Power-law Genesis: strong coupling and galileon-like vector fields
aa r X i v : . [ h e p - t h ] A p r INR-TH-2020-022
Power-law Genesis: strong coupling andgalileon-like vector fields.
P. K. Petrov a,ba
Department of Particle Physics and Cosmology, Faculty of Physics, M. V. LomonosovMoscow State University, Vorobyovy Gory, 1-2, Moscow, 119991, Russia b Institute for Nuclear Research of the Russian Academy of Sciences,60th October Anniversary Prospect, 7a, 117312 Moscow, Russia
Abstract
A simple way to construct models with early cosmological Genesis epoch is to em-ploy bosonic fields whose Lagrangians transform homogeneously under scaling trans-formation. We show that in these theories, for a range of parameters defining theLagrangian, there exists a homogeneous power-law solution in flat space-time, whoseenergy density vanishes, while pressure is negative (power-law Genesis). We find thecondition for the legitimacy of the classical field theory description of such a situation.We note that this condition does not hold for our earlier Genesis model with vectorfield. We construct another model with vector field and power-law background solutionin flat space-time, which is legitimately treated within classical field theory, violatesthe NEC and is stable. Upon turning on gravity, this model describes the early Genesisstage.
Genesis [1] is a cosmological scenario without initial singularity. In this scenario, the Universestarts its expansion from flat space-time and zero energy density at large negative times. Asthe Universe evolves, the energy density and the Hubble rate grow, and eventually reachlarge values. If gravity is described by General Relativity, then this regime requires thedomination of exotic matter which violates the Null Energy Condition, NEC (for a reviewsee [2]). Later on, the energy density of exotic matter has to be converted into the energydensity of usual matter, and the conventional cosmological evolution starts. As shown in [3],the violation of the NEC in a healthy way is possible in the context of the scalar Galileon1heories [4]. By now, numerous ways to implement the Genesis idea have been proposed,mostly in the context of theories involving scalars (see Ref. [1, 5] for an incomplete list andRef. [6] for topical review), but also in vector field models [7].A straighforward way to construct a model of the early Genesis epoch is to make use of aLagrangian which, in the absence of gravity, transforms homogeneously under scaling trans-formation: L ⇒ λ N L when π α ( x ν ) ⇒ λ s π α ( λx ν ), where π α denote the non-gravitationalfields in the model, and N and s are constant parameters. Then, quite generally, the model,still in the absence of gravity, has a spatially homogeneous solution π ∝ | t | − s as t → −∞ (power-law Genesis, see Sec. 2), for which the energy density vanishes while pressure is neg-ative. This precisely means the violation of the NEC. When gravity (described by GR) isturned on, energy density no longer stays equal to zero; instead, it increases as required forthe early Genesis stage. This mechanism has been invented in Ref. [1] (with N = 4 and s = 1) and then utilized in other contexts (see Ref. [2] for a review), including models withvector fields [7].However, within this class of models, the coefficients in the quadratic Lagrangian forperturbations about the classical solution often tend to zero as t → −∞ , which implies thatthe strong coupling energy scale also tends to zero. In such a situation, the classical treatmentmay become problematic, cf. Refs. [1, 8]. To figure out whether or not this is the case, oneshould study both qadratic and interaction terms in the Lagrangian for perturbations andfind the behavior of the strong coupling scale Λ as t → −∞ :Λ( t ) ∝ | t | − σ . This scale should be compared with the classical energy scale E cl , which is merely theevolution rate, and in the power-law Genesis case is given by E cl ( t ) ∝ | t | − . The classical treatment is legitimate provided that E cl ≪ Λ, which means σ ≤ σ = 1 is subtle: the relation E cl ≪ Λ may be valid in a restricted region ofparameter space).In this note we address this strong coupling issue in the context of the power-like mod-els described above. This is done in Sec. 2, where we show that the requirement (1.1) isequivalent to N ≤ . We note that this property does not hold for Genesis with vector field proposed in Ref. [7].Therefore, in Sec. 3 we construct another model with vector field and power-law background2olution that obeys (1.1); we determine the range of parameters in which the background isstable and violates the NEC in Minkowski space. For completeness, we also turn on gravity(in the form of GR) and describe the evolution of the scale factor at the early Genesis stage.
As outlined in Introduction, we consider the Lagrangian for M bosonic fields π α , α =1 , , ..., M , in 4d Minkowski space. Index α may either enumerate the fields (say, if π α arescalars) or be Lorentz index, or both. The Lagrangian is assumed to transform homoge-neously under scaling transformation x ν ⇒ λx ν , π α ( x ν ) ⇒ λ s π α ( λx ν ) , s = 0 . Namely, L ⇒ λ N L . (2.1)Importantly, we assume that equations of motion are second order in derivatives, even thoughthe Lagrangian may involve second derivatives of the fields. This is the case in generalizedGalileon theories [2, 4, 9] as well as in theories with Galileon-like vector fields [7].We consider for definiteness the Lagrangians which are linear combinations of the mono-mials involving n fields without derivatives, m first derivatives and l second derivatives ofthe fields (the argument goes through if one allows also for inverse powers of the fields):( π α ...π α n ) · ( ∂π γ ...∂π γ m ) · ( ∂ π ω ...∂ π ω l ) ∼ [ π ] n · [ ∂π ] m · [ ∂ π ] l . (2.2)Here ns + m ( s + 1) + l ( s + 2) = N, so that the transformaton law (2.1) holds. For a range of parameters defining the Lagrangian,there exists a homogeneous power-law solution π (0) α = β α | t | − s , (2.3)with constant β α . Indeed, the term (2.2) gives a contribution to equation of motion with totalnumber of fields equal to ( n + m + l −
1) and total number of derivatives ( m + 2 l ). Therefore,with the Ansatz (2.3), each of the M equations of motion is proportional to | t | − N + s with theproportionality coefficient being a polynomial in β α . In other words, equations of motionmake a system of M algebraic equations for M coefficients β α , which has a solution for arange of parameters entering the Lagrangian . Unless there is some symmetry that relates coefficients of different monomials (2.2) in such a way thatthis algebraic system does not have a real solution. π α = π (0) α + δπ . Ourpurpose is to determine the time-dependence of the lowest strong coupling scale in the limit t → −∞ . We begin with quadratic Lagrangian for perturbations. Since we assume thatthere are no third and higher derivatives of δπ α in the equations of motion, there are noterms with second and higher derivatives in the quadratic Lagrangian. So, the relevantterms are, schematically, ( ∂ δπ ) . The monomial (2.2) in the original Lagrangian contributesto the terms ( ∂ δπ ) in the quadratic Lagrangian with coefficients involving ( n + m + l − π (0) and ( m + 2 l −
2) derivatives acting on them. Hence, the structure ofthe quadratic Lagrangian is L (2) ⊃ | t | − N +2 s +2 ( ∂ δπ ) . This implies that canonically normalized fields are ξ α ∝ | t | − N/ s +1 δπ α . (2.4)Their mass dimension, by definition, equals 1.We now turn to the interactions between perturbations δπ . The term (2.2) inducesinteractions of the following form:[ π (0) ] n − a · [ ∂π (0) ] m − b · [ ∂ π (0) ] l − c × [ δ π ] a · [ ∂ δ π ] b · [ ∂ δ π ] c , where a + b + c ≥ . (2.5)We make use of (2.3) and (2.4) and find that in terms of canonically normalized field, thisinteraction Lagrangian is proportional to | t | N ( a + b + c − c − a × [ ξ ] a · [ ∂ξ ] b · [ ∂ ξ ] c . On dimensional grounds, the coefficient of [ ξ ] a · [ ∂ξ ] b · [ ∂ ξ ] c in the Lagrangian is E − ( a +2 b +3 c − s ,where E s is the (naive) strong interaction scale (we consider the case a + 2 b + 3 c − > E s ∝ | t | − N a + b + c − c − aa +2 b +3 c − . We require that this scale is higher than the classical energy scale t − for | t | → ∞ and get N ( a + b + c −
2) + c − aa + 2 b + 3 c − < , or ( N − a + b + c − < .
4e recall (2.5) and obtain finally N ≤ , where we include the case N = 4 in which both classical and quantum strong coupling scalesbehave as | t | − , and the quantum scale may be higher due to specific relationships betweenthe parameters in the Lagrangian, see Ref. [1] for an example. We now construct a simple vector field model which is covariant under scaling transformation A µ ( x ν ) → λ s A µ ( λx ν ), so that the Lagrangian transforms as given by (2.1), and, furthermore, N ≤ N = and s = − : L = q ( D A ρ (cid:3) A ρ + kB + lC + u ( F µν F νρ A µ,ρ + 2 A ρ,µ A ρ,ν A ,νµ ) , (3.1)where q , k , l and u are free parameters, and F µν = ∂ µ A ν − ∂ ν A µ ,D = A µ ; ν A µ A ν ,B = A µ A ν A µ ; λ A ν ; λ ,C = A µ ; τ A τ A ρ A µ ; ρ . In accordance with Sec. 2, equations of motion have a solution A bgµ = ( β | t | , , ,
0) (3.2)with constant β . This classical evolution occurs in a weak coupling regime at early times, t → −∞ .We now wish to figure out whether there exists a set of parameters q , k , l , u in theLagrangian (3.1), such that the solution (3.2) is stable and violates the NEC. By solving thefield equation, we find β = 20 u m − , where m = l + k + u.
5o see the NEC-violation, we need the expression for the energy-momentum tensor of thissolution: T µν = 2 δ ( √− g L ) √− gδg µν (cid:12)(cid:12) g ρσ = η ρσ . To this end, we consider minimal coupling to the metric, i.e., set A µ ; ν = ∇ ν A µ , (cid:3) A ρ = ∇ µ ∇ µ A ρ , D = A µ ; ν A τ A λ g µτ g νλ , etc., in curved space-time. The Lagrangian (3.1) can bewritten in the following form: L = 12 f ( D ) (cid:3) F − f ( D ) A τ ; σ A τ ; σ + L ( A µ , A λ,ν )where F = A µ A µ ,f ( D ) = qD ,L = q [ kB + lC + u ( F µν F νρ A µ,ρ + 2 A ρ,µ A ρ,ν A ,νµ )] . We find T = 0 ,T ij = pδ ij , i, j = 1 , , ,p = (cid:16) − ∂ τ f ∂ τ F + L − f A τ ; σ A τ ; σ (cid:17)(cid:12)(cid:12)(cid:12) g µν = η µν ; A µ = A bgµ . (3.3)This gives p = qu − (11 − m )(3 m − ( − t ) − , t < . (3.4)Thus, the background A bgµ violates the NEC provided that q (11 − m ) < . (3.5)Let us consider the stability of the solution A bgµ . Having in mind Ref. [10], we alsorequire subluminality of the perturbations about it. Stability conditions and conditions forthe absence of superluminal perturbations for Galileon-like vector models were derived inRef. [7]. Making use of the results of Ref. [7], it is straightforward to find that there aretwo ranges of parameters such that all these conditions together with (3.5) are satisfied for t <
0: ( I ) q > ,u = 0 , < k , − k < l < − k + 19 , II ) q > ,u = 0 ,k > , − k < l < − k + 19 , Thus, our example shows that there are stable homogeneous solutions in vector theoriesthat violate the NEC and avoid strong coupling regime at early times.
Here we construct an initial stage of the cosmological Genesis scenario, similiar to Ref. [1].To this end, we turn on gravity and assume that it is described by conventional GeneralRelativity, while the vector field is minimally coupled to metric, as described above. Im-portantly, all equations of motion, for both vector field and metric, remain second order inderivatives [7], just like in Horndeski theories.In the asymptotic past, space-time is assumed to be Minkowskian, and in accordancewith (3.3), (3.4), energy-momentum tensor vanishes as t → −∞ . At large but finite | t | , gravitational effects on the vector field evolution are negligible, so, to the leading order in M − P l , the energy density and pressure are given by (3.3), (3.4). Then the Hubble parameteris obtained from ˙ H = − πG ( ρ + p ) . We find H = 40 πGqu − ( m − m − (cid:0) − t (cid:1) − , t → −∞ . Thus, the Universe undergoes accelerated expansion characteristic of the early Genesis epoch.At this stage, perturbations about the background are stable and subluminal.
Acknowledgements
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