The general form of the coupled Horndeski Lagrangian that allows cosmological scaling solutions

Transcription

1 The general form of the coupled Horndeski Lagrangian that allows cosmological scaling solutions Adalto R. Gomes Universidade Federal do Maranhão UFMA Based on a collaboration with Luca Amendola, Heidelberg University arxiv: , JCAP ; arxiv: , JCAP III José Pĺınio Baptista School of Cosmology September 2016 Pedra Azul, ES, Brazil Support: CNPq, FAPEMA

2 Outline 1. Introduction 2. General Horndeski Lagrangian and equations of motion 3. Scaling condition and master equation 4. Solutions of the master equation 5. Conclusions

3 1 Introduction Models based on scalar fields to explain the accelerated expansion of the Universe. The main goal: suitable solutions to the background and perturbation equations of motion and to study their stability properties and their degree of independence of the initial conditions. Scalar field Lagrangian expanded by including terms coupled to gravity and terms that are general functions of the kinetic energy. The most general 4-dimensional, Lorentz covariant, local scalar-tensor theories keeping the equations of motion at second order are described by the Horndeski Lagrangian Deffayet, Deser and Esposito-Farese, 2009

4 The Horndeski Lagrangian Horndeski 1975, Deffayet et al No ghosts, No classical instabilities L H = 5 i=2 L i 1 L 2 = Kφ, X, 2 L 3 = G 3 φ, X φ, 3 L 4 = G 4 φ, XR + G 4,X [ φ 2 µ ν φ µ ν φ], 4 L 5 = G 5 φ, XG µ,ν µ ν φ G 5,X[ φ 3 3 φ µ ν φ µ ν φ+2 µ α φ α β φ β µ φ]. 6 where X = 1 2 µφ µ φ.

5 Application to specific theories De Felice, Kobayashi, Tsujikawa, PLB 2011 fr theories - L = M pl 2 2 fr J. O Hanlon PRL 1972, T. Chiba PLB 2003 K = M 2 pl 2 Rf R f, G 3 = G 5 =0, G 4 = 1 2 M plφ, φ = M pl f R. 7 Brans-Dicke theories with potential V φ K = M plω BD X φ V φ, G 3 = G 5 =0, G 4 = 1 2 M plφ. 8 Covariant Galileon without the field potential C. Deffayet et al, PRD2009. K = c 2 X, G 3 = c3 M 3X, G 4 = 1 2 M pl c4 M 6X2, G 5 = 3c 5 M 9X2 9

6 Kinetic gravity braidings C. Deffayet, O. Pujolas, I. Sawicki, A. Vikman, JCAP K = Kφ, X, G 3 = G 3 φ, X, G 4 = 1 2 M plφ, G 5 =0 10 Purely kinetic coupled gravity Gubitosi and Linder, PLB 2011 K = X, G 4 = 1 2 M pl, G 5 = λ φ Mpl 2. 11

7 An exhaustive study of the Horndeski model is very difficult due to the number of free functions. It is therefore interesting to ask whether one can find some general property without solving the equations of motion. An important class of cosmological solutions that has been studied for several models are the so-called scaling solutions, defined by the ratio of energy density Ω m /Ω φ constant. Asecondconditionthathasalsobeenoftenemployedtosimplifythetreatmentisthatthefield equation of state ω φ remains constant. Scaling solutions are particularly interesting because one can hope to employ them to avoid the problem of the coincidence between the present matter and dark energy densities, i.e. the fact that today the two density fractions Ω m, Ω φ are very similar.

8 We consider that there is only one type of matter of energy density ρ m = T 0 0,intheEinstein frame, where the energy-momentum tensor is defined by we neglect the small baryon component T µν = 2 δs m. 12 g δg µν In this frame matter is directly coupled to the scalar field through the function Qφ,where Q = 1 ρ m g δs m δφ. 13 F. Piazza and S. Tsujikawa JCAP, 2004: the most general action of the scalar quadratic kinetic term and of the field itself that contains scaling solutions must have the form with S = d 4 x g [ 1 R + Kφ, X 2 ] + S m φ, ψ i,g µν 14 Kφ, X =XgXe λφ, 15 where g an arbitrary function and λ aconstantrelatedtothecouplingbetweenmatterandscalar field.

9 L. Amendola, M. Quartin, S. Tsujikawa, I. Waga PRD, 2006: the former result has been extended for the case of variable coupling A. R. Gomes and L. Amendola JCAP, 2014: inclusion of G 3 φ, X µ µ φ in the Lagrangian, with variable coupling with solution where S = d 4 x g [ ] 1 2 R + Kφ, X G 3φ, X µ µ φ + S m φ, ψ i,g µν 16 LX, φ, φ =XgY ay 2 + ry φ. 17 Y = Xe λφ 18 and λ = Q 1+weff Ω φ w m w φ. 19

10 2 General Horndeski Lagrangian and equations of motion We consider a FLRW flat metric with ds 2 = dt 2 + A 2 tdx 2,whereAt isthescalefactorand H = A/A. Weconsiderpressurelessmatter. Varying the action with respect to g µν,anddefiningh = JCAP 2012 A/A, onegetsdefeliceandtsujikawa, where 5 i=2 5 i=2 E i = ρ m, 20 P i = 0, 21 E 2 = 2XK,X K, 22 E 3 = 6X φhg3,x 2XG 3,φ, 23 E 4 = 6H 2 G 4 +24H 2 XG 4,X + XG 4,XX 12HX φg4,φx 6H φg4,φ, 24 E 5 = 2H 3 X φ5g5,x +2XG 5,XX 6H 2 X3G 5,φ +2XG 5,φ,X. 25

11 and P 2 = K, P 3 = 2XG 3,φ + φg3,x, P 4 = 23H 2 +2ḢG 4 12H 2 XG 4,X 4HẊG 4,X 8ḢXG 4,X 8HXẊG 4,XX +2 φ +2H φg4,φ +4XG 4,φφ +4X φ 2H φg4,φx, 28 P 5 = 2X2H 3 φ +2HḢ φ +3H 2 φg5,x 4H 2 X 2 φg5,xx +4HXẊ HXG 5,φX + 2[2ḢX + HẊ+3H 2 X]G 5,φ +4HX φg5,φφ. 29 Now, in order to confront these models with SNIa observations, we isolate from the complete action a term corresponding to the Einstein-Hilbert one. Then the action is rewritten as S = d 4 x 1 g 2 R + L + S m φ, ψ i,g µν 30

12 with L = Kφ, X G 3 φ, X φ + G 4 φ, X 1 R + G 4,X [ φ 2 µ ν φ µ ν φ] 2 +G 5 φ, XG µν µ ν φ 1 6 G 5,X[ φ 3 3 φ µ ν φ µ ν φ +2 µ α φ α β φ β µ φ 31 and we write the Einstein equations in the usual form Tsujikawa, 2011 H 2 = 1 3 ρ φ + ρ m 32 and 2Ḣ = ρ m + ρ φ + p. 33 Comparing Eqs. 32, 33 with Eqs. 20, 21, we arrive to useful definitions for the energy

13 density ρ φ andpressurep ofdarkenergy: or ρ φ p 5 i=2 5 i=2 E i +3H 2, 34 P i 3H 2 +2Ḣ, 35 ρ φ 2XK,X K +6X φhg3,x 2XG 3,φ +3H 2 1 2G 4 +24H 2 XG 4,X + XG 4,XX 12HX φg4,φx 6H φg4,φ +2H 3 X φ5g5,x +2XG 5,XX 6H 2 X3G 5,φ +2XG 5,φ,X, 36 p K 2XG 3,φ + φg3,x 3H 2 +2Ḣ1 2G 4 12H 2 XG 4,X 4HẊG 4,X 8ḢXG 4,X 8HXẊG 4,XX +2 φ +2H φg4,φ +4XG 4,φφ +4X φ 2H φg4,φx 2X2H 3 φ +2HḢ φ +3H 2 φg5,x 4H 2 X 2 φg5,xx +4HXẊ HXG 5,φX + 2[2ḢX + HẊ+3H 2 X]G 5,φ +4HX φg5,φφ. 37

14 Defining ρ φ Ω φ = 3H 2, Ω m = ρ m 3H 2 38 we can rewrite the Friedman equation Eq. 32 as Ω φ +Ω m =1. 39 Now we introduce the e-folding time N =loga, sothatd/dt = Hd/dN. motion for the scalar field φ and matter can be written as where w φ = p/ρ φ. Then the equations of dρ φ dn +31+w φρ φ = ρ m Q dφ 40 dn dρ m dn +3ρ m = ρ m Q dφ dn, 41

15 3 Scaling condition and master equation The condition Ω φ /Ω m constant defines scaling solutions. This is equivalent to ρ φ /ρ m constant, or to d log ρ φ dn = d log ρ m 42 dn Also, from Eq. 39 we get that Ω φ is a constant. We also assume that on scaling solutions the equation of state parameter w φ is a constant Tsujikawa and Sami, PLB Then we get where w eff = w φ Ω φ. dφ dn = 3Ω φw φ Q 1 Q, 43 d log ρ φ = d log ρ m dn dn = 31 + w eff, 44 d log p dn = 31 + w eff, 45 We proved that, for the FRW metric, the pressure is equivalent to the original Lagrangian density: p = L

16 For FRW metric we also get R = 1 3 p. 46 w φ Ω φ Then we can write the pressure of dark energy as p = 1 fφ, X [ Kφ, X G 3 φ, X φ + G 4,X [ φ 2 φ] +G 5 φ, X φ 1 6 G 5,X[ φ 3 3 φ φ +2 φ ] 47 That is, p = px, φ, φ, φ, φ, φ with fφ, X [ 1+ ] 1 2 G 4φ, X c. 48

17 and φ µ ν φ µ ν φ 49 φ G µν µ ν φ 50 φ µ α φ α β φ β µ φ 51 Ageneralized masterequation forp = px, φ, φ, φ, φ, φis found to be log p φ + log p log φ dφ dn + log p log X d log X dn d log φ dn + log p log φ Each partial derivative was obtained, restricting the coupling to dq log p λq 2 dφ log X dq λq 2 dφ log p log φ + + log p d log φ log φ dn + log p d log φ log φ dn d log φ dn = 31 + w eff dq log p λq 2 dφ log φ dq λq 2 dφ 1 dq Q 2 dφ = c constant, leading to 2+ 2 dq λq 2 dφ log p log φ 1 log p λq φ log p log φ =1. 53

18 4 Solutions for the master equation We start with Eq. 53 rewritten as 1+ 2 dq log p Q dψ log X dq log p Q dψ log φ dq log p Q dψ log φ dq log p Q dψ log φ dq log p Q dψ log φ log p =1, 54 ψ where ψ = φ du[λqu]. 55 Now set p = XQ 2 φ GX, φ, φ, φ, φ, φ. 56

19 where G is an arbitrary function of its argument. Then for G 0weobtain dq Q dψ 2+ 1 dq Q dψ X G X dq Q dψ φ G φ + φ G φ dq Q dψ 2+ 2 dq Q dψ φ G φ G ψ φ G φ =0. 57 Here, inspired by the expression given by Eq. 47 for p, welookforsolutionsof G of the form GX, φ, φ, φ, φ, φ = 1 φ, φ, φ, φ, φ. 58 fφ, X gx, Then, Eq. 57 turns into {[ dq X f f Q dψ X f ] g ψ f [ 1+ 2 dq X g Q dψ X dq φ g Q dψ φ dq Q dψ dq g φ Q dψ φ dq g φ Q dψ φ g ψ g φ φ ]} =0. 59 For f 0thisisequivalenttoÔ1f G Ô2 g =0. Thesimplestchoiceistoimposethat g and f

20 satisfy separately the linear differential equations: Ô 1 f = 0, 60 Ô 2 g = The solution of the first equation is known. For the second one, and guided by the form of p, ishas the form of a superposition of general functions in terms of combinations of X, ψ, φ, φ, φ, φ: g = gh b +g c h c +g d h d +..., 62 with h b X, ψ = f 1b Xf 3b ψ, 63 h c X, φ = f 1c Xf 2c φ h q X, φ, ψ = f 1q Xf 3q ψf 5q φ Each particular differential equation is solved separately, with solutions depending on arbitrary con-

21 stants. For example we get g q h q =q X φe 3ψ Q 3,andsoon. The general form of p impose relations between the general constants in order to get a compatible solution for the general functions. Then, after some algebra, the general scaling Horndeski Lagrangian obtained is L H = XQ 2 gxq 2 e ψ [d 1 XQ 3 e ψ + l 1 X 2 Q 5 e 2ψ ] φ + hφ+ 12 d 2X 2 Q 4 e 2ψ + 13 l 2X 3 e 3ψ Q 6 R + d 2 XQ 4 e 2ψ + l 2 X 2 e 3ψ Q 6 [ φ 2 µ ν φ µ ν φ] +qx 2 e 3ψ Q 5 G µν µ ν φ 1 6 2qXe 3ψ Q 5 [ φ 3 3 φ µ ν φ µ ν φ +2 µ α φ α β φ β µ φ]. 68

22 5 Conclusions Integrating the Lagrangian by parts, we proved that the entire Horndeski Lagrangian is equivalent to the scalar field pressure at least in a FLRW metric. It is possible to show that there are two equivalent versions of the general Horndeski Lagrangian that allows for scaling solutions: in the form presented here, the general Horndeski functions K, G 3,G 4,G 5 are more evident, but the Lagrangian depends on the coupling Q; wealsoscaled the field to obtain the Lagrangian with a constant coupling, but with an even more intrincate structure. The existence of this particular class of solution is interesting since it could represent a solution of the coincidence problem. If the ratio Ω m /Ω φ depends on the fundamental constant of the theory, instead of on initial conditions, then the fact that it is close to unity would no longer be a surprising coincidence.