Anisotropic cosmological models in Horndeski gravity
Rafkat Galeev, Ruslan Muharlyamov, Alexei A. Starobinsky, Sergey V. Sushkov, Mikhail S. Volkov
AAnisotropic cosmological models in Horndeski gravity
Rafkat Galeev, ∗ Ruslan Muharlyamov, † Alexei A. Starobinsky,
2, 1, ‡ Sergey V. Sushkov, § and Mikhail S. Volkov
3, 1, ¶ Department of Physics, Kazan Federal University,Kremlevskaya str. 18, Kazan 420008, Russia L.D. Landau Institute for Theoretical Physics RAS, Moscow 119334, Russia Institut Denis Poisson, UMR - CNRS 7013,Universit´e de Tours, Parc de Grandmont, 37200 Tours, France
It was found recently that the anisotropies in the homogeneous Bianchi I cosmol-ogy considered within the context of a specific Horndeski theory are damped nearthe initial singularity instead of being amplified. In this work we extend the analysisof this phenomenon to cover the whole of the Horndeski family. We find that thephenomenon is absent in the K-essence and/or Kinetic Gravity Braiding theories,where the anisotropies grow as one approaches the singularity. The anisotropies aredamped at early times only in more general Horndeski models whose Lagrangianincludes terms quadratic and cubic in second derivatives of the scalar field. Suchtheories are often considered as being inconsistent with the observations because theypredict a non-constant speed of gravitational waves. However, the predicted valueof the speed at present can be close to the speed of light with any required precision,hence the theories actually agree with the present time observations. We considertwo different examples of such theories, both characterized by a late self-accelerationand an early inflation driven by the non-minimal coupling. Their anisotropies aremaximal at intermediate times and approach zero at early and late times. The earlyinflationary stage exhibits an instability with respect to inhomogeneous perturba-tions, suggesting that the initial state of the universe should be inhomogeneous.However, more general Horndeski models may probably be stable. ∗ [email protected] † [email protected] ‡ [email protected] § sergey [email protected] ¶ [email protected] a r X i v : . [ g r- q c ] M a r I. INTRODUCTION
It is usually assumed that the state of the universe close to the initial singularity should bestrongly anisotropic [1–3]. This belief is based on the fact that spatial anisotropies producein the Einstein equations terms which become dominant when one goes backwards in time.In other words, anisotropic perturbations grow to the past. When the universe expands, theanisotropy terms decrease faster than the contribution of other forms of energy subject tothe dominant energy condition, and the universe rapidly approaches a locally isotropic stateduring inflation [4], [5] (without the inflationary stage this process may require a longtimeor may not happen at all due to the possibility of recollapse). Therefore, thinking aboutthe early history of the universe, one could expect the isotropic phase of inflation to begenerically preceded by an anisotropic phase.Although this argument seems quite robust, an explicit example in which the anisotropiesin the Bianchi I homogeneous model are damped at early times instead of being amplifiedwas recently found [6] within the context of a specific Horndeski theory for a gravitatingscalar field [7]. Therefore, the initial stage of the universe in this theory is not anisotropic.It remained unclear whether the finding of [6] is generic or specific only for one particularHorndeski model. To find the answer, we extend in what follows the analysis of [6] to coverthe whole of the Horndeski family. We find that the effect of the anisotropy damping is notnecessarily present in all Horndeski theories. In particular, it is absent in the the K-essenceand/or Kinetic Gravity Braiding theories. The spatial anisotropies in such theories alwaysgrow as one approaches the singularity. However, the anisotropies are damped at early (andlate) times in more general Horndeski models whose Lagrangian includes terms quadraticand cubic in second derivatives of the scalar field. Such theories are often considered as beinginconsistent with the observations because they predict a non-constant speed of gravitationalwaves (GW) [8–10], whereas the GW170817 event shows that the GW speed is equal to thespeed of light with very high precision [11]. However, the theories actually predict the valueof the GW speed at present to be close to unity within the required precision. In addition,the theories admit stable in the future self-accelerating cosmologies. Therefore, they canperfectly agree with the current observations, and we can extrapolate them to the earlytimes as well since no observational data about the GW speed at redshifts z > . II. HORNDESKI THEORY
This is the most general theory for a gravity-coupled scalar field φ whose equations areat most of second order. The theory was first obtained in [7], but we shall use its action inthe form given in [13]: S = (cid:90) ( L + L + L + L ) √− g d x , (2.1)where L = G ( φ, X ) , L = − G ( φ, X ) (cid:50) φ , L = G ( φ, X ) R + G X ( φ, X ) (cid:2) ( (cid:3) φ ) − ( ∇ µ ∇ ν φ ) (cid:3) , L = G ( φ, X ) G µν ∇ µ ∇ ν φ − G X ×× (cid:2) ( (cid:50) φ ) − (cid:50) φ ( ∇ µ ∇ ν φ ) + 2 ( ∇ µ ∇ ν φ ) (cid:3) . (2.2)Depending on the choice of the four arbitrary functions G A ( φ, X ) (with A = 2 , , ,
5) ofthe scalar field φ and of its canonical kinetic term X = − ∇ µ φ ∇ µ φ , this determines notjust one theory but a large family of theories. One has G AX ≡ ∂G A /∂X , ( ∇ µ ∇ ν φ ) = ∇ µ ∇ ν φ ∇ ν ∇ µ φ , and ( ∇ µ ∇ ν φ ) = ∇ µ ∇ ν φ ∇ ν ∇ ρ φ ∇ ρ φ ∇ µ φ . Finally, R and G µν are theRicci scalar and the Einstein tensor.For example, setting G = G = 0, G = const and G = X − V ( φ ) yields the standardtheory of the inflaton type, a more general choice of G ( φ, X ) yields the K -essence theory[14], while including also G ( φ, X ) (cid:54) = 0 yields the Kinetic Gravity Braiding (KGB) theory[15]. The KGB theory, possible with G = G ( φ ), is the most general Horndeski model inwhich the sound speed of tensor perturbations is equal to the speed of light [8–10]. TheLagrangian of this theory contains the second derivatives of the scalar field only linearly. Ifthe Lagrangian contains also quadratic ( ∇ µ ∇ ν φ ) and/or cubic ( ∇ µ ∇ ν φ ) terms, which isthe case if G and/or G depend on X , then the GW speed is no longer constant. III. BIANCHI I MODEL
The simplest cosmological model is homogeneous and isotropic, with the metric ds = − N dt + a ( dx + dx + dx ) , (3.1)where the scale factor a, the lapse N, as well as the scalar field φ , depend only on t . Thecorresponding field equations for the theory (2.1) are explicitly shown in [13]. We make thenext step and consider the homogeneous and anisotropic Bianchi I metric, ds = − N dt + a dx + a dx + a dx , (3.2)with the three scale factors a m ( m = 1 , , φ dependingonly on t . Substituting this into (2.1) yields the reduced one-dimensional action that can bevaried with respect to a m , N and φ . Although the action contains the second derivatives, allhigher derivatives arising during the variation cancel. As a result, first varying the actionwith respect to N and a m and then imposing the gauge condition N = 1, yields the followingequations: G (cid:16) G − G X ˙ φ − G XX ˙ φ + 2 G φ ˙ φ + G Xφ ˙ φ (cid:17) = G − G X ˙ φ − G X H ˙ φ + G φ ˙ φ + 6 G φ H ˙ φ + 6 G Xφ ˙ φ H − G X H H H ˙ φ − G XX H H H ˙ φ , (3.3) G G ii − ( H j + H k ) d G dt = G − ˙ φ dG dt + 2 ddt ( G φ ˙ φ ) − ddt ( G X ˙ φ H j H k ) − G X ˙ φ H j H k ( H j + H k ) . (3.4)Here the dot denotes the t -derivative, one has H i = ˙a i / a i , and the average Hubble parameteris H = (cid:80) i =1 H i ≡ ˙a / a with a = (a a a ) / . The Einstein tensor components are G = − ( H H + H H + H H ) ,G ii = − (cid:16) ˙ H j + ˙ H k + H j + H k + H j H k (cid:17) , (3.5)where the triples of indices { i, j, k } take values { , , } , { , , } , or { , , } . In addition,we have defined G = 2 G − G X ˙ φ + G φ ˙ φ . (3.6)Varying the action (2.1) with respect to φ yields the equation which, after some rear-rangements, can be cast into the following form:1a ddt (a J ) = P , (3.7)with J = ˙ φ (cid:104) G X − G φ + 3 H ˙ φ ( G X − G Xφ )+ G ( − G X − φ G XX + 2 G φ + G Xφ ˙ φ )+ H H H (3 G X ˙ φ + G XX ˙ φ ) (cid:105) , (3.8) P = G φ − ˙ φ ( G φφ + G Xφ ¨ φ ) + RG φ + 2 G Xφ ˙ φ (3 ¨ φH − ˙ φG )+ G G φφ ˙ φ + G Xφ ˙ φ H H H , (3.9)where R is the scalar curvature, R = − G µµ .Let us parameterize the three scale factors asa = a e β + + √ β − , a = a e β + −√ β − , a = a e − β + , (3.10)hence H = H + ˙ β + + √ β − , H = H + ˙ β + − √ β − , H = H − β + , (3.11)where H = ˙a / a. The anisotropies are determined by ˙ β ± , and if they vanish, then H = H = H = H and the universe is isotropic. It will be convenient to introduce σ = ˙ β + ˙ β − . (3.12)Using these definitions, the G Einstein equation (3.3) assumes the form3 (cid:0) H − σ (cid:1) (cid:16) G − G X ˙ φ − G XX ˙ φ + 2 G φ ˙ φ + G Xφ ˙ φ (cid:17) = − G + ˙ φ G X +3 G X H ˙ φ − G φ ˙ φ − G φ H ˙ φ − G Xφ H ˙ φ + ˙ φ (5 G X + G XX ˙ φ )( H − β + ) (cid:2) ( H + ˙ β + ) − β − (cid:3) . (3.13)This equation contains only first derivatives. The remaining three Einstein equations (3.4)contain second derivatives and read (cid:0) H + 3 H + 3 σ (cid:1) G + 2 H ˙ G = − G + G φ ˙ φ + G X ˙ φ ¨ φ − ddt (cid:0) G φ ˙ φ (cid:1) + ddt (cid:104) G X ˙ φ (cid:0) H − σ (cid:1)(cid:105) +2 G X ˙ φ (cid:16) H + ˙ β − β + ˙ β − (cid:17) , (3.14) ddt (cid:104) G a ˙ β + + G X ˙ φ a (cid:16) ˙ β − − ˙ β − H ˙ β + (cid:17)(cid:105) = 0 , (3.15) ddt (cid:104) G a ˙ β − + G X ˙ φ a (cid:16) β + ˙ β − − H ˙ β − (cid:17)(cid:105) = 0 . (3.16)We notice that the two latter equations have the total derivative structure and can beintegrated once, which gives first order conditions G ˙ β + + G X ˙ φ (cid:16) ˙ β − − ˙ β − H ˙ β + (cid:17) = C + a , (3.17) G ˙ β − + G X ˙ φ (cid:16) β + ˙ β − − H ˙ β − (cid:17) = C − a , (3.18)with C + , C − being integration constants. Supplementing these two equations by the firstorder equation (3.13) and by the scalar field equation (3.7), yields a closed system of fourdifferential equations for the four functions a( t ) , β ± ( t ) and φ ( t ). The remaining equation(3.14) can be ignored, since it is automatically fulfilled by virtue of the Bianchi identities.An additional simplification is achieved if the scalar source P defined by (3.9) vanishes,since in this case the scalar field equation (3.7) also assumes the total derivative structureand can be integrated once. The source P will vanish if all four functions G A are independenton φ , in which case the theory is invariant under shifts φ → φ + φ . However, P will vanishalso if G and G are independent of φ , while G and G depend on φ only linearly , suchthat G φ = const and G φ = const . Then the scalar field equation (3.7) becomes˙ φ (cid:104) G X + 3 HG X ˙ φ + G ( − G X − φ G XX + 2 G φ )+( H − β + )[( H + ˙ β + ) − β − ](3 G X ˙ φ + G XX ˙ φ ) (cid:105) + C φ a = 0 , (3.19)with C φ being an integration constant. The problem therefore reduces in this case to fourequations (3.13),(3.17),(3.18) and (3.19) which determine algebraically the Hubble parameter H (a), the anisotropies ˙ β ± (a), and the derivative of the scalar field ˙ φ (a).To recapitulate, if there is an explicit dependence on φ , then the problem reduces tofour differential equations (3.13),(3.17),(3.18) and (3.7) to determine a( t ), β ± ( t ), φ ( t ). If thecoefficient functions G , G are φ -independent while G , G depend on φ at most linearly,then the problem reduces to four equations (3.13),(3.17),(3.18),(3.19) which determine thefunctions H (a), ˙ β ± (a) and ˙ φ (a) algebraically . The time dependence can then be restored byintegrating the equation ˙a / a = H (a).In what follows we shall not at first assume anything about the φ -dependence, but laterwe shall consider specific examples admitting the simplified description in terms of the fouralgebraic equations. Our aim is to study the anisotropies described by (3.17) and (3.18).The structure of these equations suggests considering separately two different cases, G X = 0and G X (cid:54) = 0, which will be described, respectively, in the following two Sections. IV. THE G X = 0 CASE
In this case the anisotropy equations (3.17) and (3.18) are linear in ˙ β ± and yield˙ β ± = C ± G a . (4.1)The behaviour of the anisotropies is therefore determined by the function G defined by (3.6).This definition can equivalently be viewed as the equation for G , G ( φ, X ) = 2 G ( φ, X ) − X ∂G ( φ, X ) ∂X + 2 XG φ ( φ ) , (4.2)whose solution is G ( φ, X ) = f ( φ ) √ X + g (cid:48) ( φ ) X − √ X (cid:90) G ( φ, X ) X / dX, G = g ( φ ) , (4.3)with arbitrary f ( φ ) and g ( φ ). Let us first consider the subcase where A. G = µ = const In this case Eq.(4.1) yields ˙ β ± = C ± µ a , (4.4)so that the anisotropies behave in the same way as in General Relativity: they grow asa →
0. Therefore, the initial singularity is strongly anisotropic, while at late times theanisotropies decay. Eq.(4.2) then yields G ( X ) = µ f ( φ ) √ X + g (cid:48) ( φ ) X, G = g ( φ ) . (4.5)This describes all the conventional theories. Setting f ( φ ) = g ( φ ) = 0 one can, depending onwhether G and G are included or not, distinguish the following particular cases. • G = G = G = 0, G = µ/
2. This corresponds to the vacuum General Relativity,assuming that µ = M . • G = X − V ( φ ) and G = G = 0, G = µ/
2, which defines the General Relativitywith the conventional scalar field. • G ( φ, X ) and G = G = 0, G = µ/
2, which gives the K-essence theory. • G ( φ, X ), G ( φ, X ), G = 0, G = µ/
2, which gives the KGB theory.In all of these theories the anisotropies ˙ β ± grow as one approches the initial singularity. B. G = µ ( φ ) Formulas (4.1),(4.5) still apply, with the replacement µ → µ ( φ ). Let us consider thesimplest option: G = X, G = G = 0 , G = 12 µ ( φ ) , ˙ β ± = C ± µ ( φ ) a . (4.6)Since G depends on φ , the φ -equation remains differential and the system does not reduceto algebraic equations. At the same time, the theory with the gravitational kinetic term µ ( φ ) R can be converted to the theory with the standard kinetic term µR by a conformaltransformation of the metric. This brings us back to the theories considered in the previoussubsection, where the anisotropies are always unbounded near singularity. Performing theinverse conformal transformation to pass to the original frame changes only the scale factor(and the proper time) without changing the anisotropies. Hence the latter are unboundedin the original frame too. Therefore, the choice G = µ ( φ ) does not insure the damping ofanisotropies, and we shall now consider a more complex choice. C. G = G ( X ) and G ( X ) We shall consider the theory sometimes called “kinetic inflation” [12, 16–18], [19–23]. Itcorresponds to the choice G = X − Λ , G = 0 , G = 12 ( µ + γX ) , G = 12 ( α + γ ) φ, (4.7)where Λ, µ, α are constant parameters. The constant γ is a gauge parameter which dropsout from the equations due to the relation XR + ( (cid:50) φ ) − ( ∇ µ ∇ ν φ ) = − φG µν ∇ µ ∇ ν φ + total derivative [13], which allows one to trade the G ∼ φ term in the Lagrangian (2.2) forthe G ∼ X term. In the γ = 0 gauge one has G = const and G ∼ φ , while choosing γ = − α yields G = 0.The homogeneous and isotropic cosmologies in the model (4.7) are characterized, apartfrom the late inflationary phase driven by Λ, also by an early inflationary phase with theHubble rate determined not by Λ but rather by α , so that Λ is “screened at early times”[24]. The GW speed in the theory is not constant, but its value at present is predicted tobe close to the speed of light with a very high precision [6].Injecting (4.7) to (4.2) yields G = µ + αX ⇒ ˙ β ± = C ± ( µ + αX )a . (4.8)It turns out that X = ˙ φ / → H − ˙ β − ˙ β ) (cid:18) µ + 32 α ˙ φ (cid:19) = 12 ˙ φ + Λ , (cid:16) α ( H − ˙ β − ˙ β ) − (cid:17) ˙ φ = C φ a , (cid:16) µ + α φ (cid:17) ˙ β ± = C ± a . (4.9)We shall need a dimensionless version of these equations. Let us assume that α >
0. If H and a are the present values of the Hubble parameter and of the scale factor, then settinga = a a, H = H √ y, Λ = 3 µH Ω , α = 13 H ζ ,C φ = (cid:112) µα Ω H a , C ± = µH a Q ± , ˙ φ = (cid:114) µ Ω α ψ , ˙ β ± = H s ± (4.10)reduces (4.9) to equations containing only dimensionless variables a, Y, ψ, s ± and dimension-less parameters ζ, Ω , Ω :Ω (3 Y − ζ ) ψ + Y = Ω , ( Y − ζ ) ψ = 1 a , (Ω ψ + 1) s ± = Q ± a , (4.11)where Y = y − s − s − . The solution can be expressed in the parametric form, as functionsof Y : a = Ω ( ζ − Y )( Y − ζ ) ( Y − Ω ) , ψ = 1( Y − ζ ) a , s ± = Q ± S , y = Y + s + s − , (4.12)0 ln ( a ) H / H cT2 cS2 - - - FIG. 1. The dimensionless Hubble rate √ y = H/H , the sound speeds squared in the scalar andtensor sectors against ln( a ) for the isotropic ( Q ± = 0) solution (4.12) with Ω = 0 .
73 and ζ = 60. where S = 1(Ω ψ + 1) a = ( Y − ζ ) a Ω + ( Y − ζ ) a . (4.13)When the parameter Y ranges from ζ/ , the scale factor a changes, respectively, fromzero to infinity. As one can see, the function S determining the anisotropies approaches zeroin both of these limits, hence the universe becomes isotropic not only at late times but alsoat early times. In both limits the amplitude Y reduces to y and the Hubble rate is (cid:18) H early H (cid:19) ≡ ζ ← y = (cid:18) HH (cid:19) → Ω ≡ (cid:18) H late H (cid:19) as 0 ← a → ∞ . (4.14)Therefore, the universe interpolates between the early and late isotropic inflationary stagesdriven by ζ and Ω , respectively. The present stage of the universe is highly isotropic, hence Y ≈ y = a = 1 should fulfill (4.12), which requires thatΩ = (1 − ζ ) (1 − Ω )( ζ − . (4.15)As a result, the theory actually depends only on two parameters ζ and Ω determining valuesof the two Hubble rates, apart from the anisotropy charges Q ± .Setting Q ± = 0 yields homogeneous and isotropic solutions, in which case one can applythe known formulas about describing small fluctuations. These formulas apply also foranisotropic solutions with Q ± (cid:54) = 0 at late and early times, when the solutions become1 H / HS ln ( a ) - - A = = S ( t - t ) H00 - - FIG. 2. Left: the anisotropy amplitude S and the Hubble rate H/H defined by (4.12),(4.13) withΩ = 0 . ζ = 60 and A = (cid:113) Q + Q − = 1 against ln( a ). Right: S ( t ) for A = 1 and A = 10. isotropic. The quadratic action for fluctuations is I = µ (cid:90) K (cid:18) ˙ F − c p a F (cid:19) a d x, (4.16)where F denotes the fluctuation amplitude after separating the variables and p is the spatialmomentum. The expressions for the kinetic term K and the sound speed squared c withinthe model (4.7) were derived in [6], and they agree with the earlier result obtained within thegeneric Horndeski theory [13]. It turns out that the kinetic term is always positive, both inthe tensor and scalar sectors, hence there are no ghosts. As seen in Fig.1, the sound speedsin both sectors are not constant, but they approach unity at late times. The deviationof the speed of tensor modes from unity at present is negligibly small and proportional to( H late /H early ) [6].It is also worth mentioning that, when written in the gauge where G = 0 and hence G = ( µ − αX ) /
2, the theory (4.7) can be mapped to Class I DHOST theory [25] viaa disformal transformation of the metric g µν → ˜ g µν = A ( X ) g µν + B ( X ) ∇ µ φ ∇ ν φ . Thistransformation changes the light cone, hence the sound speeds change. If the functions A, B are chosen such that B ( A − XB ) G = G X then the resulting DHOST theory will respectthe condition which insures that the GW speed is equal to the speed of light (in the languageof [26] this condition is α = α = 0; see Eq.(D.5) of that work). Therefore, the GW speedcan be made constant via the disformal transformation.2The anisotropies are s ± = Q ± S where, as seen in Fig.2, the function S is well localized,hence the anisotropies vanish both at the early and late stages of the universe and aremaximal in between. It is worth noting that, as seen in Fig.2, the anisotropies contributeto the Hubble rate and increase it. Since dt = 1 H d ln( a ) , (4.17)the proper time interval dt decreases if H increases, hence the proper time duration of theanisotropic period decreases when the anisotropy amplitude A ≡ (cid:112) Q + Q − gets larger,since H then increases. In other words, the function S ( t ) shows a more and more narrowpeak when A gets larger, as seen in Fig.2.One should emphasise that, although the anisotropies approach zero at early times, stillthe universe cannot be isotropic at this stage, since it is unstable in this limit with respect toinhomogeneous perturbations. This can be seen in Fig.1, which shows that the sound speedssquared become negative at early times. This means that the early stage of the universeshould be inhomogeneous [6].To recapitulate, the above example shows that anisotropies in the theory with G = µ + α X are damped at early times. It is possible that choosing other functions G ( X ) yieldsother models with a similar property. However, we shall now rather return to the originalanisotropy equations (3.17) and (3.18) and consider situations when the nonlinear terms inthese equations become important. V. THE G ( X ) CASE
Theories with a nontrivial G ( X ) are also characterized by a non-constant GW speed.We shall consider a theory which also shows two inflationary stages, similarly to the G ( X )model considered above. It is defined by the choice G = X − Λ , G = 0 , G = 12 µ, G = const + ξ √ X. (5.1)3Equations (3.13),(3.17)-(3.19) then reduce to3 µ ( H − ˙ β − ˙ β − ) + 4 ξ ˙ φ (2 ˙ β + − H )[( H + ˙ β + ) − β − ] = 12 ˙ φ + Λ , ˙ φ (cid:16) − ξ (2 ˙ β + − H )[( H + ˙ β + ) − β − ] (cid:17) = C φ a , (cid:16) µ ˙ β + − ξ ˙ φ ( ˙ β + H ˙ β + − ˙ β − ) (cid:17) = C + a , (cid:16) µ + ξ ˙ φ (2 ˙ β + − H )) (cid:17) ˙ β − = C − a , (5.2)all containing terms nonlinear in ˙ β ± . Their dimensionless version is obtained by settinga = a a, H = H y, Λ = 3 µH Ω , ξ = − H ζ ,C φ = (cid:112) − Ω µξH H a , C ± = µH a Q ± , ˙ φ = (cid:114) − µH ξ ψ , ˙ β ± = H s ± , (5.3)where we assume that the coupling ξ is negative , hence ζ >
0. This yields the equations3 ( y − s − s − ) + 4 ψ [( y − s + )[( y + s + ) − s − ] − ζ ] = 3Ω , (cid:0) ζ − ( y − s + )[( y + s + ) − s − ] (cid:1) ψ = √ Ω a , (cid:0) s + + ψ [ s + ys + − s − ] (cid:1) = Q + a , (cid:0) ψ ( y − s + )] (cid:1) s − = Q − a . (5.4)Consider first the isotropic case, s ± = 0 , Q ± = 0 . (5.5)Then equations (5.4) reduce to3 y + 4 ψ ( y − ζ ) = 3Ω , (4 ζ − y ) ψ = √ Ω a , (5.6)with the solution a = Ω ( ζ − y )3(4 ζ − y ) ( y − Ω ) , ψ = √ Ω a (4 ζ − y ) . (5.7)This solution again shows the early and late inflationary stages, since the Hubble parameter ζ ← y = HH → (cid:112) Ω as 0 ← a → ∞ . (5.8)Requiring the solution to pass through the a = y = 1 point yieldsΩ = 3(4 ζ − (1 − Ω )( ζ − . (5.9)4 H / H Sc c ln ( a ) - - - FIG. 3. The Hubble rate, the sound speeds squared and the anisotropy amplitude S defined by(5.15) against ln( a ) for the solution (5.7) with Ω = 0 . ζ = 5. The sound speed squared inthe scalar sector is always positive. In the linear approximation assumed in (5.15) the anisotropiesdo not contribute to the Hubble rate. Choosing Ω = 0 . ζ = 5 then yields the result shown in Fig.3. Remarkably, we see thatthe sound speed squared in the scalar sector is now always positive. The kinetic terms arealso positive, and there remains only the gradient instability in the tensor sector at earlytimes. Therefore, the theory is more stable than the G ( X ) model considered above. Thissuggests that other choices of functions G A ( X ) may perhaps give completely stable theories,but this issue requires a separate analysis.Equation (5.7) actually defines not one by two different solutions related to each othervia a → − a and ψ → − ψ , since a = ±√ a can be either positive or negative whereasthe metric contains only a and is insensitive to the sign of a . As we shall see below, theanisotropic generalizations of these two solutions will no longer be related to each other ina simple way.Let us consider anisotropic solutions of (5.4), starting from the simplest case where Q ± = 0. The simplest solution is then the isotropic one, s ± = 0 , (5.10)with a and ψ given by (5.7). In addition, since the equations are nonlinear in the anisotropies,there are also solutions with s ± (cid:54) = 0. They can be represented in the parametric form,5choosing ψ as the parameter: a = √ Ω ψ ζ ψ − ψ + 1 , y = 23 ζ ψ + 12 Ω ψ − ψ , (5.11)with the anisotropies being either s + = 12 (cid:18) y + 1 ψ (cid:19) , s − = ±√ s + , (5.12)or s + = − (cid:18) y + 1 ψ (cid:19) , s − = 0 . (5.13)The parameter ψ in (5.11) takes values in the interval [0 , ψ m ] where ψ m is the root of4 ζ ψ − ψ + 1 = 0. For example, if ζ = 0 . = 0 . ψ m = 0 .
92. When ψ increases from zero to ψ m , the scale factor a grows from zero to infinity, while the Hubbleparameter y and the anisotropy behave as follows: ∞ ← y → − Ω ψ ψ , − ∞ ← y + 1 ψ → − Ω ψ as 0 ← a → ∞ . (5.14)We see that the anisotropies s ± ∼ ( y + 1 /ψ ) do not vanish at late times but approachconstant values, unless for Ω = 0. This again provides a counterexample to the standardwisdom. Indeed, in General Relativity the Bianchi universes with a positive cosmologicalconstant always evolve toward an isotropic state at late times [4], [5]. The solution (5.11)-(5.14), although also containing a positive cosmological constant, shows just the opposite“self-anisotropizing” behaviour. It should be said that such a self-anisotropization in theHorndeski theory with a non-trivial G ( X ) was actually detected before in Ref. [27], alsowhen analyzing the Bianchi I models. We therefore shall not discuss this phenomenonanymore and simply refer to [27], since we are interested in the early time “isotropization”rather than in the late time “anisotropization”. For all other solutions that we considerin this text, apart from (5.11)-(5.14), the anisotropies always approach zero at late times.Therefore, we now return back to the isotropic solution (5.10) and consider its deformationsinduced by adding nonzero anisotropy charges Q ± .If Q ± are very small, then one can expect the anisotropies s ± to be small as well, in whichcase one can neglect all nonlinear in s ± terms in the equations. The first two equations in(5.4) contain only such terms, and neglecting them yields the equations of the isotropic casewhose solution was described above by (5.7). The last two equations in (5.4) do contain6 H / H ln (| a |) - - - - FIG. 4. The Hubble rate
H/H = y = Y − s + against ln( | a | ) for the two solutions of (5.22) withΩ = 0 . ζ = 5 for S = 0 .
02. The Hubble rate is affected by the anisotropies when the nonlinearterms are taken into account. terms linear in s ± , and keeping only these yields the solution s ± = Q ± a (1 + ψ y ) ≡ Q ± S , (5.15)with a, ψ are given by (5.7). The function S here is well localized, as seen in Fig.3, and ithas the following limits: 36 ζ Ω a ← S → a as 0 ← a → ∞ . (5.16)Therefore, the anisotropies are suppressed both at early and late times.If the charges Q ± are not small, then one cannot neglect the nonlinear in anisotropiesterms in the equations. It is not then obvious that the anisotropies will still be suppressedat early and late times. Let us therefore take the nonlinear terms into account. To simplifythe analysis, we set one of the anisotropy amplitudes and the corresponding charge to zero, s − = Q − = 0 , (5.17)while keeping s + (cid:54) = 0 and denoting Q + = (cid:112) Ω S. (5.18)It turns out that all nonlinear in s + terms in the equations can be absorbed by introducingthe new variable Y = y + s + . (5.19)7 s + ln (| a |) - - - - - - - FIG. 5. The anisotropy s + defined by (5.21) against ln( | a | ) for the two solutions shown in Fig.4,assuming the normalizaton (5.9). Then equations (5.4) reduce, without any approximation, to4 ψ ( Y − Y s + − ζ ) + 3 Y − Y s + = 3Ω ,ψ (4 ζ + 3 Y s + − Y ) = √ Ω a , ( Y ψ + 1) s + = S √ Ω a . (5.20)Their solution is a = √ Ω (6 S Y ψ − Y ψ − ψ ( Y − ζ )( Y ψ + 1) , s + = S √ Ω a ( Y ψ + 1) , (5.21)where Y and ψ are related via ψ = 34 Y − Ω ζ − Y + 32 S Y ψ ζ Y ψ + Y + 8 ζ − Y ( Y ψ + 1)( Y − ζ ) . (5.22)If S = 0 then s + = 0, Y = y , and these formulas reduce to (5.7) describing the isotropicsolution with y = Y ∈ [ √ Ω , ζ ]. If S (cid:54) = 0 then (5.22) yields the fourth order algebraicequation for ψ = ψ ( Y ). Fortunately, all of its four solutions can be found analytically.Each of them is real-valued only within a finite interval of Y , but combining these piecewisesolutions together yields two global solutions which are smooth and real-valued everywherein the interval Y ∈ [ √ Ω , ζ ]. These two solutions have opposite signs of ψ and of a .In the isotropic limit these two solutions are related by simply ψ → − ψ and a → − a ,as described above, while their Hubble rates y ( | a | ) are the same. If S (cid:54) = 0 then the two8solutions are no longer related to each other in a simple way and their Hubble rates aredifferent, as seen in Figs.4. As seen in Fig.5, the anisotropies again vanish at late and earlytimes. These nonlinear solutions were obtained for the value of the anisotropy parameterwhich is still small enough, S = 0 .
02, but increasing S does not qualitatively change thesituation. Aalready for S = 1 the anisotropy s + attains very large values in the intermediateregion, but it always approaches zero as a → , ∞ . Therefore, the anisotropies are dampedat early times also at the nonlinear level. VI. CONCLUSIONS
Summarizing the above discussion, we have studied homogeneous and anisotropic Bianchi Icosmologies within the most general Horndeski class. Our aim was to see whether the phe-nomenon of anisotropy damping previously observed within the specific Horndeski model[6] is present in other Horndeski theories as well. We have found the phenomenon to beabsent for a large class of Horndeski models in which the GW speed is constant. However,the phenomenon seems to be generically present in the more general models with nontrivial G ( X ) and/or G ( X ). The GW speed in such theories is not constant, but no contradictionwith the observation arises since the predicted value of the GW speed at present is extremelyclose to unity, whereas no observation data of the GW speed in the past are available.Such theories show gradient instabilities at early times, therefore their initial phase,although not anisotropic, cannot be isotropic either. It should therefore be inhomogeneous.At the same time, it is possible that a systematic analysis of theories with more general G ( X, φ ) and/or G ( X, φ ) may reveal models free of instabilities. In the case of nonsingularbounce-type [28] or Genesis-type [29]) cosmologies, no stable solution can exist within theHorndeski class [30], [31], although they exist within the more general DHOST models(see [32] for a review). However, we are unaware of similar no-go results for cosmologieswith an initial singularity. In fact, an explicit example of a completely stable Horndeskitheory is known, although not containing an early inflationary phase [33]. Therefore, it isnot excluded that stable cosmologies with the early and late inflationary phases may existwithin the Horndeski theory, hence their anisotropies should be damped near singularity.It should also be mentioned that, as was first observed in [6], the effect of anisotropydamping may be sensitive to the inclusion of spatial curvature.9
ACKNOWLEDGMENTS
It is a pleasure to thank Karim Noui for discussions. R.G., R.M., A.A.S. and S.V.S weresupported by the Russian Foundation for Basic Research, grant No.19-52-15008. M.S.V. waspartly supported by the CNRS/RFBR PRC grant No.289860. This work was also partiallysupported by the Kazan Federal University Strategic Academic Leadership Program. [1] V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz,
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