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 ( 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(generalized) 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 solution. 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 numerically, 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 primordial 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 genesis, 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. [1] A.A. Starobinsky,Phys.Lett. B 91, 99 (1980). [2] A. H.Guth, Phys.Rev. D 23, 347 (1981). 10 [3] K.Sato, Mon. Not. Roy.Astron.Soc. 195, 467 (1981). Phys. Rev. Lett. 110, 241303 (2013) [arXiv:1212.3607 [4] V.F.MukhanovandG.V.Chibisov,JETPLett.33,532 [hep-th]]. (1981) [Pisma Zh.Eksp. Teor. Fiz. 33, 549 (1981)]. [21] C.Deffayet,O.Pujolas,I.SawickiandA.Vikman,JCAP [5] D. Larson, J. Dunkley, G. Hinshaw, E. Komatsu, 1010, 026 (2010) [arXiv:1008.0048 [hep-th]]. M. R. Nolta, C. L. Bennett, B. Gold and M. Halpern et [22] T. Kobayashi, M. Yamaguchi and J. Yokoyama, Phys. al.,Astrophys.J.Suppl.192,16(2011)[arXiv:1001.4635 Rev.Lett.105,231302(2010)[arXiv:1008.0603[hep-th]]. [astro-ph.CO]]. [23] T.Qiu,J.Evslin,Y.F.Cai, M.LiandX.Zhang,JCAP [6] P.A.R.Adeetal.[PlanckCollaboration],Astron.Astro- 1110, 036 (2011) [arXiv:1108.0593 [hep-th]]. phys. 571, A22 (2014) [arXiv:1303.5082 [astro-ph.CO]]; [24] D. A. Easson, I. Sawicki and A. Vikman, JCAP 1111, P.A.R.Adeetal.[PlanckCollaboration],Astron.Astro- 021 (2011) [arXiv:1109.1047 [hep-th]]. phys.571, A16 (2014) [arXiv:1303.5076 [astro-ph.CO]]. [25] Y. F. Cai, D. A. Easson and R. Brandenberger, JCAP [7] A.A.Starobinsky,JETPLett.30,682(1979)[PismaZh. 1208, 020 (2012) [arXiv:1206.2382 [hep-th]]. Eksp.Teor. Fiz. 30, 719 (1979)]. [26] Y.-F.Cai,R.BrandenbergerandP.Peter,Class.Quant. [8] A. Borde and A. Vilenkin, Int. J. Mod. Phys. D 5, 813 Grav. 30, 075019 (2013) [arXiv:1301.4703 [gr-qc]]. (1996) [gr-qc/9612036]. [27] M. Osipov and V. Rubakov, JCAP 1311, 031 (2013) [9] For a comprehensive review, see, e.g., D. Battefeld [arXiv:1303.1221 [hep-th]]. and P. Peter, arXiv:1406.2790 [astro-ph.CO]; M. Nov- [28] T. Qiu, X. Gao and E. N. Saridakis, Phys. Rev. D 88, ello and S. E. P. Bergliaffa, Phys. Rept. 463, 127 no. 4, 043525 (2013) [arXiv:1303.2372 [astro-ph.CO]]. (2008) [arXiv:0802.1634 [astro-ph]]; J. L. Lehners, Phys. [29] D.Pirtskhalava,L.Santoni,E.Trincherini,P.Uttayarat, Rept. 465, 223 (2008) [arXiv:0806.1245 [astro-ph]].; JHEP 1412, 151 (2014) [arXiv:1410.0882 [hep-th]]. R.H.Brandenberger,Int.J.Mod.Phys.Conf.Ser.01,67 [30] V. A. Rubakov, Phys. Usp. 57, 128 (2014) (2011) [arXiv:0902.4731 [hep-th]]; Y. F. Cai, Sci. China [arXiv:1401.4024 [hep-th]]. Phys. Mech. Astron. 57, 1414 (2014) [arXiv:1405.1369 [31] G. W.Horndeski, Int.J. Theor. Phys. 10, 363 (1974). [hep-th]]. [32] C.Deffayet,X.Gao,D.A.SteerandG.Zahariade,Phys. [10] I. Sawicki, A. Vikman, Phys. Rev. D 87, 067301 (2013) Rev. D 84, 064039 (2011) [arXiv:1103.3260 [hep-th]]. arXiv:1209.2961 [astro-ph.CO]. [33] T. Kobayashi, M. Yamaguchi and J. Yokoyama, Prog. [11] P.Creminelli,A.NicolisandE.Trincherini,JCAP1011, Theor.Phys.126,511(2011)[arXiv:1105.5723 [hep-th]]. 021 (2010) [arXiv:1007.0027 [hep-th]]. [34] S.Nishi,T.Kobayashi,M.YamaguchiandN.Tanahashi [12] P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore, JCAP 008, 03 (2014) [arXiv:1401.1045 [hep-th]] JHEP 0612, 080 (2006) [hep-th/0606090]. [35] Z. G. Liu, J. Zhang and Y. S. Piao, Phys. Rev. D 84, [13] L. Perreault Levasseur, R. Brandenberger and 063508 (2011) [arXiv:1105.5713 [astro-ph.CO]]. A. C. Davis, Phys. Rev. D 84, 103512 (2011) [36] R. M. Wald, Phys. Rev.D 28, 2118 (1983). [arXiv:1105.5649 [astro-ph.CO]]. [37] K. E. Kunze and R. Durrer, Class. Quant. Grav. 17, [14] Y.WangandR.Brandenberger,JCAP1210,021(2012) 2597 (2000) [gr-qc/9912081]; J. K. Erickson, D. H. Wes- [arXiv:1206.4309 [hep-th]]. ley, P. J. Steinhardt and N. Turok, Phys. Rev. D 69, [15] D. A. Easson, I. Sawickib and A. Vikman, JCAP 1307, 063514 (2004) [hep-th/0312009]; B.XueandP.J. Stein- 014 (2013) [arXiv:1304.3903 [hep-th]]. hardt, Phys.Rev.D 84, 083520 (2011) [arXiv:1106.1416 [16] V. A. Rubakov, Phys. Rev. D 88, 044015 (2013) [hep-th]]. [arXiv:1305.2614 [hep-th]]. [38] J. -c. Hwang and E. T. Vishniac, Astrophys. J. 382, [17] B. Elder, A. Joyce and J. Khoury, Phys. Rev. D 89, 363 (1991); N. Deruelle and V. F. Mukhanov, Phys. 044027 (2014) [arXiv:1311.5889 [hep-th]]. Rev. D 52, 5549 (1995) [gr-qc/9503050]; R. Durrer [18] P.Creminelli,K.Hinterbichler,J.Khoury,A.Nicolisand and F. Vernizzi, Phys. Rev. D 66, 083503 (2002) [hep- E.Trincherini,JHEP1302,006(2013)[arXiv:1209.3768 ph/0203275]. [hep-th]]. [39] Y. S. Piao, Phys. Lett. B 701, 526 (2011) [19] K.Hinterbichler,A.Joyce,J.KhouryandG.E.J.Miller, [arXiv:1012.2734 [hep-th]]. JCAP 1212, 030 (2012) [arXiv:1209.5742 [hep-th]]. [20] K.Hinterbichler,A.Joyce,J.KhouryandG.E.J.Miller,

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