Generalized Galilean Genesis

0.31 MB·English

by  Sakine Nishi

RUP-15-1
Generalized Galilean Genesis
Sakine Nishi1, and Tsutomu Kobayashi1,
∗ †
1Department of Physics, Rikkyo University, Toshima, Tokyo 175-8501, Japan
The galilean genesis scenario isan alternativeto inflation inwhich theuniversestarts expanding
from Minkowski in the asymptotic past by violating the null energy condition stably. Several
concretemodels ofgalilean genesis havebeen constructedso farwithin thecontextof galileon-type
scalar-field theories. Wegive a generic, unified description of the galilean genesis scenario in terms
oftheHorndeskitheory,i.e.,themostgeneralscalar-tensortheorywithsecond-orderfieldequations.
Indoingsowegeneralizethepreviousmodelstohaveanewparameter(denotedbyα)whichresults
in controlling the evolution of the Hubble rate. The background dynamics is investigated to show
that the generalized galilean genesis solution is an attractor, similarly to the original model. We
5 alsostudythenatureofprimordialperturbationsinthegeneralized galilean genesisscenario. Inall
1
the models described by our generalized genesis Lagrangian, amplification of tensor perturbations
0
does not occur as opposed to what happens in quasi-deSitter inflation. We show that thespectral
2
indexofcurvatureperturbationsisdeterminedsolelyfromtheparameterαanddoesnotdependon

r theother details of the model. In contrast to theoriginal model, a nearly scale-invariant spectrum
a of curvatureperturbations is obtained for a specific choice of α.
M
PACSnumbers: 98.80.Cq,04.50.Kd
1
3
I. INTRODUCTION implies that
]
h dH
t >0, (1)
- dt
p Itisfairtosaythatinflation[1–3]followedbyahotBig
e Bang is a standard scenario of modern cosmology. Infla- where H is the Hubble rate and t is cosmic time. This
h tion is attractive because the period of quasi-de Sitter signals ghost instabilities in general relativity. Recently,
[ expansioninthe earlyuniverseresolvesseveralproblems however, it was noticed that in noncanonical galileon-
2 that would otherwise indicate the need for fine-tuning. type scalar-field theories the NEC can be violated sta-
v Moreover, curvature perturbations are naturally gener- bly,1 andbasedonthisidea,Creminelliet al. proposeda
3 ated from quantum fluctuations during inflation, which novel, stable alternative to inflation named galilean gen-
5 seed large-scale structure of the universe [4]. The ba- esis [11]. (See also Ref. [12].) In the galilean genesis sce-
5
sic prediction of inflation is that the primordial curva- nario, the universe is asymptotically Minkowski in the
2
ture perturbations are nearly scale-invariant, adiabatic, pastandstartsexpandingfromthislowenergystate. As
0
. and Gaussian. This is in agreementwith observationsof such, this scenario is devoid of the horizon and flatness
1 CMB anisotropies [5, 6]. Inflationary models also pre- problems. Aspects of galilean genesis have been studied
0
dictthe quantummechanicalproductionofgravitational inRefs.[13–17]andtheoriginalmodelhasbeenextended
5
1 waves [7], the detection of which would be the evidence in Refs. [18–20] to possess improved properties. See also
: for inflation. Refs.[21–29]forotherinterestingNECviolatingcosmolo-
v gies in galileon-type theories and Ref. [30] for a related
i
X Despite thesuccessofinflation,itwouldbereasonable review.
toaskwhetheronlyinflationcanbeaconsistentscenario In this paper, we introduce a unified treatment of the
r
a compatible with observations. It should also be noted galileangenesismodelsandgiveagenericLagrangianad-
that an inflationary universe is past geodesically incom- mitting the genesis solutions. This is done by using the
plete [8] and so the problem of an initial singularity still Horndeski theory [31], which is the most general scalar-
persists. Fromthisviewpoint,variousalternativescenar- tensortheorywithsecond-orderfieldequations. Ourgen-
ios have been proposed so far, such as bouncing models. eralized galilean genesis Lagrangian contains four func-
Although such models can eliminate the initial singular- tional degrees of freedom and a constant parameter de-
ity, many of them are unfortunately plagued by insta- noted α. This parameter determines the behavior of the
bilities originated from the violation of the null energy Hubble rate. For specific choices of those functions and
condition (NEC), the growth of shear, and primordial α = 1, our Lagrangian reproduces the previous models
perturbations incompatible with observations [9]. exploredinRefs.[11,18–20]. Asisoftenthecasewithin-
flationalternatives,it turns out that the galileangenesis
In the context of cosmology, the violation of the NEC
1 The NEC can be violated stability at least within linear per-
∗Email: sakine n”at”rikkyo.ac.jp turbation analysis. However, at nonlinear order, it is not clear
†Email: tsutomu”at”rikkyo.ac.jp whether therearenoinstabilities[10].
2
models in general fail to produce nearly scale-invariant InRef.[34]itwasnoticedthatthe genesissolution(5)
curvature perturbations. We show, however, that with is obtained generically in the subclass of the Horndeski
anappropriatetuningofαitispossibletohaveaslightly theory with
tilted spectrum consistent with observations.
TheHorndeskitheorywasdevelopedaboutfortyyears G =e4λφg (Y), G =e2λφg (Y),
2 2 3 3
ago [31] and was revived recently as the generalized M2
galileon theory [32]. The equivalence of the two theories G4 = Pl +e2λφg4(Y), G5 =e−2λφg5(Y), (6)
2
was proven for the first time in Ref. [33]. The action of
the Horndeski theory is given in the generalized galileon where each g (i=2,3,4,5) is an arbitrary function of
i
form by
Y :=e 2λφX. (7)
−
S = d4x√ g( + + + ), (2)
2 3 4 5
− L L L L ThisextendstheLagrangiangiveninRef.[16]toinclude
Z
the Horndeskifunctions G andG . The Lagrangian(4)
4 5
with
and the DBI conformal galileon theory are included in
=G (φ,X), = G (φ,X)✷φ, the general framework defined by (6) as specific cases.
2 2 3 3
L L − In this paper, we further generalize (6) and consider
=G (φ,X)R+G (✷φ)2 ( φ)2 ,
4 4 4X µ ν
L − ∇ ∇
=G (φ,X)Gµν (cid:2)φ 1G (✷φ)3 (cid:3) G2 =e2(α+1)λφg2(Y), G3 =e2αλφg3(Y),
5 5 µ ν 5X
L ∇ ∇ − 6 M2
−3✷φ(∇µ∇νφ)2+2(∇µ∇(cid:2)νφ)3 , (3) G4 = 2Pl +e2αλφg4(Y), G5 =e−2λφg5(Y), (8)
where R is the Ricci scalar, Gµν is the Einste(cid:3)in tensor, where α (> 0) is a new dimensionless parameter. The
and each Gi(φ,X)(i=2,3,4,5)is an arbitrary function fourfunctions, g2, g3, g4,andg5, arearbitraryaslongas
of the scalar field φ and X := gµν∂µφ∂νφ/2. We use severalconditionspresentedinthissectionandinSec.IV
−
the notation GiX to denote ∂Gi/∂X. aresatisfied. Weassume,however,thatg4(0)=0,sothat
Theplanofthispaperisasfollows. Inthenextsection, G M2/2 as Y 0. The Horndeski theory with (8)
we present a generic Lagrangian that admits the gener- ad4m→itsthPelfollowing→generalized galilean genesissolution:
alized galilean genesis solution. In Sec. III we analyze
the background evolution analytically and numerically 1 1 h
eλφ , H 0 ( <t<0),(9)
andshowthatthe generalizedgalileangenesissolutionis ≃ λ√2Y ( t) ≃ ( t)2α+1 −∞
the dynamical attractor for a wide range of initial con- 0 − −
ditions. Primordialtensorandscalarperturbationsfrom for large t, where Y and h are positive constants. We
0 0
| |
generalized galilean genesis are studied in Sec. IV, and see that Y Y for this background. The parameter α
0
≃
the curvatonmechanismin the genesisscenariois briefly in the Lagrangian results in controlling the evolution of
discussed in Sec. V. We draw our conclusions in Sec. VI. the Hubble rate. The scale factor is given by
1 h
0
a 1+ , (10)
II. GENERALIZED GENESIS SOLUTIONS ≃ 2α( t)2α
−
The original model of galilean genesis is constructed and hence the solutiondescribes the universe that starts
by using the Lagrangianof the form [11, 18] expanding from Minkowski in the asymptotic past, sim-
ilarly to the original galilean genesis solution which cor-
M2 responds to the case of α = 1. The “slow-expansion”
= PlR+f e2λφX +f X2+f X✷φ, (4)
L 2 1 2 3 modelconsideredinRef.[35]isreproducedbytakingthe
particularfunctionsg withα=2. Wethusobtainaone-
i
where f1, f2, f3,and λ areconstants. (We havechanged parameter family of the generalized genesis solutions as
notations of Refs. [11, 18].) The above Lagrangian has an alternative to inflation. Note that, although the evo-
the genesis solution, lution of the scale factor is very different from quasi-de
Sitter, the universe in this scenario is also accelerating:
const const
eλφ , H ( <t<0), (5) ∂t(aH) > 0, and hence fluctuation modes will leave the
≃ t ≃ ( t)3 −∞
horizon during the genesis phase.
− −
Substituting Eq. (9) to the background equa-
for large t. (We have a degree of freedom to shift the
| | tions (A1)–(A3) and picking up the dominant terms at
origin of time: t t t .) The scale factor is given
by a 1 + const→/( t−)2,0describing the universe that large |t|, we have
≃ −
starts expanding from singularity-free Minkowski in the
e2(α+1)λφρˆ(Y ) 0, (11)
asymptotic past. The same genesis solution can also be E ≃ 0 ≃
obtained from the DBI conformal galileons [19, 20]. 2 (Y )H˙ +e2(α+1)λφpˆ(Y ) 0, (12)
0 0
P ≃ G ≃
3
where III. BACKGROUND EVOLUTION
ρˆ(Y) := 2Yg g 4λY (αg Yg ), (13)
2′ − 2− 3− 3′ To see whether or not the generalized genesis solu-
pˆ(Y) := g 4αλYg
2 3 tion presented in the previous section is an attractor,
−
+8(2α+1)λ2Y(αg4−Yg4′), (14) we trace the background evolution starting from generic
G(Y) := MP2l−4λY (g5+Yg5′), (15) initial conditions.
an overdot stands for differentiation with respect to t,
and a prime for differentiation with respect to Y. The A. Analytic argument
constant Y is determined as a root of
0
ρˆ(Y )=0, (16) Let us begin with a simplified discussion neglecting
0
gravity,i.e., the effect of the cosmic expansion [16]. It is
and then h0 is determined from Eq. (12) as convenient to introduce a new variable
1 pˆ(Y )
h0 =−2(2α+1)(2λ2Y )α+1 (Y0). (17) ψ :=e−λφ (>0). (21)
0 0
G
In terms of ψ we have Y = ψ˙2/(2λ2). For any homoge-
As will be seen shortly, this background is stable for
(Y )>0. Therefore, the above NEC violating solution neous solutions the scalar-field equation of motion (A4)
0
iGs possible provided that with the functions (8) can be written as
pˆ(Y )<0. (18) d
0 ψ 2(α+1)ρˆ(Y) =0. (22)
−
dt
Aswillbedemonstratedinthenextsection,thegener- h i
alized genesis solution will develop a singularityH Integrating this, we obtain
→∞
at some t = t , as in the original genesis model.
sing
We therefore assume that the genesis phase is matched ρˆ(Y)=Cψ2(α+1), (23)
onto the standard radiation-dominated universe before
t=t ,ignoringforthemomentthedetailofthereheat- where C is an integration constant. Equation (23) de-
sing
ing process. In conventional general relativity, matching fines a curve in the (ψ,ψ˙) space for each C, as shown in
two different phases can be done by imposing that the Fig. 1. With an initial condition (ψ,ψ˙ ) away from the
i i
Hubble parameter is continuous across the two phases. genesis solution, the integration constant is determined
However,thematchingconditionsaremodifiedingeneral as C =ψ−2(α+1)ρˆ(ψ˙2/2λ2). If ψ˙ < 0 initially, the scalar
scalar-tensortheories as second-derivatives of the metric i i
field rolls along the curve toward ψ 0, i.e., ρˆ 0.
andthe scalarfieldaremixedinthe fieldequations. The → →
Hence, this solution approach to one of the genesis solu-
modified matching condition [34] reads tions which are denoted as horizontal lines (ψ˙ = const)
e(2α+1)λφ Y0 inthe(ψ,ψ˙)plane. Ifψ˙ >0initially,thescalarfieldrolls
MP2lHrad = G(Y0)H − 2 2yg3′(y)dy the opposite way along the curve and goes further away
Z0 from the genesis solutions. This is the time reversal of
+2λφ˙e2αλφ(αg4−Y0g4′p), (19) the ψ˙ <0 solutions.
The above analytic argument implies that the genesis
and we require that the subsequent radiation-dominated
solution is the attractor for initial conditions such that
universe is expanding: Hrad > 0. This condition trans- ψ˙ <0( (eλφ)˙>0). In the next subsection we perform
lates to ⇔
numericalcalculations to show that this is basically true
Y0 g evenifonetakesintoaccountoftheeffectofgravity. The
3
g 2λY g +(2α+1)λ Y dy >0. (20)
− 2− 0 3 0 √y numerical analysis also allows us to see the final fate of
p Z0 the genesis solutions for which the effect of the cosmic
It is easy to see that in the case of α=1 all the expres- expansion cannot be ignored.
sionspresentedabovereproducethepreviousresults[34].
Beforeclosingthis section,letus emphasizethat(gen-
eralized) galilean genesis has the Minkowski phase only B. Full numerical analysis
in the asymptotic past. The true Minkowski spacetime
solution corresponds to the special case of Y = 0, i.e.,
Inthe Horndeskitheorywith (8)the Friedmannequa-
φ=const. TheY =0solutionisfoundonlyifg (0)=0.
2 tion can be written as
One may wonder if the true Minkowski vacuum (Y =0)
in our neighborhood begins to expand to form a genesis = e2(α+1)λφρˆ(Y)+6Hφ˙e2αλφc (Y)
1
universe (Y = Y > 0). This is forbidden because the E
two different stab0le solutions cannot be interpolated, as 3H2 c2(Y)+e2αλφd2(Y) +2H3φ˙e−2λφc3(Y)
−
argued in Ref. [16]. (See, however, Ref. [17].) = 0, (24)
(cid:2) (cid:3)
4
3
2
1
Ψ 0
-1
Genesis
-2
-3
0 200 400 600 800
Ψ
FIG. 1: Examples of the curves defined by Eq. (23). Hori- FIG.2: Numericalresultsofthebackgroundevolutionforthe
zontal dashed lines correspond to thegenesis solutions. model with g2 = −Y +Y2, g3 = Y, and g4 = g5 = 0. The
parameters are given by MPl =1, λ=1, and α=1.
where
1.5
c = Yg 2αλg +2(3 2α)λYg +4λY2g ,(25)
1 3′ − 4 − 4′ 4′′
c2 = MP2l−12λYg5−28λY2g5′ −8λY3g5′′, (26) 1.0
c = 5Yg +2Y2g , (27)
3 5′ 5′′
d2 = 2g4−8Yg4′ −8Y2g4′′. (28) 0.5 Genesis
Equation (24) is exact and hence can be used even if
Ψ 0.0
the background evolution is away from the genesis solu-
tion. Similarly, one can substitute Eq. (8) to the evolu-
-0.5
tion equation =0 and the scalar-fieldequation of mo-
P
tiontowritestraightforwardlytheexactequationsforthe
background. The resultant equations are integrated nu- -1.0
merically, giving the backgroundevolutionstarting from
generic initial conditions. -1.5
Given the initial conditions (φ(t ),φ˙(t )), the initial
0 0
0 200 400 600 800
value for H is determined from the Friedmann equa-
Ψ
tion(24). Therefore,theinitialvalues(φ(t ),φ˙(t ))must
0 0
be chosenin such a waythat Eq.(24) admits a realroot
H. Equation (24) is quadratic in H if g = 0 and cubic FIG.3: Numericalresultsofthebackgroundevolutionforthe
5
if g =0. In both cases, the discriminant for eλφ 1 model g2 = −Y +3Y2−Y3, g3 =Y, and g4 =g5 = 0. The
5
is giv6en by D ≪ parameters are given by MPl =1, λ=1, and α=2.
=e2(α+1)λφc (Y)ρˆ(Y)+ (e2(2α+1)λφ). (29)
2
D O genesis is the attractor for ψ˙ < 0. At late times where
In the g =0 case, the initial data (φ(t ),φ˙(t )) must lie ψ 1, the numerical solutions are no longer approxi-
5 0 0 ≪
intheregionwhere 0issatisfied. Intheg =0case, mated by Eq. (9), and within a finite time the Hubble
5
D≥ 6
theFriedmannequationhasatleastonerealrootforany rateH diverges. AstheFriedmannequationisquadratic
(φ(t ),φ˙(t )). inH,wehavetwobranchesofthesolutions,oneofwhich
0 0
ConcretenumericalexamplesarepresentedinFigs.2– may be contracting initially (H < 0). The cosmological
4. In Figs. 2 and 3 we show the cases where the Fried- evolutionnevertheless approachesthe same genesis solu-
mann equation is quadratic in H. The shaded regions tion and the trajectories in the (ψ,ψ˙) space are almost
( < 0) cannot be accessed because H would be imagi- indistinguishable.
D
nary there. In Fig. 2 we have one genesis solution, while The behavior of the models with g = 0 is more com-
5
6
wehavetwoinFig.3. Inbothcases,generalizedgalilean plicated, as illustrated in Fig. 4. In the white region, we
5
3 logical background equations with the spatial curvature
K in the Horndeskitheory,whicharesummarizedin the
Appendix A.
2
Let us take an initial condition such that H is suffi-
cientlysmallintheequationofmotionforφand eλφ ˙>
1 0. Then,inauniversewithK =0theequationofmotion
6 (cid:0) (cid:1)
for φ can be written as
Ψ 0 d (Y) (Y)
Genesis e2(α+1)λφρˆ(Y) e2αλφK4 K5 =0, (30)
dt − a2 − a2
(cid:20) (cid:21)
-1
where
-2 K4(Y) := 6(g4−2Yg4′)K, (31)
(Y) := 12λY (g +Yg )K. (32)
K5 − 5 5′
-3
0 200 400 600 800 Even if e2(α+1)λφρˆ e2αλφ 4, 5 at the initial moment,
Ψ thecurvatureterms∼becomeKsmKallerrelativetotheρˆterm
asthescalarfieldrolls. Thus,wehavethesameattractor
solution Y =Y satisfying ρˆ(Y )=0, i.e., eλφ ( t) 1.
FIG. 4: Numerical results of the background evolution for 0 0 −
∼ −
g2 =−Y+Y2,g3=Y,g4 =0,andg5 =−Y. Theparameters Along this attractor, the evolution equation reads
are given byMPl =1, λ=1, and α=1.
K
2 (Y )H˙ +e2(α+1)λφpˆ(Y )+ M2 +4λY g (Y ) 0,
G 0 0 Pl 0 5 0 a2 ≃
have >0andsotherearethreepossiblechoicesforthe (cid:2) (cid:3) (33)
D
initial value of H. Two of the three branches converge
to the genesis solution similarly to the g = 0 case, as where we assumed that H˙ H2. Equation (33) im-
5 ≫
shown as the black lines in Fig. 4. Also in this case we plies that the curvature term becomes subdominant as
find H within a finite time. However, the remain- the scalar field rolls, and as a result H˙ is determined by
ing one→br∞anch never converges to the genesis solution. the pˆterm, recovering the evolution of the genesis back-
The corresponding examples are shown as the gray lines ground. Thus, the flatness problem is resolved in the
inFig. 4. Inthe shadedregion,we have <0 andthere genesis model.
D
is only one possible initial value for H at each point,
which corresponds to the latter branch. Therefore, the
D. Anisotropy
generalized galilean genesis solution can be a dynamical
attractorfortheinitialdatainthewhite ( >0)region.
D
We thus conclude that the galilean genesis solution is In conventional cosmology, an initial anisotropy is
the attractor provided that ψ˙ < 0 ( (eλφ)˙ > 0) ini- wiped out during inflation [36]. However, in alternative
tially, though the situation in the pr⇔esence of g is in- scenarios such as bouncing cosmology, it is often prob-
5
volved. In the inflationary scenario, usually it does not lematic that the initial anisotropy grows in a contract-
matter which direction the scalar field rolls initially, but ing phase [37] (see however [26]). In this subsection we
the universe must be expanding initially. In contrast to will show that adding the initial anisotropy on the gen-
the case of inflation, the galilean genesis scenario allows eralizedgalileangenesissolutiondoes not destabilize the
both for expanding and contracting universes at the ini- backgroundevolution.
tialmoment,whilethe scalarfieldmustrollinaparticu- We consider the Kasner metric
lar direction initially. As far as we have investigated nu-
merically, all the solutions develop a singularity H ds2 = dt2+a2 e2θ1(t)dx2+e2θ2(t)dy2+e2θ3(t)dz2 (,34)
→∞ −
at some time t=tsing in the future. In passing, we have h i
checkedthatthenumericalexamplesinFigs.2and4sat- where it is convenient to write
isfythestabilityconditionspresentedinthenextsection.
θ =β +√3β , θ =β √3β , θ = 2β .(35)
1 + 2 + 3 +
− − − −
If the deviations from the genesis background are not
C. Spatial curvature
large, it follows from Eqs. (B4) and (B5) that
We haveso farneglectedthe spatialcurvature. Inthis d
β˙ 2e 2λφφ˙Y g β˙2 β˙2 = 0, (36)
subsection,letusjustifythisassumptionbyshowingthat dt G +− − 0 5′ +− −
the spatial curvature does not interfere with the evolu- h d (cid:16) (cid:17)i
tion of the genesis background. We will use the cosmo- dt Gβ˙−+4e−2λφφ˙Y0g5′β˙+β˙− = 0. (37)
h i
6
In the models with g =0, this simply gives B. Scalar perturbations
5′
β˙ , β˙ const, (38)
+ The quadratic action for the curvature perturbation ζ
− ∼
in the unitary gauge is given by
so that the initial anisotropydilutes as θ ( t). In the
i
∼ −
models with g5′ 6=0, we have the following possibilities: S(2) = dtd3xa3 ζ˙2 c2s( ζ)2 , (45)
ζ GS − a2 ∇
1 √3 1 √3 Z (cid:20) (cid:21)
(β˙ ,β˙ )=(0,0),(b,0),( b, b),( b, b),(39)
+ − −2 2 −2 − 2 where with some manipulation GS and c2s in the genesis
phase are written as
where
(2α+1)λξ2(Y ) 2
b:= 2e−2λGφφ˙Y0g5′ ∼(−t)−1. (40) GS = 2(cid:20) Y0ξ′(Y0) 0 (cid:21) ρˆ′(Y0)e−2αλφ, (46)
ξ (Y )pˆ(Y )
In this case, for nonzero β˙ the initial anisotropy can c2s = ξ′(Y0)ρˆ(Y0), (47)
grow logarithmically: θ l±n( t). However, this should 0 ′ 0
i
∼ −
be compared with lna ( t) 2α; we see that the log- with
−
∼ −
arithmic growth of θ does not spoil the genesis back-
i Y (Y)
ground. ξ(Y):= G . (48)
− pˆ(Y)
Equations(46)and(47)showthat ( t)2α andc2 =
IV. PRIMORIDAL PERTURBATIONS GS ∝ − s
const. It follows from Eqs. (18) and (43) that ξ(Y )>0.
0
We thus find that stability against scalar perturbations
Let us now study the behavior of primordial tensor
is guaranteed if
and scalar perturbations around the generalized genesis
background to obtain predictions of our scenario as well ρˆ(Y )>0, (49)
′ 0
astoimposestabilityconditions. Todoso,weutilizethe
ξ′(Y0)<0, (50)
general quadratic action for cosmological perturbations
in the Horndeski theory derived in Ref. [33]. are fulfilled. We can choose the functional degrees of
freedom so that this is possible.
Let us evaluate the power spectrum of ζ. To simplify
A. Tensor perturbations the notation, it is convenient to write = ( t)2α,
S
G A −
where is a constant deduced from Eq. (46), the value
A
The quadratic action for tensor perturbations h in of which depends on the model, i.e., α and the concrete
ij
the genesis phase is given by form of g (Y). The equation of motion derived from the
i
action (45) is given by
1 c2
Sh(2) = 8 dtd3xa3G(Y0) h˙2ij − at2(∇hij)2 , (41) ζ¨ + 2αζ˙ +c2k2ζ =0, (51)
Z (cid:20) (cid:21) k t k s k
where
wherewe movedto the Fourierspace. This equationcan
M2 +4λY g (Y ) be solved to give
c2 = Pl 0 5 0 (42)
t (Y )
G 0 ζ = 1 π ( t)νH(1)( c kt), ν := 1 α, (52)
andnotethata 1. Itcanbeseenthatstabilityagainst k 2 2 (Y0) − ν − s 2 −
≃ r A
tensor perturbations is assured if
where H(1) is the Hankel function of the first kind and
ν
(Y0)>0, (43) thepositivefrequencymodeshavebeenchosen. Onlarge
G
MP2l+4λY0g5(Y0)>0, (44) scales, |cskt|≪1, we have
are satisfied. ζk Ak+Bk( t)1−2α, (53)
≃ −
Since both (Y ) and c2 are constantduring the gene-
G 0 t where
sisphase,thetensorperturbationsareeffectivelylivingin
Minkowskiwithout regardto α and the concrete formof π Γ(ν)
A := i2ν 1 (c k) ν, (54)
g (Y),andconsequentlyamplificationofquantumfluctu- k − s −
i − 2 π
ations does not occur as opposed to the case of quasi-de r A
π 1 icos(πν)Γ( ν)
Sitter inflation. This means that no detectable primor- Bk := 2−ν−1 −
dial gravitational waves are generated from our generic r2A(cid:20)Γ(ν+1) − π (cid:21)
class of the genesis models. (c k)ν. (55)
s
×
7
If0<α<1/2,thesecondterminEq.(53)decaysasis V. CURVATON
commontousualcosmologies,leavingtheconstantmode
at late times. Thus, in this case the power spectrum is In the previous section we have seen that the nearly
given by scale-invariant spectrum for curvature perturbations is
22ν 4c 2νΓ2(ν) possible only in the case of α 2. In the other cases we
Pζ(k)= − π−s3 k3−2ν, (56) need to consider an alternativ≃e mechanism such as the
A curvatoninordertoobtainascale-invariantspectrum. In
and the spectral index is found to be
thissection,westudyslightlyinmoredetailthecurvaton
coupled to a conformal metric, the basic idea of which
n =2α+3, (57)
s
was proposed earlier in Ref. [11]. A similar mechanism
yieldingabluespectrumincompatiblewithobservations. was proposed in Ref. [14].
Thecaseofα>1/2ismoresubtle,becausethesecond To make a scale-invariant power spectrum, we intro-
term in Eq. (53) grows and dominates on large scales. duce a curvaton field σ coupled to the conformalmetric,
This is what happens in the original galilean genesis
model(α=1)[11]. Toextractthelate-timeamplitudeof gˆµν =e2βλφgµν, (63)
ζ , let us consider the following situation. Suppose that
k where β is a constant parameter which is assumed to be
the genesis phase terminates at t = t and is matched
end close to unity, β 1. Assuming the simplest potential,
onto some other phase. We assume that the scalar field ≃
we consider the following action for σ:
ishomogeneousonthet=t hypersurface. Inthesub-
end
sequentphase,thecurvatureperturbationonlargescales 1 1
S = d4x gˆ gˆµν∂ σ∂ σ m2σ2 . (64)
may be written as σ µ ν
− −2 − 2
Z (cid:20) (cid:21)
ζ =C D ∞ dt′ , (58) Theconformalmpetric(63)impliesthattheeffectivescale
k k− kZt a3(t′)GS(t′) factor for the curvaton is eβλφ ∼ (−t)−β with β ≃ 1, so
where we do not specify (t) for t > t , but assume that σ lives effectively in a quasi-de Sitter spacetime.
S end
that Eq. (58) gives the Gconstant and decaying modes Theequationsofmotionforthehomogeneouspartσ =
and hence the integral converges. The late-time ampli- σ0(t) is given by
tude is givenby C . The matching conditions [34] imply
k σ¨ +(2βλφ˙ +3H)σ˙ +e2βλφm2σ =0. (65)
that ζ and ζ˙ are continuous across the two phases 0 0 0
k S k
G
(cf. [38]). It is then straightforward to obtain Ck = On the genesis background, one can ignore H
Ak + Bk(−tend)1−2α(1+I) ≃ Bk(−tend)1−2α(1+I), (−t)−(2α+1) relative to λφ˙ ∼(−t)−1, leading to ∼
where
2β m2
:=(2α 1) ∞ a3(tend)A(−tend)2α dt′ (59) σ¨0− t σ˙0+ [λ√2Y ( t)]2βσ0 =0. (66)
I − a3(t) (t) t 0 −
Ztend ′ GS ′ | end|
The effective Hubble rate for the curvaton is λ√2Y .
is independent of k. We may thus use the estimate ∼ 0
For the “light” curvaton with
(k) (k) , (60)
Pζ ∼C× Pζ |t=tend m2 λ2Y , (67)
0
with beingsomek-independentfactor. Thepowerspec- ≪
trumCevaluated at the end of the genesis phase is given we thus have σ0 const and the other independent so-
≃
by lution decays quickly.
The energy density and pressure of σ are given by
2 2ν 4c2νΓ2( ν)
Pζ(k)|t=tend = − −π3s − |tend|4νk3+2ν, (61) ρ = 1e2βλφσ˙2+ 1e4βλφm2σ2 ( t) 4β, (68)
A σ 2 0 2 0 ∼ − −
so that
1 1
p = e2βλφσ˙2 e4βλφm2σ2 ( t) 4β. (69)
ns =5 2α. (62) σ 2 0 − 2 0 ∼ − −
−
Although the overall amplitude depends on the details Equations (11) and (12) imply that the dominant part
of the model construction, the spectral index depends of the cosmological background equations grows as
only on α and not on the concrete form of gi(Y). We (−t)−2(α+1). Thus, in order for the (initially subdom∼-
have an exactly scale-invariant spectrum for α = 2, and inant) curvaton not to spoil the genesis background as
this is in sharp contrast to the original galilean genesis time proceeds, we require that
model having α = 1, which produces a blue-tilted spec-
α+1 2β. (70)
trumofcurvatureperturbations. Aparticularrealization ≥
of α = 2 is found in Ref. [35, 39], where the same con- The fluctuation of the curvaton, δσ(t,x), obeys
clusion is reached. Taking α = 2.02, one can obtain the
nearly scale-invariant, but slightly red-tilted, spectrum δ¨σ 2βδ˙σ 2δσ+ m2 δσ =0. (71)
with ns ≃0.96. − t −∇ [λ√2Y0(−t)]2β
8
Neglecting the mass term, this can be solved in the ofinitialconditions. Inparticular,we haveseenthat the
Fourier space to give spatialcurvature and aninitial anisotropydo not hinder
the evolution of the genesis phase.
δσ = √π λ 2Y β( t)β+1/2H(1) ( kt), (72) We have then studied the primordial perturbations
k 2 0 − β+1/2 − from the generalized galilean genesis models. From the
(cid:16) p (cid:17) quadraticactionsforcosmologicalperturbationswehave
where the positive frequency modes have been chosen.
imposed several stability conditions on the functions in
Thus,the powerspectrumofthe curvatonfluctuationsis
our generic Lagrangian. In contrastto the case of quasi-
de Sitter inflation, tensor fluctuations are not amplified
23β 2λ2βYβΓ2(β+1/2)
(k)= − 0 k2 2β, (73) in the genesis phase in all the galilean genesis models
Pδσ π3 −
we have constructed, and hence no detectable primor-
dial gravitational waves are expected. The evolution of
and we find
the curvature perturbation ζ depends on the parameter
n =3 2β. (74) α and has turned out to be more interesting. In the
s
− case of α > 1/2, ζ grows on large scales, as in the orig-
In the case of β = 1, the effective scale factor for the inal galilean genesis model (α = 1) [11]. The tilt of the
curvaton is that of exact de Sitter, and hence the power power spectrum at the end of the genesis phase is given
spectrum is exactly scale-invariant, as is expected. Tak- by n = 5 2α, irrespective of the other details of the
s
−
ing β = 1.04 we obtain n = 0.96. The curvaton model. Thus, we have a slightly red-tilted spectrum for
s
fluctuations can be converted into adiabatic ones after α&2. In the case of α<1/2, the constant mode domi-
the genesis phase, where σ behaves as a conventional natesonlargescalesasinconventionalcosmology. Inthis
scalar field in a true expanding universe, in the same case, the power spectrum has been shown to be always
way as the usual curvaton field in the inflationary sce- blue-tilted. We have also discussed the possibility of the
narios. Note, however, that due to the restriction (70) curvaton mechanism in the generalized galilean genesis
thepresentcurvatonmechanismworksonlyforthemod- scenario.
els with α 2 n (>1). Wehaveignoredthereheatingprocessinourscenario.
s
≥ −
It would be interesting to explore how the universe re-
heatsandhowmatteriscreatedattheendofgeneralized
VI. CONCLUSIONS galileangenesis. Since the Lagrangiandefined by (8) ex-
cludes a cosmological constant, it is not clear how the
In this paper, we have extended the galilean genesis genesis phase is connected finally to the late-time uni-
models [11, 18–20]andconstructedagenericLagrangian versedescribedbytheΛCDMmodel. Thesearetheopen
from the Horndeski theory that admits the generalized questions.
galilean genesis solution. In generalized galilean gene-
sis, the universe starts expanding from Minkowski in a
singularity free manner with the increasing Hubble rate Acknowledgments
H ( t) (2α+1), where α (> 0) is a new constant pa-
−
∼ −
rameter in the Lagrangian. We have investigated the We would like to thank A. Vikman for useful com-
background evolution and shown that the generalized ments. This workwassupportedinpartbyJSPSGrant-
galileangenesis solutionis the attractorfor a wide range in-Aid for Young Scientists (B) No. 24740161(T.K.).
Appendix A: Cosmological background equations in the Horndeski theory
The cosmologicalbackgroundequations in the Horndeski theory are given by [33]
=0, =0, (A1)
E P
where
:= 2XG G +6Xφ˙HG 2XG 6H2G +24H2X(G +XG ) 12HXφ˙G 6Hφ˙G
2X 2 3X 3φ 4 4X 4XX 4φX 4φ
E − − − − −
+2H3Xφ˙(5G +2XG ) 6H2X(3G +2XG ), (A2)
5X 5XX 5φ 5φX
−
:= G 2X G +φ¨G +2 3H2+2H˙ G 12H2XG 4HX˙G 8H˙XG 8HXX˙G
2 3φ 3X 4 4X 4X 4X 4XX
P − − − − −
(cid:16) (cid:17) (cid:16) (cid:17)
+2 φ¨+2Hφ˙ G +4XG +4X φ¨ 2Hφ˙ G 2X 2H3φ˙ +2HH˙φ˙ +3H2φ¨ G
4φ 4φφ 4φX 5X
− −
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
4H2X2φ¨G +4HX X˙ HX G +2 2(HX)˙+3H2X G +4HXφ˙G . (A3)
5XX 5φX 5φ 5φφ
− −
(cid:16) (cid:17) (cid:2) (cid:3)
9
The equation of motion for φ takes the form
1 d
a3J = P , (A4)
a3dt φ
where (cid:0) (cid:1)
J := φ˙G +6HXG 2φ˙G +6H2φ˙(G +2XG ) 12HXG
2X 3X 3φ 4X 4XX 4φX
− −
+2H3X(3G +2XG ) 6H2φ˙(G +XG ), (A5)
5X 5XX 5φ 5φX
−
and
P := G 2X G +φ¨G +6 2H2+H˙ G
φ 2φ 3φφ 3φX 4φ
−
(cid:16) (cid:17) (cid:16) (cid:17)
+6H X˙ +2HX G 6H2XG +2H3Xφ˙G . (A6)
4φX 5φφ 5φX
−
(cid:16) (cid:17)
The above equations are for the spatially flat background. In the open (K = 1) and closed (K = 1) cases, the
−
corresponding equations are given by
3 K K
T T
G =0, + F =0, (A7)
E − a2 P a2
and
1 d 6K 3K ∂
a3 J + (φ˙G +HXG φ˙G ) =P + FT, (A8)
a3dt a2 4X 5X − 5φ φ a2 ∂φ
(cid:26) (cid:20) (cid:21)(cid:27)
where
:= 2 G 2XG X Hφ˙G G , (A9)
T 4 4X 5X 5φ
G − − −
h (cid:16) (cid:17)i
:= 2 G X φ¨G +G . (A10)
T 4 5X 5φ
F −
h (cid:16) (cid:17)i
The cosmologicalbackgroundequations for the open and closed models are derived for the first time in this paper.
Appendix B: Anisotropic Kasner universe in the Horndeski theory
We derive the basic equations governing the evolution of an anisotropic Kasner universe in the Horndeksi theory.
We consider the following metric:
ds2 = N2dt2+a2 e2(β++√3β−)dx2+e2(β+−√3β−)dy2+e−4β+dz2 . (B1)
−
h i
Substituting this to the Horndeski action, we obtain
S =S +S , (B2)
iso aniso
where S is identical to the action for the homogeneous and isotropic metric and S is given by
iso aniso
6a3 Hφ˙ 4a3
S = dtd3x G 2XG XG +XG β˙2 +β˙2 Xφ˙G β˙3 3β˙ β˙2 . (B3)
aniso " N 4− 4X − N2 5X 5φ! + − − N3 5X +− + − #
Z (cid:16) (cid:17) (cid:16) (cid:17)
Note that X here should be understood as X =φ˙2/2N2.
Varying the above action with respect to β and setting N =1, we obtain
±
d
a3 β˙ 2Xφ˙G β˙2 β˙2 = 0, (B4)
dt GT +− 5X +− −
n dh (cid:16) (cid:17)io
a3 β˙ +4Xφ˙G β˙ β˙ = 0. (B5)
T 5X +
dt G − −
n h io
Varying the action (B2) with respect to N, a, and φ, one can also derive the cosmological evolution equations with
shear contributions.
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