Table Of ContentCERN-PH-TH/2011-012, YITP-11-9
A covariant approach
to general field space metric
in multi-field inflation
2
1
0 Jinn-Ouk Gong 1 and Takahiro Tanaka 2
∗ †
2
n
a ∗ Theory Division, CERN
J
CH-1211 Gen`eve 23, Switzerland
4
1
† Yukawa Institute for Theoretical Physics, Kyoto University
]
O Kyoto 606-8502, Japan
C
h. January 17, 2012
p
-
o
r
t Abstract
s
a
[ We present a covariant formalism for general multi-field system which enables us to
3 obtain higher order action of cosmological perturbations easily and systematically. The
v effects of the field space geometry, described by the Riemann curvature tensor of the field
9
space, are naturally incorporated. We explicitly calculate up to the cubic order action
0
8 which is necessary to estimate non-Gaussianity and present those geometric terms which
4
have not yet known before.
.
1
0
1
1
:
v
i
X
r
a
[email protected]
[email protected]
1 Introduction
Inflation [1] is currently the leading candidate to lay down the necessary initial conditions for
the successful hot big bang evolution of the universe [2]. The most recent observations from the
cosmic microwave background (CMB) are consistent with the predictions of the inflationary
paradigm [3]: the universe is homogeneous and isotropic with vanishing spatial curvature, and
the primordial scalar perturbation is dominantly adiabatic and follows almost perfect Gaussian
statistics with a nearly scale invariant power spectrum. Thus, any small deviation from these
predictions would provide crucial information for us to distinguish different models of inflation.
Especially, the non-linearities in the primordial perturbation have received an extensive interest
nowadays in the light of upcoming precise cosmological observations. For example, while the
current bound on the non-linear parameter f [4] is constrained to be f . (100) from
NL NL
| | O
the Wilkinson Microwave Anisotropy Probe observation on the CMB [3], the Planck satellite
can probe with better precision to detect f = (5) [5]. The sensitivity may be even further
NL
| | O
improved from the observations on large scale structure [6].
The absence of the relevant scalar field which can support inflation in the standard model
(SM) of particle physics3 demands that inflation be described in the context of the theories
beyond the SM. Typically there are plenty of scalar fields which can contribute to the infla-
tionary dynamics [10]. Further, in multi-field system we can obtain interesting observational
signatures which deviate from the predictions of the single field models of inflation and can be
detected in near future, such as isocurvature perturbation [11] or non-Gaussianity [12]. Thus
we have both theoretical and phenomenological motivations to develop a complete formulation
of general multi-field inflation.
An important point in multi-field system is that in the field space which generally has non-
trivial field space metric, the scalar fields play the role of the coordinate. Naturally, as we do in
general relativity, it is preferable to formulate the dynamics in the field space in the coordinate
independent manner. That is, we need a covariant formulation of multi-field inflation which
allowsustodescribetheinflationarydynamicswitharbitraryfieldspace. However, moststudies
on multi-field inflation, especially regarding non-linear perturbations, are based on trivial field
space[13]ornon-covariantdescription[14,15,16]. Theexisting studieswithcovariantapproach
to general field space metric are mostly on linear perturbation theory [17].
In this note, we develop a fully covariant formulation of non-linear perturbations in general
multi-field inflation. Along with the covariance for general field space, it allows us to obtain
arbitrary higher order action of cosmological perturbations easily and systematically. We con-
sider the matter Lagrangian which is a generic function of the field space metric G with I
IJ
and J being generic field space indices, kinetic function ∂µφI∂ φJ and the fields [18]. This form
µ
includes not only the matter Lagrangian with the standard canonical kinetic term but also
more generic ones motivated from high energy theories, such as the Dirac-Born-Infeld (DBI)
type [19].
This note is outlined as follows. In Section 2 we set up the geodesic equation to describe
the field fluctuation around the background trajectory. In Section 3, we consider pure matter
Lagrangianandpresent acovariant formulationto describe the fieldfluctuations upto arbitrary
3 It was recently suggested that the SM Higgs field can play the role of the inflaton provided that it is
non-minimally coupled to gravity [7]. However the unitarity of the simplest Higgs inflation appears to be
controversial. See e.g. Refs. [8] and [9] and references therein for different points of view on this issue.
1
order. The extension to include gravity follows in Section 4 and we explicitly compute the
perturbed action up to cubic order. We also discuss the genuine multi-field effects briefly. We
conclude in Section 5. Technical details to compare with the previously known non-covariant
description are presented in the Appendix.
2 Issue of mapping
To begin with, first let us consider how to describe the physical field fluctuation δφI in the field
space in a covariant manner. We can think of the background field trajectory parametrized
by a single parameter, usually taken as the cosmic time t: φI = φI(t). The real physical field
0 0
in a fixed gauge φI incorporates quantum fluctuations δφI around this background trajectory.
However, the fluctuations δφI are coordinate dependent, and hence they are not covariant.
These two points, φI(t) and φI, can be connected by a unique geodesic with respect to the
0
field space metric G as long as their separation is sufficiently small. This geodesic can be
IJ
specified by the initial point φI and its initial velocity, which we denote by QI. This situation
0
is depicted in Figure 1. Hence, the issue here is the “mapping” beyond linear order between
the finite displacement δφI φI φI and a vector QI living in the tangent space at φI. Let us
≡ − 0 0
parametrize the geodesic trajectory in the field space by λ, which runs from 0 to ǫ > 0: λ = 0
and λ = ǫ correspond to φI and φI, respectively. Here ǫ is a parameter introduced to count the
0
order of perturbation just for a bookkeeping purpose, and hence it is set to unity at the end of
calculation.
Figure1: Aschematicfigureshowingaphysical fieldφI inthefieldspacearoundthebackground
trajectory φI(t). The geodesic connecting φI and φI is parametrized λ, which runs from 0 to ǫ.
0 0
2
Denoting the covariant differentiation in λ-direction by D D/dλ, the geodesic equation
λ
≡
for φI(λ) is written as
d2φI dφJ dφK
D2φI = +ΓI = 0, (1)
λ dλ2 JK dλ dλ
and the initial conditions are
φI =φI , (2)
λ=0 0
(cid:12) dφI
D φI(cid:12) = = QI . (3)
λ λ=0 dλ (cid:12)
(cid:12) (cid:12)λ=0
(cid:12) (cid:12)
(cid:12)
Now, we expand φI(λ = ǫ) as a power series with respect to ǫ from λ = 0 as
dφI 1 d2φI 1 d3φI
φI(λ = ǫ) = φI + ǫ+ ǫ2 + ǫ3 + . (4)
λ=0 dλ (cid:12) 2! dλ2 (cid:12) 3! dλ3 (cid:12) ···
(cid:12) (cid:12)λ=0 (cid:12)λ=0 (cid:12)λ=0
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
Note that the derivatives with respect to λ here are not covariant ones. Thus, we can trade
quadratic and higher derivatives with single derivatives by means of the geodesic equation (1).
Namely, we can replace a quadratic derivative with
d2φI dφJ dφK
= ΓI , (5)
dλ2 − JK dλ dλ
and the third order derivative with
d3φI dφLdφJ dφK
= ΓI ΓM ΓI , (6)
dλ3 LM JK − JK;L dλ dλ dλ
(cid:0) (cid:1)
and so on. Thus, we can write (4) as
1 1
φI(λ = ǫ) = φI +QIǫ ΓI QJQKǫ2 + ΓI ΓM ΓI QJQKQLǫ3 + , (7)
0 − 2 JK 6 LM JK − JK;L ···
(cid:0) (cid:1)
which we can continue up to arbitrary non-linear order. In the end, setting ǫ = 1, we obtain
1 1
φI φI δφI = QI ΓI QJQK + ΓI ΓM ΓI QJQKQL + . (8)
− 0 ≡ − 2 JK 6 LM JK − JK;L ···
(cid:0) (cid:1)
If we truncate (8) at linear order, we can identify δφI and QI. Then, we do not have to pay
attention to the difference between them. However, when we consider non-linear perturbations,
we have to distinguish them clearly. Only when we write the equations in terms of QI, they
can be expressed in a covariant manner.
3 General matter Lagrangian
Now, let us consider the general effective matter Lagrangian P, which is a function of the field
space metric G , kinetic function XIJ = gµν∂ φI∂ φJ/2 and φI, i.e.
IJ µ ν
−
P = P(G ,XIJ,φI). (9)
IJ
3
NotethatwedonotrestrictthekineticfunctiontoafunctionofX = G XIJ, i.e. alltheindices
IJ
contracted with the metric. This is because we may have a term like G G XIJXKL, as is
IK JL
typical in the multi-field DBI inflation. Also we do not consider higher derivative terms such as
(cid:3)φI, which usually leads to ghost except for some special combinations such as Galileon[20]. In
this section, we treat the spacetime metric g as a given background, but it is not necessarily
µν
spatially homogeneous. Inclusion of the metric perturbation will be discussed in the succeeding
section.
We consider that the fields φI contained in P are all functions of λ, a parameter along the
geodesic in the field space. Then, P as a whole is a function of λ and is a scalar with respect
to the field space indices. We expand P in terms of the parameter λ, and set λ to ǫ, to obtain
1 1
P = P + D P ǫ+ D2P ǫ2 + D3P ǫ3 + , (10)
|λ=0 λ |λ=0 2! λ λ=0 3! λ λ=0 ···
(cid:12) (cid:12)
(cid:12) (cid:12)
where we have used the fact that an ordinary derivative of a field space scalar is identical to a
covariant one. First let us consider the linear variation, D P. At this stage, we find
λ
∂P ∂P
D P = D XIJ + D φI , (11)
λ ∂XIJ λ ∂φI λ
without any subtle issue. Here we have used D G = 0 that follows from the definition of the
λ IJ
covariant differentiation, and we have also assumed that a derivative of P with respect to XIJ
is automatically symmetrized, i.e.
∂P 1 ∂P ∂P
+ P . (12)
∂XIJ → 2 (cid:18)∂XIJ ∂XJI(cid:19) ≡ hIJi
However, from the quadratic variation, we find that our notation becomes a little uncom-
fortable. Explicitly, we can write
D2P = D2XIJP +D XIJ D P +D2φIP +D φI (D P ) , (13)
λ λ hIJi λ λ hIJi λ ,I λ λ ,I
(cid:0) (cid:1)
where the third term vanishes due to the geodesic equation of φI. The difficulty is in the second
and the last terms: how to write the covariant derivatives of the derivative of P? In fact, we
can easily come to know that the differentiation of P with respect to XIJ should be understood
as an ordinary one because XIJ is not a coordinate in the field space but a tensor living in the
tangent space. On the other hand, the differentiation with respect to φI should be understood
as a covariant one because φI is a coordinate of the field space: any differentiation in the field
space necessarily incorporates parallel transport.
While the above considerations are legitimate, it is very uncomfortable to have covariant
and ordinary differentiations mixed. Moreover, covariant and ordinary differentiations do not
commute. We have P coming from the second term of (13) and P coming from
hIJi;K ;KhIJi
the last term, but they are not the same. Explicitly, P = P ΓL P ΓL P .
hIJi;K ;KhIJi − IK hLJi − JK hILi
Thereforeif werewriteoneexpression withtheother, theresult containstheChristoffel symbols
and is not manifestly covariant.
To avoid this mess, we consider an alternative description. We assume that P depends on
φI only through field space tensors such as fJ1···Jna(φI), where the subscript a is introduced
a
to discriminate different kinds of such tensors. One most important example is the potential
4
V(φI), which is a field space scalar. Here we are assuming that there is no spacetime derivatives
of fields in fJ1···Jna(φI). With this, first let us consider a single derivative. From
a
P = P G ,XIJ,fJ1···Jna(φI) , (14)
IJ a
(cid:2) (cid:3)
a single derivative with respect to λ is easily calculated as
∂P ∂P ∂P
D P =D G +D XIJ + D fJ1···Jna
λ λ IJ∂GIJ λ ∂XIJ Xa λ a ∂faJ1···Jna
=D XIJP + fJ1···Jna D φIP , (15)
λ hIJi a ;I λ {J1···Jna}a
Xa
where we have defined P ∂P/∂fJ1···Jna. Nowthedifferentiations ofP areallordinary
{J1···Jna}a ≡ a
ones, and those of fJ1···Jna are all covariant ones. In this way, we can straightforwardly write
a
up to cubic order expansion of the general matter Lagrangian P with respect to λ as
1 1
P =P +P δXIJ +P δf + P δXIJδXKL +P δXIJδf + P δf δf
λ=0 hIJi a a hIJihKLi hIJia a ab a b
| 2! 2!
1 1
+ P δXIJδXKLδXMN + P δXIJδXKLδf
hIJihKLihMNi hIJihKLia a
3! 2!
1 1
+ P δXIJδf δf + P δf δf δf + , (16)
hIJiab a b abc a b c
2! 3! ···
where we have assumed that the field space tensors fJ1···Jna are all scalars for simplicity, and
a
introduced the following notations
∂P
P , (17)
a
≡∂f
a
∞ ǫn
δXIJ DnXIJ , (18)
≡ n! λ |λ=0
Xn=1
∞ ǫn
δf Dnf . (19)
a ≡ n! λ a|λ=0
Xn=1
With the aid of the geodesic equation (1), it is trivial to find that the derivatives of f are given
a
by
D f =f QI , (20)
λ a|λ=0 a;I
D2f =f QIQJ , (21)
λ a λ=0 a;IJ
D3f (cid:12) =f QIQJQK . (22)
λ a(cid:12)λ=0 a;IJK
(cid:12)
(cid:12)
Obtaining the derivatives of XIJ with respect to λ needs some manipulation. First, we should
understand that∂ φI isavector living inthetangent space. Hence, thecovariant differentiation
µ
of ∂ φI is given by
µ
dφI dφK dφI
D ∂ φI = ∂ +ΓI ∂ φJ D . (23)
λ µ µ dλ JK µ dλ ≡ µ dλ
5
When we recursively act the covariant differentiation D , we need the commutator between
λ
D and D . The necessary commutation relation can be derived in the same manner as in the
µ λ
derivation of the geodesic deviation equation, e.g. for an arbitrary vector VI,
dφK
[D ,D ]VI = RI VJ ∂ φL. (24)
λ µ JKL dλ µ
Then, we obtain
D XIJ = gµνD Q(I∂ φJ), (25)
λ λ=0 − µ µ 0
(cid:12)
D2XIJ(cid:12) = gµν R(I ∂ φJ)∂ φMQKQL +D QID QJ , (26)
λ λ=0 − KLM µ 0 ν 0 µ ν
h i
(cid:12)
D3XIJ(cid:12) = gµν R(I ∂ φJ)∂ φMQNQKQL +R(I ∂ φJ)QKQLD QM
λ λ=0 − KLM;N µ 0 ν 0 KLM µ 0 ν
h
(cid:12)
(cid:12) +3R(I D QJ)∂ φMQKQL , (27)
KLM ν µ 0
i
whereparenthesesovertheindicesdenotesymmetrization. Herewehavewrittendownexplicitly
how the inverse metric gµν is contained in the expressions for the later convenience when we
consider metric perturbations.
4 Gravity
4.1 General arguments
Until now, we have only considered matter Lagrangian and treated the metric as a given back-
ground. But to describe real physics we must take into account the dynamics of gravitational
degrees of freedom: additional 4 scalar, 4 vector, and 2 tensor degrees of freedom. Here, scalar,
vector and tensor are those with respect to the three dimensional isometry. However, not all
of them are physical. The fictitious gauge degrees of freedom can be removed by imposing
appropriate gauge conditions. Here in this note we choose the flat gauge as we will explain
immediately below, neglecting the vector and tensor degrees of freedom. Their contributions,
especially those of tensor perturbations, to the higher order correlation functions of the curva-
ture perturbation enter only through loop corrections, which are highly suppressed.
At the beginning, we have n+4 scalar variables: n from n scalar field components, 4 from
the metric. Since there are 1 temporal and 1 spatial gauge transformations in the scalar sector,
we can eliminate 2 of them. In the flat gauge, we impose the conditions that the perturbations
of three dimensional spatial metric on each time slice vanish. The remaining metric degrees of
freedom are perturbations of the lapse function and the shift vector. We denote them by ξα
symbolically. Further, by solving 2 constraint equations, we can also remove the remaining two
degrees of freedom ξα, so that after all n degrees of freedom are left. Namely, we can write all
the metric degrees of freedom solely in terms of the field fluctuations δφI.
First let us formally expand the metric fluctuations δξα in ǫ as
ξα(λ = ǫ) = ξα +ξα ǫ+ξα ǫ2 + . (28)
0 (1) (2) ···
The constraint equations are simply given by the variation of the action with respect to ξα,
δS
= 0. (29)
δξα
6
When we expand the action with respect to ξα , the n-th order term in ξα, we find
(n)
δS 1 δ2S
S = S + ξα + ξα ξβ + . (30)
|ξ(αn)=0 δξα (cid:12)(cid:12) (n) 2 δξα δξβ (cid:12)(cid:12) (n) (n) ···
(n)(cid:12)ξµ =0 (n) (n)(cid:12)ξµ =0
(cid:12) (n) (cid:12) (n)
(cid:12) (cid:12)
Then, writing (29) as
δS δ2S 1 δ3S
= ξβ ξβ ξγ + = (ǫn), (31)
δξα(cid:12) − δξβδξα(cid:12) (n) − 2 δξβδξγδξα(cid:12) (n) (n) ··· O
(cid:12)ξµ =0 (cid:12)ξµ =0 (cid:12)ξµ =0
(cid:12) (n) (cid:12) (n) (cid:12) (n)
(cid:12) (cid:12) (cid:12)
we find that both the second and the third terms on the right hand side of (30) are (ǫ2n).
O
Hence, when we want to know the action to, say, the cubic order in ǫ, the second and higher
order of ξα are not necessary. To obtain the linear order of ξα, we only need to solve the
constraint equations (29) expanded up to linear order in ǫ,
δS
= 0. (32)
(cid:18)δξα(cid:19)
(1)
Plugging the solution for ξα of the above constraint equations back into the action, we obtain
(1)
the action written in terms of the field perturbation QI.
4.2 Explicit calculations
Now let us move onto more explicit computations. We consider a general matter Lagrangian
which describes multi-field system minimally coupledto Einstein gravity intheArnowitt-Deser-
Misner form [21],
m2 1
S = d4xN√γ Pl R(3) + Ei Ej E2 +P , (33)
j i
Z (cid:26) 2 (cid:20) N2 − (cid:21) (cid:27)
(cid:0) (cid:1)
where R(3) is the 3-curvature scalar constructed from the spatial metric γ , and
ij
1
E γ˙ N N , (34)
ij ij i|j j|i
≡ 2 − −
(cid:0) (cid:1)
with a vertical bar denoting a covariant differentiation with respect to γ . The gauge we choose
ij
is, as advertised, the so-called flat gauge, in which the spatial metric γ is unperturbed, i.e.
ij
γ = a2δ , (35)
ij ij
which completely fixes both spatial slicing and temporal threading beyond linear level [22], as
long as one neglects the vector and tensor perturbations. In this gauge, we separate the action
into the gravity and matter sectors,
S = S(G) +S(M), (36)
7
with
a3m2
S(G) = d4x Pl Ei Ej E2
j i
Z 2N −
(cid:0) (cid:1)
a3m2 1
= d4x Pl 3H2 +2HNi + N Ni,j +N Nj,i 2Ni Nj , (37)
,i i,j i,j ,i ,j
Z N (cid:20)− 4 − (cid:21)
(cid:0) (cid:1)
S(M) = d4xa3NP . (38)
Z
We choose the background values of the metric variables, which we associate with a subscript
(0), as N = 1 and Ni = 0, corresponding to the Friedmann-Lemaˆıtre-Robertson-Walker
(0) (0)
model written using the cosmological time coordinate. As we have explained above, to obtain
the cubic order action, we only need to keep the linear order for the metric perturbations ξα.
In the action (36), therefore we set
N =1+N ǫ, (39)
(1)
Ni =Ni ǫ. (40)
(1)
4.2.1 Action expansion including metric perturbations
It is straightforward to write down the gravity part of the action. All we need to do is just to
plug the expansions (39) and (40) into the gravity action (37). To the cubic order, we have
S(G) = d4xa3m2 1 N ǫ+N2 ǫ2 N3 ǫ3
Z Pl − (1) (1) − (1)
(cid:2) (cid:3)
1
3H2 +2HNi ǫ+ N(1)Ni,j +N(1)Nj,i 2Ni Nj ǫ2 . (41)
×(cid:20)− (1),i 4 i,j (1) i,j (1) − (1),i (1),j (cid:21)
(cid:16) (cid:17)
The expansion of the matter Lagrangian is a little more non-trivial. However, by assump-
tion, our matter Lagrangian contains the spacetime metric only through XIJ. Therefore, all
we have to do is just to replace the expression for δXIJ to the one that explicitly includes the
expansion with respect to metric perturbations. As the inverse metric gµν is given by
1
gµν∂ ∂ = (∂ Nj∂ )2 +γij∂ ∂ , (42)
µ ν t j i j
−N2 −
(25), (26) and (27) are more explicitly written down as
1
D XIJ = D Q(Iφ˙J), (43)
λ |λ=0 N2 t 0
D2XIJ = 1 eR(I φ˙J)φ˙MQKQL +D QID QJ γij∂ QI∂ QJ , (44)
λ |λ=0 N2 KLM 0 0 t t − i j
h i
D3XIJ = 1 R(I φ˙J)φ˙MQNQKQeL +Re(I φ˙J)QKQLD QM
λ |λ=0 N2 KLM;N 0 0 KLM 0 t
(cid:2)
+3R(I φ˙MQKQLD QJ) , e (45)
KLM 0 t
(cid:3)
e
where we have defined
D D Nj∂ . (46)
t t j
≡ −
e
8
More explicitly expanding the perturbation of XIJ in terms of ǫ, we obtain
XIJ = XIJ +XIJǫ+XIJǫ2 +XIJǫ3 + , (47)
0 (1) (2) (3) ···
with
XIJ = N φ˙Iφ˙J +D Q(Iφ˙J), (48)
(1) − (1) 0 0 t 0
3
XIJ = N2 φ˙Iφ˙J 2N D Q(Iφ˙J) Nj ∂ Q(Iφ˙J)
(2) 2 (1) 0 0 − (1) t 0 − (1) j 0
1
+ D Q(ID QJ) γij∂ Q(I∂ QJ) +R(I φ˙J)φ˙MQKQL , (49)
2 t t − i j KLM 0 0
h i
XIJ = 2N3 φ˙Iφ˙J +3N2 D Q(Iφ˙J) Nj ∂ Q(ID QJ)
(3) − (1) 0 0 (1) t 0 − (1) j t
N D Q(ID QJ) +R(I φ˙J)φ˙MQKQL 2Nj ∂ Q(Iφ˙J)
− (1) t t KLM 0 0 − (1) j 0
h i
1
+ 3R(I D QJ)φ˙MQKQL +R(I φ˙J)D QMQKQL +R(I φ˙J)φ˙MQNQKQL .
6 KLM t 0 KLM 0 t KLM;N 0 0
h i
(50)
4.2.2 Linear order action
First we consider the first order terms, where we can extract the background equations of
motion. Collecting the results that we have obtained in the preceding sections, the first order
action becomes
S = d4xa3 3m2 H2 +P P φ˙Iφ˙J N +P D QIφ˙J +P f QI , (51)
1 Z h(cid:16) Pl 0 − hIJi 0 0(cid:17) (1) hIJi t 0 a a;I i
where by P we denote the matter Lagrangian with the background quantities substituted.
0
What we can immediately see is that we can derive two equations of motion by varying with
respect to the lapse perturbation N and the field fluctuation QI. Taking a variation of (51)
(1)
with respect to N , we obtain
(1)
1
H2 = P φ˙Iφ˙J P , (52)
3m2 hIJi 0 0 − 0
Pl (cid:16) (cid:17)
which is the background Friedmann equation. We can also immediately obtain the equation of
the background field φI as
0
1
D a3P φ˙J = P f , (53)
a3 t hIJi 0 a a;I
(cid:16) (cid:17)
or more explicitly,
P +P φ˙Kφ˙L D φ˙J + 3HP +P f φ˙K φ˙J P f = 0. (54)
hIJi hIKihJLi 0 0 t 0 hIJi hIJia a;K 0 0 − a a;I
(cid:16) (cid:17) (cid:16) (cid:17)
9
