Cosmological constraints on extended Galileon models

Abstract

The extended Galileon models possess tracker solutions with de Sitter
attractors along which the dark energy equation of state is constant during the
matter-dominated epoch, i.e. w_DE = -1-s, where s is a positive constant. Even
with this phantom equation of state there are viable parameter spaces in which
the ghosts and Laplacian instabilities are absent. Using the observational data
of the supernovae type Ia, the cosmic microwave background (CMB), and baryon
acoustic oscillations, we place constraints on the tracker solutions at the
background level and find that the parameter s is constrained to be s=0.034
(-0.034,+0.327) (95% CL) in the flat Universe. In order to break the degeneracy
between the models we also study the evolution of cosmological density
perturbations relevant to the large-scale structure (LSS) and the
Integrated-Sachs-Wolfe (ISW) effect in CMB. We show that, depending on the
model parameters, the LSS and the ISW effect is either positively or negatively
correlated. It is then possible to constrain viable parameter spaces further
from the observational data of the ISW-LSS cross-correlation as well as from
the matter power spectrum.

Full text

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arXiv:1112.1774v1 [astro-ph.CO] 8 Dec 2011
Cosmological constraints on extended Galileon models
Antonio De Felice1,2and Shinji Tsujikawa3
1TPTP & NEP, The Institute for Fundamental Study,
Naresuan University, Phitsanulok 65000, Thailand
2Thailand Center of Excellence in Physics, Ministry of Education, Bangkok 10400, Thailand
3Department of Physics, Faculty of Science, Tokyo University of Science,
1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
(Dated: December 9, 2011)
The extended Galileon models possess tracker solutions with de Sitter attractors along which the
dark energy equation of state is constant during the matter-dominated epoch, i.e. wDE = −1 − s,
where s is a positive constant. Even with this phantom equation of state there are viable parameter
spaces in which the ghosts and Laplacian instabilities are absent. Using the observational data of
the supernovae type Ia, the cosmic microwave background (CMB), and baryon acoustic oscillations,
we place constraints on the tracker solutions at the background level and find that the parameter s
is constrained to be s = 0.034+0.327
between the models we also study the evolution of cosmological density perturbations relevant to
the large-scale structure (LSS) and the Integrated-Sachs-Wolfe (ISW) effect in CMB. We show that,
depending on the model parameters, the LSS and the ISW effect is either positively or negatively
correlated. It is then possible to constrain viable parameter spaces further from the observational
data of the ISW-LSS cross-correlation as well as from the matter power spectrum.
−0.034(95 % CL) in the flat Universe. In order to break the degeneracy
I.INTRODUCTION
The main target of the dark energy research over the next few years or so is to distinguish between the Λ-Cold-Dark-
Matter (ΛCDM) model and dynamical models with time-varying equations of state wDE. From the observational data
of WMAP7 combined with baryon acoustic oscillations (BAO) [1] and the Hubble constant measurement [2], Komatsu
et al. [3] derived the bound wDE= −1.10±0.14 (68 % CL) for the constant equation of state. Adding the supernovae
type Ia (SN Ia) data provides tighter constraints on wDE, but still the phantom equation of state (wDE < −1) is
allowed by the joint data analysis [3]. This property persists for the time-varying dark energy equation of state with
the parametrization such as wDE= w0+ wa(1 − a) [4], where a is the scale factor [5].
In the framework of General Relativity (GR) it is generally difficult to construct theoretically consistent models of
dark energy which realize wDE< −1. In quintessence [6] with a slowly varying scalar-field potential, for example, the
field equation of state is always larger than −1. A ghost field with a negative kinetic energy leads to wDE< −1 [7],
but such a field is plagued by a catastrophic instability of the vacuum associated with the spontaneous creation of
ghost and photon pairs [8].
In modified gravitational theories it is possible to realize wDE < −1 without having ghosts and Laplacian-type
instabilities (see Refs. [9]). In f(R) gravity, where the Lagrangian f is a function of the Ricci scalar R, the dark
energy equation of state crosses the cosmological constant boundary (wDE = −1) [10–13] for the viable models
constructed to satisfy cosmological and local gravity constraints [10–12, 14]. This is also the case for the Brans-Dicke
theory [15] with a field potential which accommodates the chameleon mechanism [16] to suppress the propagation of
the fifth force [17]. In modified gravity models of dark energy based on the chameleon mechanism (including f(R)
gravity), the effective potential of a scalar degree of freedom needs to be carefully designed to pass cosmological and
local gravity constraints [18].
There is another class of modified gravity models of dark energy in which a nonlinear self-interaction of a scalar
degree of freedom φ can lead to the recovery of GR in a local region through the Vainshtein mechanism [19]. The
representative models of this class are those based on the Dvali-Gabadadze-Porrati (DGP) braneworld [20] and the
Galileon gravity [21] (see Refs. [22, 23] for the implementation of the Vainshtein mechanism in these models). The
nonlinear interaction of the form (∂φ)2?φ, which appears from the brane-bending mode in the DGP model [22], gives
rise to the field equation invariant under the Galilean shift ∂µφ → ∂µφ +bµin the flat spacetime. This was extended
to more general field Lagrangians satisfying the Galilean symmetry in the limit of the Minkowski spacetime [21, 24].
The cosmology based on the covariant Galileon or on its modified versions has been studied by many authors
[25, 26]. In Refs. [27, 28] the dynamics of dark energy was investigated in the presence of the full covariant Galileon
Lagrangian. In this model the solutions with different initial conditions converge to a common trajectory (tracker).
Along the tracker the dark energy equation of state wDEchanges as −7/3 (radiation era) → −2 (matter era) → −1
(de Sitter era) [27–30]. There exists a viable parameter space in which the ghosts and Laplacian instabilities are
absent. However, the joint analysis based on the observational data of SN Ia, CMB, and BAO shows that the tracker

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solution is disfavored because of the large deviation of wDEfrom −1 during the matter era [30, 31]. The solutions
that approach the tracker only at late times are allowed from the combined data analysis [31].
As an extension of the covariant Galileon model, Deffayet et al. [32] obtained the most general Lagrangian in
scalar-tensor theories with second-order equations of motion. In four dimensions the corresponding Lagrangian is
of the form (1) with the four functions (2)-(5) given below. In fact this is equivalent to the Lagrangian found by
Horndeski [33] more than 3 decades ago [34, 35]. The conditions for the avoidance of ghosts and Laplacian instabilities
were recently derived in Ref. [36] in the presence of two perfect fluids (non-relativistic matter and radiation).
The covariant Galileon corresponds to the choice K = −c2X, G3 = c3X/M3, G4 = M2
3c5X2/M9in Eqs. (2)-(5), where ci’s are dimensionless constants, X = −∂µφ∂µφ/2, Mplis the reduced Planck mass,
and M is a constant having the dimension of mass. Kimura and Yamamoto [30] studied the model with the functions
K = −c2X, G3= c3M1−4nXn(n ≥ 1), G4= M2
during the matter era is given by wDE= −1−s with s = 1/(2n−1) > 0. At the background level this is equivalent to
the Dvali-Turner model [37], which can be consistent with the observational data for n larger than the order of 1. If
we consider the evolution of cosmological perturbations, the LSS tends to be anti-correlated with the late-time ISW
effect. This places the tight bound on the power n, as n > 4.2 × 103(95% CL) [38], in which case the dark energy
equation of state is practically indistinguishable from that in the ΛCDM model.
In Ref. [36] the present authors proposed more general extended Galileon models with the functions K =
−c2M4(1−p2)
masses Mi’s are fixed by the Hubble parameter at the late-time de Sitter solution with˙φ =constant. For the powers
p2= p, p3= p + (2q − 1)/2, p4 = p + 2q, p5= p + (6q − 1)/2, where p and q are positive constants, there exists
a tracker solution characterized by H˙φ2q=constant. During the matter-dominated epoch one has wDE= −1 − s,
where s = p/(2q), along the tracker. This covers the model of Kimura and Yamamoto [30] as a specific case (p = 1,
q = n − 1/2, c4= 0, c5= 0). In the presence of the nonlinear field self-interactions in G4and G5, the degeneracy
of the background tracker solution for given values of p and q is broken by considering the evolution of cosmological
perturbations. Hence the ISW-LSS anti-correlation found in Refs. [30, 38] for c4= c5= 0 should not be necessarily
present for the models with non-zero values of c4and c5.
In this paper we first place constraints on the tracker solution in the extended Galileon models by using the recent
observational data of SN Ia, CMB, and BAO. The bound on the value s = p/(2q) is derived from the background
cosmic expansion history with/without the cosmic curvature K. We then study the evolution of cosmological density
perturbations in the presence of non-relativistic matter to break the degeneracy of the tracker solution at the back-
ground level. We will show that the LSS and the ISW effect are either positively or negatively correlated, depending
on the parameters c4and c5. This information should be useful to distinguish between the extended Galileon models
with different values of c4and c5from observations.
pl/2 − c4X2/M6, G5 =
pl/2, and G5= 0, in which case the dark energy equation of state
2
Xp2, G3 = c3M1−4p3
3
Xp3, G4 = M2
pl/2 − c4M2−4p4
4
Xp4, and G5 = 3c5M−(1+4p5)
5
Xp5, where the
II.BACKGROUND FIELD EQUATIONS
We start with the following Lagrangian
L =
5
?
i=2
Li,
(1)
where
L2= K(X),
L3= −G3(X)?φ,
L4= G4(X)R + G4,X[(?φ)2− (∇µ∇νφ)(∇µ∇νφ)],
L5= G5(X)Gµν(∇µ∇νφ) − (G5,X/6)[(?φ)3− 3(?φ)(∇µ∇νφ)(∇µ∇νφ) + 2(∇µ∇αφ)(∇α∇βφ)(∇β∇µφ)]. (5)
K and Gi(i = 3,4,5) are functions in terms of the field kinetic energy X = −∂µφ∂µφ/2 with Gi,X≡ dGi/dX, R is
the Ricci scalar, and Gµν is the Einstein tensor. If we allow the φ-dependence for the functions K and Gi as well,
the Lagrangian (1) corresponds to the most general Lagrangian in scalar-tensor theories [32, 33]. In order to discuss
models relevant to dark energy we also take into account the perfect fluids of non-relativistic matter and radiation
(with the Lagrangians Lmand Lrrespectively), in which case the total 4-dimensional action is given by
ˆ
(2)
(3)
(4)
S =d4x√−g(L + Lm+ Lr).
(6)

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In the following we focus on the extended Galileon models [36] in which K and Giare given by
K = −c2M4(1−p2)
where Mplis the reduced Planck mass, ci and pi (i = 2,3,4,5) are dimensionless constants, and Mi(i = 2,3,4,5)
are constants having the dimension of mass. In the flat Universe it was shown in Ref. [36] that tracker solutions
characterized by the condition H˙φ2q= constant (q > 0 and a dot represents a derivative with respect to cosmic time
t) are present for
2
Xp2,G3= c3M1−4p3
3
Xp3,G4= M2
pl/2−c4M2−4p4
4
Xp4,G5= 3c5M−(1+4p5)
5
Xp5, (7)
p2= p,p3= p + (2q − 1)/2,p4= p + 2q ,p5= p + (6q − 1)/2.
(8)
The covariant Galileon [24] corresponds to p = 1 and q = 1/2, i.e. p2= p3= 1, p4= p5= 2.
We will extend the analysis to the general Friedmann-Lemaître-Robertson-Walker (FLRW) background with the
cosmic curvature K:
ds2= −dt2+ a2(t)
?
dr2
1 − Kr2+ r2(dθ2+ sin2θdφ2)
?
,
(9)
where a(t) is the scale factor. The closed, flat, and open geometries correspond to K > 0, K = 0, and K < 0,
respectively. For the theories given by the action (6) the dynamical equations of motion are
3H2M2
(3H2+ 2˙H)M2
˙ ρm+ 3Hρm= 0,
˙ ρr+ 4Hρr= 0,
˙ ρK+ 2HρK= 0.
pl= ρDE+ ρm+ ρr+ ρK,
(10)
pl= −PDE− ρr/3 + ρK/3,
(11)
(12)
(13)
(14)
Here H ≡ ˙ a/a, ρK ≡ −3KM2
respectively, and
pl/a2, ρm and ρr are the energy densities of non-relativistic matter and radiation
ρDE≡ 2XK,X− K + 6H˙φXG3,X− 6H2˜G4+ 24H2X(G4,X+ XG4,XX) + 2H3˙φX(5G5,X+ 2XG5,XX),
PDE≡ K − 2X¨φG3,X+ 2(3H2+ 2˙H)˜G4− 4(3H2X + H˙X + 2˙HX)G4,X− 8HX˙XG4,XX
−2X(2H3˙φ + 2H˙H˙φ + 3H2¨φ)G5,X− 4H2X2¨φG5,XX,
where˜G4≡ G4− M2
From Eqs. (10) and (11) we find that there exists a de Sitter solution characterized by˙H = 0 and¨φ = 0. In order
to discuss the cosmological dynamics we introduce the dimensionless variables [36]
(15)
(16)
pl/2 = −c4M2−4p4
4
Xp4.
r1≡
?xdS
x
?2q?HdS
H
?1+2q
,r2≡
??
x
xdS
?21
r3
1
?p+2q
1+2q
,Ωr≡
ρr
3H2M2
pl
,
(17)
where x ≡˙φ/(HMpl), and the subscript “dS” represents the quantities at the de Sitter solution. We relate the
masses Mi (i = 2,··· ,5) in Eq. (7) with HdS, as M2 ≡ (HdSMpl)1/2, M3 ≡ (H−2p3
(H−2p4
dS
M2−2p4
pl
)1/(2−4p4), and M5 ≡ (H2+2p5
the coefficients c2and c3are related with c4and c5, via
dS
M1−2p3
pl
)1/(1−4p3), M4 ≡
dS
M2p5−1
pl
)1/(1+4p5). The existence of de Sitter solutions demands that
c2=3
2
?
2
x2
dS
?p
(3α − 4β + 2),c3=
√2
2p + q − 1
?
2
x2
dS
?p+q
[3(p + q)(α − β) + p] ,
(18)
where
α ≡4(2p4− 1)
3
?x2
dS
2
?p4
c4,β ≡ 2√2p5
?x2
dS
2
?p5+1/2
c5.
(19)
The density parameter of dark energy, ΩDE≡ ρDE/(3H2M2
pl), can be expressed as
ΩDE=r
p−1
2q+1
1
r2
2
?
r1
?r1
?12(α − β)(p + q) + 4p − r1(2p − 1)(3α − 4β + 2)?− 3α(2p + 4q + 1)?+4β(p+3q+1)
?
. (20)

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From Eq. (10) it follows that ΩDE+ Ωm+ Ωr+ ΩK= 1, where Ωm≡ ρm/(3H2M2
The autonomous equations for r1, r2, and Ωrare written in terms of r1, r2, Ωr, α, β, p, q. As in the case of the
flat Universe [36] one can show that there is a fixed point for the differential equation of r1characterized by
pl) and ΩK≡ ρK/(3H2M2
pl).
r1= 1,
(21)
From the definition of r1 in Eq. (17) this corresponds to the tracker solution where H˙φ2q=constant. Along the
tracker the autonomous equations for r2and Ωrare
r′
2=(p + 2q)(Ωr+ 3 − 3r2− ΩK)
pr2+ 2q
r=2q(Ωr− 1 − 3r2− ΩK) − 4pr2
pr2+ 2q
r2,
(22)
Ω′
Ωr,
(23)
where a prime represents the derivative with respect to N = lna. Combining these equations, we obtain the integrated
solution
r2= c1a4(1+s)Ω1+s
r
,s =
p
2q,
(24)
where c1 is a constant. For the theoretical consistency the parameter s is positive [36]. Since r2 ∝ H−2(1+s), the
quantity r2grows toward the value 1 at the de Sitter solution. Along the tracker the density parameter (20) is given
by
ΩDE= r2=1 − Ωm,0− Ωr,0− ΩK,0
Ω1+s
r,0
e4(1+s)NΩ1+s
r
,
(25)
where the subscripts “0” represent the values today (the scale factor a0= 1, i.e. N0= lna0= 0). Using the relation
ΩK/Ωr= (ΩK,0/Ωr,0)e2Nas well, Eq. (23) reads
Ω′
r= −1 − Ωr+ ΩK,0e2NΩr/Ωr,0+ (1 − Ωm,0− Ωr,0− ΩK,0)(3 + 4s)e4(1+s)NΩ1+s
1 + (1 − Ωm,0− Ωr,0− ΩK,0)se4(1+s)NΩ1+s
r
/Ω1+s
r,0
r
/Ω1+s
r,0
Ωr.
(26)
This equation can be solved as
1 − Ωm,0− Ωr,0− ΩK,0
Ω1+s
r,0
e4(1+s)NΩ1+s
r
+Ωm,0
Ωr,0
eNΩr+ Ωr+ΩK,0
Ωr,0
e2NΩr= 1,
(27)
which is nothing but the relation ΩDE+Ωm+Ωr+ΩK= 1. From Eq. (25) the dark energy density parameter evolves
as ΩDE∝ H−2(1+s)and hence
H
H0
=
?ΩDE,0
ΩDE
?1/[2(1+s)]
.
(28)
Since it is not generally possible to solve Eq. (27) for Ωrin terms of N (apart from some specific values of s such as
s = 1), we numerically integrate Eq. (26) and find the expression of H/H0by using Eqs. (25) and (28).
Along the tracker the dark energy equation of state wDE ≡ PDE/ρDE and the effective equation of state weff =
−1 − 2˙H/(3H2) are given by
wDE= −3 + s(3 + Ωr− ΩK)
3(1 + sr2)
,weff= −r2(3s + 3 − sΩK) − Ωr
3(1 + sr2)
.
(29)
In the early cosmological epoch (r2≪ 1) these reduce to wDE≃ −1 − s(3 + Ωr− ΩK)/3 and weff ≃ Ωr/3. During
the matter era in which {Ωr,|ΩK|} ≪ 1 it follows that wDE≃ −1−s < −1 (for s > 0) and weff≃ 0. At the de Sitter
fixed point (r2= 1) with Ωr = ΩK = 0 one has wDE= weff = −1. In Fig. 1 we plot the evolution of wDE versus
the redshift z = a0/a − 1 in the open Universe with ΩK,0= 0.1 for the model parameters p = 1, q = 5/2, α = 3,
β = 1.45 (i.e. s = 0.2). The tracker is shown as a solid curve, along which wDEchanges as −1.267 (radiation era) →
−1.2 (matter era) → −1 (de Sitter era). The effect of the cosmic curvature ΩKbecomes important only for the late
Universe, which affects the luminosity distance in the SN Ia observations.

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Figure 1. Evolution of wDE versus the redshift z for the open Universe with ΩK,0 = 0.1 for p = 1, q = 5/2, α = 3, β = 1.45.
The solid curve corresponds to the tracker solution with the initial conditions r1 = 1, r2 = 1.0 × 10−30, Ωr = 0.99987,
ΩK = 4.0×10−12at log10(z +1) = 7.245. In the cases (A) and (B) the initial conditions are chosen to be (A) r1 = 1.0×10−2,
r2 = 1.0 × 10−23, Ωr = 0.99985, ΩK = 5.0 × 10−12at log10(z + 1) = 7.210, and (B) r1 = 3.0 × 10−6, r2 = 1.0 × 10−10,
Ωr = 0.9998, ΩK = 1.15 × 10−11at log10(z + 1) = 6.967, respectively.
If the solutions start from the regime r1≪ 1, the evolution of wDEis different from Eq. (29) before they reach the
tracker. For r1≪ 1 and r2≪ 1, wDEand weff are approximately given by
wDE≃ −1 + Ωr− ΩK
2(2p + 6q − 1),weff≃1
3Ωr.
(30)
In Fig. 1 we show the variation of wDE for p = 1, q = 5/2, α = 3, β = 1.45 with two different initial conditions
satisfying r1≪ 1. In both cases the density parameter ΩKtoday is ΩK,0= 0.1. The cases (A) and (B) correspond to
the early and late trackings, respectively. For smaller initial values of r1the tracking occurs later. As estimated by
Eq. (30), wDEstarts from the value wDE≃ −1/16 in the deep radiation era. If the solutions do not reach the tracker
during the matter era (as in the case (B) in Fig. 1), wDEtemporally approaches the value −1/32.
In Ref. [31] it was shown that the tracker for the covariant Galileon (s = 1) is disfavored from observations, but
the late-time tracking solution is allowed from the data. This property comes from the fact that for the late-time
tracker the deviation of wDEfrom −1 is not significant. For s ≪ 1 even the tracker is expected to be allowed from
observations. In such cases the solutions starting from the initial conditions with r1≪ 1 should be also compatible
with the data (because even for s = 1 the late-time tracking solution is allowed). In the following sections we will
focus on the tracker solution to discuss the background observational constraints and the evolution of cosmological
perturbations.
III.OBSERVATIONAL CONSTRAINTS ON THE EXTENDED GALILEON MODELS
In this section we place observational constraints on the tracker solution from the background cosmic expansion
history. We use three data sets: 1) the CMB shift parameters (WMAP7) [3]; 2) the BAO (SDSS7) [1]; 3) and the
SN Ia (Constitution) [39]. The total chi-square χ2
totfor all three combined data sets will be calculated on a grid
representing a chosen set of available parameters. We then find the minimum on this grid, and consequently find the
1σ and 2σ contours.
In order to integrate Eq. (26), once a set of model parameters (in this case not only s, but also Ωr,0, Ωm,0, and ΩK,0)
is given, we have the choice of one initial condition, that is Ωr,i≡ Ωr(Ni). In principle it is possible to solve Eq. (26)
backwards for a given value of Ωr(0) = Ωr,0, but we find that the integrated results are prone to numerical instabilities.