[Taiwan] Galilean Creation€¦ · Masahide Yamaguchi (Tokyo Inst. Tech.) Jun’ichi Yokoyama (RESCEU, Univ. of Tokyo) JCAP 1507 (2015) 07, 017 [1504.05710] Everything About Gravity

Galilean Creation ofthe Inflationary Universe

Tsutomu KobayashiRikkyo University

Based on work with Masahide Yamaguchi (Tokyo Inst. Tech.) Jun’ichi Yokoyama (RESCEU, Univ. of Tokyo) JCAP 1507 (2015) 07, 017 [1504.05710]

Everything About Gravity Dec. 15 2015 Taiwan

小林努

立教大学

1. Introduction & Motivation––– Singularity-free, early-time completion of inflation

2. ADM decomposition of scalar-tensor theories ––– General framework

3. The Lagrangian 4. A concrete example

Talk Plan




Talk Plan

Inflation

Almost perfect standard scenario

However, singular in the past ( ), if the null energy condition (NEC) is satisfied:

Introduction

H ! 1

NEC violating scenarios

Bouncing models, etc.

Not so successful as inflation…

Usually, NEC violation implies instabilities (ghost/gradient)

H = �4⇡G(⇢+ p) < 0

Galilean GenesisStable, NEC violating cosmology

Galileon

Genesis solution –– Starting the Universe from Minkowski

Perturbations are stable

Nicolis, Rattazzi, Trincherini (2009)

L =M2

Pl

2R +

12e2��(��)2 + c2(��)4 + c3(��)2��

Creminelli, Nicolis, Trincherini (2010)

a ' 1 +

const

(�t)2, H ' const

(�t)3(t ⇠ �1)

Cubic Galileon

2nd-order field equations

A new proposalSingularity-free, early-time completion of inflation!

Pirtskhalave et al. (2014); TK, Yamaguchi, Yokoyama (2015)

Galilean GenesisH

t

Inflation



Galilean GenesisH

t

Inflation

L =M2

Pl

2R +

12e2��(��)2 + c2(��)4 + c3(��)2��



Galilean GenesisH

t

Inflation

L =M2

Pl

2R +

12e2��(��)2 + c2(��)4 + c3(��)2��

L =M2

Pl

2R+

1

2(@�)2 + c2(@�)

4 + c3(@�)2⇤�

Kinetically driven G-inflation



Galilean GenesisH

t

Inflation

L =M2

Pl

2R +

12e2��(��)2 + c2(��)4 + c3(��)2��

L =M2

Pl

2R+

1

2(@�)2 + c2(@�)

4 + c3(@�)2⇤�

Kinetically driven G-inflationPirtskhalave et al. (2014)

L =M2

Pl

2R+

1

2

e2��

1 + e2��(@�)2 + · · ·

Instability at the transition

0 500 1000 1500-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

t

c2s

!2 = c2sk2

Sound speed squared of the curvature perturbation ⇣

Dispersion relation:

c2s < 0

Galilean Genesis Inflation

Rapid exp growth at high k

MotivationHigher spatial derivative comes to the rescue!

No higher time derivatives (Ostrogradski’s ghost)

Can we construct a theory of a scalar field and gravity that admits the stable evolution of Galilean Genesis + Inflation + Graceful exit?

Negative is acceptable for a short periodc2s

!2 = c2sk2 +

✏

a2k4, ✏ > 0

Perturbations are stabilized at high k




Talk Plan

Horndeski theoryThe most general single-scalar-tensor theory with 2nd-order field equations:

4 arbitrary functions of and

No term because field equations are of 2nd order

Need further generalization

Lp�g

= G2(�, X)�G3(�, X)⇤�+G4(�, X)R

+G4,X

⇥(⇤�)2 � (rµr⌫�)

2⇤

+G5(�, X)Gµ⌫rµr⌫�� 1

6G5,X

⇥(⇤�)3 + . . .

⇤

Horndeski (1974); Deffayet, et al. (2011); TK, Yamaguchi, Yokoyama (2011)

X = �1

2gµ⌫@µ�@⌫��

k4





Lp�g

= G2(�, X)�G3(�, X)⇤�+G4(�, X)R

+G4,X

⇥(⇤�)2 � (rµr⌫�)

2⇤

+G5(�, X)Gµ⌫rµr⌫�� 1

6G5,X

⇥(⇤�)3 + . . .

⇤


X = �1

2gµ⌫@µ�@⌫��

k4

This Lagrangian was obtained as a generalization of the Galileon by Deffayet, et al. (2011). The most general 2nd-order theory was derived in a different form by Horndeski (1974) (and had been forgotten). The equivalence of the two theories was shown for the first time by TK, Yamaguchi, Yokohama (2011). The generalized Galileon is therefore the most general 2nd-order scalar-tensor theory.

Remark





Lp�g

= G2(�, X)�G3(�, X)⇤�+G4(�, X)R

+G4,X

⇥(⇤�)2 � (rµr⌫�)

2⇤

+G5(�, X)Gµ⌫rµr⌫�� 1

6G5,X

⇥(⇤�)3 + . . .

⇤


X = �1

2gµ⌫@µ�@⌫��

k4

ADM decomposition of scalar-tensor theories

Take as constant time hypersurfaces� = const

G(�, X) = G(�(t), �2(t)/2N2) = A(t,N)

R = R(3) +KijKij �K2 + · · ·

rµr⌫� ⇠ Kij

Lapse function

ADM decomposition of scalar-tensor theories

Take as constant time hypersurfaces� = const

G(�, X) = G(�(t), �2(t)/2N2) = A(t,N)

R = R(3) +KijKij �K2 + · · ·

rµr⌫� ⇠ Kij

LNp�

= A2(t,N) +A3(t,N)K +B4(t,N)R(3)

� (B4 +NB4,N )�K2 �K2

ij

�+B5(t,N)G(3)

ij Kij + · · ·

t N

Horndeski in the ADM form/in the unitary gauge


Lapse function

Beyond HorndeskiCounting the degrees of freedom…

The Hamiltonian depends nonlinearly on

Second-class constraint, , eliminating only 1 dof

N

C(N, �ij ,⇡ij) = 0

�, �ij

Propagating dofs

Gleyzes, et al. (2014)

LNp�

= A2(t,N) +A3(t,N)K +B4(t,N)R(3)

�(B4 +NB4,N )(K2 �K2ij) + · · ·




N

C(N, �ij ,⇡ij) = 0

�, �ij

Propagating dofs


More general theoryretains the same structure, yielding single scalar and transverse-traceless gravitons

Higher derivatives in field equations

LNp�

= A2(t,N) +A3(t,N)K +B4(t,N)R(3)

+A4(t,N)K2 � eA4(t,N)KijKij + · · ·

Gao (2014)

Arbitrary functions




N

C(N, �ij ,⇡ij) = 0

�, �ij

Propagating dofs


More general theoryretains the same structure, yielding single scalar and transverse-traceless gravitons

Higher derivatives in field equations

LNp�

= A2(t,N) +A3(t,N)K +B4(t,N)R(3)

+A4(t,N)K2 � eA4(t,N)KijKij + · · ·

Gao (2014)

GeneratingMinimal extensionis enough to get in the dispersion relation

k4

LNp�

= A2(t,N) +A3(t,N)K +B4(t,N)R(3)

+A4(t,N)��K2 �K2

ij

�

k4� : const

(c.f. Horava gravity)

GeneratingMinimal extensionis enough to get in the dispersion relation

k4

LNp�

= A2(t,N) +A3(t,N)K +B4(t,N)R(3)

+A4(t,N)��K2 �K2

ij

�

k4� : const

(c.f. Horava gravity)

Curvature perturbation

✏ / �� 1

L(2)⇣ =

Na3FS

c2s

"⇣2

N2+ ⇣

✓c2s

@2

a2� ✏

@4

a4

◆⇣

#




Talk Plan

The LagrangianSpecifying amounts to defining a concrete theory

Assume separable functions:

Given the evolution of and is determined from the Euler-Lagrange equations,

A2(t,N), A3(t,N), · · ·

f = f(t)with

f(t), a2(N), a3(N), N(t) a(t)

↵ > 0( is a constant parameter)A2 = M42 f

�2(↵+1)a2(N)

A3 = M33 f

�(2↵+1)a3(N)

A4 = �B4 = �M2Pl/2

�N : C(N, a, a) = 0

�a : E(N, N, a, a, a) = 0

Designing f(t)

Designing f(t)

0.0 0.2 0.4 0.6 0.8

0.0

0.5

1.0

1.5

2.0

2.5

Suppose you want the following

a(t)

a(t)

Designing f(t)

0.0 0.2 0.4 0.6 0.8

0.0

0.5

1.0

1.5

2.0

2.5


a(t)

a(t)

-10 -8 -6 -4 -2 0

0.0

0.5

1.0

1.5

2.0

You can design to reproduce your

f(t)

f(t) a(t)

Designing f(t)

0.0 0.2 0.4 0.6 0.8

0.0

0.5

1.0

1.5

2.0

2.5


a(t)

a(t)

-10 -8 -6 -4 -2 0

0.0

0.5

1.0

1.5

2.0

You can design to reproduce your

f(t)

f(t) a(t)

f(t)Find for Galilean Genesis + Inflation + Graceful exit

Genesis phaseForthe (generalized) Galilean Genesis solution is obtained,

Need for stability

f = const⇥ (�t) � 1

N ' N0 (= const)

a ' 1 +

const

(�t)2↵

A3(t,N)K

⇠ G3(�, X)⇤�

Nishi, TK (2015)

Galileon interaction is essential for the stable NEC violation

Inflation and graceful exitForthe de Sitter inflationary solution is obtained,

f = const

N = Ninf (= const)

H = Hinf (= const)

f / t1/(↵+1)

Forthe Universe exits from inflation,

N = Nexit

(= const)

H / 1/t

Our choicef(t)

t

/ �t

/ t1/(↵+1)

Galilean Genesis Inflation Graceful exit




Talk Plan

A concrete examplea2(N) = � 1

N2+

1

3N4, a3(N) =

�

N3, f(t) =

8

The equation of motion for the canonically normalizedvariable uk :=

√2GSaζk during inflation is of the form

d2uk

dτ2+

!ω2 −

2

τ2

"uk = 0, (67)

where

ω2 = c2sk2 + ϵ2k4τ2, (68)

with c2s = FS/GS and ϵ := HinfH1/2S /G1/2

S being dimen-sionless constants. Here, we have introduced the confor-mal time τ (< 0) defined by adτ = Ndt. The dispersionrelations of this form have been studied in the context ofinflation, e.g., in Refs. [42, 43]. The positive frequencymodes are given by [43]

uk =e−πc2

s/8ϵWic2

s/4ϵ,3/4(−iϵk2τ2)

(−2ϵk2τ)1/2, (69)

where Wκ,m is the Whittaker function. Taking the limitτ → 0, the power spectrum of the curvature perturbationcan be calculated as

Pζ =H2

inf

2GSc3s

1

F (c2s/ϵ), (70)

where

F (x) :=4

πx−3/2eπx/4 |Γ(5/4− ix/4)|2 . (71)

Even in the presence of the k4 term in the dispersionrelation, the power spectrum is scale-invariant in thecase of exact de Sitter inflation. Since we have F → 1as x → ∞, we recover the result of generalized G-inflation [29] in the limit ϵ → 0. For x ≪ 1 we haveF ≃ (4/π)|Γ(5/4)|2x−3/2, so that one can take the limitc2s → 0 smoothly to get

Pζ →πH2

inf

8GS |Γ(5/4)|2ϵ3/2. (72)

We have approximated the inflationary phase as exactde Sitter. If we consider a background slightly differentfrom de Sitter by incorporating weak time dependence inf , we would be able to obtain a tilted spectrum of ζ.

C. Graceful Exit

After inflation, we have GT ≃ M2Pl, FT = βM2

Pl,

FS ≃ βM2Pl−λ1 + 1 + ℓm/2

λ1 − 1 + ℓ, (73)

GS ≃ M2Pl

3λ1 − 1

λ1 − 1 + ℓ, (74)

HS ≃β2M2

Pl

H2

λ1 − 1

3λ1 − 1

ℓ

λ1 − 1 + ℓ, (75)

FIG. 2: The background evolution of (a) the Hubble param-eter H and (b) the lapse function N around the Genesis-deSitter transition.

where to simplify the expression we introduced

ℓ := −4

3

(Nea2)′

(N2e a

′2)

′. (76)

Recalling that we have been imposing λ1 > 1, all of thesecoefficients are positive provided that ℓm > 2(λ1 − 1).This condition can be written equivalently as

Nea′2(N2

e a′2)

′< −

1

2(λ1 − 1) (< 0). (77)

V. A CONCRETE EXAMPLE

Let us provide a concrete Lagrangian exhibiting theGenesis-de Sitter transition. The Lagrangian is charac-terized by

a2 = −1

N2+

N20

3N4, a3 =

γ

N3, (78)

where N0 (> 0) and γ (> 0) are constants. We take a4 =a5 = 0, B4 = M2

Pl/2, and B5 = 0. We also take λ1 > 1 toguarantee the stability. This corresponds to the (λ1 > 1generalization of the) unitary gauge description of theLagrangian considered in Ref. [13]. In the Genesis stage

9

FIG. 3: (a) The sound speed squared, FS/GS , and (b) thecoefficient of k4 (divided by GS) around the Genesis-de Sittertransition.

we have

N = N0, (79)

p = −!2M4

2

3N20

+ (2α+ 1)γ

N40

M33 |f0|

"< 0. (80)

Since λ1 > 1 and (N0a′2)′ = 2/N2

0 > 0, we see thatGS > 0 and HS > 0. We also see that

FS

M2Pl

=2

λ1 − 1

#γM3

3 |f0|M4

2N20

−1

3(2α+ 1)

$

− 1, (81)

and hence it is easy to satisfy FS > 0 during the Genesisphase by choosing the parameters appropriately.A numerical example of the Genesis-de Sitter transi-

tion is illustrated in Figs. 2 and 3. Our numerical calcu-lation was performed as follows: we solve the evolutionequations P = 0 and dE/dt = 0 with initial data (H,N)satisfying E = 0, and confirm that the constraint E = 0is satisfied at each time step. In the numerical calcula-tion, the parameters are given by MPl = M2 = M3 = 1,α = 1, λ1 = 1+ 10−3, N0 = 1, and γ = 10. The functionf(t) is taken to be

f =f02

!t−

ln(2 cosh(st))

s

"+ f1, (82)

with f0 = −10−1, f1 = 10, and s = 2 × 10−3. Thebackground evolution is shown in Fig. 2. The evolution

FIG. 4: The background evolution of (a) the Hubble parame-terH and (b) the lapse function N around the end of inflation.

of the sound speed squared, FS/GS , and the coefficientof k4 in the dispersion relation is shown in Fig. 3. Aspointed out in Ref. [26], c2s flips the sign at the transi-tion. The sound speed squared is positive except in thisfinite period. During the Genesis and subsequent de Sit-ter phases we have GS > 0 and HS > 0, and therefore wemay conclude that this model is stable.Although we have thus obtained the stable example of

the Genesis-de Sitter transition, the simple example (78)is not completely satisfactory if one would want successfulgravitational reheating. Indeed, the condition (45) im-plies that x := (Ne/N0)2 < 1, but m−4 = −2x/(1−x) <0 for such x. This problem can be evaded easily by thefollowing small deformation of a2:

a2 = −1

N2+

1 + 5∆2

3

N20

N4−∆2N

40

N6, (83)

where ∆ is a parameter smaller than 1/5. The condi-tion (45) now reads (1 − x)(x − 5∆2) > 0, i.e., 5∆2 <x < 1, while

m− 4 =2(∆+ x)(∆ − x)

(1− x)(x − 5∆2)(84)

is positive for 5∆2 < x < ∆. The stability condi-tion further restricts the allowed ranges of x and ∆.The necessary condition for stability is Nea′2/(N

2e a

′2)

′ <0 [see Eq. (77)]. This translates to 1 + 5∆2 −

H

N

Genesis to inflation The end of inflation

A concrete example

Genesis to inflation The end of inflation9

FIG. 3: (a) The sound speed squared, FS/GS , and (b) thecoefficient of k4 (divided by GS) around the Genesis-de Sittertransition.

we have

N = N0, (79)

p = −!2M4

2

3N20

+ (2α+ 1)γ

N40

M33 |f0|

"< 0. (80)

Since λ1 > 1 and (N0a′2)′ = 2/N2

0 > 0, we see thatGS > 0 and HS > 0. We also see that

FS

M2Pl

=2

λ1 − 1

#γM3

3 |f0|M4

2N20

−1

3(2α+ 1)

$

− 1, (81)

and hence it is easy to satisfy FS > 0 during the Genesisphase by choosing the parameters appropriately.A numerical example of the Genesis-de Sitter transi-

tion is illustrated in Figs. 2 and 3. Our numerical calcu-lation was performed as follows: we solve the evolutionequations P = 0 and dE/dt = 0 with initial data (H,N)satisfying E = 0, and confirm that the constraint E = 0is satisfied at each time step. In the numerical calcula-tion, the parameters are given by MPl = M2 = M3 = 1,α = 1, λ1 = 1+ 10−3, N0 = 1, and γ = 10. The functionf(t) is taken to be

f =f02

!t−

ln(2 cosh(st))

s

"+ f1, (82)

with f0 = −10−1, f1 = 10, and s = 2 × 10−3. Thebackground evolution is shown in Fig. 2. The evolution

FIG. 4: The background evolution of (a) the Hubble parame-terH and (b) the lapse function N around the end of inflation.

of the sound speed squared, FS/GS , and the coefficientof k4 in the dispersion relation is shown in Fig. 3. Aspointed out in Ref. [26], c2s flips the sign at the transi-tion. The sound speed squared is positive except in thisfinite period. During the Genesis and subsequent de Sit-ter phases we have GS > 0 and HS > 0, and therefore wemay conclude that this model is stable.Although we have thus obtained the stable example of

the Genesis-de Sitter transition, the simple example (78)is not completely satisfactory if one would want successfulgravitational reheating. Indeed, the condition (45) im-plies that x := (Ne/N0)2 < 1, but m−4 = −2x/(1−x) <0 for such x. This problem can be evaded easily by thefollowing small deformation of a2:

a2 = −1

N2+

1 + 5∆2

3

N20

N4−∆2N

40

N6, (83)

where ∆ is a parameter smaller than 1/5. The condi-tion (45) now reads (1 − x)(x − 5∆2) > 0, i.e., 5∆2 <x < 1, while

m− 4 =2(∆+ x)(∆ − x)

(1− x)(x − 5∆2)(84)

is positive for 5∆2 < x < ∆. The stability condi-tion further restricts the allowed ranges of x and ∆.The necessary condition for stability is Nea′2/(N

2e a

′2)

′ <0 [see Eq. (77)]. This translates to 1 + 5∆2 −

10

FIG. 5: (a) The sound speed squared, FS/GS , and (b) thecoefficient of k4 (divided by GS) around the end of inflation.

√1− 5∆2 + 25∆4 < x < ∆ < (4−

√11)/5 ≃ 0.137, lead-

ing to m < 24/5 = 4.8. Note that the small deformationof a2 with ∆ ! 0.1 does not change the background andperturbation dynamics of the Genesis and inflationaryphases.To illustrate the final stage of inflation, let us take

f =

!fα+11 +

v

2

"t+

ln (2 cosh(s′t))

s′

#$1/(α+1)

, (85)

where the origin of time is shifted so that the end of in-flation is given by t ∼ 0. In the numerical plots pre-sented in Figs. 4 and 5, the parameters are given bys′ = 10−2, v = 6, and ∆ = 0.05, while the other pa-rameters are taken to be the same as the previous exam-ple of the Genesis-de Sitter transition. It is found thatm ∼ 4.5 > 4. Again, we see that c2s < 0 in the finite pe-riod around the transition. However, GS and HS remainpositive all through the inflation and subsequent stages.

VI. DISCUSSION AND CONCLUSION

In this paper, we have introduced a generic descriptionof Galilean Genesis in terms of the ADM Lagrangian andconstructed a concrete realization of inflation precededby Galilean Genesis, i.e., the scenario in which the uni-verse starts from Minkowski spacetime in the asymptotic

past and is connected smoothly to the inflationary phasefollowed by the graceful exit. Our model utilizes the re-cent extension of the Horndeski theory, which has thesame number of propagating degrees of freedom as theHorndeski theory and thus can avoid Ostrogradski insta-bilities. This approach allows us to cover the backgroundand perturbation evolution in all the three phases withthe same single Lagrangian, as opposed to the effectivefield theory approach. In our scenario, the sound speedsquared during the transition from the Genesis phase toinflation becomes negative for a short period. However,thanks to the nonlinear dispersion relation arising fromthe fourth-order derivative term in the quadratic action,modes with higher momenta are stable and the growthrate of perturbations with smaller momenta is finite andunder control. It should also be noted that the soundspeed of the primordial perturbations can be smaller thanunity by choosing the parameter of the model appropri-ately.

Although we have constructed our inflation model inorder to resolve the initial singularity and possible trans-Planckian problems by incorporating Galilean Genesisphase before inflation, we could make use of our modelto realize the original Galilean Genesis scenario, which isan alternative to inflationary cosmology, simply by takingvanishingly short period of inflation there. As discussedin the Appendix, the sound speed squared becomes nega-tive at the transition also in this case, but the instabilitiesare relevant only for small k modes thanks to the k4 termin the dispersion relation. Thus, the transition from theGenesis phase to the reheating stage is described in ahealthy and controllable manner.

In fact, it would be fair to say that such a cosmol-ogy works quite well among the proposed alternatives toinflation, because, in contrast with the bouncing cosmol-ogy, in which all the would-be decaying modes in theexpanding universe such as vector fluctuations and spa-tial anisotropy severely increase in an undesirable man-ner, the Genesis solution is an attractor and generationof nearly scale-invariant curvature perturbation is alsopossible with an appropriate choice of model parameters[18]. Since no first-order tensor perturbation is generatedin this type of scenarios, detection of tensor perturbationwith its amplitude larger than 10−10 would be a smokinggun of inflation.

Acknowledgments

This work was supported in part by the JSPS Grant-in-Aid for Scientific Research Nos. 24740161 (T.K.),25287054 and 26610062 (M.Y.), 23340058 and 15H02082(J.Y.).

c2s

✏

!2 = c2sk2 +

✏

a2k4

c2s < 0 c2s < 0

✏ > 0✏ > 0

Stable!Stable!

Dispersion relation of :⇣

SummarySingularity-free, early-time completion of inflation–– The Universe starts expanding from Minkowski in the asymptotic past

General framework of scalar-tensor theories –– Higher spatial derivatives to circumvent instabilities

A stable concrete example capturing the evolution from pre-inflationary Galilean Genesis to the exit from inflation

Documents

[Taiwan] Galilean Creation€¦ · Masahide Yamaguchi (Tokyo Inst. Tech.) Jun’ichi Yokoyama (RESCEU, Univ. of Tokyo) JCAP 1507 (2015) 07, 017 [1504.05710] Everything About Gravity