element Method(Springer, 2012), ISBN 978-3-642-23098-1.
[36] L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692 (1988).[37] M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95, 71 (1996), astro-ph/9507001.[38] C. Gordon, D. Wands, B. A. Bassett, and R. Maartens, Phys. Rev. D 63, 023506 (2000), astro-ph/0009131.[39] G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974).[40] C. Deffayet, X. Gao, D. Steer, and G. Zahariade, Phys. Rev. D 84, 064039 (2011), 1103.3260.[41] T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Prog. Theor. Phys. 126, 511 (2011), 1105.5723.[42] A. De Felice and S. Mukohyama, JCAP 04, 028 (2016), 1512.04008.[43] A. E. Gümrükçüoğlu, S. Mukohyama, and T. P. Sotiriou, Phys. Rev. D 94, 064001 (2016), 1606.00618.
I IntroductionII Two-field modelA Two-field model with linear kinetic terms1 Equivalent description of P(X)2 Two-field completion by adding kinetic term for extra scalar3 Reconstruction of v(f)V()4 Extension to P(,X)
B Two-field model with DBI-type kinetic terms1 Equivalent description of P(X)2 Two-field completion by adding kinetic term for extra scalar3 Reconstruction of v(f)V()4 Extension to P(,X)
C Comparison with other proposals
III Caustic avoidanceA Planar symmetric configurationB Setup of numerical calculationC Results
IV Cosmological applicationsA FLRW backgroundB Cosmological perturbationsC EFT expansion in terms of gauge-invariant energy density perturbation
V Summary and discussion acknowledgementA Change of variables with derivatives and derivation of its action1 Relation to Canonical Transformation
References