Issue 
A&A
Volume 555, July 2013



Article Number  A53  
Number of page(s)  18  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201321256  
Published online  01 July 2013 
Experimental constraints on the uncoupled Galileon model from SNLS3 data and other cosmological probes
^{1}
CEA,Centre de Saclay, Irfu/SPP,
91191
GifsurYvette,
France
email:
jeremy.neveu@cea.fr
^{2}
Center for Astrophysics and Space Astronomy, University of
Colorado, Boulder,
CO
803090389,
USA
^{3}
LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRSIN2P3, 4 place
Jussieu, 75252
Paris Cedex 05,
France
^{4}
Laboratoire de Physique Théorique d’Orsay, Bâtiment 210,
Université ParisSud 11, 91405
Orsay Cedex,
France
^{5}
𝒢ℝεℂ𝒪, Institut d’Astrophysique de Paris, UMR 7095CNRS, Université Pierre et Marie
CurieParis 6, 98bis boulevard
Arago, 75014
Paris,
France
Received:
7
February
2013
Accepted:
25
April
2013
Aims. The Galileon model is a modified gravity theory that may provide an explanation for the accelerated expansion of the Universe. This model does not suffer from instabilities or ghost problems (normally associated with higherorder derivative theories), restores local General Relativity – thanks to the Vainshtein screening effect – and predicts latetime acceleration of the expansion.
Methods. We derive a new definition of the Galileon parameters that allows us to avoid having to choose initial conditions for the Galileon field. We tested this model against precise measurements of the cosmological distances and the rate of growth of cosmic structures.
Results. We observe a weak tension between the constraints set by growth data and those from distances. However, we find that the Galileon model remains consistent with current observations and is still competitive with the ΛCDM model, contrary to what was concluded in recent publications.
Key words: supernovae: general / cosmology: observations / dark energy
© ESO, 2013
1. Introduction
The discovery of the accelerated expansion of the Universe (Riess et al. 1998; Perlmutter et al. 1999) led cosmologists to introduce dark energy to explain our Universe. Adding a cosmological constant (Λ) to Einstein’s General Relativity is the simplest way to interpret observational data. However, even if adding a new fundamental constant is satisfactory, the value of Λ obtained from numerous measurements results in significant finetuning and coincidence problems. Thus, there is theoretical motivation to find alternative explanations, such as modified gravity models.
The Galileon model is just such a formulation. It was first proposed by Nicolis et al. (2009) as a general theory involving a scalar field, hereafter called π, and a secondorder equation of motion invariant under a Galilean shift symmetry (∂_{μ}π → ∂_{μ}π + b_{μ}, where b_{μ} is a constant vector). This symmetry was first noticed in braneworld theories such as the DGP model of Dvali et al. (2000). The DGP model has the advantage of providing a selfaccelerating solution to explain the expansion of the Universe, but it is plagued by ghost and instability problems. Galileon theories are a generalization of the DGP model that avoid these problems. The Galileon model was derived in a covariant formalism by Deffayet et al. (2009). It was also shown that this model forms a subclass of the general tensorscalar theories involving only up to secondorder derivatives originally found by Horndeski (1974).
In a fourdimension spacetime, only five Lagrangian terms are possible when forming an equation of motion for π invariant under the Galilean symmetry. Therefore, the Galileon Lagrangian has only five parameters. In the Galileon theory, as in the DGP theory, a screening mechanism called the Vainshtein effect (Vainshtein 1972) arises near massive objects due to nonlinear derivative selfcouplings of the π field. These ensure that the Galileon fifth force is screened near massive objects, and preserves General Relativity on local scales where it has been experimentally tested to high precision. However, this screening is only effective below a certain distance from massive objects (the Vainshtein radius) that depends on the mass of the object and on the values of the Galileon parameters (Burrage & Seery 2010). Experimental constraints on the Galileon parameters based on local tests of gravity have been proposed by Brax et al. (2011) and Babichev et al. (2011).
Recently, the Galileon model has been tested against observational cosmological data by Appleby & Linder (2012b), Okada et al. (2013), and Nesseris et al. (2010). These authors tend to reject the Galileon model because of tensions between growthofstructure constraints and the other cosmological probes. The evolution of the Universe in the Galileon theory is based on differential equations involving the π field, which requires one to set initial conditions, and the above studies resorted to different methods for setting these initial conditions. In this work, we avoid this problem by introducing a new parametrization of the Galileon model that renders it independent of initial conditions. Combined with theoretical constraints derived in Appleby & Linder (2012a) and De Felice & Tsujikawa (2011), we compare our model with cosmological observables, and find that the Galileon model is not significantly disfavored by current observations.
We used the most recent measurements of Type Ia supernovae (SN Ia) luminosity distances, the cosmic microwave background (CMB), and baryon acoustic oscillations (BAO). The highestquality SN Ia sample currently available is the SNLS3 sample described in Guy et al. (2010), Conley et al. (2011), and Sullivan et al. (2011). For the CMB, we used the observables from WMAP7 (Komatsu et al. 2011) and the set of BAO distances of the BOSS analysis (Sánchez et al. 2012). The growth of structures is an important probe for distinguishing modified gravity models from standard cosmological models such as ΛCDM, so it has to be used carefully. In this work, we used fσ_{8}(z) measurements from several surveys, corrected for the AlcockPaczynski effect.
Section 2 provides the Galileon equations used to compute the evolution of the Universe and the theoretical constraints imposed on the Galileon field. Section 3 describes the likelihood analysis, data samples, and the computing of cosmological observables. Section 4 gives the constraints on the Galileon model derived from data, and Sect. 5 discusses these results and their implications. We conclude in Sect. 6.
2. Cosmology with Galileons
2.1. Lagrangians
The Galileon model is based on the assumption that the scalar field equation of motion is invariant under Galilean symmetries: ∂_{μ}π → ∂_{μ}π + b_{μ}, where b_{μ} is a constant four vector. By imposing this symmetry, Nicolis et al. (2009) showed that there are only five possible Lagrangian terms L_{i} for the Galileon model action. The covariant formulation of the Galileon Lagrangian was derived in Deffayet et al. (2009). In this paper we start with this covariant action with the parametrization of Appleby & Linder (2012a): (1)with L_{m} the standardmatter Lagrangian, M_{P} the Planck mass, R the Ricci scalar, and g the determinant of the metric. The c_{i}s are the arbitrary dimensionless parameters of the Galileon model that weight the different terms. The Galileon Lagrangians have a covariant formulation derived in Deffayet et al. (2009): (2)where M is a mass parameter defined as , where H_{0} is the current value of the Hubble parameter. With this definition the c_{i}s are dimensionless.
L_{2} is the usual kinetic term for a scalar field, while L_{3} to L_{5} are nonlinear couplings of the Galileon field to itself, to the Ricci scalar R, and to the Einstein tensor G_{μν}, providing the necessary features for modifying gravity and mimicking dark energy. L_{1} is a tadpole term that acts as the usual cosmological constant, and may furthermore lead to vacuum instability because it is an unbounded potential term. Therefore, in the following we set c_{1} = 0.
Appleby & Linder (2012a) proposed additional direct linear couplings to matter to add to the action: a linear coupling to matter and a derivative coupling to matter L_{G} = c_{G}∂_{μ}π∂_{ν}πT^{μν}/(M_{P}M^{3}), which arises in some braneworld theories (see e.g. Trodden & Hinterbichler 2011), where T^{μν} is the matter energymomentum tensor. These couplings may modify the physical origin of the accelerated expansion of the Universe. Without coupling, the Universe is accelerated only because of the backreaction of the metric to the energymomentum tensor of the scalar field, and the Galileon acts as a dark energy component. If the Galileon is coupled directly to matter, instead, it can give rise to accelerated expansion in the Jordan frame, while the Einsteinframe expansion rate is not accelerating. In that case, the cosmic acceleration stems entirely from a genuine modified gravity effect. In this work, we do not consider these optional extensions to the theory, so the Einstein frame and Jordan frame coincide. For more information about the Einstein and Jordan frames, see e.g. Faraoni et al. (1999).
Action 1 leads to three differential equations: two Einstein equations ((00) temporal component and (ij) spatial component) coming from the variation of the action with respect to the metric g_{μν}, and the scalar field equation of motion from the variation of the action with respect to the π field. The equations are given explicitly in Appendix B of Appleby & Linder (2012a). With these three differential equations the evolution of the Universe and the dynamics of the field can be computed.
To solve the cosmological equations, we chose the FriedmannLemaîtreRobertsonWalker (FLRW) metric. With no direct couplings, the functions to compute are the Hubble parameter H = ȧ/a (with a the cosmic scale factor), and x = π′/M_{P}, with a prime denoting d/dlna (see Appleby & Linder 2012a and Sect. 2.3).
2.2. Initial conditions
To compute the solutions of the above equations, we need to set one initial condition for x. We arbitrarily chose to define this initial condition at z = 0, which we denote x_{0} = x(z = 0). Unfortunately, we have no prior information about the value of the Galileon field or its derivative at any epoch. Fortunately, x_{0} can be absorbed by redefining the c_{i}s as follows: This redefinition allows us to avoid treating x_{0} as an extra free parameter of the model^{1}. Doing so, the s remain dimensionless, and the initial conditions are simple: (6)Note that the (00) Einstein equation could also be used as a constraint equation to fix x_{0} (see Appendix A) given a set of cosmological parameters c_{i}s, and . If we were to adapt this, we would observe a degeneracy between the parameters: the same cosmological evolution can be obtained with small c_{i}s and a high x_{0}, or with high c_{i}s and a small x_{0}. In other words, different sets of parameters {c_{i},x_{0}} produce the same cosmology, i.e., the same ρ_{π}(z), which is undesirable. Our parametrization avoids this problem by absorbing the degeneracy between the c_{i}s and x_{0} into our s.
2.3. Cosmological equations
To compute cosmological evolution in the Galileon model, we assume for simplicity that the Universe is spatially flat, in agreement with current observations. We used the FriedmannLemaîtreRobertsonWalker (FLRW) metric in a flat space: (7)When writing the cosmological equations, we can mix the (ij) Einstein equation and the π equation of motion to obtain the following system of differential equations for and : with as derived in the formalism of Appleby & Linder (2012a), but using our normalization for the c_{i}s. We obtain the same equations except that the c_{i}s are changed into s, and that we have a different treatment for the initial conditions. Equations (8) and (9) depend only on the s and . The radiation energy density in Eq. (14) is computed from the usual formula with N_{eff} = 3.04 the standard effective number of neutrino species (Mangano et al. 2002). The photon energy density at the current epoch is given by (where, as usual, h = H_{0}/(100 km s Mpc^{1}) for T_{CMB} = 2.725 K.
2.4. Perturbation equations
To test the Galileon model predictions for the growth of structures, we also need the equations describing density perturbations. We followed the approach of Appleby & Linder (2012a) for the scalar perturbation. Appleby & Linder (2012a) performed their computation in the frame of the Newtonian gauge, for scalar modes in the subhorizon limit, with the following perturbed metric: (16)In this context, the perturbed equations of the (00) Einstein equation, the (ij) Einstein equation, the π equation of motion, and the equation of state of matter are in the quasistatic approximation where δy = δπ/M_{P} is the perturbed Galileon, , ρ_{m} is the matter density, and δ_{m} = δρ_{m}/ρ_{m} is the contrast matter density. κ_{i}s are the same as in Appleby & Linder (2012a), but rewritten following our parametrization: With Eqs. (17) to (20), we can obtain a Poisson equation for ψ, with an effective gravitational coupling that varies with time and depends on the Galileon model parameters s: with G_{N} Newton’s gravitational constant. These equations can be used to compute the growth of matter perturbations in the frame of the Galileon model (see Sect. 3.2.4). Tensorial perturbations modes also exist, and are studied in Sect. 2.5.4.
2.5. Theoretical constraints
With so many parameters, it is necessary to restrict the parameter space theoretically before comparing the model to data. The theoretical constraints arise from multiple considerations: the (00) Einstein equation, requiring positive energy densities, and avoiding instabilities in scalar and tensorial perturbations.
2.5.1. The (00) Einstein equation and
Because we used only the (ij) Einstein equation and the π equation of motion to compute the dynamics of the Universe (Eqs. (8) and (9)), we are able to use the (00) Einstein equation as a constraint on the model parameters: (28) More precisely, we used this constraint both at z = 0 to fix one of our parameters and, at other redshifts, to check the reliability of our numerical computations (see Sect. 3.1). The parameter we chose to fix at z = 0 is (29)We chose to fix based on the other parameters because allowing it to float introduces significant numerical difficulties when solving Eqs. (8) and (9), since it represents the weight of the most nonlinear term in these equations. As is fixed given h, our parameter space has been reduced to and .
2.5.2. Positive energy density
We require that the energy density of the Galileon field be positive from z = 0 to z = 10^{7} (see Sect. 3.2.2 and Appendix B). At every redshift in this range, this constraint amounts to (30)This constraint is not really necessary for generic scalar field models. But as we will see in the following, it has no impact on our analysis because the other theoretical conditions described below are stronger.
2.5.3. Scalar perturbations
As suggested by Appleby & Linder (2012a), outside the quasistatic approximation the propagation equation for δy leads to two conditions, which we again checked from z = 0 to z = 10^{7} to ensure the viability of the linearly perturbed model:

1.
a noghost condition, which requires a positive energy for theperturbation (31)

2.
a Laplace stability condition for the propagation speed of the perturbed field (32)
2.5.4. Tensorial perturbations
We also addrd two conditions derived by De Felice & Tsujikawa (2011) for the propagation of tensor perturbations. Considering a traceless and divergencefree perturbation δg_{ij} = a^{2}h_{ij}, these authors obtained identical perturbed actions at second order for each of the two polarisation modes h_{⊕} and h_{⊗}. For h_{⊕}(34)with Q_{T} and c_{T} as defined below. From that equation, we extracted two conditions in our parametrization that have to be satisfied (again from z = 0 to z = 10^{7}):
These conditions allowed us to reduce our parameter space significantly. The Galileon model contains degeneracies between the s, as pointed out in e.g. Barreia et al. (2012). The above theoretical constraints and our new parametrization allowed us to break degeneracies between the parameters that would make it difficult to converge to a unique bestfit with current cosmological observations. As an example, the tensorial theoretical conditions lead to a significant reduction of the parameter space (see dark dotted regions in Fig. 2), so that closed probability contours are obtained.
3. Likelihood analysis method and observables
In the following, we define a scenario to be a specific realisation of the cosmological equations for a given set of parameters .
To perform the likelihood analysis, the method used in Conley et al. (2011) for the analysis of SNLS data^{2} was adapted to the Galileon model. For each cosmological probe, a likelihood surface ℒ was derived by computing the χ^{2} for each visited scenario: . The way h is treated is described in Sect. 3.2.2. Then we report the mean value of the marginalized parameters as the fit values of and the s.
3.1. Numerical computation method
To compute numerical solutions to Eqs. (8) and (9), we used a fourthorder RungeKutta method to compute and iteratively starting from the current epoch, where the initial conditions for and are specified (see 2.2), and propagating backwards in time to higher z. We used a sufficiently small step size in z to avoid numerical divergences. This is challenging because of the significant nonlinearities in our equations. To determine the step size, we therefore required that Eq. (28), normalized by , be satisfied at better than 10^{5} for each step.
At each step of the computation, we also checked that all previously discussed theoretical conditions were satisfied (Eqs. (30)–(32), (35), and (36)). Cosmological scenarios that fail any of these conditions were rejected and their likelihood set to zero. The result of these requirements is shown e.g. in Fig. 2 as dark dotted regions. Equation (30) concerns a negligible number of Galileon scenarios, but the four other constraints lead to a significant reduction of the parameter space.
3.2. Data
Here we describe the cosmological observations we used in our analysis. Special care was taken to choose data that do not depend on additional cosmological assumptions.
3.2.1. Type Ia supernovae
The SN Ia data sample used in this work is the SNLS3 sample described in Conley et al. (2011). It consists of 472 wellmeasured supernovae from the SNLS, SDSS, HST, and a variety of lowz surveys.
A Type Ia supernova with intrinsic stretch s and color has a restframe Bband apparent magnitude m_{B} that can be modeled as follows: (37)where is the Hubbleconstant free luminosity distance, which in a flat Universe is given by (38)z_{hel} and z_{CMB} are the SN Ia redshift in the heliocentric and CMB rest frames, respectively, “cosmo” represents the cosmological parameters of the model. α and β are parameters describing the lightcurve widthluminosity and colorluminosity relationships for SNe Ia. ℳ_{ℬ} is defined as ℳ_{ℬ} = M_{B} + 5log _{10}c/H_{0} + 25, where M_{B} is the restframe absolute magnitude of a fiducial () SN Ia in the Bband, and c/H_{0} is expressed in Mpc. α,β and ℳ_{ℬ} are nuisance parameters that are fit simultaneously with the cosmological parameters. As in Conley et al. (2011) and Sullivan et al. (2011), we allowed for different ℳ_{ℬ} in galaxies with the host galaxy stellar mass below and above 10^{10} M_{⊙} to account for relations between SN Ia brightness and host properties that are not corrected for via the standard s and relations. When computing Type Ia supernova distance luminosities in Sect. 4, we neglect the radiation component in , since all measurements are restricted to redshifts below 1.4 where the effects of radiation density are negligible.
Systematic uncertainties must be treated carefully when using SN Ia data, because they depend on α and β and due to covariances between different supernovae. We followed the treatment of Conley et al. (2011) and Sullivan et al. (2011).
3.2.2. Cosmological microwave background
The CMB is a powerful probe to constrain the expansion history of the Universe because it gives highredshift cosmological observables. The power spectrum provides much information on the content of the Universe and the relations between the different fluids, as long as we are able to model the thermodynamics of these fluids before recombination. The Galileon model does not modify the standard baryonphoton flux physics as long as the Galileon field does not couple directly to matter, as is assumed in this work. Thus, the usual formulae and predictions used in the standard analysis of the CMB power spectrum remain valid.
The positions of the acoustic peaks can be quantified by three observables: {l_{a},R,z_{∗}} (see e.g. Komatsu et al. 2011 and Komatsu et al. 2009), where l_{a} is the acoustic scale related to the comoving sound speed horizon, R is the shift parameter related to the distance between us and the last scattering surface, and z_{∗} is the redshift of the last scattering surface. These quantities are derived from the angular diameter distance, which in a flat space is given by (39)and from the comoving sound speed horizon: (40) is the usual normalized sound speed in the baryonphoton fluid before recombination: (41)where is the baryon energy density parameter today.
With the above definitions, the acoustic scale l_{a} is given by (42)and the shift parameter R by (43)z_{∗} is given by the fitting formula of Hu & Sugiyama (1996): According to Hu & Sugiyama (1996), formula (44) is valid for a wide range of and .
BAO measurements.
To compare these observables with the sevenyear WMAP data (WMAP7), we followed the numerical recipe given in Komatsu et al. (2009). The key point of this recipe is that for each cosmological scenario, must be minimized over h and , which appear in Eq. (44) and in the computation of through (see Eq. (14)).
An important feature to note is that we have to solve Eqs. (8) and (9) from a = 1 to a = 0 to compute the CMB observables. Numerically, however, we cannot reach a = 0 (z = ∞) because of numerical divergences. To avoid them, we carried out these computations up to a = 10^{7} and then linearly extrapolated the value of the integral to a = 0 (for more details on the reliability of this approximation see Appendix B). Thus, the theoretical constraints of 2.5 were checked from a = 1 to a = 10^{7}.
Finally, because CMB observables depend explicitly on H_{0}, we imposed a Gaussian prior on its value, h = 0.737 ± 0.024 as measured by Riess et al. (2011) from lowredshift SNe Ia and Cepheid variables.
The WMAP7 recommended bestfit values of the CMB observables are (47)with the corresponding inverse covariance matrix: (48)from Komatsu et al. (2011). As pointed out by Nesseris et al. (2010), the uncoupled Galileon model fulfils the assumptions required in Komatsu et al. (2009) to use these distance priors, namely a FLRW Universe with the standard number of neutrinos and a dark energy background with negligible interactions with the primordial Universe. Once the observables {l_{a},R,z_{∗}} were computed in a cosmological scenario, we built the difference vector: (49)and computed the CMB contribution to the total χ^{2} : (50)
3.2.3. Baryonic acoustic oscillations
BAO distances provide information on the imprint of the comoving sound horizon after recombination on the distribution of galaxies. The BAO observable is defined as y_{s}(z) = r_{s}(z_{d})/D_{V}(z), where r_{s} is the comoving sound horizon at the baryon drag epoch redshift z_{d}, and D_{V}(z) is the effective distance (Eisenstein et al. 2005) given by (51)z_{d} is computed using the Eisenstein & Hu (1998) fitting formula: This formula remains valid for a Galileon field not coupled to matter.
Therefore BAO distances depend on h and as the CMB observables so we followed the same recipe as previously mentioned to compute them, including the H_{0} prior from Riess et al. (2011). We also made the same approximation as for the CMB to compute r_{s}. The minimization over h and was performed independently for CMB and BAO when their individual constraints are derived and simultaneously when combined constraints were computed.
We used the dataset of distances derived from galaxy surveys as published in the SDSSIII BOSS cosmological analysis (Anderson et al. 2012 and Sánchez et al. 2012) to avoid redshift overlaps in the measurements (see Table 1).
For a cosmological constraint derived from BAO distances alone, the BAO contribution to the total χ^{2} is given by (55)where we added a Gaussian prior on when dealing with this probe alone.
When BAO and CMB probes were combined, we computed their contributions to the χ^{2} simultaneously to avoid overcounting the Hubble constant prior. Therefore, the combined contribution is (56)
Growth data.
Cosmological constraints on the Galileon model from the SNLS3 sample.
3.2.4. Growth rate of structures
The cosmological growth of structures is a critical test of the Galileon model, as noted by many authors (see Linder 2005 for example). It is a very discriminant constraint for distinguishing dark energy and modified gravity models. Many models can mimic ΛCDM behavior for the expansion history of the Universe, but all modify gravity and structure formation in a different manner.
In linear perturbation theory, the growth of a matter perturbation δ_{m} = δρ_{m}/ρ_{m} is governed by the equation (57)But as argued in Linder (2005) and as used in Komatsu et al. (2009), it is better to study the growth evolution with the function g(a) ≡ D(a)/a ≡ δ_{m}(a)/(aδ_{m}(1)). In the Galileon case, the Newton constant is replaced by as given in Eq. (27). The g(a) is obtained by solving the following secondorder differential equation (58)A natural choice for the initial conditions is g(a_{initial}) = 1 and dg/da_{ainitial} = 0 (Komatsu et al. 2009), where a_{initial} is 0.001 ≈ 1/(1 + z_{∗}). We checked that our results do not depend on this choice as long as a_{initial} is taken between 10^{2} and 10^{5}.
Measurements of the rate of growth of cosmic structures from redshift space distortions can be expressed in terms of f(a) = dlnD(a)/dlna or fσ_{8}(a), where σ_{8} is the normalization of the matter power spectrum. fσ_{8}(a) is known to be less sensitive to the overall normalization of the power spectrum model used to derive the measurements (Song & Percival 2009). Accordingly this is the observable we chose in this work. To predict fσ_{8}(a) in our analysis, we solved Eq. (58) to obtain g(a), from which we deduced f(a) and D(a), and we computed σ_{8}(a) in the following way (Samushia et al. 2012a): (59)where (60)and is the present value of the CMB power spectrum normalization published by Komatsu et al. (2011) in the framework of the ΛCDM model. Equation (60) states that the normalization of the CMB power spectrum at decoupling is the same in the ΛCDM and Galileon models, which is consistent with our assumption that the CMB physics is not modified by the Galileon presence. This equation holds if D(a) has no scale dependence, which is the case in both models in the linear regime. Equation (59) takes into account the different growth histories since recombination in the two models.
However, standalone fσ_{8}(a) measurements extracted from observed matter power spectra usually use a fiducial cosmology, which assumes General Relativity. This hypothesis is no longer necessary when taking into account the AlcockPaczynski effect (Alcock & Paczynski 1979) in the power spectrum analysis. This results in joint measurements of fσ_{8}(a) and the AlcockPaczynski parameter F(a) ≡ c^{1}D_{A}(a)H(a)/a, which are to be preferred when constraining modified gravity models (see e.g. Beutler et al. 2012 and Samushia et al. 2012b). Note that Eqs. (8) and (9) are all we need to predict F(a) in the Galileon model.
The measurements of fσ_{8}(z) and F(z) used in this work are summarized in Table 2. To compare these with our model, we first solved Eqs. (8) and (9) from a = 1 to a_{initial} to obtain values of , F(a) and , and then solved Eq. (58) from a_{initial} to a = 1, which provides us with fσ_{8}(z) predictions.
Galileon model bestfit values from different data samples.
Because F(z) and fσ_{8}(z) measurements are correlated, a covariance matrix C_{GoS} was built using data presented in Table 2. Moreover, our fσ_{8} prediction relies on the WMAP7 measurement of σ_{8}(a = 1) (Eq. (60)), so the WMAP7 experimental uncertainty is also propagated to the diagonal and offdiagonal terms of C_{GoS}. Then a vector V_{GoS} containing all predictions at each z_{i} was built (61)The contribution of the growth rate of structures to the total χ^{2} is then (62)with ΔV_{GoS} = V_{GoS} − ⟨ V_{GoS} ⟩, where ⟨ V_{GoS} ⟩ contains the measurements of Table 2.
Note that Eq. (14) requires a value for , and hence in principle this equation should be simultaneously solved with the BAO and CMB constraints using the same prior on H_{0}. However, we found that this has essentially no effect on our χ^{2}. Therefore, we set here h to the value derived from the H_{0} measurements of Riess et al. (2011) to accelerate the computation.
4. Results
In the following we present the results of the experimental constraints on the Galileon model derived from the cosmological probes.
4.1. SN constraints
Results from SN Ia data are presented in Fig. 2 and Table 3.
4.1.1. SN results
Despite the large number of free parameters in the model, we obtained closed probability contours in any twodimensional projection of the parameter space. We observed strong correlations between the s, especially between and .
We note that the bestfit value for is compatible with the current constraints obtained in the ΛCDM or FWCDM models. The s are found to be globally of the order of ≈− 1. From the bestfit values of the parameters, we derived the value of using Eq. (29) and find , including systematic uncertainties.
In the following we discuss the impact of fixing the nuisance parameters α and β and the effect of systematics on the bestfit values.
Fig. 1
Confidence contours for the SN nuisance parameters α and β when marginalizing over all other parameters of the model. Dashed red contours represent 68.3%, 95.4%, and 99.7% probability contours for the ΛCDM model. Filled blue contours are for the Galileon model. Note that they are nearly identical, the Galileon one is just 2.8% wider, which is likely due to larger steps in α and β. See Table 3 for numerical values. 
Fig. 2
Experimental constraints on the Galileon model from SNLS3 data alone. To represent the fourdimensional likelihood , six twodimensional contours for each pair of the Galileon model parameters are presented, after marginalizing over , , α, β, and the remaining Galileon parameters. The filled dark, medium, and lightblue contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. The contours include statistical and all identified systematic uncertainties. The dark dotted regions correspond to scenarios rejected by theoretical constraints, as described in the text. Labels in these regions indicate the main cause for excluding the scenarios. 
Fig. 3
Experimental constraints on the Galileon model from WMAP7+BAO+H0 data. To represent the fourdimensional likelihood , six twodimensional contours for each pair of the Galileon model parameters are presented after marginalizing over the left over Galileon parameters. The filled dark, medium, and lightgreen contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. Dark dotted regions correspond to scenarios rejected by theoretical constraints. 
4.1.2. Impact of nuisance parameters
When marginalizing over the cosmological parameters, the bestfit values of the SN nuisance parameters α, β, , and in the Galileon context are identical to those published for the ΛCDM model, as shown in Fig. 1 and Table 3. This is a truly important point to note. It means that the modeling of the SN Ia physics contained in these nuisance parameters is adequate for these two cosmological models despite their differences.
In principle, the correct method to use when analyzing SN Ia data is to scan and marginalize over the nuisance parameters. However, once the bestfit values of α and β are known, keeping them fixed to their bestfit values in any study using the same SN sample has a negligible impact on our results (see Table 3). In the Galileon case, the contour areas decrease by only 0.7% and have the same shape as in Fig. 2. For future studies with the SNLS3 sample in the ΛCDM or Galileon models, our analysis therefore demonstrates that it is reasonable to keep the nuisance parameters fixed to the values published the SNLS papers.
4.1.3. Impact of systematic uncertainties
From the results in Table 3, we note that the identified systematic uncertainties shift the bestfit values of the Galileon parameters by less than their statistical uncertainties. With systematics included, the area of the inner contours increases by about 53%. This is less than what is observed in fits to the ΛCDM or FWCDM models (103% and 80% respectively, see Conley et al. 2011).
4.2. Combined CMB, BAO, and H_{0} constraints
The results using CMB, BAO, and H_{0} data are presented in Fig. 3 and Table 4.
The combined WMAP7+BAO+H0 data provide a very powerful constraint on , but no tighter constraints on the than SNe Ia alone. is, as for the SNLS3 sample, close to the current best estimates for this parameter in the standard cosmologies, but this time with very sharp error bars competitive with the most recent studies on other cosmological models. However, the bestfit values are similar to those predicted with the SNLS3 sample.
To use the WMAP7+BAO+H0 data, h and have to be minimized for each explored Galileon scenario. Minimized values of these parameters are collected in the histograms of Fig. 4 for the subset of the scenarios that fulfilled the theoretical constraints. Values for the bestfit scenarios are reported in Table 4. For the Galileon model, the h distribution has a mean of 0.65 with a dispersion of 0.06, compatible with the H_{0} prior. The constraint on h is slightly lower than the Riess et al. (2011) value, but the same behavior is obtained for the ΛCDM model using the same program and data. The central value for the distribution is fully compatible with the WMAP7 value, for the Galileon and the ΛCDM model. However, in the Galileon model, values below 0.22 are much more disfavored.
For completeness, we present in Fig. 5 examples of results obtained from the WMAP7+H0 and BAO+H0 probes separately. Both plots were obtained with a minimization on h and , but a Gaussian prior on was added for the BAO (see Eq. (55)). We used the WMAP7 constraint for that prior because Fig. 4 shows that the Galileon model is consistent with it.
4.3. Growthofstructure constraints
Results using growth data are presented in Fig. 6 and Table 4, and are commented on in detail in Sect. 5.
Growth data and cosmological distances provide consistent values for the s. The bestfit value from growth data, , is below that from the other probes, but is still compatible at the 1.5σ level. This is the main difference between the two types of probes.
However, the use of growth data in cosmology deserves some comments. In our work, as in many others, different assumptions about the importance of nonlinearities in structure formation are made in the theoretical predictions and in the experimental extraction of growth data from the measured matter power spectrum.
As noted in Sect. 2.4, our theoretical predictions are derived in the linear regime and using a quasistatic approximation. While Barreia et al. (2012) confirmed that the latter is valid in the Galileon model, using only the linear regime is restrictive. As an example, this may be the origin of the divergences in that appear in some Galileon scenarios, as noted by Appleby & Linder (2012b). Going beyond the linear perturbation theory may change our predictions and thus could modify the result of our analysis.
Fig. 4
Minimized values of h and for a large subset of tested scenarios, in ΛCDM (red dashed histogram) and in the Galileon cosmology (blue filled histogram). Dashed black bands represent the measurements of H_{0} from Riess et al. (2011) and from Komatsu et al. (2011). Only scenarios with χ^{2} < 200 enter these histograms to deal only with pertinent scenarios. Note that both models give values of h and that agree with the measurements. 
Fig. 5
Experimental constraints on the Galileon parameters and from WMAP7+H0 data (top panel) and from BAO+H0 data (bottom panel). The fourdimensional likelihood has been marginalized over and . The filled dark, medium, and lightgreen contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. Dark dotted regions correspond to scenarios rejected by theoretical constraints. A Gaussian prior on based on the WMAP7 value has been added to the BAO+H0 fit. 
To estimate this effect, we tried to identify at which scale nonlinearities start to matter and checked whether this value is outside the range of scales taken into account in the growthofstructure measurements. As an example, WiggleZ measurements of fσ_{8} are derived using a nonlinear growthofstructure model (Jennings et al. 2011) and encompass all scales k < 0.3 h Mpc^{1}. In this model, the frontier between the linear and the nonlinear regimes is k ≈ 0.03 h Mpc^{1}. Other measurements in Table 2 include scales up to k ≈ 0.2−0.4 h Mpc^{1} as well. On the other hand, there is no prediction in the Galileon model that goes beyond the linear regime. However, estimates of the scale at which nonlinear effects appear exist in similar modified gravity models. Numerical simulations of the Chameleon screening effect for theories show that nonlinearity effects can be significant at scales k ≈ 0.05 h Mpc^{1} (see Brax et al. 2012; Jennings et al. 2012 and Li et al. 2013). Other simulations of the Vainshtein effect in the DGP model show that significant differences between the linear and nonlinear regimes appear for scales k > 0.2 h Mpc^{1} (Schmidt 2009). Unlike the DGP model, the Galileon model we considered does not contain a direct scalarmatter coupling ~ that is usually considered as an essential ingredient of the Vainshtein effect. However, Babichev & EspositoFarese (2013) showed that even if the Galileon field is not directly coupled to matter, the cosmological evolution of the Galileon field gives rise to an induced coupling of about 1, because of the Galileonmetric mixing. Therefore, the Vainshtein effect is expected to operate approximately at the same scales as in the DGP model in the model we considered.
This means the lack of nonlinear effects in our perturbation equations, and hence in our predictions for fσ_{8} in the Galileon model, is likely to have a significant impact on the constraints we derived from growth measurements, since the latter accounted partially for nonlinear effects.
4.4. Full combined constraints
Results from all data are presented in Fig. 7. Table 4 presents the bestfit values for the Galileon model parameters. The derived value is (63)Note that negative values are preferred for the s at the 1σ level. Moreover, the Galileon h bestfit values are compatible with the Riess et al. (2011) measurement.
We carried out an a posteriori check to identify which scenarios present a significant amount of early dark energy. At decoupling, Ω_{π}(z_{∗}) > 10%Ω_{r}(z_{∗}) only for viable scenarios with and . This check can be made after comparing theory with data because only data can provide values for h and . For Galileon scenarios with , which is the region favoured by data, we found no significant early dark energy.
4.5. Analysis of the bestfit scenario
What does the bestfit scenario (derived from all data; the last line of Table 4) look like? Because ρ_{π} can be defined from the (00) Einstein equation, a Galileon pressure P_{π} can be defined from the (ij) Einstein equation: (64)Combining ρ_{π} and P_{π}, an equation of state parameter w_{π}(z) = P_{π}(z)/ρ_{π}(z) can be built for the Galileon “fluid”. We can also construct an equation for Ω_{π}(z) using . The evolution of w_{π}(z), Ω_{π}(z) and for the Galileon bestfit scenario is shown in Figs. 8 and 9.
Fig. 6
Experimental constraints on the Galileon model from growth data (red) and from SNLS3+WMAP7+BAO+H0 combined constraints (dashed). The filled dark, medium, and lightcolored contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. Dark dotted regions correspond to scenarios rejected by theoretical constraints. 
Fig. 7
Combined constraints on the Galileon model from SNLS3, WMAP7+BAO+H0, and growth data. The filled dark, medium, and lightyellow contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. Dark dotted regions correspond to scenarios rejected by theoretical constraints. 
ΛCDM bestfit values from different data samples.
Fig. 8
Evolution of the Ω_{i}(z) (left) and of w(z) (right, solid curve) for the bestfit Galileon model from all data (last row of Table 4). As a comparison, the dashed orange line gives w(z) for the bestfit scenario from SN data alone. 
4.5.1. Cosmic evolution
The left panel of Fig. 8 shows that for the bestfit scenario, radiation, matter, and dark energy (here the Galileon) dominate alternatively during the history of the Universe, as in any standard cosmological model. These three epochs are also visible in the evolution of w(z). Moreover, the bestfit scenario evolves in the future toward the de Sitter solution w = −1, which is an attractor of the Galileon model (De Felice & Tsujikawa 2010). In the region 0 < z < 1, where SNe tightly constrain dark energy, w(z) deviates significantly from − 1, its ΛCDM value. Note that in the fit with SNe alone, the deviation is less pronounced, with an average value of − 1.09 in 0 < z < 1, which is compatible with the fitted value of w in constant w dark energy models, as published in Conley et al. (2011).
During matter domination, dark energy contributes about 0.4% to the massenergy budget at z = 10. For comparison, in a standard ΛCDM model dark energy contributes only 0.2% at this redshift (assuming a flat ΛCDM model with ). In the same way, dark energy contributes 0.04% at z_{∗} in the Galileon bestfit scenario, whereas for ΛCDM Ω_{Λ} = 10^{9} at z_{∗}. In our bestfit Galileon scenario, dark energy is more present throughout the history of the Universe than in the ΛCDM model, but is still negligible during the matter and radiation eras.
Figure 9 shows the evolution of for the bestfit scenario and for the growthdata bestfit scenario. Both curves show deviations from 1 at redshifts around 0. Particularly, the divergence near the current epoch suggests that we should push the Galileon predictions for fσ_{8} beyond the linear regime, as already advocated in Sect. 4.3.
4.5.2. Comparison with ΛCDM
In Fig. 10 and Table 5, bestfit values for the ΛCDM parameters are presented using the same analysis tools and observables. Interestingly, even in the ΛCDM model there is tension between growth data and other probes. The bestfit value is similar in both models, but the h value departs more from the H_{0}Riess et al. (2011) measurement. As far as the χ^{2}s are concerned, SNe Ia provide a good agreement with both models. CMB+BAO+H0 data are more compatible with the Galileon model, reflecting the better agreement on the h minimized value. Yet growthofstructure data agree better with the ΛCDM model. Finally, due to the poorer fit to growth data in the Galileon model, the difference in χ^{2} is Δχ^{2} = 10.2. This indicates that the Galileon model is slightly disfavored with respect to the ΛCDM model, despite having two extra free parameters.
Because we are comparing two models with a different number of parameters and complexity, other criteria than comparing χ^{2}s can be helpful. A review of the selection model criterion is provided in Liddle (2007). Because our study leads to the full computation of the likelihood functions, we can use precise criteria such as the Bayes factor (see Beringer et al. 2012; John & Narlikar 2002; Kass & Raftery 1995 and Liddle 2009) or the deviance information criterion (DIC, see Spiegelhalter et al. 2002 and Kunz et al. 2006). The Akaike information criterion (AIC) and the Bayesian information criterion (BIC) criteria used in Nesseris et al. (2010) are approximations of the first two using only the maximum likelihood and not the whole function. Hereafter we restrict the discussion to the DIC criterion.
The DIC criterion is based on the computation of the deviance likelihoods Dev(θ) = −2log p(Dθ) + C (with C a constant not important for DIC evaluation). p(Dθ) is the computed likelihood function ℒ(θ) of the model. An effective number of parameters is derived with the expectation values for θ and the mean deviance likelihood value: (65)where p(θD) is the posterior probability density function for a vector θ of parameters of the tested model, knowing the data D: (66)p(D), the probability to obtain the data D, is also called the marginal likelihood because it can be computed using the summation over all θs: (67)Note that if the priors are flat, p(θD) is just the likelihood function ℒ(θ) normalized to 1. In our case, prior(θ) is a flat prior reflecting the theoretically allowed volume in the scanned parameter space. We checked that the DIC criterion is not sensitive to the exact definition of the prior, which makes it a robust tool.
Then . The model with the smallest DIC is favored by the data. In our study, we obtained DIC_{Galileon} − DIC_{ΛCDM} = 12.25 > 0. Again, the Galileon model is slightly disfavored by data against the ΛCDM model. The DIC criterion just reflects the Δχ^{2} and does not penalize the Galileon model so much because of its higher number of free parameters.
In the future, provided the tension between growthofstructure data and distances does not increase after more precise measurements of the observables used in this paper are included, new observables will be necessary to distinguish between the two models. A promising way would be to exploit, e.g., the ISW effect as discussed in Kobayashi et al. (2010).
Fig. 9
Evolution of for the bestfit scenario from growth data only (dashed orange line) and from all data (blue solid line). 
Fig. 10
Experimental constraints on the ΛCDM model from SNLS3 data (blue), growth data (red), BAO+WMAP7+H0 data (green), and all data combined (yellow). The black dashed line indicates the flatness condition Ω_{m} + Ω_{Λ} = 1. 
Fig. 11
Experimental constraints on the FWCDM model from SNLS3 data (blue), growth data (red), BAO+WMAP7+H0 data (green), and all data combined (yellow). 
FWCDM bestfit values from different data samples.
4.5.3. Comparison with FWCDM
For consistency with our assumption about flatness, we also present a comparison with the effective FWCDM model, a model with a constant dark energy equation of state parameter w in a flat Universe (see Table 6 and Fig. 11). The data set points toward a value of w below − 1, which is consistent with the Galileon bestfit scenario (see Fig. 8).
However, the difference in χ^{2} is the same as for the ΛCDM model, Δχ^{2} = 10.2, and the DIC criterion gives DIC_{Galileon} − DIC_{FWCDM} = 12.16 > 0. Here again, the Galileon model is not significantly disfavored.
5. Discussion
In this section we compare our results with other recent publications on the same subject.
Appleby & Linder (2012b) concluded that the uncoupled Galileon model is ruled out by current data since their bestfit yielded Δχ^{2} = 31 compared with the bestfit ΛCDM model. In addition, they obtained a long narrow region of degenerate scenarios with nearly the same likelihood. In our case, the bestfit has Δχ^{2} = 10.2, we obtained enclosed contours in all projections and a clear minimum.
Although we used the same expansion and perturbation equations as Appleby & Linder (2012b), there are differences between the two works. We used a parametrization of the model, which makes our study independent of initial conditions for x, while they set x_{i} = x(z_{i} = 10^{6}) by imposing a ρ_{π}(z_{i}) which varied in their parameter scan. This requires one to solve a fifthorder polynomial equation in x_{i} – and hence one is forced to choose one of the five solutions – or to assume one of the four terms is dominant in the (00) Einstein equation. In any case, this leads to a parameter space that is different than the one we explored. Another difference arises from the theoretical constraints that are used to restrict the parameter space to viable scenarios only. Our set of theoretical constraints is larger because we also used tensorial constraints, which proved to be very powerful. This also leads to a different explored parameter space.
In De Felice & Tsujikawa (2010), the rescaling of the Galileon parameters was performed with a de Sitter solution instead of using x_{0}, as in this paper. This led to relations fixing their “” and “” coefficients as a function of their “” and “” coefficients (denoted α and β in their study), but required two initial conditions to compute the cosmological evolution. Those were also fitted using experimental data. With this parametrization and without growth constraints, Nesseris et al. (2010) found bestfit values for their “” and “” of the same sign and same order of magnitude as in our work, despite our different parametrizations. A second paper by Okada et al. (2013) included redshift space distortion measurements and ruled out the Galileon model at the 10σ level.
The first difference with respect to our work is the treatment of the initial conditions and the use of an extra theoretical constraint to avoid numerical instabilities during the transition from the matter era to the de Sitter epoch. This reduces the parameter space with respect to that explored in our work. As stated above, a better modeling of including nonlinear effects should be conducted instead of discarding scenarios with such instabilities.
Second, Okada et al. (2013) used fσ_{8} measurements not corrected for the AlcockPaczynski effect. Moreover, to make their fσ_{8} predictions in the Galileon model, Okada et al. (2013) set the normalization of σ_{8} today to the WMAP7 σ_{8}(z = 0) measurement, which was obtained in a cosmological fit to the ΛCDM model. This normalization led to the following σ_{8}(z) evolution: (68)This assumes that the Galileon theory predicts a matter power spectrum similar to that of ΛCDM at z = 0, which is not guaranteed (Barreia et al. 2012). In contrast, we used the WMAP7 σ_{8} measurement to set the normalization at decoupling z ≈ z_{∗} (see Eq. (59)). Thus we took into account the different growth histories between the ΛCDM and the Galileon models (Eq. (60), which is different from Eq. (68)). We can compare our bestfit scenarios for these two models with the fσ_{8} and F measurements. Figures 12 and 13 show the result of this comparison. The agreement with the data is good in both models. In particular, the Galileon model does not exhibit a discrepancy as strong as was found in Fig. 3 of Okada et al. (2013).
Fig. 12
fσ_{8}(z) measurements from different surveys (6dFGRS, 2fFGRS, SDSS LRG, BOSS, and WiggleZ) compared with predictions for the ΛCDM model (with parameters of Table 5 – dashed purple line) Galileon scenarios. The solid blue line stands for the bestfit Galileon scenario using all data, whereas the orange dashed line stands for the bestfit Galileon scenario using growth data only. 
Fig. 13
F(z) measurements from different surveys (BOSS and WiggleZ) compared with the prediction for the ΛCDM model (with parameters of Table 5 – dashed purple line) and for Galileon scenarios. The solid blue line stands for the bestfit Galileon scenario using all data, whereas the orange dashed line stands for the bestfit Galileon scenario using growth data only. 
6. Conclusion
We have confronted the uncoupled Galileon model with the most recent cosmological data. We introduced a renormalization of the Galileon parameters by the derivative of the Galileon field normalized to the Planck mass to break some degeneracies inherent to the model. Theoretical conditions were added to restrict the analysis to viable scenarios only. This allowed us to break the parameter degeneracies that otherwise would have prevented us from obtaining enclosed probability contours. In particular, the conditions on the tensorial propagation mode of the perturbed metric proved to be very helpful.
We used a grid search technique to explore the Galileon parameter space. Our data set encompassed the SNLS3 SN Ia sample, WMAP7 {l_{a},R,z_{∗}} constraints, BAO measurements, and growth data with the AlcockPaczynski effect taken into account. We found . The final χ^{2} is slightly above that of the ΛCDM model due to a poorer fit to the growth data.
The bestfit Galileon scenario mimics a ΛCDM model with the three periods of radiation, matter, and dark energy domination, with an evolving dark energy equation of state parameter w(z), and an effective gravitational coupling . Predictions for the latter are possible only in the linear regime, which may have an impact on our results derived from growth data because the latter were computed using a nonlinear theory. A more precise theoretical and phenomenological study should be conducted to fairly compare the Galileon model with these data.
Our bestfit is more favorable to the Galileon model than other recent results. The main difference between our treatment and those works lies in the treatment of initial conditions. We also tried to make as few assumptions and approximations as possible when computing observable quantities. Finally, when using growth data, we took care to choose measurements that were derived in a modelindependent way. In the future, a study considering precise predictions of the full power spectra as suggested by Barreia et al. (2012) would provide more stringent tests of the validity of the Galileon model.
Acknowledgments
We thank Philippe Brax for introducing us to the Galileon theory, and Christos Charmoussis, Cédric Deffayet, JeanBaptiste Melin, and Marc Besançon for fruitful discussions about the Galileon model. We also thank Chris Blake for useful advice on the use of the WiggleZ measurements. The work of Eugeny Babichev was supported in part by grant FQXiMGA1209 from the Foundational Questions Institute.
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Appendix A: Instability of probability contours
Fig. A.1
Experimental constraints on the Galileon model from SNLS3 data for different ranges in c_{i}s using the method developed in Appendix A, with α and β fixed to their ΛCDM bestfit values from Sullivan et al. (2011). The fourdimensional likelihood (c_{5} fixed to 0 here) is marginalized over , to visualize the contour plots. The filled dark, medium, and lightblue contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. 
Instead of absorbing the initial condition x_{0} in the redefinition, we can be tempted to fix it using the (00) Einstein equation at z = 0 for each scenario: (A.1)To find x_{0}, a fifthorder polynomial equation is to be solved, which can lead to five complex solutions. A reasonable choice is to keep only the scenarios that give a unique real solution.
Fig. A.2
Left panel: evolution of l_{a} with a_{min} without the linear interpolation as described in the text for a subset of Galileon scenarios. Note that most scenarios approach the WMAP7 measurement l_{a} ≈ 300. Right panel: correction to l_{a} for different values of a_{min} and for the same subset of scenarios as in the left panel. The dashed line is the value of σ_{la}, the WMAP7 measurement error on l_{a}. 
The system of differential Eqs. (8) and (9) adopts an unusual behavior. Referring to Fig. A.1, the shape of the probability contours remains unchanged regardless of the limits of the scanned parameter space. In other words, the likelihood surface is invariant when the limits of the explored parameter space are proportionally changed. The model seems to exhibit a scale invariance allowing data to be fitted regardless of the boundaries of the explored parameter range. Moreover, we cannot obtain contours well enclosed in any explored parameter space: the likelihood surface has an infinite valley of minimum χ^{2} instead of a unique minimum.
Equation (A.1) shows that small c_{i}s produce a high x_{0}, and high c_{i}s a low x_{0}. Nevertheless, the theoretical constraints of Sect. 2.5 cannot favor or disfavor high c_{i}s or x_{0} because they also contain this correspondence between the c_{i}s and x_{0}. Accordingly, for different sets of c_{i}s, identical cosmological scenarios are computed regardless of the scale of the c_{i}s: the important point is that these equivalent scenarios have the same Ω_{π}(z), whether this is due to high or small c_{i}s and as a consequence have the same evolution and then the same χ^{2}.
Thus, a scale choice has to be made to fix the likelihood surface, but this choice has not to be arbitrary. A solution is provided in 2.2 by absorbing x_{0} into new parameters s. This new parametrization absorbs a degree of freedom and allows us to use the (00) Einstein equation to fix . This may be the origin of the degeneracy in χ^{2} reported in Sect. III of Appleby & Linder (2012a).
Appendix B: Approximation for l_{a} computation
The computation of l_{a} (see Eq. (42)) requires the evolution of the cosmological model from today to a = 0 (see Eq. (40)). In the Galileon context, the nonlinear evolution equations require increasing precision and finer steps when approaching the limit a → 0. In addition, it is physically questionable to extrapolate the Galileon model up to the very first instants of the Universe.
Therefore our iterative computation is stopped at a certain a_{min} close to 0, without affecting significantly the final value of l_{a}. Let a_{min − 1} be the step before a_{min} where the cosmological equations are computed, and f(a) the integrand function of r_{s}(z_{∗}). Although the integral is stopped at a = a_{min}, we can compensate this approximation by a linear interpolation of the integral: (B.1)In the left panel of Fig. A.2, we present the evolution of l_{a} with a_{min} without the linear interpolation for a subset of Galileon scenarios. The smooth evolution with a_{min} allows us to consider the linear interpolation as a reasonable assumption. Moreover, for a_{min} ≲ 10^{6}, the value of l_{a} changes less than the WMAP7 measurement error σ_{la} = 0.76, as shown in the right panel of Fig. A.2. Based on these results, we decide to use a_{min} = 10^{7}, which provides a correction on l_{a} an order of magnitude below σ_{la}.
All Tables
All Figures
Fig. 1
Confidence contours for the SN nuisance parameters α and β when marginalizing over all other parameters of the model. Dashed red contours represent 68.3%, 95.4%, and 99.7% probability contours for the ΛCDM model. Filled blue contours are for the Galileon model. Note that they are nearly identical, the Galileon one is just 2.8% wider, which is likely due to larger steps in α and β. See Table 3 for numerical values. 

In the text 
Fig. 2
Experimental constraints on the Galileon model from SNLS3 data alone. To represent the fourdimensional likelihood , six twodimensional contours for each pair of the Galileon model parameters are presented, after marginalizing over , , α, β, and the remaining Galileon parameters. The filled dark, medium, and lightblue contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. The contours include statistical and all identified systematic uncertainties. The dark dotted regions correspond to scenarios rejected by theoretical constraints, as described in the text. Labels in these regions indicate the main cause for excluding the scenarios. 

In the text 
Fig. 3
Experimental constraints on the Galileon model from WMAP7+BAO+H0 data. To represent the fourdimensional likelihood , six twodimensional contours for each pair of the Galileon model parameters are presented after marginalizing over the left over Galileon parameters. The filled dark, medium, and lightgreen contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. Dark dotted regions correspond to scenarios rejected by theoretical constraints. 

In the text 
Fig. 4
Minimized values of h and for a large subset of tested scenarios, in ΛCDM (red dashed histogram) and in the Galileon cosmology (blue filled histogram). Dashed black bands represent the measurements of H_{0} from Riess et al. (2011) and from Komatsu et al. (2011). Only scenarios with χ^{2} < 200 enter these histograms to deal only with pertinent scenarios. Note that both models give values of h and that agree with the measurements. 

In the text 
Fig. 5
Experimental constraints on the Galileon parameters and from WMAP7+H0 data (top panel) and from BAO+H0 data (bottom panel). The fourdimensional likelihood has been marginalized over and . The filled dark, medium, and lightgreen contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. Dark dotted regions correspond to scenarios rejected by theoretical constraints. A Gaussian prior on based on the WMAP7 value has been added to the BAO+H0 fit. 

In the text 
Fig. 6
Experimental constraints on the Galileon model from growth data (red) and from SNLS3+WMAP7+BAO+H0 combined constraints (dashed). The filled dark, medium, and lightcolored contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. Dark dotted regions correspond to scenarios rejected by theoretical constraints. 

In the text 
Fig. 7
Combined constraints on the Galileon model from SNLS3, WMAP7+BAO+H0, and growth data. The filled dark, medium, and lightyellow contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. Dark dotted regions correspond to scenarios rejected by theoretical constraints. 

In the text 
Fig. 8
Evolution of the Ω_{i}(z) (left) and of w(z) (right, solid curve) for the bestfit Galileon model from all data (last row of Table 4). As a comparison, the dashed orange line gives w(z) for the bestfit scenario from SN data alone. 

In the text 
Fig. 9
Evolution of for the bestfit scenario from growth data only (dashed orange line) and from all data (blue solid line). 

In the text 
Fig. 10
Experimental constraints on the ΛCDM model from SNLS3 data (blue), growth data (red), BAO+WMAP7+H0 data (green), and all data combined (yellow). The black dashed line indicates the flatness condition Ω_{m} + Ω_{Λ} = 1. 

In the text 
Fig. 11
Experimental constraints on the FWCDM model from SNLS3 data (blue), growth data (red), BAO+WMAP7+H0 data (green), and all data combined (yellow). 

In the text 
Fig. 12
fσ_{8}(z) measurements from different surveys (6dFGRS, 2fFGRS, SDSS LRG, BOSS, and WiggleZ) compared with predictions for the ΛCDM model (with parameters of Table 5 – dashed purple line) Galileon scenarios. The solid blue line stands for the bestfit Galileon scenario using all data, whereas the orange dashed line stands for the bestfit Galileon scenario using growth data only. 

In the text 
Fig. 13
F(z) measurements from different surveys (BOSS and WiggleZ) compared with the prediction for the ΛCDM model (with parameters of Table 5 – dashed purple line) and for Galileon scenarios. The solid blue line stands for the bestfit Galileon scenario using all data, whereas the orange dashed line stands for the bestfit Galileon scenario using growth data only. 

In the text 
Fig. A.1
Experimental constraints on the Galileon model from SNLS3 data for different ranges in c_{i}s using the method developed in Appendix A, with α and β fixed to their ΛCDM bestfit values from Sullivan et al. (2011). The fourdimensional likelihood (c_{5} fixed to 0 here) is marginalized over , to visualize the contour plots. The filled dark, medium, and lightblue contours enclose 68.3, 95.4, and 99.7% of the probability, respectively. 

In the text 
Fig. A.2
Left panel: evolution of l_{a} with a_{min} without the linear interpolation as described in the text for a subset of Galileon scenarios. Note that most scenarios approach the WMAP7 measurement l_{a} ≈ 300. Right panel: correction to l_{a} for different values of a_{min} and for the same subset of scenarios as in the left panel. The dashed line is the value of σ_{la}, the WMAP7 measurement error on l_{a}. 

In the text 
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