Bi-galileon theory I: motivation and formulation

0.27 MB·English

by  Antonio Padilla

Bi-galileon theory I: motivation and formulation
Antonio Padilla,∗ Paul M. Saffin,† and Shuang-Yong Zhou‡
School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
(Dated: January 26, 2011)
We introduce bi-galileon theory, the generalisation of the single galileon model introduced by
Nicolis et al. The theory contains two coupled scalar fields and is described by a Lagrangian that
is invariant under Galilean shifts in those fields. This paper is the first of two, and focuses on
the motivation and formulation of the theory. We show that the boundary effective theory of the
cascading cosmology model corresponds to a bi-galileon theory in the decoupling limit, and argue
that this is to be expected for co-dimension 2 braneworld models exhibiting infra-red modification
of gravity. We then generalise this, by constructing the most general bi-galileon Lagrangian. By
1
coupling one of the galileons to the energy-momentum tensor, we pitch this as a modified gravity
1
theory in which the modifications to General Relativity are encoded in the dynamics of the two
0
galileons. We initiate a study of phenomenology by looking at maximally symmetric vacua and
2
theirstability,developingelegantgeometrictechniquesthattriviallyexplainwhysomeofthevacua
n have to be unstable in certain cases (eg DGP). A detailed study of phenomenology appears in our
a
companion paper.
J

5
2

] I. INTRODUCTION
h
t
-
p When Urbain Le Verrier first noticed that the slow precession of Mercury’s orbit could not be explained
e using perturbative Newtonian gravity and the known planets, he postulated the existence of a hitherto
h unseen dark planet, which he called Vulcan [1]. Of course, thanks to Einstein, we now know that what was
[
reallyrequiredwasanewtheoryofgravity,specifically,theGeneralTheoryofRelativity[2]. 150yearslater,

4 flat rotations curves of galaxies [3] and accelerated cosmic expansion [4] have led most of us to follow Le
v Verrier’s approach and to postulate the existence of dark matter [5] and dark energy [6] respectively. For
4 dark matter, bullet cluster observations suggest that Le Verrier’s approach is probably the right one [7].
2 The Le Verrier approachto dark energy is less compelling. Whereas extensions of the Standard Model offer
4
some very good dark matter candidates, they have been nothing short of hopeless in offering a satisfactory
5
candidate for dark energy. It is certainly worth taking a lesson from history and exploring the possibility
.
7 that a new theory of gravity may be required to explain the observed cosmic acceleration.
0
In order to play a role in the accelerated expansion, we need gravity to deviate from General Relativity
0 (GR) at large scales, and in particular at the scale of the cosmic horizon H−1 1026m. Such deviations
1 0 ∼
shouldnotbefeltallthewaydowntoarbitrarilyshortdistancesbecauseGeneralRelativityisverysuccessful
:
v in describingclassicalgravityup to the scale of the SolarSystem, at about1015m. Modifications ofGeneral
i
X Relativity are most easily understood at the level of linearised theory. In GR, the gravitational force is
mediated by a massless spin 2 particle carrying two polarisationdegrees of freedom. In modified gravity we
r
a expectadditionaldegreesoffreedomtobe present. Forexample,inPauli-Fierzmassivegravitythe graviton
carries 5, rather than 2, polarisation degrees of freedom [8]. Clearly if they are to play a role in accelerated
expansionwithoutspoilingthephenomenologicalsuccessesofGRatshorterdistances,theadditionaldegrees
of freedom should be suppressed up to Solar System scales, but not on cosmological scales. This should
occur without introducing any new pathological modes. In massive gravity, the longitudinal graviton mode
is problematic since it gives rise to the so called vDVZ discontinuity in the graviton propagator, even at
scaleswherethegravitonmassisnegligible[9]. However,thismodecanbe”screened”onSolarSystemscales
thanks to non-linearinteractionsandthe Vainshteinmechanism[10], butonly atthe expense ofintroducing
∗[email protected]
†paul.saffi[email protected]
‡[email protected]
2
a new pathology in the form of the Boulware-Deserghost [11, 12].
Extra dimensions and the braneworld paradigm [13] have opened up some intriguing possibilities for
modifying gravity at large distances (see, for example, [14–17]). The DGP model [14] is probably the
most celebrated example of this. The model admits two distinct sectors: the normal branch and the self-
acceleratingbranch. The latter generatedplenty of interest since it gaverise to cosmic accelerationwithout
the need fordarkenergy[18]. However,fluctuations aboutthe self-acceleratingvacuumwerefound to suffer
from ghost-like instabilities, ruling out this sector of the theory [19, 20]. Although the normal branch is
less interesting phenomenologically, it is fundamentally more healthy and is the closest thing we have to a
consistent non-linear completion of massive gravity. Here the graviton is a resonance of finite width H , as
0
opposed to a massive field. At short distances, r H−1 the brane dynamics does not feel the width of
the resonance and the theory resembles 4D GR. At≪large0distances, r > H−1, however, the theory becomes
0
five dimensional as the resonance effectively decays into continuum K∼aluza-Klein modes. The Vainshtein
mechanismworkswellonthenormalbranchofDGP,screeningthelongitudinalgravitonwithoutintroducing
any new pathological modes, in contrast to massive gravity [12, 21].
Much of the interesting dynamics is contained in the properties of the longitudinal scalar, π. One can
identify this mode with the brane bending mode. As this mode only excites the extrinsic curvature of the
brane,andnotthe braneorbulk geometry,itbecomesstronglycoupledwellbelowthe 4D Planckscale[20].
It is precisely this strong coupling that allows the Vainshtein mechanism to take effect as non-linearities
become important at larger than expected distances, helping to shut down the scalar on Solar System
scales[12,23]. We canisolatethedynamicsofthescalarbytakingtheso-calleddecouplinglimitaroundflat
space, in which the 4D and 5D Planck scales go to infinity, but the strong coupling scale is held fixed. In
this limit the tensor mode, h , decouples from the scalar, π, to all orders [20, 22]. The π-Lagrangian now
µν
contains a plethora of information about both branches of solution, which can be studied much more easily
than in the full theory.
Now an important feature of the π-Lagrangian is that it exhibits galileon invariance [24]. That is, the
action is invariant under π π+b xµ+c, where b and c are constant. This essentially follows from the
µ µ
→
fact that the brane bending contribution to the extrinsic curvature is given by K ∂ ∂ π. Equivalently,
µν µ ν
≃
Poincare transformations in the bulk should not affect the physics of the brane bending mode. If the brane
position is given by y = π(x), then after an infinitesimal Poincare transformation, xa xa +ǫa xb +υa
b
→
the brane position is given by y π(x)+ǫy xµ+υy. Since the physics of the brane bending mode should
µ
≃
remain unaffected it is clear that it should exhibit galileon invariance.
Motivated by the DGP model, Nicolis et al [24] developed the idea of galileoninvariance and wrote down
themostgeneralLagrangianforasinglegalileondegreeoffreedom(see[25]forsomeearlyworkalongsimilar
lines). Remarkably the theory contains only five free parameters, enabling the authors, and others [26], to
complete a thorough analysis. Although the Galilean symmetry is broken when we include coupling to
gravity[27], covariantcompletionsofthe generalgalileonactioncanbe identified withthe positionmodulus
of a probe DBI brane [28]. In any event, it seems clear that any co-dimension one brane model with large
distancedeviationsfromGR,willindeedadmitagalileondescriptionaroundflatspace,insomeappropriate
limit. This follows fromthe discussionof the brane bending mode inthe previous paragraph,andfrom[23],
where itis arguedthatany viable infra-redmodificationofgravityleadsto strongcoupling,andis therefore
expected to admit a non-trivial decoupling limit.
Recently there has been plenty of interest in branes of higher co-dimension and the role they might play
in modifying gravityat cosmologicalscales [29–38]. An example of this is the cascadingDGP model [29], in
which one has DGP branes within DGP branes [29–31]. In the 6D version one has a co-dimension 2 DGP
3-brane intersecting a co-dimensionone DGP 4-brane [29]. Higher co-dimensionbrane models seem to have
extra features, such as degravitation[30, 39], or perhaps even self-tuning [29, 32–34], in which the geometry
ofthe3-braneisinsensitivetoalarge3-branevacuumenergy. Nowtospecifythepositionofaco-dimension-
N brane we need to specify the values of N “bulk” components: y = π (x),y = π (x),...,y = π (x).
1 1 2 2 N N
For a viable model giving rise to large distance deviations from GR, there should be strong coupling and
thereforeanappropriatelimitinwhichthe branebending modesdecouple fromthe tensormode aroundflat
space. Then the physics of each brane bending mode should be unaffected by Poincare transformations in
the bulk, and we conclude that we are left with an N-galileon theory.
In this paper we will focus on the case of two coupled galileons, which we will call bi-galileon theory. Of
course, this has particular relevance to co-dimension 2 brane models. Indeed, in section II, we will show
3
explicitly that the 5D cascading cosmology model is one example of a bi-galileon theory. This model is
derived from the original 6D cascading gravity model [29, 30] by computing the boundary effective field
theoryin5dimensions,andtakingasuitabledecouplinglimit. Althoughadetailedstudyofphenomenology
will appear elsewhere [41], this paper is devoted to the motivation and formulation of the most general
bi-galileon theory. In section III we will pitch our theory as a modification of gravity by coupling one of
the galileon fields to the energy-momentum tensor. We will not couple the graviton directly to the scalars,
which amounts to ignoring their backreaction on to the geometry, as was done for the case of the single
galileon [24]. In section IV we will construct the general bi-galileontheory. We will also state the extension
to any number of coupled galileon fields. We initiate a study of phenomenology in section V by looking at
maximally symmetric vacua and their stability. Section VI contains a half-time analysis.
II. EXAMPLE: CASCADING COSMOLOGY
There have been a number of attempts to extend the DGP model to higher dimensions. Although the
earliest of these ran into problems with ghosts and instabilities [40], the recently developed cascading mod-
els [29–31] have been more successful. These models exhibit degravitation and/or self-tuning, and seem to
be free from ghosts, at least if the 3-brane tension is large enough. In this section we will compute the 4D
boundary effective theory for the cascading cosmology model [31]. By taking the appropriate limit we will
decouple the scalar sector and show that it corresponds to a bi-galileon theory.
The cascading cosmology model [31] is closely related to the 6D cascading DGP model [29]. The latter
contains a DGP 3-brane sitting on a DGP 4-brane in 6 dimensions. The action is given by
M4 M3 M2
S = 6 d6x√ g R + 5 d5x√ g R + 4 d4x√ g R . (1)
cascading 6 6 5 5 4 4
2 − 2 − 2 −
Zbulk Z4−brane Z3−brane
If we have a hierarchyof scales M <M <M , then the Newtonian potential cascades from 1/r to 1/r2 to
6 5 4
1/r3 as we move further away from the source. The behaviour crosses over from 4D to 5D at a scale m−1
5
and from 5D to 6D at a scale m−1, where
6
2M3 2M4
m = 5 , m = 6 . (2)
5 M2 6 M3
4 5
In the original model [29] both crossover scales are taken to be of order the cosmic horizon, so that m
5
m H 10−33 eV. Taking M M 1018 GeV, it follows that M 10 MeV and M meV. Not∼e
6 0 4 pl 5 6
∼ ∼ ∼ ∼ ∼ ∼
that bulk quantum gravity corrections are expected to kick in at M−1 0.1 mm.
6 ∼
Now, one can integrate out the bulk, just as in [20], to obtain the effective 5D theory. It turns out that
the longitudinal mode of the 5D gravitonbecomes strongly coupled at a scale
Λ =(m4M3)1/7 10−16 eV, (3)
6 6 5 ∼
or,equivalently,atdistancesoforderΛ−1 106km. Nowatdistancesmuchshorterthanthecosmichorizon,
6 ∼
but muchlargerthanthe millimeter scale,we canhappily take M , M ,whilst holdingΛ fixed. This
5 6 6
→∞
correspondsto the decoupling limit in the effective 5D theory [29, 31]. The cascading cosmologymodel [31]
represents a non-linear completion of this decoupled effective theory1. The action is given by
S = M3 d5x√ g e−3π/2R α(∂π)2(cid:3) π (4)
5 − 5− 5
ZM
α 1 M2
2M3 d4x√ q e−3π/2K+ (∂ π)2 π+ ( π)3 + 4 d4x√ q R , (5)
− 5 − 2 µ Ln 3 Ln 2 − 4
Z∂M (cid:20) (cid:18) (cid:19)(cid:21) Z∂M
1 Thisnon-linearcompletioniscertainlynotuniquebutonecanreasonablyexpectitcapturethesalientfeaturesoftheoriginal
6D theory.
4
where the full 5D spacetime is made up of two copies of , identified across a common boundary, ∂ ,
given by the 3-brane. Here g dxadxb and q dxµdxν arMe the bulk and brane metric respectively, wMith
ab µν
corresponding Ricci scalars R and R . The extrinsic curvature of the brane is given by K = 1 q
where is the Lie derivative5withresp4ectto the inwardpointingnormal, na, to∂ . The buµlkνscal2aLrnπ(µxν)
n
is the 5LD longitudinal mode and the parameter α= 27 . M
4m26
If we choose coordinates xa =(y,xµ), so that the brane is at y =0 and the bulk extends into y >0, then
it is convenient to write the bulk metric in ADM coordinates
ds2 =g dxadxb =N2dy2+q (dxµ+Nµdy)(dxν +Nνdy), (6)
ab µν
where N is the lapse function and Nµ is the shift vector. Then
1 1
K = ∂ q 2D N , π = (∂ NµD )π, (7)
µν 2N y µν − (µ ν) Ln N y− µ
(cid:0) (cid:1)
where D is the covariant derivative with respect to the brane metric q .
µ µν
To calculate the boundary effective action, we essentially follow steps completely analogous to the ones
taken in [20]. In order to integrate out the bulk fields we must choose a gauge. We therefore set
g =η +h , q =η +h , (8)
ab ab ab µν µν µν
and add a bulk gauge fixing term
M3 3 3
S = 5 d5x (F + ∂ π)(Fa+ ∂aπ), (9)
bulk,gf a a
− 2 2 2
ZM
where
1
Fa =∂ hab ∂ah, h=ηabh . (10)
b ab
− 2
Classically this imposes the gauge F = 3∂ π. To quadratic order we have
a −2 a
1 1 3 9
δ S = M3 d4xdy hab hηab (cid:3) h + π(cid:3) h+ π(cid:3) π
2 5 4 − 2 5 ab 4 5 8 5
Z (cid:18) (cid:19)
1 1 1 3 1
+M3 d4x hµν h ηµν ∂ h + Nµ∂ N + π∂ h (h ∂ h +h ∂ h )
5 4 − 2 4 y µν 2 y µ 4 y 4− 8 yy y 4 4 y yy
Zy=0 (cid:18) (cid:19)
1 3 9 1 3
+ h ∂ h + π∂ h + π∂ π+N fµ ∂µh + ∂µπ
yy y yy y yy y µ yy
8 4 8 − 2 2
(cid:18) (cid:19)
M2 1 1 1
+ 4 d4x hµν h ηµν (cid:3) h + f fµ, (11)
4 4 µν µ
2 4 − 2 2
Zy=0 (cid:18) (cid:19)
where
1
fµ =∂ hµν ∂µh , h =ηµνh , (12)
ν 4 4 µν
− 2
and we have re-defined N =h . The bulk equations of motion
µ µy
(cid:3) h =0, (cid:3) π =0 (13)
5 ab 5
have solutions
h =e−y∆h , π =e−y∆π , (14)
ab ab y=0 y=0
| |
where ∆ = √ (cid:3) . Now we integrate out the bulk and fix the residual gauge freedom left on the brane by
4
−
adding a brane gauge fixing term,
M3
S = 4 d4x (f +m N )(fµ+m Nµ). (15)
brane,gf µ 5 µ 5
− 4
ZM
5
This gauge fixing eliminates the mixing between N and f and we are left with
µ µ
M2 1 1 1
δ S = 4 d4x hµν h ηµν ((cid:3) m ∆)h m Nµ(∆+m )N
2 4 4 5 µν 5 5 µ
2 4 − 2 − − 2
Zy=0 (cid:18) (cid:19) (cid:20)
1 1 1
+ N ∂µ(h 3π) (h 3π)∆h+ (h +3π)∆(h +3π) . (16)
µ yy yy yy yy
2 − − 4 − 8
(cid:21)
We can diagonalise this action by making the following field re-definitions
1 2
h =h˜ +(π+φ)η , N =N˜ + ∂ (π+φ), h =3π (∆+m )(π+φ). (17)
µν µν µν µ µ µ yy 5
m − m
5 5
This gives
M2 1 1 1
δ S = 4 d4x h˜µν h˜ ηµν ((cid:3) m ∆)h˜ m N˜µ(∆+m )N˜
2 4 4 5 µν 5 5 µ
2 4 − 2 − − 2
Zy=0 (cid:18) (cid:19)
3 3
+ φ((cid:3) m ∆)φ π((cid:3) +2m ∆)π. (18)
4 5 4 5
2 − − 2
Note that the π field is a ghost. This can be avoidedby increasing the vacuum energy on the brane [29]. In
addition, to quadratic order, the matter coupling is given by
1 1
δ S = d4xh Tµν = d4x h˜ Tµν +(π+φ)T . (19)
2 matter µν µν
2 2
Zy=0 Zy=0
We now consider higher order terms. Schematically it is clear from Eq. 5 that these will take the form
d4x M3∆(π)a1(h )b1(N )c1(h )d1, (20)
5 µν µ yy
Z
d4x M3α∆3(π)a2(h )b2(N )c2(h )d2, (21)
5 µν µ yy
Z
d4x M2∆2(h )b3, (22)
4 µν
Z
where a ,...,d are non-negative integers satisfying a +b +c +d 3. Note that we must have a 1
i i i i i i 2
≥ ≥
sincetermsoftheformEq. 21stemfromtheboundaryπ termgiveninEq. 5. Aftercanonicallynormalising
the fields, it is easy enough to check that the largest contribution comes from a term like Eq. 20, with
a = b = 0, c +d = 3. This term is exactly like the cubic term computed in [20], and comes from the
1 1 1 1
following term in the action (5)
M3 d5x√ g e−3π/2R 2M3 d4x√ qe−3π/2K. (23)
5 − 5− 5 −
ZM Z∂M
Since we want a =0, we can switch off e−3π/2, and write this term in ADM form,
1
M3 d4xdy√ qN(R +K2 K Kµν). (24)
5 − 4 − µν
Z
Of course, we also want b = 0, so we need to switch off h . This means q = η , and so R = 0 and
1 µν µν µν 4
K = 1∂ N . Given that N 1+ 1h , the cubic order interaction term is given by
µν −N (µ ν) ≈ 2 yy
M3
5 d4xdyh (∂ Nµ)2 ∂ N ∂µNν . (25)
− 2 yy µ − (µ ν)
Z
(cid:2) (cid:3)
In the limit where m 0, the strongly interacting mode can be parametrized in the bulk as
5
→
h 0, N ∂ Ψ, h 2∂ Ψ, π 0, (26)
µν µ µ yy y
→ ∼ ∼ →
6
where(cid:3) Ψ=0. Thissatisfiesthebulkequationsofmotion,aswellasthegaugechoiceF = 3∂ π. Indeed,
5 a −2 a
the strongly interacting mode correspondsto a trivial bulk geometry,as expected [20]. Furthermore, we see
from Eq. 17 that the boundary values for the fields go like N 1 ∂ (π+φ), and h 2 ∆(π+φ) so
µ ∼ m5 µ yy ∼−m5
we take Ψ=e−∆y π+φ . Eq. 25 can now be written as
m5
(cid:16) (cid:17)
M3
M3 d4xdy ∂ Ψ ((cid:3) Ψ)2 (∂ ∂ Ψ)2 = 5 d4xdy ∂ [∂ Ψ∂µΨ(cid:3) Ψ]
− 5 y 4 − µ ν 2 y µ 4
Z Z
(cid:2) (cid:3) M3
= 5 d4x ∂ (π+φ)∂µ(π+φ)(cid:3) (π+φ). (27)
−2m3 µ 4
5 Z
Nowif we canonicallynormalise,sothatπ = 2 πˆ andφ= 2 φˆ , thenwe see thatthe cubic interaction
3M4 3M4
term goes like q q
1 1
d4x ∂ (πˆ+φˆ)∂µ(πˆ+φˆ)(cid:3) (πˆ+φˆ), (28)
− 3√6 Λ3 µ 4
(cid:18) (cid:19) 5 Z
where
Λ =(m2M )1/3 10−13eV. (29)
5 5 4 ∼
This represents the scale at which the 4D scalars becomes strongly coupled, and corresponds to a distance
Λ−1 1000km. We can now take the formal limit2
5 ∼
T
µν
M ,M ,M ,T , Λ =const, Λ =const, =const. (30)
4 5 6 µν 5 6
→∞ M
4
Notethatthisrequiresthatm ,m 0. Itiseasytoseethattheleadingordermattercouplingremains. In
5 6
→
contrast,the largeinteractionterm Eq.28 is the only interactionterm to survive this limit. As a result, the
tensor h˜ andvector N˜ decouple fromone another and fromthe scalarsπ and φ to allorders. Interms of
µν µ
the canonically normalised fields, the action is given by
1 1 1 1 1
S = d4x hˆµν hˆ ηµν (cid:3) hˆ Nˆµ∆Nˆ + φˆ(cid:3) φˆ πˆ(cid:3) πˆ
dec 4 4 µν µ 4 4
2 − 2 − 2 2 − 2
Z (cid:18) (cid:19)
1 1 Tµν 1 T
∂ (πˆ+φˆ)∂µ(πˆ+φˆ)(cid:3) (πˆ+φˆ)+hˆ + (πˆ+φˆ) , (31)
−(cid:18)3√6(cid:19)Λ35 µ 4 µν M4 √6 M4
3
where ˆh = M4˜h and Nˆµ = M2N˜µ. Focussing on the scalar sector, the relevant piece of the action is
µν 2 µν 5
given by
1 1 1 1 1 T
S = d4x φˆ(cid:3) φˆ πˆ(cid:3) πˆ ∂ (πˆ+φˆ)∂µ(πˆ+φˆ)(cid:3) (πˆ+φˆ)+ (πˆ+φˆ) . (32)
π,φ Z 2 4 − 2 4 −(cid:18)3√6(cid:19)Λ35 µ 4 √6 M4
The resulting equations of motion are explicitly Galilean invariant,
2 1 1 T
(cid:3) φˆ (∂ ∂ (πˆ+φˆ))2 ((cid:3) (πˆ+φˆ))2 + =0, (33)
4 −(cid:18)3√6(cid:19)Λ35 µ ν − 4 √6M4
h i
2 1 1 T
(cid:3) πˆ (∂ ∂ (πˆ+φˆ))2 ((cid:3) (πˆ+φˆ))2 + =0. (34)
− 4 −(cid:18)3√6(cid:19)Λ35 h µ ν − 4 i √6M4
2 In taking this limitwe inevitably project out the pure DGP scenario, since that would require us to take M6 ≡0,Λ6 ≡0,
whichis,ofcourseincompatiblewithEq. 30.
7
That is, they are invariant under
πˆ πˆ+b xµ+c, φˆ φˆ+˜b xµ+c˜, (35)
µ µ
→ →
for constant b ,˜b ,c and c˜ [24]. Thus we confirm what we expected. Cascading cosmology admits a bi-
µ µ
galileon description in the full decoupling limit. The limit corresponds to taking the Planck masses to
infinity but holding the 4D and 5D strong coupling scales fixed. The latter correspond to 1000 km and
1000000 km respectively, whilst the Planck lengths in the original model are all submillimeter. In taking
the decoupling limit we have sent all the next largestinteractions to zero,so our descriptionis only valid at
scales where these additional interactions may be ignored, up to the horizon scale. These interactions may
have an important role to play in understanding various aspect of the full theory (eg. Vainshtein effects),
but since our goal was merely to demonstrate that a bi-galileon description exists in principle at certain
scales, we do not concern ourselves with that here.
Theπ,φ-LagrangianassociatedwithEq.32istheanalogueoftheπ-LagrangianinDGPgravity[20,22]. It
shouldcapturelocalphysicsonthe3-brane,correspondingtonormalisablefluctuationsinthe5Dbulk. Note
that this means it cannot capture the self tuning behaviour of the full 6D theory. This is because the self
tuning solutions correspond to non-local, non-normalisable fluctuations about the vacuum of the effective
5D theory described by the cascading cosmology model (5). We could, of course, consider normalisable
fluctuationsaboutanon-trivialselftuningsolution. Weexpectthatthis wouldalsogiveabi-galileontheory
in the appropriate decoupling limit. For sufficiently large 3-brane tension it would actually correspond to a
ghost-free bi-galileon theory [29], unlike the example given here. However, a detailed analysis is outside of
the scope of this paper.
III. BI-GALILEON MODIFICATION OF GRAVITY
The cascading cosmology model described in the previous section is an example of a modification of
General Relativity by two additional galileon fields, π and φ. In a generic theory of modified gravity, the
amplitude for the exchange of one graviton between two conserved sources, T and T′ , is given by
µν µν
=T DµναβT′ , (36)
A µν αβ
where Dµναβ is the gravitonpropagator. Now, for General Relativity, the amplitude is given by
2 1 1 1
= Tµν T′ T T′ , (37)
AGR −M2 (cid:3) µν − 2 (cid:3)
pl (cid:18) 4 4 (cid:19)
where T =Tµ . We will be interested in the case where gravity gets modified by two additional scalars so
µ
that locally we have
1 1
δ = =T T′+T T′. (38)
A A−AGR α (cid:3) α (cid:3)
1 4 2 4
Such a theory can be described by the following action
M2 α α 1
S = d4x pl g˜R(g˜) 1π (cid:3) π 2π (cid:3) π + h˜ Tµν +π T +π T +interactions, (39)
1 4 1 2 4 2 µν 1 2
2 − − 2 − 2 2
Z
p
where g˜ = η +h˜ . The fluctuation h˜ is identified with the graviton in General Relativity since it
µν µν µν µν
satisfies the same linearised field equations. Therefore, for a given source and boundary conditions, the
solution h˜ coincides with the linearised GR solution. This statement is true to all orders in the limit,
µν
α Tµν
M , α , Tµν , i =const, =const, (40)
pl i →∞ M2 M
pl pl
8
where the scalars decouple from ˜h . The physical graviton differs from its GR counterpart by a scalar
µν
fluctuation
h =h˜ +2(π +π )η , (41)
µν µν 1 2 µν
since the physical metric seen by matter is given by g =η +h .
µν µν µν
Let us consider the decoupling limit (40), with the additional assumption that the strength of all scalar
interactions are held fixed. This basically means we are neglecting the backreaction of the scalars on the
geometry,so we canconsiderthem as fields onMinkowskispace. Note thatwe retainscalarself-interactions
becausetheyareessentialforbi-galileoncorrectionstoGRtobeextrapolatedfromO(10−5)inthesolarsys-
temtoO(1)atcosmologicalscales. SignificantmodificationstogravityatHubbledistanceswouldinevitably
have a role to play in the dark energy problem. The action is given by
M2 1
S = d4x plh˜µν h˜ + h˜ Tµν +π T +π T + , (42)
− 4 E µν 2 µν 1 2 Lπ1,π2
Z
where h˜ = 1(cid:3) h˜ 1h˜η +... is the linearised Einstein tensor, and the Lagrangian only
E µν −2 4 µν − 2 µν Lπ1,π2
dependsonthescalars(cid:16)π1andπ2. A(cid:17)swecanseefromEq.31,theactionforthecascadingcosmologycertainly
takes this form in the decoupling limit. In Eq. 31 the scalar action is Galilean invariant. To generalise this,
we are interested in bi-galileon modifications of gravity for which is the Lagrangian for a bi-galileon
theory. This means that if π π +(b ) xµ+c then Lπ1,π2+total derivative.
i → i i µ i Lπ1,π2 →Lπ1,π2
Giventhe formofthe physicalgraviton(41), it is convenientto define new galileonfields π =π +π and
1 2
ξ =π π , so that only one of the scalars couples to matter. The action now takes the form
1 2
−
M2 1
S = d4x pl˜hµν h˜ + h˜ Tµν +πT + , (43)
µν µν π,ξ
− 4 E 2 L
Z
where corresponds to a bi-galileon theory. The physical graviton is now
π,ξ
L
h =˜h +2πη . (44)
µν µν µν
Although only π couples to matter directly, ξ couples to matter indirectly, through its mixing with π. It is
possible for the two scalars to be mixed at quadratic order, as well as at higher order.
GiventhatwehavetakenallourfieldstopropagateonMinkowskispace,onemightbeforgivenforthinking
that we cannot say anything about cosmology, or perturbations about cosmological solutions, including de
Sitter vacua. Fortunately, this is not the case. We can think of any spatial flat cosmological spacetime as a
local perturbation around Minkowski space [24]. In this context, local means local in both space and time,
at sub-horizondistances and overa sub-Hubble time. If we take here to be ~x=0 and now to be t=0, then
for ~x H−1 and t H−1, we have [24]
| |≪ | |≪
1 1
ds2 = dτ2+a(τ)2d~y2 1 H2 ~x2+ (2H˙ +H2)t2 ( dt2+d~x2), (45)
− ≈ − 2 | | 2 −
(cid:20) (cid:21)
wheretheHubblescaleH anditstimederivativeH˙ maybeevaluatedatt=0. Thisisaperturbationabout
Minkowski space written in Newtonian gauge
ds2 (1+2Φ)dt2+(1 2Ψ)d~x2, (46)
≈− −
where the Newtonian potentials are
1 1
Φ= H2 ~x2+ (2H˙ +H2)t2, Ψ= Φ. (47)
−4 | | 4 −
For a given cosmological fluid, the corresponding GR solutions have Hubble parameter H . Since h˜
GR µν
agrees with the linearised GR solution, we have h˜ = 2Φ ,h˜ =2Φ δ , where
tt GR ij GR ij
−
1 1
Φ = H2 ~x2+ (2H˙ +H2 )t2. (48)
GR −4 GR| | 4 GR GR
9
Now in our modified theory the physical Hubble parameter (associated with h ) differs from the corre-
µν
sponding GR value, H =H . Due to Eq. 44, we have a non-trivial scalar
GR
6
π =Φ Φ . (49)
GR
−
The other scalar, ξ, follows from the equations of motion. Note that a Galilean transformation π π+
b xµ+c merely corresponds to a coordinate transformation xµ xµ cxµ+ 1(x xνbµ 2b xνxµ)→in the
µ → − 2 ν − ν
physical metric.
Of particular interest are maximally symmetric vacua, for which H and H are constant. Indeed, if the
GR
vacuum energy is σ, we have H2 =σ/3M2. It is easy to check that one scalar now takes the particularly
GR pl
simple form π¯ = 1k x xµ where k = H2 H2 . In principle the other scalar can be anything, but we
−4 π µ π − GR
willtypicallyassumethat ittakesthe sameformonmaximallysymmetricsolutions,sothat ξ¯= 1k x xµ.
−4 ξ µ
Ultimately, we would like to promote our galileon theory to a fully covariant theory of gravity, coupled
to a pair of scalar fields. Whilst this will inevitably break the Galilean invariance [27], we should expect
to recover the galileon description in the decoupling limit M . One does have to be slightly careful
pl
→ ∞
not to introduce any higher derivatives by accident when covariantising the theory, but this can be done
using additionalnon-minimalcouplings [27]. As a steptowardsthis, let us firsttransformthe action(43) to
“Jordan” frame by rewriting everything in terms of the physical metric (44). This gives
M2 1
S = d4x plhµν h M2π(∂ ∂ hµν ∂2h)+ h Tµν + ˆ , (50)
− 4 E µν − pl µ ν − 2 µν Lπ,ξ
Z
where ˆ = 3M2π∂2π. The covariantcompletion of this theory will take the form
Lπ,ξ Lπ,ξ− pl
M2
S = d4x√ g pl(1 2π)R+ + (51)
cov matter scalar
− " 2 − L L #
Z
where [g;Ψ ] is the matter Lagrangian, describing matter fields, Ψ , minimally coupled to gravity.
matter n n
L
The scalar contribution, [g;π,ξ], is constructed as follows: take ˆ and let η g , ∂ ,
scalar π,ξ µν µν µ µ
L L → → ∇
thenaddnon-minimalcouplingstocurvaturetocompensateforthe higherderivativetermsintheequations
of motion.
Actually,this completionisusefulinthatitenablesustoestablishthe validityofourgalileondescription.
Recall that this neglects the backreaction of the scalars onto the geometry. From Eq. 51, we see that the
scalars contribute an energy-momentum tensor
2 δ
Tµν [g;π,ξ]= d4x√ g (52)
scalar √ gδg − Lscalar
− µν Z
The gravitationalfield equations arising from the covariantaction (51) are given by
M2 1
pl(1 2π)Gµν +M2( µ ν gµν(cid:3))π (Tµν +Tµν )=0 (53)
2 − pl ∇ ∇ − − 2 scalar
Although the galileon description includes non-linear effects at the level of the scalar field equations, it
neglectsthematthelevelofthegravityequation. ExpandingEq. 53aboutMinkowskispace,g =η +h ,
µν µν µν
we find that
M2 1 1
pl hµν +M2 ∂µ∂ν ηµν∂2 π Tµν[η;Ψ ]= Tµν [η;π,ξ]+... (54)
2 E pl − − 2 n 2 scalar
(cid:0) (cid:1)
where “...” denotes terms that are (h) suppressed relative to the terms on the LHS of Eq. 54. The
O
gravity equation in the galileon description corresponds to neglecting everything on the RHS of Eq. 54.
Whenisthis justified? Forasmallgravitonfluctuation,wecanconsistentlyneglectthosethe termsthatare
(h) suppressed, denoted by “...”. However,even then, the backreaction of the scalars on to the geometry
O
can be significant. In other words, it is not clear that the energy momentum of the scalars, evaluated on
10
Minkowski, can be consistently neglected. To justify neglecting this backreaction we conservatively require
that Tµν [η,;π,ξ] M2 hµν.
scalar ≪ plE
Provided this condition holds, we will see that our bi-galileon theory allows for a rich and interesting
phenomenology, to be studied in detail in our companion paper [41]. As we have just discussed, we can
describecosmologicalsolutions,suchasdeSitterspace,aslocalperturbationsaboutMinkowski. Wewillalso
be interestedin fluctuationsaboutthese solutions,andinparticularaboutnon-trivialmaximallysymmetric
vacua. These just correspond to fluctuations in the galileon fields about their background values. As we
mentionedearlier,thenon-linearityofthegalileonactionwillbeessentialtothesuccessofescapingthelocal
gravity tests and producing interesting cosmological effects simultaneously.
IV. CONSTRUCTING THE BI-GALILEON THEORY
Letus now constructthe mostgeneralbi-galileontheory. Ourtask is to find the mostgeneralLagrangian
describing two scalars, , that is invariant under the Galilean transformation
π,ξ
L
π π+b xµ+c, ξ ξ+˜b xµ+c˜, (55)
µ µ
→ →
This amounts to requiring that the corresponding equations of motion are built exclusively out of second
derivativeterms∂ ∂ π, ∂ ∂ ξ contractedwiththebackgroundMinkowskimetric. Single,orzeroderivative
µ ν α β
contributionstothe equationsofmotionbreakthe Galileaninvariance,whereashigherderivativesintroduce
extra degrees of freedom and ghosts.
The Galilean symmetry greatly constrains the structure of the Lagrangian. For example, under an in-
finitesimal constant shift π π+c, the Lagrangian varies as
π,ξ
→ L
∂
π,ξ
+ L c. (56)
π,ξ π,ξ
L →L ∂π
However,as this shift happens to be a Galileantransformation,we know that anychange inthe Lagrangian
must be a total derivative. It follows that ∂ /∂π = ∂ µ, for some µ. Now the π equation of motion
π,ξ µ
L J J
goes like
δ ∂ ∂
d4x = Lπ,ξ ∂ Lπ,ξ ... (57)
π,ξ µ
δπ L ∂π − ∂(∂ π) −
Z (cid:20) µ (cid:21)
which is clearly a total derivative ∂ [ µ ∂ /∂(∂ π)+...]. Indeed, the π equation of motion is a total
µ π,ξ µ
J − L
derivative at each order in π and ξ. We can apply a similar argument to the ξ equation of motion.
We can now formulate the problem as follows: at (m,n)-th order (m-th order in π and n-th order in ξ),
what is the most general total derivative built out of second derivative terms? One obvious total derivative
is given by
=(m+n)!δµ1 ...δµm δρ1...δρn (∂ ∂ν1π)...(∂ ∂νmπ) (∂ ∂σ1ξ)...(∂ ∂σnξ). (58)
Em,n [ν1 νm σ1 σn] µ1 µm ρ1 ρn
It is easy to see all such terms with m + n > 4 vanish in four dimensions. We will now write out
all the non-vanishing explicitly in 4D. For notational convenience, we suppress the Lorentz indices
m,n
E
and separate Lorentz scalars by ‘’ when it is confusing. For example, we have ∂∂π∂∂ξ ∂∂π∂∂ξ
· · ≡