gravity(covariant theory,2012)

7/31/2019 Gravity(Covariant Theory,2012)

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arXiv:1201.4630v2

[gr-qc]18Oct2012

Inflation in general covariant theory of gravity

Yongqing Huang a, Anzhong Wang a,b, and Qiang Wu baGCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA

b Institute for Advanced Physics & Mathematics,Zhejiang University of Technology, Hangzhou 310032, China

(Dated: October 19, 2012)

In this paper, we study inflation in the framework of the nonrelativistic general covariant theory

of the Horava-Lifshitz gravity with the pro jectability condition and an arbitrary coupling constant. We find that the Friedmann-Robterson-Walker (FRW) universe is necessarily flat in such asetup. We work out explicitly the linear perturbations of the flat FRW universe without specifyingto a particular gauge, and find that the perturbations are different from those obtained in generalrelativity, because of the presence of the high-order spatial derivative terms. Applied the generalformulas to a single scalar field, we show that in the sub-horizon regions, the metric and scalarfield are tightly coupled and have the same oscillating frequencies. In the super-horizon regions,the perturbations become adiabatic, and the comoving curvature perturbation is constant. We alsocalculate the power spectra and indices of both the scalar and tensor perturbations, and expressthem explicitly in terms of the slow roll parameters and the coupling constants of the high-orderspatial derivative terms. In particular, we find that the p erturbations, of both scalar and tensor,are almost scale-invariant, and, with some reasonable assumptions on the coupling coefficients, thespectrum index of the tensor perturbation is the same as that given in the minimum scenario inGR, whereas the index for scalar perturbation in general depends on and is different from thestandard GR value. The ratio of the scalar and tensor power spectra depends on the high-orderspatial derivative terms, and can be different from that of GR significantly.

PACS numbers: 04.50.Kd; 98.80.-k; 98.80.Bp

I. INTRODUCTION

The Horava-Lifshitz (HL) theory of quantum gravity,proposed recently by Horava [1], motivated by the Lif-shitz scalar field theory in solid state physics [2], hasattracted a great deal of attention, due to its several re-markable features [3, 4]. The HL theory is based on theperspective that Lorentz symmetry should appear as an

emergent symmetry at long distances, but can be funda-mentally absent at short ones [5]. In the latter regime,the system exhibits a strong anisotropic scaling betweenspace and time,

x x, t zt, (1.1)where z 3 in the (3 + 1)-dimensional spacetime [1, 6].At long distances, high-order curvature corrections be-come negligible, and the lowest order terms R and take over, whereby the Lorentz invariance is expectedto be accidentally restored, where R denotes the 3-dimensional Ricci scalar of the hypersurfaces t = Con-stant, and the cosmological constant.

Because of the anisotropic scaling, the gauge symmetryof the theory is broken down to the foliation-preservingdiffeomorphism, Diff(M, F),

t = f(t), xi = i(t,x), (1.2)

Electronic address: [email protected] address: [email protected] address: [email protected]

for which the lapse function N, shift vector Ni, and 3-spatial metric gij transform as

N = kkN + N f + Nf,Ni = Nkik + kkNi + gikk + Nif + Nif ,gij = ij + ji + fgij, (1.3)

where f

df/dt,

i denotes the covariant derivative

with respect to gij, Ni = gikNk, and gij gij t, xkgij

t, xk

, etc. From these expressions one can see that

N and Ni play the role of gauge fields of the Diff(M, F).Therefore, it is natural to assume that N and Ni inheritthe same dependence on space and time as the corre-sponding generators [1],

N = N(t), Ni = Ni(t, x), (1.4)

which is often referred to as the projectability condition.Due to the Diff(M, F) diffeomorphisms (1.2), one

more degree of freedom appears in the gravitational sec-tor - a spin-0 graviton. This is potentially dangerous,and needs to decouple in the IR regime, in order to beconsistent with observations. Whether this is possible ornot is still an open question [3, 7]. In particular, spher-ically symmetric static spacetimes were studied in [4],and shown that the spin-0 graviton indeed decouples af-ter nonlinear effects are taken into account, an analogueof the Vainshtein effect in massive gravity [8]. Along thesame direction, considerations in cosmology were given in[911]. In particular, in [10, 11] a fully nonlinear analysisof superhorizon cosmological perturbations was carriedout, by adopting the so-called gradient expansion method
http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2http://lanl.arxiv.org/abs/1201.4630v2mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://lanl.arxiv.org/abs/1201.4630v2


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[12]. It was found that the relativistic limit is continuous,and general relativity (GR) is recovered in two differentcases: (a) when only the dark matter as an integrationconstant is present [10]; and (b) when a scalar field andthe dark matter as an integration constant are present[11].

Another very promising approach is to eliminate thespin-0 graviton by introducing two auxiliary fields, the

U(1) gauge field A and the Newtonian prepotentail ,by extending the Diff(M, F) symmetry (1.2) to includea local U(1) symmetry [13],

U(1) Diff(M, F). (1.5)Under this extended symmetry, the special status of timemaintains, so that the anisotropic scaling (1.1) can stillbe realized, and the theory is kept power-counting renor-malizable. Meanwhile, because of the elimination of thespin-0 graviton, its IR behavior can be significantly im-proved. Under the Diff(M, F), A and transform as,

A = iiA + f A + fA,

= f + ii. (1.6)

Under the local U(1) symmetry, the fields transform as

A = Nii, = ,Ni = Ni, gij = 0, N = 0, (1.7)

where is the generator of the local U(1) gauge symme-try. For the detail, we refer readers to [13, 14].

The elimination of the spin-0 graviton was done ini-tially in the case = 1 [13, 14], but soon generalized tothe case with any [1517], where denotes a couplingconstant that characterizes the deviation of the kinetic

part of action from the corresponding one given in GRwith GR = 1 (For the analysis of Hamiltonian consis-tency, see [13, 18]). To avoid the strong coupling prob-lem, one may follow Blas, Pujolas and Sibiryakov (BPS)[19] to introduce an energy scale M that satisfies thecondition [17],

M < , (1.8)

where M is the suppression energy of the sixth-orderderivative terms, and is the would-be strong couplingenergy scale, given by,

c13/2

| 1|5/4Mpl, (1.9)

where is related to the Planck mass Mpl throughEq.(2.6), and c1, defined in (B.7), represents the couplingof a scalar field with the gauge field A. In the case with-out the projectability condition, the observed alignmentof the rotation axis of the Sun with the ecliptic requiresM 1015 GeV [19]. Similar considerations have notbeen carried out in the current version of the HL theory,and the up bound of M is unknown. From the above

expression, it is clear that cannot be precisely equalto one, in order for the BPS mechanism to work eitherwithout [19] or with [17] the projectability condition.

It is remarkable to note that the elimination ofthe spin-0 graviton can be also realized in the non-projectability case with the extended symmetry (1.5)[20, 21]. In addition, the number of independent cou-pling constants can be significantly reduced (from more

than 70 [22] to 15), by simply imposing the softly break-ing detailed balance condition, while the theory still re-mains power-counting renornalizable and has a healthyIR limit.

In this paper, we study inflation of a scalar field in theHorava and Melby-Thompson (HMT) setup with the pro-

jectability condition [13] and an arbitrary coupling con-stant [15]. Specifically, after a brief review of the theoryin Sec. II, we first show that the FRW universe is neces-sarily flat, when it is filled with (multi-) scalar, vector orfermionic fields in Sec. III.A. Then, in the second part ofSec. III we present the general linear scalar perturbationswithout specifying to a particular gauge or specific mat-

ter fields, while in the third part of it, we consider severalpossible gauge choices. Unlike the case without the U(1)symmetry [23], some gauges used in GR [24], such as thelongitudinal gauge, now become possible, because of theU(1) gauge freedom. In Sec. IV, we first consider theflat FRW background, and show clearly that the slow-roll conditions imposed in GR are also needed here, inorder to obtain enough e-fold to solve the problems suchas horizon, monopole, domain walls, and so on [25]. Inaddition, in this section we also show that in the super-horizon regions, the perturbations become adiabatic, andthe comoving curvature perturbation is constant. In Sec.V, we show explicitly that in the sub-horizon regions, themetric and scalar field are tightly coupled and have the

same oscillating frequencies, while in the super-horizonregions, the perturbations are almost scale-invariant. Itis remarkable that a master equation for the scalar per-turbations exists, in contrast to the case without the U(1)symmetry [23]. In Sec. VI, we calculate the power spec-tra and indices of both scalar and tensor perturbationsin the slow-roll approximations, by using the uniform ap-proximation [26]. We express them explicitly in terms ofthe slow roll parameters and the coupling constants ofthe high-order spatial derivative terms. Finally, in Sec.VII we present our main conclusions.

II. GENERAL COVARIANT THEORY WITHAN ARBITRARY CONSTANT

In this section, we shall give a very brief introductionto the HMT setup with the projectability condition and = 1. For detail, we refer readers to [13, 15, 16].

The total action of the theory can be written as,

S = 2

dtd3xN

gLK LV + L + LA + L


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+2LM

, (2.1)

where g = det(gij), and

LK = KijKij K2,L = Gij

2Kij + ij

,

LA =

A

N2g R,L = 1 2 + 2K. (2.2)Here gijij , g is a coupling constant, the Ricciand Riemann tensors Rij and R

ijkl all refer to the 3-

metric gij, and

Kij =1

2N(gij + iNj + jNi) ,

Gij = Rij 12

gijR + ggij. (2.3)

LM is the Lagrangian of matter fields, which is a scalarnot only with respect to the Diff(M, F) symmetry (1.2),but also to the U(1) symmetry (1.7). LV is an arbitraryDiff()-invariant local scalar functional built out of thespatial metric, its Riemann tensor and spatial covariantderivatives, without the use of time derivatives. Assum-ing that the highest order derivatives are six, and that thetheory respects the parity and time-reflection symmetrythe most general form ofLV is given by [27, 28],

LV = 2g0 + g1R + 12

g2R2 + g3RijR

ij

+1

4

g4R

3 + g5R RijRij + g6R

ijR

jkR

ki

+

1

4 g7RR + g8 (iRjk) iRjk , (2.4)

where the coupling constants gs (s = 0, 1, 2, . . . 8) are alldimensionless, and

=1

22g0, (2.5)

is the cosmological constant. The relativistic limit in theIR, on the other hand, requires,

g1 = 1, 2 = 116G

=M2pl

2, (2.6)

where G is the Newtonian constant.Note the difference between the notations used here

and the ones used in [13, 15] 1. In this paper, we shalluse directly the notations and conventions defined in[14, 16, 23] without further explanations. Then, the fieldequations are given in Appendix A.

1 In particular, we have = HMT, Kij = KHMTij , A =

aHMT, g = HMT, Gij = HMTij , where quantities with super

indice HMT are the ones used in [13].

III. COSMOLOGICAL PERTURBATIONS

In this section, we first give a brief review of the FRWuniverse, and then argue that it must be flat in the frame-work of the HMT generalization. This is a very impor-tant implication. In fact, one of the main motivations ofinflation was to solve the flatness problem [25]. In the sec-ond part of this section, we consider scalar perturbations

without restricting ourselves to a particular gauge. Weshall closely follow the presentation given in [16], whichwill be referred to as Paper I. To see the differences, wepresent our formulas closely parallel to those given in GR[24], and point out the similarities and differences when-ever they raise.

A. Flatness of the FRW Universe

The homogeneous and isotropic universe is describedby,

N = 1, Ni = 0, gij = a2(t)ij , (3.1)

where ij = ij

1 + 14r22

, with r2 x2+y2+z2, =0, 1. As in Paper I, we use symbols with bars to de-note the quantities of the background in the (t ,x,y,z)-coordinates. Using the U(1) gauge freedom of Eq.(1.7),on the other hand, we can set

= 0. (3.2)

Then, we find

Kij = a2Hij, Rij = 2ij ,

FijA =2A

a4ij, Fij = 0, F

i = 0,

Fij =ij

a2

+

a2+

212

a4+

1223

a6

, (3.3)

and

LK = 3

1 3H2, L = 0 = L,LA = 2A

g 3

a2 ,

LV = 2 6a2

+ 1212a4

+ 2423a6

, (3.4)

where H = a/a and

1 3g2 + g32

, 2 9g4 + 3g5 + g64

. (3.5)

The super-momentum constraint (A.2) is satisfied iden-tically, provided that Ji = 0, while the Hamiltonian con-


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straint (A.1) yields 2,

1

2

31H2+

a2=

8G

3+

3+

212

a4+

423

a6, (3.6)

where Jt 2. On the other hand, Eqs.(A.4) and (A.5)yield, respectively,

Hg a2 = 8G3 J, (3.7)3

a2 g = 4GJA, (3.8)

while the dynamical equation (A.7) reduces to

1

2

3 1 a

a= 4G

3( + 3 p) +

1

3 21

2

a4

823

a6+

1

2A

a2 g

, (3.9)

where ij = p gij.The conservation law of the momentum (A.14) is satis-

fied identically, while the one of the energy (A.13) reducesto,

+ 3H( + p) = AJ. (3.10)

From Eqs.(3.7) and (3.8), one can see that when

JA = 0 = J, (3.11)

the universe is necessarily flat, k = 0 = g. This is truefor the case where the source is a scalar field [ 16], as canbe seen from Eqs.(B.12) and (B.13) given in the nextsection, where both JA and J are proportional to thespatial gradients of the scalar field . This can be easilygeneralized to the case with multi-scalar fields.

In general, the coupling of the gauge field A and theNewtonian prepotential to a matter field n is givenby [15],

dtd3x

gZ(n, gij , k)(A A), (3.12)

where A is defined as

A + Nii + 12

N2, (3.13)

and Z is the most general scalar operator under the full

symmetry of Eq.(1.5), with its dimension

[Z] = 2. (3.14)

2 Since now the Hamiltonian constraint is a global one, one caninclude a dark matter component as an integration constant,as first noted in [29]. For the sake of simplicity, in this paper weshall not consider this possibility, and it is not difficult to showthat our mainly conclusions are equally applicable to this case.

For a vector field (A0, Ai), we have [A0] = 2, [Ai] = 0[28]. Then, we find

Z(A0, Ai, gij , k) = KBiBi, (3.15)where K is an arbitrary function of AiAi, and

Bi =1

2

jkig

Fjk , iBi = 0, (3.16)

with Fij jAi iAj . This can be easily generalizedto several vector fields, (A

(n)0 , A

(n)i ), for which we have

Z( A0, Ai, gij, k) =m,n

KmnB(m)i B(n)i, (3.17)

where Km,n is an arbitrary function of A(k)iA(l)i . Then,it is easy to show that in the FRW background, we have

JA = 0, because B(m)i = 0 [30], as can be seen from

Eq.(3.16). With the gauge choice (3.2), one can alsoshow that J = 0. Therefore, an early universe domi-nated by vector fields is also necessarily flat. This can be

further generalized to the case of Yang-Mills fields [31].For fermions, on the other hand, their dimensions are[n] = 3/2 [32]. Then, Z(n, gij, k) cannot be a func-tional of n. Therefore, in this case JA and J vanishidentically.

Although we cannot exhaust all the matter fields, withthe special form of the coupling given by Eq.(3.12), it isquite reasonable to argue that the universe is necessarily

flat for all cosmologically viable models in the HMT setup.Therefore, in the rest of this paper, we shall consider onlythe flat FRW universe, i.e.,

= 0 = g, (3.18)

for which Eq.(3.11) holds.

B. Linear Perturbations

As mentioned previously, to solve the strong couplingproblem, one needs to impose the condition (1.8). Onceit is satisfied, one can safely carry out the linear per-turbations. With this in mind, as usual, we study theseperturbations in terms of the conformal time , where =

dt/a(t). Under this coordinate transformation,the fields transform as,

N = a

N , Ni

= a

Ni

, gij = gij,A = aA, = , (3.19)

where the quantities with tildes are the ones defined inthe coordinates (t, xi). With these in mind, we write thelinear scalar perturbations of the metric in the form,

N = a, N i = a2B,i,

gij = 2a2

ij E,ij

,

A = A + A, = + , (3.20)


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where A = aA and = . Quantities with hats denotethe ones of the background in the coordinates (, xi).Under the gauge transformations (1.2), they transformas

= H0 0 , = + H0,B = B + 0 , E = E ,

= 0, A = A 0A 0

A, (3.21)

where f = 0, i = ,i, H a/a, and a primedenotes the ordinary derivative with respect to . Underthe U(1) gauge transformations, on the other hand, wefind that

= , E = E, = , B = B a

, = + , A = A , (3.22)where = . Then, the gauge transformations of thewhole group U(1) Diff(M, F) will be the linear com-bination of the above two. Out of the six unknowns, onecan construct three gauge-invariant quantities [16],

= 1a

a 1

a 2 Ha , = +

Ha

a , = A +

a

Aa a

, (3.23)

where E B. For the background, we have chosenthe gauge (3.2), for which Eq.(3.23) reduces to

= 1a

a ,

= Ha

a,

= A +

A

a

a , ( = 0). (3.24)

Then, for the general perturbations (3.20), we have

Kij = a

ij ,ij+ H (2 + ) ij 2E,ij,Rij = ,ij + 2ij. (3.25)

Thus, to first-order the Hamiltonian and momentum con-straints become, respectively,

d3x

2 1

2

3 1H3( + H) 2

4Ga2

= 0, (3.26)

(3 1) + H+ (1 )2 1a

= 8Gaq+ (), (3.27)

where

12

Jt, Ji 1a2

q,i, (3.28)

q,i = ijq,j, and () is an integration function. In GR,it is usually set to zero [24]. However, in the presentcase, since = (), another interesting choice is () =(31)H, which will cancel the second term in the left-hand side of Eq.(3.27).

On the other hand, the linearized equations (A.4) and(A.5) reduce, respectively, to

2H2 + 1 23 + H 1a

2

a

= 8Ga3J, (3.29)

2 = 2Ga2JA, (3.30)

while the linearly perturbed dynamical equations can bedivided into the trace and traceless parts. The trace partreads,

+ 2H + H + 2H + H2 1

32

+ 2H 2

3(3 1)

1 +

1a2

2 +2a4

4

2

ga3(3 1) 2A

2E 3 3/2

ga

3(3 1)2A3 2E+ 3 + A+

2

3(3 1)a 2

A A + H+

1(3 1)a

2

+ H = 8Ga23 1 p, (3.31)

where

1 8g2 + 3g32

, 2 8g7 3g84

,

p P+ 23

p2E, GR + 2pE,

ij =1

a2 P+ 2 pij + ,,, = ,ij 1

3ij2. (3.32)

The traceless part is given by

+ + 2H + 1a2

1 +

2a2

2

2

1a

A A H = 8Ga2GR + G(),

(3.33)


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where G() is another integration function. Again, inGR it is set to zero [24]. But, similar to the momen-tum constraint (3.27), one can also choose G() = sothat the second term in the left-hand side of the aboveequation is canceled.

The conservation laws (A.13) and (A.14) to first orderare given, respectively, by,

d3x2a + 3H (P+ ) + 2pH2E+ ( + p)

2E 3 J AJA + 3HJA 3HJAA A+ AJA

3 2E = 0, (3.34)q + 3Hq ap 2a

32GR = I(), (3.35)

where I() is another integration function of only. InGR, it is usually chosen to be zero [24].

This completes the general description of linear scalarperturbations in the flat FRW background in the frame-work of the HMT setup with any given [15], withoutchoosing any specific gauge for the linear perturbations.However, before closing this section, let us consider somepossible gauges.

C. Gauge Choices

To consider the gauge choices, we first note that

= (, x), 0 = 0().

Then, from Eqs.(3.21) and (3.22) one immediately finds

that the spatially flat gauge = 0 = E [24] is impossiblein the HMT setup. Since = (), a natural gauge forthe time sector is

= 0, (3.36)

for which 0 is uniquely fixed up to a constant C,

0() =1

a()

a()()d +

C

a(). (3.37)

Then, depending on the choices of and , we can have

various different gauges.

1. Longitudinal Gauge

The longitudinal gauge in GR is defined as [24],

E = 0 = B, (3.38)

which is impossible in the HL theory without the U(1)symmetry [23]. However, with the U(1) gauge freedom,

Eqs.(3.21) and (3.22) show that now this gauge becomespossible with the choice,

= E, = a(B E + 0), (3.39)

where 0 is given by Eq.(3.37). It should be noted thatthis gauge is fundamentally different from that given inGR [24], because now we also have = 0.

2. Synchronous Gauge

In GR, the synchronous gauge is defined as [24],

= 0 = B. (3.40)

However, this is already implied in the above longitudinalgauge. With the extra U(1) gauge freedom , we canfurther require,

(i)

= 0, or (ii)

A = 0. (3.41)

The former will be referred to as the Newtonian syn-chronous gauge, while the latter the Maxwell synchronousgauge. For the Newtonian synchronous gauge, and aregiven by

(, x) =

B + 0 +

1

a

0 d + D(x),

(, x) = 0 , (3.42)

where D(x) is an arbitrary function of xi only. For theMaxwell synchronous gauge, they are given by

(, x) = A 0Ad + D1(x),(, x) =

B + 0

a

d + D2(x), (3.43)

where D1(x) and D2(x) are other two arbitrary functionsofxi only. From the above one can see that none of themcan fix the gauge uniquely.

3. Quasilongitudinal Gauge

In [14, 16], the gauge,

= E = = 0, (3.44)was used. With this gauge, we find that

(, x) = E(, x), (, x) = 0 (, x), (3.45)

and 0 is given by Eq.(3.37). Thus, in this case the gaugefreedom of Eqs.(3.21) and (3.22) are also uniquely deter-mined up to the constant C, similar to the longitudinalgauge (3.38).


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Note that instead of choosing the above gauge, one canalso choose

= E = A = 0, (3.46)for which we have

(, x) = E(, x),

(, x) = A 0A d + D3(x), (3.47)where D3(x) is another integration function of x

i only.Thus, unlike the gauge (3.44), now the gauge is fixed onlyup to a constant C and an arbitrary function D3(x).

To be distinguish from the one defined in the case with-out the U(1) symmetry [23], we shall refer the gauge(3.44) to as the Newtonian quasilongitudinal gauge, andEq.(3.46) the Maxwell quasilongitudinal gauge.

IV. INFLATION OF A SCALAR FIELD

In Appendix B, we construct the action for a singlescalar field. In this section, we apply the perturbationsdeveloped in Sec. III to study inflationary models of sucha scalar field. To this goal, let us first consider the slow-roll conditions.

A. Slow-Roll Inflation

For the flat FRW background, we find that

Jt =

2f12 2 + V() 2,Ji = J = JA = 0,

ij = f a2

1

22 V()

ij a2pij , (4.1)

where V() V()/f. Then, Eqs.(3.6) - (3.8) and (3.10)yield g = 0 and

H2 =8G

3

1

22

+ V()

+

3, (4.2)

where

G 2f G3 1 ,

2

3 1 . (4.3)

On the other hand, Eq.(B.17) reduces to,

+ 3H + V = 0. (4.4)

Eqs.(4.2) and (4.4) are identical to these given in GR [24],

if one identifies G and to the Newtonian and cosmologi-cal constants, respectively. As a result, all the conditions

for inflationary models obtained in GR are equally ap-plicable to the current case, as long as the background isconcerned. In particular, the slow-roll conditions,

V, |V| 1, (4.5)need to be imposed in order to get enough e-fold, where

V

M

2

pl2 V

V2 = 3 1

2fV,

V M2pl

V

V

=

3 12f

V, (4.6)

with M2pl 1/(8G), and V and V are the ones definedin GR [25].

However, due to the presence of high-order spatialderivatives, the perturbations will be dramatically dif-ferent, as to be shown below.

B. Linear Perturbations

In this section, in order for the formulas developedbelow to be applicable to as many cases as possible, weshall not restrict ourselves to any specific gauge. Then,to first-order we find that

= f

a2

+ V4a4

4 + V,

q =f

a, J A =

2c1a2

2,

J =1

a3

c1 + c1H f

2 + c1

2

,

p = fa2 ( ) V,

= 2pE, GR = 0. (4.7)Hence, Eqs. (3.26) - (3.30) read, respectively,

d3x

2 1

2

3 1H3( + H) 2

=

d3x4G

f

+V4a2

4

+ a2V, (4.8)(3 1) + (1 )2

1a

= 8Gf, (4.9)

2H + 1 3 1a

2

a

= 8G

c1 + c1H f

+ c1

, (4.10)

= 4Gc1. (4.11)


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8

Note that in writing Eq.(4.9), we had chosen () =(3 1)H. It is also interesting to note that, unlikethe case without the U(1) symmetry [34], now the metricperturbation is proportional to . It is this differencethat leads to a master equation, as to be shown below.Without the U(1) symmetry, this is in general impossible[34].

The trace and traceless parts of dynamical equation

read, respectively,

+ 2H + H + 2H + H2 13

2

+ 2H 2

3(3 1)

1 +

1a2

2 +2a4

4

2

+2

3(3 1)a 2

A A + H+

1(3 1)a

2

+ H=

8G

3

1

f( ) a2V

, (4.12)

+ + 2H + 1a2

1 +

2a2

2

2

1a

A A H = 0, (4.13)

where in writing Eq.(4.13) we had set G() = . Theenergy conservation law now takes the form,

d3xa2

f + 2Hf + a2V + 2a2V

f

2E 3

A

(ac1)

a

=

d3x 4

V4 +

V4 V4H

, (4.14)

The momentum conservation is identically satisfied,while the Klein-Gordon equation becomes

f

+ 2H 3 + 2

+ 2a2V +

a2V 2 =

2

a

2A (c1 c2) c1 + f + c1A

+ 2V1 V2 + V4a2 2 V6a4 42, (4.15)which can be rewritten as a perturbed energy balanceequation,

+ 3H( + p) ( + p)

3 2 1f

2(v + B)

=

1

f( + p)QHMT, (4.16)

where

QHMT =V4

a224 +

1

2V1 2 V6

a44

1a2

2V2 + V

4 +

V4H

2

2

+

1

a22 A (c1 c2) c1

+f + c1A

,

q a( + p)(v + B). (4.17)

C. Uniform Density Perturbation

Under the gauge transformations (3.21) and (3.22), and transforma, respectively, as

= 0,

= 0. (4.18)

Therefore, the quantity defined by

+ H

, (4.19)

is gauge-invariant. In GR it is often referred to as thegauge-invariant perturbation on uniform-density hyper-surfaces. It can be shown that it obeys the evolutionequation,

= Hpnad + p

+1

3

QHMT 2

2

f(v + B)

, (4.20)

where the non-adiabatic pressure perturbation is definedas

pnad p p = pGRnad + pHMTnad , (4.21)with,

pGRnad 2

3a2

2 +

H

( )

( H)

= 2aV

3H

a

( )

a

,

pHMTnad

1 +2

H

V43a4

4. (4.22)

Note that Eq.(4.20) is quite similar to that of the casewithout the U(1) symmetry [34], and the only differenceis the inclusion of the U(1) gauge field A and Newtonianprepotential in QHMT, as one can see from Eq.(4.17)given above and Eq.(4.3) given in [34]. But, these termsvanish in the super-horizon region. As a result, all theconclusions obtained in [34] in this region are equallyapplicable to the present case. In particular, the pertur-bations in this region are adiabatic during the slow-rollinflation, as in GR.


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9

D. Comoving Curvature Perturbation

On the other hand, the comoving curvature perturba-tion, defined by

R = + H

, (4.23)

is gauge-invariant even in the HL theory. From its defi-

nition, it can be shown that R satisfies the equation,

R = HS+ H H2

+ + H, (4.24)

where the dimensionless intrinsic entropy perturbation Sis defined as

S

H2

= 3H2V

pGRnad, (4.25)

where to get the last step Eq.(4.22) was used. In terms ofR the super-momentum constraint (4.9) can be writtenin the form,

R = HS+ 13 1 2 a , (4.26)

which reduces to R = HSon all scales in the relativisticlimit 1. Thus, in the slow-roll approximations andneglecting the spatial gradients on large scales, we obtainthe same conclusion as that given in [34], namely, thecomoving curvature perturbation has two modes on largescales, a constant mode and a rapidly decaying mode,given by

R C1 + C2

d

a2. (4.27)

In addition, unlike that in GR where the local Hamil-

tonian constraint enforces adiabaticity on large scales,in HMT setup it is the slow-roll evolution ( = 0, or, = H) that leads to rapidly decaying entropy per-turbations at late times.

Note that we could also find the first-order equationfor Sby using the Klein-Gordon equation, which can bewritten in the form,

S +

2

+ H

S= 1f

[f3H]

+2

(1 + 2V1) 2f

3 1 (

a)

+

2

a2A (c1 c2) + c1A c1

2 4

a2

V2 + V4

+ V6a2

2

. (4.28)

Thus, in the large scales (neglecting all the spatial gradi-ent terms), the first term in the right-hand side is func-tion of only (Recall that = ()). Then, the corre-sponding entropy equation depends only on time on theselarge-scales.

V. SCALAR PERTURBATIONS IN SUB- ANDSUPER-HORIZON SCALES

So far, we have not chosen any gauge. In this section,we shall restrict ourselves to the Newtonian quasilongi-tudinal gauge defined by Eq.(3.44) in Sec. III.C, i.e.,

= E = = 0. (5.1)

Then, Eqs.(4.8) - (4.15) can be cast in the forms ofEqs.(B.19) - (B.26). From Eqs.(B.21), (B.23), (B.25),we can express , B and A in terms of , and thensubmit them into Eq.(B.26), we obtain a master equationfor , which can be written as

+ P + Q = F2, (5.2)where

0 f + 4Gc12

|c2|,

P 10

(0 + 2H0) ,

Q 10a2V + 4Gc1c1 2|c2| 8G 1 f2f c1

4Gc1a2V

3 +c1

f|c2| 1|c2|

,

F 10

1 + 2V1 + 2A(c1

c2) 4Gc12

1 A 2

a2

V2 + V

4 + 2G1c1

2

2

2a4

V6 + 2G2c12

4

, (5.3)

with

c2 1

1 3 . (5.4)Setting

= exp

1

2

Pd

u, (5.5)

one can write Eq.(5.2) in the momentum space in theform,

uk + 2kuk = 0, (5.6)

where

2k = k2Fk 1

4

2P + P2 4Q ,

Fk 10

1 + 2V1 + 2A(c1

c2) 4Gc12

1 A+

2k2

a2

V2 + V4 + 2G1c1

2

2k4

a4

V6 + 2G2c12

. (5.7)


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10

Note that the above hold only for = 1. When = 1,we have a first-order equation for

+

c1(c1 f) = 0, (5.8)

which has the general solution,

= expc1 f c1d 1(x), ( = 1), (5.9)where 1(x) is an arbitrary function of x only. Since inthis paper we are mainly interested in the case = 1, inthe following we shall not consider this case further.

Also, for the field to be stable in the UV regime, thecondition

V6 + 2G2c12 < 0, (5.10)

has to be satisfied. To study Eq.(5.6) further, we considerthe sub- and super-horizon scales, separately.

A. Sub-Horizon Scales

In this region, we have k H, and the dispersionrelation reduces to,

2k 2k6

0a4

V6 + 2G2c12

. (5.11)

With the extreme slow-roll condition, we have a 1Hand H, V6, c1 Constants. Then, from Eq. (5.6) wefind that

uk

eik . (5.12)

Unlike the case without the U(1) symmetry [34], themetric perturbations and B now oscillate with thesame frequency as , as one can see from Eqs.(B.21)and (B.23). Therefore, they are always coupled to thescalar field modes.

B. Super-Horizon Scales

In this region, we have k H, and to the order of k2,we find that

2k k2

01 + 2V1 + 2A(c1 c2) 4Gc12 1 A

+Q 2P + P2

4. (5.13)

In the extreme slow-roll and massless limit ( 0 V, V 0), we obtain the following solution

uk = D1H

1 +

k22

20

1 + 2V1

+ 2A(c1 c2) 4Gc12

1 A

+ D22

1 k

22

100

1 + 2V1

+ 2A(c1 c2) 4Gc12

1 A

D1a + D22, (5.14)where the first term represents a constant perturbation,while the second term represents a decaying mode. Then,we find that

D1 D2H3, 4Gc1,k2B 12Gc1|c|2 D2

2. (5.15)

In terms of the gauge-invariant quantities (3.24), we ob-tain

k = k HBk,k = HBk + Bk,k k = H2k22

H2k222 1

k

+ H

Ak Ak

. (5.16)

Thus, like in the case without the U(1) symmetry [34],the dynamical evolution now leads to = 0 at latetimes ( 0).

VI. POWER SPECTRA AND INDICES OF

SCALAR AND TENSOR PERTURBATIONS

To calculate the spectra and indices of scalar and ten-sor perturbations with the slow-roll approximations, weshall use the uniform approximation, proposed recentlyin [26], and applied to the studies of tensor perturbationsin the HL theory without the U(1) symmetry in [35, 36].We shall closely follow the treatment presented in [36].In particular, for perturbations given by,

vk = [g(k, ) + q()]vk, (6.1)

where q() = 1/42, and vk is the canonically normal-ized field, the corresponding power spectrum and index

at leading order of the uniform approximation are givenas [36],

Pv(k)|k0 k3

22|vk|2k0

= limk0

k3 exp

2D(k, )

42a2

g(k, ), (6.2)

nv 1 d ln Pvd ln k

k0

, (6.3)


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11

where

D(k, ) (k)

g(k, )d, (6.4)

and (k) denotes the turning point g(k, ) = 0. Notethat in writing the above expressions, we assumed thatthere is only one turning point, that is, we consider onlythe case where g(k, ) = 0 has only one real root. For

detail, see [36]. In the following, we shall apply the aboveto the cases of scalar and tensor perturbations.

A. Power Spectrum and Index of ScalarPerturbations

With the help of the master equation (5.2) and thedefinition of gauge-invariant R in (4.23), the second orderaction reads,

S(2) =1

2

dd3xa2h2

0R2 4R2 1(iR)2

2(2R)2 3(i2R)2, (6.5)where

0 = f + 4Gc21/|c2|,

1

1 + 2V1 + 2A(c1 c2) 4Gc21(1 A)

,

2 2a2

V2 + V4 + 2Gc

211

,

3 2a4

V6 + 2Gc212

,

4 0Q 0h2

h2+

a20hh

a2h2

h 4Gc1 + H 1 = R . (6.6)After introducing the variable

v zR, z2 a2h20, (6.7)the action is normalized to

S(2) =1

2

dd3x

(v)2 1

0(iv)

2 m2effv2

12

dd3x

20

(2v)2 +30

(i2v)2

.

(6.8)

Here m2eff is defined to be

m2effz

z 4

0. (6.9)

Going through the quantization procedure as describedin Appendix C, the classical equation of motion for modefunctions vk are

vk +

2k + m2eff

vk = 0, (6.10)

where

2k =k2

0

1 + 2k

2 + 3k4

. (6.11)

Looking at the expressions of the coefficients, wesee that they contain terms of c1, c2, V1, V2, V4, V6 and A.Now go back to the Lagrangian describing the inflaton, Eqs.(B.2) and (B.9), one can see that V1, V2, V4, V6 all

stem from the potential term V, while c1 and c2 appearthrough the first line of (B.9), which can also be taken asa potential term. (Note that the second and the thirdline of (B.9) correspond to modifications of dynamicalcoupling terms due to the presence of .). Therefore, wecould assign their respective slow-roll parameters de-scribing their time evolution during inflation in a mannersimilar to V. However, unlike V, which appears in thebackground equation (4.2), these potential terms arenot constrained by the background equations. As an ap-proximation, we assume here that the time dependenceof c1, c2, V1, V2, V4, V6 and A are at least second order interms of the slow roll parameters. Since we only consider

the first order approximations in this paper, they can betaken as constant throughout inflation. This also leavesd0/d dc1/d = 0.

With the above assumptions, it can be shown that h2

relating and R is of order O(V). In fact, from itsdefinition,

h2 =

4Gc1 +

H

2=

H

2 1 +

c1

2M2plH

2

= 2M2plV

1 +

c1

2M2plH

2. (6.12)

Sincec1

2M2plH=

c12Mpl

12Mpl

H=

c12Mpl

V 1,

(6.13)where |c1| M Mpl [17], we have, to first order ofthe slow-roll parameters,

h2 2M2plV. (6.14)On the other hand, a() (1 + V)/(H), which leadsto

m2eff

2 3V + 9V2

+m2

2. (6.15)

Here the first term comes from z/z and is the same asthat from GR under the above assumptions, whereas thesecond term introduces new effects,

m2 10

3 (0 1) V +

2f2

1 60

V

2 1

c1

2Mpl

V

. (6.16)


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12

We see that the these are in general functions of. Sinceits the time-dependence of meff which breaks the exactscale-invariance, we would expect that observations onthe power index, which will be derived below, place con-straints on the value of .

The function g(k, ) defined through Eqs.(6.1) and(6.10) is now given by

g(k, ) = k2

y2a20 a2y2 + a4y4 + a6y6 , (6.17)

where

y k,a20

1

4 m2eff2 =

9

4 3V + 9V + m2,

a2 10

1 + 2V1 + 2A(c

1 + c2) 4Gc21(1 A)

,

a4 10

2H2(1 2V)

V2 + V4 + 2Gc

211

,

a6

1

0 2H4(1 4V) V6 + 2Gc212 . (6.18)Thus the power spectrum of R is given by,

PR(k) =H2(1 2V)|y0|3

420a0h2

2

e

2a0exp

a4y40 + a6y

60

3a0

lim

y0

y

y0

32a0, (6.19)

where y0 is defined to be the turning point of g(k, ),namely,

a6y60 + a4y

40 + a2y

20 a20 = 0. (6.20)

Clearly, y0 is independent of k, as ans are. Then, wefind

nR 1 = 2V 6V 23

m2. (6.21)

Due to correction term m2, the spectrum index of thescalar perturbation does not reproduce the GR value ingeneral. In particular, we see that it depends on the value1/( 1) (Note that 0 is also a function of 1). Onemay worry that in the relativistic limit at low energy 1 these terms will diverge, hence breaking the near-scale-invariance of the spectrum. However, during inflation,

we are in a region where UV physics dominates, thus thevalue of is expected far away from its relativistic fixedpoint at that time.

We note that even when the index can restore thestandard value in GR,3 it is a consequence of our as-sumption that all the potential terms c1, c2, V1, V2, V4 and

3 Mathematically, in the limit c1 = 0, 0 1 and f2/30(1) 1, the standard result can be restored.

V6 are time-independent. Comparing the definition ofz given here in Eq.(6.7) with the one given in GR, wecan see some extra terms of c1 appear. By our assump-tion, dc1/d = 0, this makes d0/d = 0 and dh

2/d d( /H)/d, which leaves the term z/z exactly the sameas that of a single field in GR. Further more, in themodified dispersion relation (6.11), the term 1/0 cor-responds to the relativistic case, and the ones 2/0 and

3/0 are induced by Lorentz-symmetry-breaking effects,which are assumed to be time-independent. If, how-ever, these potential terms are evolving with time dur-ing inflation, one needs to take into account of the time-dependence of the dispersion relation and of the varyingeffective mass [37]. The same arguments also apply tothe studies of tensor spectrum and index.

We would also like to note that, the exact form ofscale-dependence of the scalar spectrum depends on theinstant when its evaluated [38], and can receive furthercorrections when we incorporate a second order uniformapproximation [26]. Whats more, from an observationalpoint of view, as long as the scale-dependence is not bro-ken severely, the connection between tensor-scalar-ratioand slow-roll parameters is more important than the tiltitself.

Setting the slow roll parameters to zero exactly, thepower spectrum given above can be put in the simpleform,

PR(k) =4H4|y0|3

32e30 2

exp

2

9

a4y

40 + a6y

60

. (6.22)

In the relativistic limit (a2 = 0 = 1, a4 = a6 = c1 = 0,and gs = 0, (s = 2,..., 8), this yields the well-known result

obtained in GR [25],

PGRR =18

e3

H22

2, (6.23)

except for the factor 18/e3 0.896. This difference inmagnitude is due to the way we normalize the powerspectrum in the uniform approximations. As shown later,the same factor also appears in the expression for thepower spectrum of tensor perturbations, so that the ratioof them does not depend on this factor.

To estimate the effect from higher curvature terms onthe power spectrum, let us first write the dispersion re-lation (6.11) in the form,

2k b1k2 + b2k4

a2M2A+ b3

k6

a4M4B, (6.24)

where [16, 17]

MA |g3|1/2Mpl, MB |g8|1/4Mpl, (6.25)and

b1

1 + 2V1 + 2A(c1 + c2) 4Gc21(1 A)

/0,


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13

b2 2 [2 + 4 + 23] /0,b3 2 [6 + 78] /0, (6.26)

with

2 V2M2A, 4 V4M2A, 6 V6M4B,23 c

21M

2A

2M4pl(8g2 + 3g3),

78 c21M

4B

M6pl(8g7 3g8). (6.27)

Since g2 and g3 all both the coefficients of the fourth-order derivative terms, as one can see from Eq.(2.4), itis quite reasonable to assume that g2 and g3 are in thesame order, g3/g2 O(1). Similarly, one can argue thatgs/g4 O(1) for s = 5, 6, 7, 8, as all of these terms arethe coefficients of the sixth-order derivative terms. Forthe sake of simplicity, we further assume MA MB =M. Taking c1 c2 M, from Eq.(1.8) we find M Mpl|c|1/2 [17]. Then, from Eq.(6.6) we obtain

0 f() + 12

M

4/5 O(1), (6.28)as f() O(1). To determine the scales ofVn, we assumethat bn defined in Eq.(6.26) are all of order 1, i.e.,

bn O(1), (n = 1, 2, 3), (6.29)which is a reasonable assumption, considering the phys-ical meanings of the energy scales MA and MB. In fact,one can define MA and MB so that b2 = b3 = 1 precisely,as originally defined in [17]. To have b1 = 1, one canproperly choose V1. On the other hand, since A is un-

determined, and for the sake of simplicity, we further setA = 0.With all the above assumptions, we find that the func-

tion g(k, ) now reads,

g(k, ) =k2

y2

9

4 y2 1 + HLy2 + 2HLy4 ,

HL H2

M2. (6.30)

Thus, depending on the energy scale H when inflationoccurs, one can have different turning point y0. In thefollowing we consider only two limits, HL 1 and HL 1. In addition, in writing Eq.(6.30) we have set bn = 1 =0 precisely. General expressions without setting bn = 1can be found in Appendix D.

1. HL 1

When HL 1, to its second order, we find that

y20 9

4

1 9

4HL +

81

162HL

, (6.31)

for which the power spectrum is given by,

PR(k) PGRR

1 94

HL +729

1282HL

.

(6.32)

It is interesting to note that the condition HL 1 isequivalent to

V() 32

(3 1)

Mpl

2M4pl, (HL 1). (6.33)

2. HL 1

When HL 1, to find the turning point y0, we firstwrite g(k, ) given by Eq.(6.30) in the form,

g(k, ) = 2HL

k2

y2

94

2HL

y2 2HL

+ HLy2 + y4

, (6.34)

where HL

1/HL

1. Then we find the perturbativesolution

y20

3HL2

2/3 1 1

3

4HL

9

1/3 2

9

4HL

9

2/3,

(6.35)for which the power spectrum takes the form,

PR(k) PGRR (k)4HL

e

9

1 1

2

4HL

9

1/3 .

(6.36)

Thus, if the inflation happened way above the scale M,

the spectrum will be suppressed by the factor M2/H

2

,comparing with that of GR.

B. Power Spectrum and Index of Tensor

Perturbations

The tensor perturbations can be written in the form[23, 36]

gij = a2 (ij + hij) , (6.37)

where hij is traceless and transverse, i.e., hi i = 0 =jhij . For a single scalar inflaton, the anisotropic stress

is zero, so the tensor perturbations are source-free. In theADM formalism, with the results of constraint equationsderived in Section II, it can be shown that the secondorder action is given by

S(2) =1

2

dd3x

2a2

2

(hij)

2 (1 A) (chij)2

g32a2

2hij2 g8

4a4

c2hij2

. (6.38)


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14

Defining the following expansion in the momentum space[25],

hij =

d3k

(2)3

s=+,

sij(k)hsk()e

ikx, (6.39)

where ii = kiij = 0 and sij(k)s

ij(k) = 2ss , the aboveaction becomes

S(2) =

s=+,

dd3k

a2

22

(h

sk )

2 (1 A)k2(hsk)2

g3k4

2a2(hsk)

2 g8k6

4a4(hsk)

2

. (6.40)

To make the action canonically normalized, we introducevsk

by

vsk

ahsk

. (6.41)

Then, the action (6.40) becomes

S(2) =1

2

s=+,

dd3k

(vsk )

2 (2k + m2eff)(vsk)2

,

(6.42)but now with

2k = (1 A)k2 +g3k4

2a2+

g8k6

4a4,

m2eff = a

a. (6.43)

One can see that each spin state of the tensor perturba-tion acts like a scalar. After the quantization procedureprescribed in Appendix C, the classical equation of mo-tion for the mode functions again read

vk + (2k + m

2eff)vk = 0, (6.44)

where in writing the above equation, we had droppedthe super indices s, and 2k and m

2eff are now defined

by Eq.(6.43). From the above, we can directly read offg(k, ) for tensor perturbations,

g(k, ) =k2

y2

9

4(1 + 2V)

(1 A)y2

+g3H

2

2(1

2V)y

4 +g8H

4

4(1

4V)y

6,y k. (6.45)

Thus, its turning point y20 (k)2 satisfies the cubicequation

9

4(1 + 2V) = ( 1 A)y20 +

g3H2

2(1 2V)y40

+g8H4

4(1 4V)y60. (6.46)

Then the dimensionless spectrum and index for the ten-sor perturbations can be defined as [25],

PT(k)|k0 4 k3

22|vk|22a2

k0

, (6.47)

nT d ln P2T

d ln k

k0

. (6.48)

Here the factor of 4 accounts for the two spin states.Again, assuming that the gauge field A is constant dur-

ing inflation,

PT(k) =16H2|y0|3

32e32

exp

2H2y4092

g3 + g8y

20

H2

2

, (6.49)

nT = 2V. (6.50)In the relativistic limit, Eq.(6.49) yields the well-known

results obtained in GR [25]

PGRT (k) =18

e32H2

2M2pl. (6.51)

Because of the normalization of the power spectrum inthe uniform approximation, a difference of a factor 18/e3

also appears in the tensor perturbations.To study the effect of high order curvature terms, fol-

lowing what we did for the scalar perturbations, we con-sider the two cases HL 1 and HL 1, separately.

1. HL 1

In this case, the power spectrum (6.49) takes the form

PT(k) PGRT (k)

1 92

HL +729

322HL

. (6.52)

Then, from Eqs.(6.32) and (6.52), we find that the scalar-tensor ratio is given by

r PT(k)PR(k)

16V

1 94

HL +2187

1282HL

.(6.53)

For the general case, see Eq.(D.2).

2. HL 1

When HL 1, from Eq.(6.45) we find that y20 is givenby,

y20

3HL4

2/3 1 1

6

16HL

9

1/3 1

18

16HL

9

2/3 , (6.54)


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15

and the power spectrum takes the form,

PT(k) PGRT (k)2e1/2HL

9

1 1

4

16HL

9

1/3 .

(6.55)

Then, the combination of it with Eq.(6.36) yields

r 8V1 + 2 22/34

4HL9

1/3 . (6.56)

For arbitrary bn, see Eq.(D.4).

VII. CONCLUSIONS

In this paper, we have studied inflation driven by asingle scalar field in the HMT setup [13] with the pro-

jectability condition and an arbitrary coupling constant [15]. Because of the particular coupling of matter fields(3.12), in Sec. III.A we have been able to show that theFRW universe is necessarily flat for (multi-) scalar, vectorand fermionic fields. It is quite reasonable to argue thatthis should be true for all the viable cosmological models4. Therefore, the HMT setup provides a built-in mecha-nism to solve the flatness problem. However, to solve thehorizon problem inflation may or may not be needed [29],although to solve other problems, such as monopole anddomain walls [25], inflation with the slow-roll conditionsseems still required.

After first developing the general formulas of lin-ear scalar perturbations without specifying a particulargauge and matter fields in Sec. III.B, we have investi-gated several possible gauge choices in Sec. III.C, and

found that, unlike the case without the U(1) symmetry[23], now various gauge choices become possible, includ-ing the generalized longitudinal gauge, synchronousgauge, and quasilongitudinal gauge.

Applied the general formulas to a single scalar field,in Sec. IV we have first shown that the flat FRW uni-verse has the same dynamics as that given in GR. As aresult, all the results obtained in GR are also applicablehere in the HMT setup, as far as only the background isconcerned, including the slow roll conditions. Then, wehave found that in the super-horizon regions, the pertur-bations become adiabatic, and the comoving curvatureperturbation is constant, though for a different reason

from that in GR.In Sec. V, we have shown that a master equation [cf.Eq.(5.2)] for the scalar perturbations exists, in contrastto the case without the U(1) symmetry [23]. In addition,we have also shown explicitly that in the sub-horizon

4 It should be noted that this conclusion is based on the couplingof matter fields to the gauge field A and Newtonian prepotential, proposed in [15].

regions, the metric and scalar field are tightly coupledand have the same oscillating frequencies.

We have also calculated the power spectra and spec-trum indices of both the scalar and tensor perturbationsin the slow-roll approximations (Sec. VI), by using theuniform approximations [26], and expressed them explic-itly in terms of the slow roll parameters and the couplingconstants of high order curvature terms. Weve found

that, with some reasonable conditions on the couplingcoefficients c1,2 and Vi [cf. the discussions presented afterEq.(6.23) and Eqs.(6.28) and (6.29)], the spectrum indexof the tensor perturbations is the same as the value givenin GR, whereas the index of the scalar perturbation is afunction of and can be different from the standard GRvalue. The power spectra are in general different fromthose of GR. For more general cases, the power spectrumPR and the ratio r are given in Appendix D. We havealso found that inflation in the HMT setup produces allthe observational features of the universe [40]. Therefore,as far as slow-roll inflation is concerned, the HL theoryare consistent with observations.

Acknowledgements: The work of AW was sup-ported in part by DOE Grant, DE-FG02-10ER41692 andNNSFC 11075141; and QW was supported in part byNNSFC grant, 11047008.

Appendix A: Field Equations

For the action (2.1), the Hamiltonian and momentumconstraints are given, respectively, by,

d3x

g

LK + LV Gijij

1

2

= 8G

d3x

g Jt, (A.1)

j

ij Gij

1 gij2 = 8GJi, (A.2)where

Jt 2 (NLM)N

, Ji NLMNi

,

ij (NLK)gij

= Kij + Kg ij. (A.3)

Variation of the action (2.1) with respect to and A

yield,

Gij

Kij + ij

+

1 K+ = 8GJ, (A.4)

R 2g = 8GJA, (A.5)

where

J LM

, JA 2 (NLM)A

. (A.6)


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16

On the other hand, the dynamical equations now read5,

1

N

g

g

ij Gij 1 gij,t

= 2 K2ij + 2KKij 2N

k(ikNj)

+ kNk

N

ij

(1 )Fk

g

ij 21 K+ ij + Kij+ 2(1 )(ij) (K+ )

+1

N(iNj)

+1

2

LK + L + LA + L

gij

+ Fij + Fij + FijA + 8G

ij , (A.7)

where

K2

ij KilKjl , f(ij) (fij + fji) /2, and

Fij 1g

gLVgij

=8

s=0

gsnS (Fs)

ij ,

Fij =

3n=1

Fij(,n),

Fi =

K+ 2i + N

i

N,

FijA =1

N

ARij ij gij

A

,

(A.8)

with nS

= (2, 0, 2, 2, 4, 4, 4, 4, 4). (Fs)ij andFij(,n) are given by [14, 23],

(F0)ij = 12

gij,

(F1)ij = 12

gijR + Rij ,

(F2)ij = 12

gijR2 + 2RRij 2(ij)R

+2gij2R,(F3)ij = 1

2gijRmnR

mn + 2RikRkj 2k(iRj)k

+2R

ij+ g

ijmnRmn,

(F4)ij = 12

gijR3 + 3R2Rij 3(ij)R2

+3gij2R2,

5 Note that the dynamical equations given here differ from thosegiven in [16] because here we took Ni as the fundamental variableinstead ofNi as what we did in [16]. They are both self-consistentif Ni and Ni are used consistently.

(F5)ij = 12

gijRRmnRmn + RijR

mnRmn

+2RRkiRkj (ij) (RmnRmn)

2n(iRRj)n + gij2 (RmnRmn)+2 (RRij) + gijmn (RRmn) ,

(F6)ij = 12

gijRmn R

npR

pm + 3R

mnRniRmj

+ 322 RinRnj + 32 gijkl RknRln

3k(i

Rj)nRnk

,

(F7)ij = 12

gij(R)2 + (iR) (jR) 2Rij2R+2(ij)2R 2gij4R,

(F8)ij = 12

gij (pRmn) (pRmn) 4Rij+ (iRmn) (jRmn) + 2 (pRin)pRnj

+2n(i2Rj)n + 2n

Rnm(iRmj)

2n Rm(ji)Rmn 2n Rm(inRmj)gijnm2Rmn, (A.9)

and

Fij(,1) =1

2

2K+ 2

Rij

2

2Kjk + jk

Rik

2

2Kik + ik

Rjk

2g R

2Kij + ij

,

Fij(,2) =

1

2k2Gk(ij) Gij 2NkN + k+

2

NGk(ikNj),

Fij(,3) =1

2

2k(ifj)k 2fij klfkl gij ,

(A.10)

where

fij =

2Kij + ij

12

2K+ 2

gij

.

(A.11)

The stress 3-tensor ij is defined as

ij =2

g

gLM

gij. (A.12)

The conservation laws of energy and momentum ofmatter fields read, respectively,

d3x

g

gkl

kl 1g

gJt,t

+2Nk

N

g

gJk,t


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17

2 J AN

g

(

gJA),t

= 0, (A.13)

kik 1N

g

(

gJi),t Jk

N(kNi iNk)

NiN

kJk + Ji JA2N

iA = 0. (A.14)

Appendix B: Scalar Fields

How matter couples with gravity in the HMT setuphas not yet been worked out in the general case. In thisAppendix, we shall consider the coupling of a scalar field with gravity by the prescription given in [15]. Thecase with detailed balance condition softly breaking wasstudied in [33].

A. Coupling of a Scalar Field

When the scalar field coupled only with N, Ni, gij,the most general action takes the form [28, 34],

S(0)

N, Ni, gij;

=

dtd3xN

gL(0)

N, Ni, gij;

,

(B.1)where

L(0) =f()

2N2

Nii2

V,

V = V () +

1

2+ V1 ()

()2 + V2 () P21

+V3 () P31 + V4 () P2 + V5 () ()2P2+V6()

P1

P2, (B.2)

with V() and Vn() being arbitrary functions of , and

Pn n. (B.3)Note that in the kinetic term we added a factor f(),which is an arbitrary function of , subjected to the re-quirements: (i) The scalar field must be ghost-free in allthe energy scales, including the UV and IR. (ii) In the IRlimit, the scalar field has a well-defined velocity, whichshould be equal or very closed to its relativistic value.(iii) The stability condition in the IR requires [33],

f() > 0. (B.4)

To couple with the gauge field A and the Newtonianprepotential , we make the replacement [15],

S(0)

N, Ni, gij ;

S

N, Ni, gij, A , ;

, (B.5)

where

S

N, Ni, gij, A , ;

SA (, A)

+ S(0)

N,

Ni + Ni, gij ; , (B.6)

with

SA

dtd3x

g

c1() + c2()2A A.

(B.7)

Thus, the action can be cast in the form,

S = dtd3xNgL, (B.8)where

L = L(0) + L(A,) ,L(A,) =

A AN

c1 + c22

fN

Niikk

+f

2

kk2, (B.9)with

L(0) given by Eq.(B.2). Then, we find that

Jt = 2

f

2N2

Nkk2

+ V

c1 + c222

+fkk2, (B.10)

Ji =f

N

Nk + Nkki

+

c1 + c22i, (B.11)

J = 1Nggc1 + c22,t

1N

i

f

Nk + Nkki+

c1 + c22

Ni + Ni

, (B.12)

JA = 2

c1 + c22, (B.13)

ij = (0)ij +

ij , (B.14)

where

(0)ij = gij

L(0) + kV,1 + V,2k + V,2k

+

1 + 2V1 + 2V5P2

(i) (j)2(iV,1j) 2(iV,2j)2(iV,2j),

ij = gij

L(A,)

1

Nk

c1

A Ak


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18

+2(A A)

N

c1ij + c2ij

+

c1 + c22ij

+2f

N

Nk + Nkk(ij)

+2

N(i

c1

A A

j)

, (B.15)

and

V,1 VP1 = 2V2P1 + 3V3P

21 + V6P2,

V,2 VP2 = V4 + V5()

2 + V6P1. (B.16)

On the other hand, the variation of the action (B.8)with respect to yields the following generalized Klein-Gordon equation,

f

N

g

g

N

Nk + Nk

k

,t

=f

N2i

Nk + Nkk Ni + Ni

+2gij

N(i

j)

(A A)c1

(A A)c2j)

+A A

N

c1 + c

2

2+ i

1 + 2V1 + 2V5P2i

V, (V,1) 2 (V,2) , (B.17)where c1

dc1()/d, and

V, V

= V + V1()2 + V2P21 + V3P31+V4P2 + V5()2P2 + V6P1P2. (B.18)

B. Linear Perturbations under the Newtonian

Quasilongitudinal Gauge

Under the gauge (5.1), Eqs.(4.8) - (4.15) can be castin the forms,

d3x2 123 1H3 + 2B= 4G

d3x

f +

a2V +V4a2

4

,(B.19)

d3xa2

f + 2Hf + a2V 3f

A

(ac1)

a

=

d3x 4

V4

+

V4 V4H

, (B.20)

(3 1) (1 )2B = 8Gf, (B.21)2H + 1 3 + 2B

= 8G

c1 + c1H f

+ c1

, (B.22)

= 4Gc1, (B.23)

+ 2H + 13

2

B + 2HB 2

3(3 1)

1 +

1a2

2 +2a4

4

2

+2

3(3 1)a 2

A A=

8G

3 1

f a2V

, (B.24)

B + 2HB+ 1a2

1 +

2a2

2

2

1

aA A = 0, (B.25)f

+ 2H 3 + 2B + a2V= 2

1

2+ V1 V2 + V

4

a22 V6

a44

2

+1

a2

2A (c1 c2) + c1A

. (B.26)

Recall A = aA. It can be shown that Eqs.(B.22) and(B.24) are not independent, and can be obtained from theothers. Therefore, in the present case there are four in-dependent differential equations, (B.21), (B.23), (B.25),and (B.26), for the four unknowns, , B, A and .

Appendix C: Quantization of scalar perturbations

This part summarizes the discussions given in [25].6 Toquantize the scalar field, suppose we have the normalizedaction of second order

S(2) =1

2

dd3x

v2 (, n)v2

, (C.1)

where (, n) is in general time-dependent explicitlyand n = 2, 4, 6,.... Now promote the field v andits conjugate momentum to operators,

v v(, x) =

d3k

(2)3

vk()ake

ikx + vk()ak

eikx

.

(C.2)

6 Note that here we did not consider any modifications of the com-mutation relations [39].


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19

If we define the Fourier image ofv(, x) as

v(, x) =

d3k

(2)3vk()e

ikx, (C.3)

this is equivalent to say that

vk() vk() = vk()ak + vk()ak. (C.4)vk() are called the mode functions. They satisfy thesecond order classical equation of motion (EoM)

vk + (, k2n)vk = 0. (C.5)

The canonical commutation relation between quantumfield vk and its conjugate momentum v

k

is,

0| [vk, vk] |0 = i. (C.6)If we want to have

[ak, ak

] = (2)33(k k), (C.7)

the norm (Wronskian) has to be

vkvk v

k vk = i. (C.8)

Besides the normalization condition, we need anotherboundary condition to determine the mode functionscompletely. Usually this is obtained by requiring thatthe vacuum state to be the ground state of the Hamilto-nian back in the far past when the mode is deep insidehorizon

H|0 = E0 |0 , (C.9)

where the vacuum is defined as ak |0 = 0. Since we haveH =

1

2

v2k + v

2k

, (C.10)

this requires vk = i

vk if |0 is the ground state.Thus

vk = Cei d , (C.11)

where the positive frequency branch is selected to ensurethe positivity of normalization, and C will be determinedby the normalization condition (C.8).

Appendix D: General expressions for power spectra

Here we give the more general expressions of spectrawithout setting b1 = b2 = b3 = 1 = 0. MA = MB = M(and thus by (6.25) g23 = g8) and A = 0 is still assumed.In the limiting case when HL (H/M)2 1, we find

PR(k) PGRR1

(b1)2/30

1 +

c1

2M2plH

2

1 9b24b21

HL +81(17b22 8b1b3)

128b412HL

,

(D.1)

r 16V(b1)2/30

1 +c1

2M2plH

2

1 +9(b2 2b21)

4b21HL

+81(36b41 17b22 + 8b1b3)

128b412HL

. (D.2)

In the limit HL 1, we obtain

PR(k) PGRR4e

1

2 HL

9

b30

1 +

c1

2M2plH

2

1 b2

2b3 4b3

9HL

1

3

, (D.3)r 16V

b302

1 +

c1

2M2plH

2

1

1

4 b2

2b3

b34

1/316

9HL

13

.

(D.4)

Clearly, the magnitude of the ratio r are dependent onthe values of b1 and b3.

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