Generalized multi-Galileons, covariantized new terms, and the no-go theorem for non-singular cosmologies

0.14 MB·

by  Shingo Akama

RUP-17-1
Generalized multi-Galileons, covariantized new terms, and the no-go theorem for
non-singular cosmologies
Shingo Akama1,∗ and Tsutomu Kobayashi1,†
1Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
It has been pointed out that non-singular cosmological solutions in second-order scalar-tensor
theories generically suffer from gradient instabilities. We extend this no-go result to second-order
gravitationaltheorieswithanarbitrarynumberofinteractingscalarfields. Ourprooffollowsdirectly
fromtheactionofgeneralizedmulti-Galileons,andthusisdifferentfromandcomplementarytothat
basedontheeffectivefieldtheoryapproach. Severalnewtermsforgeneralizedmulti-Galileons ona
flat background were proposed recently. We find a covariant completion of them and confirm that
theydo not participate in theno-go argument.
7
1 PACSnumbers: 98.80.Cq,04.50.Kd
0
2
I. INTRODUCTION with multiple scalar fields or higher derivative theories
n
beyond Horndeski. The latter way is indeed success-
a
J ful within the Gleyzes-Langlois-Piazza-Vernizzi scalar-
Inflation[1–3]isanattractivescenariobecauseitgives
1 a naturalresolutionofthe horizonandflatnessproblems tensor theory [27–29], as pointed out in Refs. [26, 30]
1 based on the effective field theory (EFT) of cosmologi-
in standard Big Bang cosmology and accounts for the
cal perturbations [31]. Gradient instabilities can also be
origin of density perturbations that are consistent with
] cured if higher spatial derivative terms arise in the ac-
h observationssuchasCMB.However,there arecriticisms
tionforcurvatureperturbations[16, 17, 32]. Thisoccurs
t that even inflation cannot resolve the initial singular-
- in a more general framework [33, 34] than [27] including
p ity [4] and the trans-Planckianproblem for cosmological
e perturbations [5]. Alternative scenarios such as bounces Hoˇrava gravity [35]. In some cases it is possible, even
h without such general frameworks, that the strong cou-
and Galilean Genesis have therefore been explored by a
[ pling scale cuts off the instabilities [36].
number of authors (see, e.g., Ref. [6] for a review).

1
To avoidthe initial singularity,there mustbe aperiod
v
6 in which the Hubble parameter H is an increasing func- The purpose of the present paper is to show that, in
2 tionoftime. Thisindicatesaviolationofthenullenergy contrasttothecaseofthehigherderivativeextension,the
9 condition(NEC), possibly causingsome kindof instabil- no-go theorem for non-singular cosmologies still holds in
2 ities. It is easy to show that NEC-violating cosmological general multi-scalar-tensor theories of gravity. In a sub-
0 solutionsareindeedunstableifthe universeisfilledwith class of the generalized multi-Galileon theory [37], the
1. ausualscalarfieldoraperfectfluid. However,thisisnot same conclusion as in the single-field case was obtained
0 the case ifthe underlyingLagrangiandepends onsecond in [38]. It was found in [26] that the no-go theorem
7 derivativesofa scalarfield[7], andonecanconstructex- can also be extended to the EFT of multi-field models
1 plicitly a stable cosmologicalphase in which the NEC is in which a shift symmetry is assumed for the entropy
v: violated in the Galileon-type scalar-fieldtheory [8–10]. mode [39]. (See Ref. [40] for the EFT of multi-field in-
i Nevertheless, this does not mean that such non- flation without the shift symmetry.) In this paper, we
X
singular cosmological solutions are stable at all times in provide a new proof which follows directly from the full
r
a theentirehistory;ithasbeenknownthatgradientinsta- action of the generalized multi-Galileon theory.
bilitiesoccuratsomemomentinmanyconcreteexamples
(see, e.g., Refs. [11–17]), and in some cases the instabili-
ties show up evenin the far future after the NEC violat- Thispaperisorganizedasfollows. Inthenextsection,
ing stage [18–20]. Recently, it was shown that this is a we give a brief review on the generalized multi-Galileon
generic nature of non-singular cosmological solutions in theory and extend the proof of the no-go theorem for
the Horndeski/generalized Galileon theory [21–23], i.e., non-singularcosmologiestomulti-fieldmodels. Recently,
in the most general scalar-tensor theory having second- severalnewtermswerefoundthatarenotincludedinthe
order field equations, provided that graviton geodesics generalized multi-Galileon theory but still yield second-
are complete [24–26]. order field equations [41]. To keep the proof as general
Astheno-goresultisobtainedinthesingle-fieldHorn- as possible, we show in Sec. III that the main result is
deskitheory,onecouldevadethisbyconsideringtheories not changed by the addition of these new terms. In do-
ing so, we find a covariant completion of the flat-space
action of Ref. [41]. In Sec. IV we give a comment on the
(in)completeness of graviton geodesics viewed from the
∗Email: s.akama”at”rikkyo.ac.jp original (non-Einstein) frame. We draw our conclusions
†Email: tsutomu”at”rikkyo.ac.jp in Sec. V.
2
II. NO-GO THEOREM IN GENERALIZED +2∇ ∇ φI∇ν∇λφJ∇ ∇µφK , (2)
µ ν λ
MULTI-GALILEON THEORY (cid:1)
where
A. Generalized multi-Galileon theory
1
XIJ :=− gµν∂ φI∂ φJ, (3)
µ ν
2
The most general single-scalar-tensor theory whose
1 ∂G ∂G
field equations are of second order is given by the Horn- G := + . (4)
,hIJi 2(cid:18)∂XIJ ∂XJI(cid:19)
deski action [21]. To begin with, let us review briefly
how the same theory was rediscoveredin a different way
In order for the field equations to be of second order, it
starting from the Galileon theory. The Galileon theory
is required that
is a scalar-fieldtheory on a fixed Minkowskibackground
having the Galilean shift symmetry, ∂ φ → ∂ φ + b ,
µ µ µ G :=G , G :=G , (5)
3IJK 3I,hJKi 4IJKL 4,hIJi,hKLi
and second-order field equations [42]. To make the met-
G :=G , G :=G , (6)
ric dynamical and consider an arbitrary spacetime, one 5IJK 5I,hJKi 5IJKLM 4IJK,hLMi
cancovariantizetheGalileontheorybyreplacing∂ with
µ
are symmetric in all of their indices I, J, .... In what
∇ , but this procedure induces higher derivative terms
µ
follows we will write G as G . It is obvious that
in the field equations due to the noncommutativity of 4,hIJi 4IJ
G =G .
the covariant derivative. However, the resulting higher 4IJ 4JI
The multi-scalar-tensor theory described by the La-
derivative terms can be removed by introducing non-
grangian (2) seems very general and includes the ear-
minimal derivative coupling to the curvature. The co-
lier works [50, 51] and more recent ones [38, 52–55] as
variant multi-Galileon theory is thus obtained [43]. Now
specific cases. However, in contrast to the case of the
the Galileanshift symmetryis lostandwhatis moreim-
single Galileon, it is not the most general multi-scalar-
portantis the second-ordernature of the field equations,
tensor theory with second-order field equations. Indeed,
asitguaranteestheabsenceofOstrogradskiinstabilities.
as demonstrated in [56], the multi-DBI Galileon the-
One can further generalize the covariantGalileontheory
by promotingX :=−gµν∂ φ∂ φ/2 inthe actionto arbi- ory [57] is not included in the above one. To date, no
µ ν
complete multi-field generalization of the Horndeski ac-
traryfunctionsφandX whileretainingthesecond-order
tionhasbeenknown. TakingthesameapproachasHorn-
field equations [22]. This yields the Lagrangian
deski did rather than starting from the multi-Galileon
L=G (X,φ)−G (X,φ)✷φ+G (X,φ)R theory,theauthorsofRef.[58]obtainedthemostgeneral
2 3 4
∂G second-order field equations of bi-scalar-tensor theories,
+ 4 (✷φ)2−(∇µ∇νφ)2 +G5(X,φ)Gµν∇µ∇νφ butdeducingthecorrespondingactionandextendingthe
∂X
(cid:2) (cid:3) bi-scalarresultto the caseofmorethantwoscalarshave
1∂G
− 5 (✷φ)3−3✷φ(∇ ∇ φ)2+2(∇ ∇ φ)3 , notbeensuccessfulsofar. Wewillcomebacktothisissue
µ ν µ ν
6 ∂X
(cid:2) (cid:3)(1) in the next section in light of the recent result reported
in [41].
where R is the Ricci scalar and Gµν is the Einstein ten- Althoughthegeneralizedmulti-Galileontheoryisthus
sor. Interestingly, it can be shown that this Lagrangian not the most general one, it is definitely quite general
is equivalent to the one obtained by Horndeski in an ap- and so we choose to use the Lagrangian (2). This is one
parently different form [23], and therefore is the most of the best ways one can do at this stage to draw some
general one having second-order field equations. general conclusions on cosmology of multiple interacting
The multi-field generalization can proceed in the fol- scalar fields, and is considered as complementary to the
lowing way. In Refs. [44–49], the Galileons on a fixed approachbased on the effective field theory of multifield
Minkowski background was generalized to multi-field inflation [26].
models,whoseactionisafunctionalofN scalarfields φI
(I =1, 2, ..., N)andtheirderivativesoforderuptotwo.
Covariantizingthe multi-Galileons and introducing arbi- B. Stability of a non-singular universe in
trary functions of the scalar fields and their first deriva- generalized multi-Galileon theory
tives so that no higher derivative terms appear in the
field equations, one can arrive at the generalized multi-
We now show that the no-go theorem in [25] can be
Galileon theory, the Lagrangian of which is given in an
extended to the case of the generalized multi-Galileon
analogous form to Eq. (1) by [37]
theory.
L=G (XIJ,φK)−G (XIJ,φK)✷φL+G (XIJ,φK)R The quadratic actions for perturbations around a flat
2 3L 4
Friedmann backgroundhave been calculated in [56]. For
+G4,hIJi ✷φI✷φJ −∇µ∇νφI∇µ∇νφJ tensor perturbations hij(t,~x) we have
(cid:0) 1 (cid:1)
+G5L(XIJ,φK)Gµν∇µ∇νφL− 6G5I,hJKi S(2) = 1 dtd3xa3 G h˙2 − FT(∇~h )2 , (7)
× ✷φI✷φJ✷φK −3✷φ(I∇ ∇ φJ∇µ∇νφK) h 8Z (cid:20) T ij a2 ij (cid:21)
µ ν
(cid:0)
3
where This follows from the background equations, and corre-
sponds in the minimally coupled single-field case to the
G :=2 G −2XIJG −XIJ(Hφ˙KG −G ) familiar equation
T 4 4IJ 5IJK 5I,J
h i
(8) φ˙2+2M2H˙ =0. (17)
Pl
and It is required for the stability of the scalar sector that
thematricesK=(K )andD =(D )mustbepositive
IJ IJ
F :=2 G −XIJ(φ¨KG +G ) . (9) definite. Hence, a non-singular cosmological solution is
T 4 5IJK 5I,J
h i free from gradient instabilities if, for every non-zero col-
umn vector v,
Here we defined G :=∂G/∂φI. Stability requires
,I
vTDv >0, (18)
G >0, F >0, (10)
T T
where vT the transpose of v. Now, let v be
at any moment in the whole cosmologicalhistory.
φ˙1
To study scalar perturbations in multi-field mod-
els, it is convenient to use the spatially flat gauge.  φ˙2 
v = . . (19)
The quadratic action for scalar perturbations is of the .
 . 
form [56]  φ˙N 
 
S(2) = 1 dtd3xa3 K Q˙IQ˙J − 1 D ∇~QI ·∇~QJ Then, Eq. (18) reads
Q 2Z (cid:20) IJ a2 IJ
vTDv =2XIJD >0. (20)
IJ
−M QIQJ +2Ω QIQ˙J , (11)
IJ IJ (cid:21) Using Eqs. (13), (14), and (16) and doing some manipu-
lation, one finds
where QI’s are the perturbations of the scalar fields de-
1dξ
fined by XIJD =H2 −F >0, (21)
IJ T
(cid:18)a dt (cid:19)
φI =φ¯I(t)+QI(t,~x). (12) where
The explicitexpressionsfor the matrices KIJ, MIJ, and ξ := aGT2. (22)
Ω can be found in [56], but are not necessary for the Θ
IJ
followingdiscussion. Sincegradientinstabilitiesmanifest The remaining part of the proof is parallel to that in
most significantly at high frequencies, only the structure the Horndeski case [25], because the structure of the in-
of the matrix DIJ is crucial to our no-go argument. We equality (21) is identical to the single-field counterpart.
will use the fact that DIJ is given by [56] In a non-singular universe, Θ never diverges because it
is composed of H and φI as given in Eq. (15) and we
D =C − J(IBJ) + 1 d aBIBJ , (13) require that the functions G2, G3I, ... in the underly-
IJ IJ
Θ adt(cid:18) 2Θ (cid:19) ing Lagrangian remain finite in the entire cosmological
history.1 We also have aG2 > 0 which comes from the
T
where CIJ is the matrix satisfying the identity stability of the tensor perturbations.2 Therefore, ξ can-
not cross zero. From Eq. (21) we have
C XIJ =2H G˙ +HG −Θ˙ −HΘ−H2F , (14)
IJ T T T dξ
(cid:16) (cid:17) >aF >0, (23)
T
dt
with
indicating that ξ is a monotonically increasing function
Θ:=−φ˙IXJKG3IJK +2HG4 of t. Integrating Eq. (23) from some ti to tf, we obtain
−8HXIJ G +XKLG tf
4IJ 4IJKL ξ(t )−ξ(t)> aF dt′. (24)
+2φ˙IXJK(cid:0)G4IJ,K +φ˙IG4,I (cid:1) f i Zti T
−H2φ˙IXJK 5G +2XLMG
5IJK 5IJKLM
+2HXIJ 3G(cid:0) +2XKLG . (cid:1) (15)
5I,J 5IJK,L
(cid:0) (cid:1) 1 Our postulate on this point is different from that adopted in
The explicit expressions for JI and BI in Eq. (13) are Ref.[20],inwhichsingularfunctionsareintroducedintheunder-
also unimportant, but we will use the equation [56] lyingLagrangiantoobtainnon-singularcosmological solutions.
2 Our postulate on this point is different from that adopted in
φ˙IJ +φ¨IB +2H˙G =0. (16) Ref.[59],inwhichallthecoefficients inthequadraticactionfor
I I T cosmologicalperturbations vanishatthesamemoment.
4
(Weadmitthatξdivergesatsomet whereΘ=0occurs. III. COVARIANTIZED NEW TERMS FOR
∗
Inthiscase,t andt aretakentobesuchthatt <t <t MULTI-GALILEON THEORY
i f i f ∗
or t <t <t .) If lim ξ =const, we take t →−∞
∗ i f t→−∞ i
in Eq. (24) and obtain Very recently, the author of Ref. [41] proposed new
terms for scalar multi-Galileon theory that are not in-
tf cludedintheexistingmulti-GalileonLagrangianbutgive
aFTdt′ <ξ(tf)−ξ(−∞)<∞. (25) rise to second-order field equation. The Lagrangians for
Z
−∞ these “extended” multi-Galileons are given by [41, 44]
Similarly, if limt→∞ξ =const then we take tf → ∞ to Lext1 =A[IJ][KL]Mδνµ11νµ22νµ33∂µ1φI∂µ2φJ∂ν1φK∂ν2φL
get ×∂ ∂ν3φM, (28)
µ3
L =A δµ1µ2µ3µ4∂ φI∂ φJ
∞ ext2 [IJ][KL](MN) ν1ν2ν3ν4 µ1 µ2
Z aFTdt′ <ξ(∞)−ξ(ti)<∞. (26) ×∂ν1φK∂ν2φL∂µ3∂ν3φM∂µ4∂ν4φN, (29)
ti
L =A δµ1µ2µ3µ4∂ φI∂ φJ∂ φK
ext3 [IJK][LMN]O ν1ν2ν3ν4 µ1 µ2 µ3
Thus, we conclude that a non-singular cosmological so- ×∂ν1φL∂ν2φM∂ν3φN∂ ∂ν4φO, (30)
µ4
lution in the generalized multi-Galileon theory is stable
where the coefficients A , ... are arbitrary func-
in the entire history provided that either [IJ][KL]M
tionsofφI andXIJ. Thesecoefficientsareantisymmetric
in indices inside [ ] and symmetric in indices inside ( ).
t ∞
aF dt′ or aF dt′ (27) In order for the field equations to be of second order, we
T T
Z−∞ Zt require that
A , A ,
is convergent. (If Θ = 0 occurs, both of the above inte- [IJ][KL]M,hNOi [IJ][KL](MN),hOPi
grals must be convergent.) As is argued in Refs. [26, 30] A , (31)
[IJK][LMN]O,hPQi
andalsoinSec.IVofthe presentpaper,the convergence
of the above integrals signals some kind of pathologies are symmetric in underlined indices.
in the tensor perturbations. If one prefers to avoid this TheLagrangians(28)–(30)arethoseforscalarfieldson
pathology, all non-singular cosmological solutions in the fixed Minkowski specetime. Let us explore a covariant
generalized multi-Galileon theory are inevitably plagued completion of the above flat-space multi-scalar theory.
with gradient instabilities. To make the metric dynamical, we first promote ∂µ to
∇ . It is easy to see that this procedure is sufficient for
Onemightexpectnaivelythat,inthepresenceofmul- µ
L and L :
tiple interactingscalarfields, adominantfieldcantrans- ext1 ext3
fer its energy to another field or matter before the in- L′ =A δµ1µ2µ3∇ φI∇ φJ∇ν1φK∇ν2φL
ext1 [IJ][KL]M ν1ν2ν3 µ1 µ2
stability of the former showsup, and thus the instability
×∇ ∇ν3φM, (32)
can be eliminated. We have shown that this is not the µ3
case in the generalized multi-Galileon theory. L′ =A δµ1µ2µ3µ4∇ φI∇ φJ∇ φK
ext3 [IJK][LMN]O ν1ν2ν3ν4 µ1 µ2 µ3
The same conclusion was reached using the EFT of ×∇ν1φL∇ν2φM∇ν3φN∇ ∇ν4φO, (33)
µ4
multi-field cosmologies, in which a shift symmetry is as-
sumed for the entropy mode [26]. Our proof is different have second-order equations of motion for the metric
from,andcomplementaryto,thatbasedontheEFT.The and scalar fields. However, the simple covariantization
EFT approach amounts to writing all the terms allowed of Lext2,
by symmetry, which leads to the theory of cosmological
L =A δµ1µ2µ3µ4∇ φI∇ φJ
perturbationsonagivenbackground. Therefore,theadi- cext2 [IJ][KL](MN) ν1ν2ν3ν4 µ1 µ2
abatic and entropy modes are decomposed by construc- ×∇ν1φK∇ν2φL∇ ∇ν3φM∇ ∇ν4φN, (34)
µ3 µ4
tion inthe EFT.In contrast,ourguiding principle is the
yields higher derivative terms in the field equations. To
second-ordernatureofthefieldequations,andsowestart
cancelsuchterms,weaddacounterterm,i.e.,acoupling
from the general action of second-order multiple scalar-
to the curvature tensor L . It turns out that the
tensor theories that governs the perturbation evolution curv2
appropriate Lagrangianis the following:
as well as the background dynamics. It should be no-
ticed that we have not performed the adiabatic/entropy L =B δµ1µ2µ3µ4
decomposition, as it is unnecessary for our no-go argu- curv2 [IJ][KL] ν1ν2ν3ν4
ment. Although the relation between the second-order ×Rν3ν4µ3µ4∇µ1φI∇µ2φJ∇ν1φK∇ν2φL, (35)
theory and the EFT of cosmological perturbations has
where
been clarified in the single-field case [60], to date, it is
not obvious how the EFT of multi-field cosmology is re- 1
B = A (36)
lated to the generalized multi-Galileon theory. [IJ][KL],hMNi 2 [IJ][KL](MN)
5
must be imposed. Thus, we find that the covariantcom- of a non-singular cosmological solution with the conver-
pletion of L is given by gentintegralwasobtainedforthefirsttimeinthesingle-
ext2
field context. Later, the authors of Ref. [61] followed
L′ext2 =Lcurv2+Lcext2 (37) Ref. [25] and presented another example.
One can move to the “Einstein frame” for tensor per-
where A[IJ][KL](MN) =2B[IJ][KL],hMNi and turbations from the original frame (7) by performing a
disformal transformation [62]. This is because one has
B[IJ][KL]MNOP :=B[IJ][KL],hMNi,hOPi (38) twoindependent functions oft inperforminga disformal
transformation which can be fitted to make F and G
T T
is symmetric in underlined indices. into their standard forms: F → M2, G → M2. It
T Pl T Pl
One can check that the multi-DBI Galileon theory at is clearly explained in Ref. [26] that because gravitons
leadingorderinXIJ expansion[56]isobtainedbytaking propagatealongnullgeodesicsinthe Einsteinframeand
the integral
B =const×(δ δ −δ δ ), (39)
[IJ][KL] IK JL IL JK
aF dt (44)
though it seems extremely difficult to see explicitly that T
Z
thecompleteLagrangianforthemulti-DBIGalileons[57]
can be reproduced by choosing appropriately the func- is nothing but the affine parameter of the null geodesics
tions in the above Lagrangians. in the Einstein frame, the convergent integral (27) im-
Now the question is how the additional terms plies past (future) incompleteness of graviton geodesics
(seealsoRef.[30]). Thismaysignalsomekindofpathol-
Lext :=L′ext1+L′ext2+L′ext3 (40) ogy in the tensor perturbations, though it is not obvious
whether the incompleteness of null geodesics in a disfor-
changethestabilityofcosmologicalsolutions. Obviously, mally related frame causes actual problems.
Lext does not change the background equations due to Let us rephrase this potential pathology of gravitons
antisymmetry. We see that, in the quadratic actions for without invoking the disformal transformation. The
scalarandtensor perturbations,only the CIJ coefficients equation of motion for the tensor perturbation hij de-
are modified as follows: rived from the action (7) can be written in the form
CIJ →CIJ +CIeJxt, (41) ZµνDµDνhij =0, (45)
with where
Cext :=32H −A XKLφ˙M +2HB XKL F3/2
IJ [IK][JL]M [IK][JL] Z dxµdxν =− T dt2+a2(F G )1/2δ dxidxj,
(cid:0)+4HB XKLXMN , (42) µν G1/2 T T ij
[IK][JL],hMNi T
(cid:1) (46)
and no other terms are affected by the addition of L .
ext
Since XIJCext = 0 due to antisymmetry, XIJD re- and D is the covariant derivative associated with the
IJ IJ µ
mains the same even if one adds L : “metric” Z . Equation (45) shows that graviton paths
ext µν
can be interpreted as null geodesics in the effective ge-
XIJD →XIJD . (43) ometry defined by Z . It turns out that the affine pa-
IJ IJ µν
rameter λ of null geodesics in the metric Z is givenby
µν
Therefore, the new terms proposed in Ref. [41] does not dλ = aF dt. Therefore, the incompleteness of graviton
T
change the no-go argument. geodesicscanbe made manifest evenwithoutworkingin
ThenewtermLextvanishesforthehomogeneousback- the Einstein frame.
ground, which implies that L contributes only to the
ext
entropymodes atthe levelofperturbations. This is con-
sistent with the result of [26], where it can be seen using V. SUMMARY
the EFT that the instability occurs in the adiabatic di-
rection. Inthispaper,wehaveshownthatallnon-singularcos-
mologicalsolutionsareplaguedwithgradientinstabilities
in the multi-field generalizationofscalar-tensortheories,
IV. GRAVITON GEODESICS if the graviton geodesic completeness is required. This
extends the recent no-go arguments of Refs. [24, 25, 38].
Wehavethusseenthatwithinthemulti-fieldextension Wehavegivenadirectproofusingthegeneralizedmulti-
of the generalized Galileons, non-singular cosmological Galileon action, so that our proof is different from and
solutions are possible only if either integral in Eq. (27) complementary to that obtained from the effective field
is convergent, as in the single-field Horndeski case. In theory of cosmological fluctuations [26]. Several new
Ref. [25], this fact was noticed and a numerical example termsformulti-Galileonsonaflatbackgroundwerefound
6
recently [41]. We have covariantized these terms and sions. This work was supported in part by MEXT-
shown that the inclusion of them does not change the Supported Program for the Strategic Research Founda-
no-go result. tionat Private Universities,2014-2017,andby the JSPS
Grants-in-Aid for Scientific Research No. 16H01102 and
No. 16K17707(T.K.).
Acknowledgments
We thank Yuji Akita, Norihiro Tanahashi, Masahide
Yamaguchi, and Shuichiro Yokoyama for helpful discus-
[1] A.H.Guth,“TheInflationaryUniverse: APossibleSolu- 151 (2014) [arXiv:1410.0882 [hep-th]].
tion to the Horizon and Flatness Problems,” Phys. Rev. [17] T. Kobayashi, M. Yamaguchi and J. Yokoyama,
D 23, 347 (1981). “Galilean Creation of the Inflationary Universe,” JCAP
[2] A.A.Starobinsky,“A NewTypeofIsotropic Cosmolog- 1507, no. 07, 017 (2015) [arXiv:1504.05710 [hep-th]].
ical Models Without Singularity,” Phys. Lett. B 91, 99 [18] T.Qiu,J.Evslin,Y.F.Cai,M.LiandX.Zhang,“Bounc-
(1980). ing Galileon Cosmologies,” JCAP 1110, 036 (2011)
[3] K.Sato,“FirstOrderPhaseTransitionofaVacuumand [arXiv:1108.0593 [hep-th]].
ExpansionoftheUniverse,”Mon.Not.Roy.Astron.Soc. [19] D. A. Easson, I. Sawicki and A. Vikman, “G-Bounce,”
195, 467 (1981). JCAP 1111, 021 (2011) [arXiv:1109.1047 [hep-th]].
[4] A. Borde and A. Vilenkin, “Singularities in inflationary [20] A. Ijjas and P. J. Steinhardt,“Classically stable nonsin-
cosmology: A Review,” Int. J. Mod. Phys. D 5, 813 gular cosmological bounces,” Phys. Rev. Lett. 117, no.
(1996) [gr-qc/9612036]. 12, 121304 (2016) [arXiv:1606.08880 [gr-qc]].
[5] J. Martin and R. H. Brandenberger, “The TransPlanck- [21] G.W.Horndeski,“Second-orderscalar-tensorfieldequa-
ianproblemofinflationarycosmology,”Phys.Rev.D63, tions in a four-dimensional space,” Int. J. Theor. Phys.
123501 (2001) [hep-th/0005209]. 10, 363 (1974).
[6] D. Battefeld and P. Peter, “A Critical Review of Clas- [22] C. Deffayet, X. Gao, D. A. Steer and G. Zahariade,
sical Bouncing Cosmologies,” Phys. Rept. 571, 1 (2015) “From k-essence to generalised Galileons,” Phys. Rev.
[arXiv:1406.2790 [astro-ph.CO]]. D 84, 064039 (2011) [arXiv:1103.3260 [hep-th]].
[7] V.A.Rubakov,“TheNullEnergyConditionanditsvio- [23] T. Kobayashi, M. Yamaguchi and J. Yokoyama, “Gen-
lation,”Phys.Usp.57,128(2014)[arXiv:1401.4024[hep- eralized G-inflation: Inflation with the most general
th]]. second-order field equations,” Prog. Theor. Phys. 126,
[8] P. Creminelli, A. Nicolis and E. Trincherini, “Galilean 511 (2011) [arXiv:1105.5723 [hep-th]].
Genesis: An Alternative to inflation,” JCAP 1011, 021 [24] M. Libanov, S. Mironov and V. Rubakov, “Generalized
(2010) [arXiv:1007.0027 [hep-th]]. Galileons: instabilitiesofbouncingandGenesiscosmolo-
[9] C.Deffayet,O.Pujolas, I.Sawicki andA.Vikman,“Im- gies and modified Genesis,” JCAP 1608, no. 08, 037
perfect Dark Energy from Kinetic Gravity Braiding,” (2016) [arXiv:1605.05992 [hep-th]].
JCAP 1010, 026 (2010) [arXiv:1008.0048 [hep-th]]. [25] T. Kobayashi, “Generic instabilities of nonsingular cos-
[10] T. Kobayashi, M. Yamaguchi and J. Yokoyama, “G- mologies in Horndeski theory: A no-go theorem,” Phys.
inflation: Inflation driven by the Galileon field,” Phys. Rev.D94, no.4, 043511 (2016) [arXiv:1606.05831 [hep-
Rev.Lett.105,231302(2010)[arXiv:1008.0603[hep-th]]. th]].
[11] Y.F.Cai,D.A.EassonandR.Brandenberger,“Towards [26] P. Creminelli, D. Pirtskhalava, L. Santoni and
a Nonsingular Bouncing Cosmology,” JCAP 1208, 020 E. Trincherini, “Stability of Geodesically Complete
(2012) [arXiv:1206.2382 [hep-th]]. Cosmologies,” JCAP 1611, no. 11, 047 (2016)
[12] M.Koehn,J.L.LehnersandB.A.Ovrut,“Cosmological [arXiv:1610.04207 [hep-th]].
super-bounce,” Phys. Rev. D 90, no. 2, 025005 (2014) [27] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi,
[arXiv:1310.7577 [hep-th]]. “Healthy theories beyond Horndeski,” Phys. Rev. Lett.
[13] L. Battarra, M. Koehn, J. L. Lehners and B. A. Ovrut, 114, no. 21, 211101 (2015) [arXiv:1404.6495 [hep-th]].
“Cosmological Perturbations Through a Non-Singular [28] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, “Ex-
Ghost-Condensate/Galileon Bounce,” JCAP 1407, 007 ploring gravitational theoriesbeyondHorndeski,”JCAP
(2014) [arXiv:1404.5067 [hep-th]]. 1502, 018 (2015) [arXiv:1408.1952 [astro-ph.CO]].
[14] T. Qiu and Y. T. Wang, “G-Bounce Inflation: To- [29] M. Zumalac´arregui and J. Garc´ıa-Bellido, “Transform-
wards Nonsingular Inflation Cosmology with Galileon ing gravity: from derivative couplings to matter to
Field,”JHEP1504,130(2015)[arXiv:1501.03568[astro- second-order scalar-tensor theories beyond the Horn-
ph.CO]]. deski Lagrangian,” Phys. Rev. D 89, 064046 (2014)
[15] Y. Wan, T. Qiu, F. P. Huang, Y. F. Cai, H. Li and [arXiv:1308.4685 [gr-qc]].
X. Zhang, “Bounce Inflation Cosmology with Standard [30] Y. Cai, Y. Wan, H. G. Li, T. Qiu and Y. S. Piao,
Model Higgs Boson,” JCAP 1512, no. 12, 019 (2015) “The Effective Field Theory of nonsingular cosmology,”
[arXiv:1509.08772 [gr-qc]]. arXiv:1610.03400 [gr-qc].
[16] D. Pirtskhalava, L. Santoni, E. Trincherini and P. Utta- [31] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan
yarat, “Inflation from Minkowski Space,” JHEP 1412, and L. Senatore, “The Effective Field Theory of Infla-
7
tion,”JHEP0803,014(2008)[arXiv:0709.0293[hep-th]]. Galileons,” Class. Quant. Grav. 28, 204003 (2011)
[32] P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore, [arXiv:1104.2088 [hep-th]].
“Starting theUniverse: StableViolation of theNullEn- [49] V.Sivanesan,“Generalizedmultiple-scalarfieldtheoryin
ergy Condition and Non-standard Cosmologies,” JHEP Minkowskispace-timefreeofOstrogradskighosts,”Phys.
0612, 080 (2006) [hep-th/0606090]. Rev. D 90, no. 10, 104006 (2014) [arXiv:1307.8081 [gr-
[33] X. Gao, “Unifying framework for scalar-tensor the- qc]].
ories of gravity,” Phys. Rev. D 90, 081501 (2014) [50] T. Damour and G. Esposito-Farese, “Tensor multiscalar
[arXiv:1406.0822 [gr-qc]]. theories of gravitation,” Class. Quant. Grav. 9, 2093
[34] X. Gao, “Hamiltonian analysis of spatially covari- (1992).
ant gravity,” Phys. Rev. D 90, 104033 (2014) [51] M. Horbatsch, H. O. Silva, D. Gerosa, P. Pani,
[arXiv:1409.6708 [gr-qc]]. E. Berti, L. Gualtieri and U. Sperhake, “Tensor-multi-
[35] P.Horava,“QuantumGravityataLifshitzPoint,”Phys. scalar theories: relativistic stars and 3 + 1 decompo-
Rev.D 79, 084008 (2009) [arXiv:0901.3775 [hep-th]]. sition,” Class. Quant. Grav. 32, no. 20, 204001 (2015)
[36] M. Koehn, J. L. Lehners and B. Ovrut, “Nonsingu- [arXiv:1505.07462 [gr-qc]].
lar bouncing cosmology: Consistency of the effective [52] A. Naruko, D. Yoshida and S. Mukohyama, “Gravita-
description,” Phys. Rev. D 93, no. 10, 103501 (2016) tional scalar-tensor theory,”Class. Quant.Grav.33, no.
[arXiv:1512.03807 [hep-th]]. 9, 09LT01 (2016) [arXiv:1512.06977 [gr-qc]].
[37] A. Padilla and V. Sivanesan, “Covariant multi-galileons [53] C. Charmousis, T. Kolyvaris, E. Papantonopoulos
and their generalisation,” JHEP 1304, 032 (2013) and M. Tsoukalas, “Black Holes in Bi-scalar Exten-
[arXiv:1210.4026 [gr-qc]]. sions of Horndeski Theories,” JHEP 1407, 085 (2014)
[38] R.KolevatovandS.Mironov,“Oncosmological bounces [arXiv:1404.1024 [gr-qc]].
andLorentzianwormholesinGalileontheorieswithextra [54] E. N. Saridakis and M. Tsoukalas, “Cosmology in new
scalar field,” arXiv:1607.04099 [hep-th]. gravitational scalar-tensor theories,” Phys. Rev. D 93,
[39] L. Senatore and M. Zaldarriaga, “The Effective Field no. 12, 124032 (2016) [arXiv:1601.06734 [gr-qc]].
Theory of Multifield Inflation,” JHEP 1204, 024 (2012) [55] E. N. Saridakis and M. Tsoukalas, “Bi-scalar modified
[arXiv:1009.2093 [hep-th]]. gravityandcosmologywithconformalinvariance,”JCAP
[40] T. Noumi, M. Yamaguchi and D. Yokoyama, “Effec- 1604, no. 04, 017 (2016) [arXiv:1602.06890 [gr-qc]].
tive field theory approach to quasi-single field inflation [56] T. Kobayashi, N. Tanahashi and M. Yamaguchi, “Mul-
and effects of heavy fields,” JHEP 1306, 051 (2013) tifield extension of G inflation,” Phys.Rev. D 88, no. 8,
[arXiv:1211.1624 [hep-th]]. 083504 (2013) [arXiv:1308.4798 [hep-th]].
[41] E. Allys, “New terms for scalar multi-galileon models, [57] S. Renaux-Petel, S. Mizuno and K. Koyama, “Pri-
applicationtoSO(N)andSU(N)grouprepresentations,” mordial fluctuations and non-Gaussianities from multi-
arXiv:1612.01972 [hep-th]. field DBI Galileon inflation,” JCAP 1111, 042 (2011)
[42] A.Nicolis,R.RattazziandE.Trincherini,“TheGalileon [arXiv:1108.0305 [astro-ph.CO]].
as a local modification of gravity,” Phys. Rev. D 79, [58] S. Ohashi, N. Tanahashi, T. Kobayashi and M. Yam-
064036 (2009) [arXiv:0811.2197 [hep-th]]. aguchi,“Themostgeneralsecond-orderfieldequationsof
[43] C. Deffayet, G. Esposito-Farese and A. Vikman, “Co- bi-scalar-tensortheoryinfourdimensions,”JHEP1507,
variant Galileon,” Phys. Rev. D 79, 084003 (2009) 008 (2015) [arXiv:1505.06029 [gr-qc]].
[arXiv:0901.1314 [hep-th]]. [59] A. Ijjas, “Cyclic anamorphic cosmology,”
[44] C. Deffayet, S. Deser and G. Esposito-Farese, “Arbi- arXiv:1610.02752 [astro-ph.CO].
traryp-formGalileons,” Phys.Rev.D82,061501(2010) [60] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, “Es-
[arXiv:1007.5278 [gr-qc]]. sential Building Blocks of Dark Energy,” JCAP 1308,
[45] A. Padilla, P. M. Saffin and S. Y. Zhou, “Bi-galileon 025 (2013) [arXiv:1304.4840 [hep-th]].
theoryI:Motivation and formulation,” JHEP1012, 031 [61] A. Ijjas and P. J. Steinhardt, “Fully stable cosmologi-
(2010) [arXiv:1007.5424 [hep-th]]. calsolutionswithanon-singularclassicalbounce,”Phys.
[46] A.Padilla,P.M.SaffinandS.Y.Zhou,“Multi-galileons, Lett. B 764, 289 (2017) [arXiv:1609.01253 [gr-qc]].
solitonsandDerrick’stheorem,”Phys.Rev.D83,045009 [62] P.Creminelli,J.Gleyzes,J.Noren˜aandF.Vernizzi,“Re-
(2011) [arXiv:1008.0745 [hep-th]]. silience of the standard predictions for primordial ten-
[47] A. Padilla, P. M. Saffin and S. Y. Zhou, “Bi-galileon sor modes,” Phys.Rev.Lett.113,no.23, 231301 (2014)
theory II: Phenomenology,” JHEP 1101, 099 (2011) [arXiv:1407.8439 [astro-ph.CO]].
[arXiv:1008.3312 [hep-th]].
[48] M. Trodden and K. Hinterbichler, “Generalizing