We model the linear and nonlinear growth of large scale structure in the Cubic Galileon gravity model, by running a suite of N-body cosmological simulations using the {\tt ECOSMOG} code. Our simulations include the Vainshtein screening effect, which reconciles the Cubic Galileon model with local tests of gravity. In the linear regime, the amplitude of the matter power spectrum increases by $\sim 25%$ with respect to the standard $Λ$CDM model today. The modified expansion rate accounts for $\sim 20%$ of this enhancement, while the fifth force is responsible for only $\sim 5%$. This is because the effective unscreened gravitational strength deviates from standard gravity only at late times, even though it can be twice as large today. In the nonlinear regime ($k \gtrsim 0.1 h\rm{Mpc}^{-1}$), the fifth force leads to only a modest increase ($\lesssim 8%$) in the clustering power on all scales due to the very efficient operation of the Vainshtein mechanism. Such a strong effect is typically not seen in other models with the same screening mechanism. The screening also results in the fifth force increasing the number density of halos by less than 10%, on all mass scales. Our results show that the screening does not ruin the validity of linear theory on large scales which anticipates very strong constraints from galaxy clustering data. We also show that, whilst the model gives an excellent match to CMB data on small angular scales ($l \gtrsim 50$), the predicted integrated Sachs-Wolf effect is in tension with Planck/WMAP results.
We obtain a constraint on the parameters of a generalized cubic Galileon gravity model exhibiting the Vainshtein mechanism by using multi-wavelength observations of the Coma Cluster. The generalized cubic Galileon model is characterized by three parameters of the turning scale associated with the Vainshtein mechanism, and the amplitude of modifying a gravitational potential and a lensing potential. X-ray and Sunyaev-Zeldovich (SZ) observations of the intra-cluster medium are sensitive to the gravitational potential, while the weak-lensing (WL) measurement is specified by the lensing potential.…
The cubic Galileon model containing the lowest non-trivial order action of the full Galileon action can produce the stable late-time cosmic acceleration. This model can have a significant role in the growth of structures. The signatures of the cubic Galileon model in the structure formation can be probed by the weak lensing statistics. Weak lensing convergence statistics is one of the strongest probes to the structure formation and hence it can probe the dark energy or modified theories of gravity models. In this work, we investigate the detectability of the cubic Galileon model from the $ Λ$C…
The maximum size of a bound cosmic structure is computed perturbatively as a function of its mass in the framework of the cubic galileon, proposed recently to model the dark energy of our Universe. Comparison of our results with observations constrains the matter-galileon coupling of the model to $0.033\lesssim α\lesssim 0.17$, thus improving previous bounds based solely on solar system physics.
In this paper we apply the tools of the dynamical systems theory in order to uncover the whole asymptotic structure of the vacuum interactions of a galileon model with a cubic derivative interaction term. It is shown that, contrary to what occurs in the presence of background matter, the galileon interactions of vacuum appreciably modify the late-time cosmic dynamics. In particular, a local late-time attractor representing phantom behavior arises which is inevitably associated with a big rip singularity. It seems that the gravitational interactions of the background matter with the galileon sc…
We investigate the linearly and quadratically coupled cubic Galileon models that include linear potentials. These models may explain the late-time acceleration. In these cases, we need two equations of state parameter named the native and effective equations of state to test whether the universe is accelerating or not because there is coupling between the cold dark matter and Galileon. It turns out that there is no transition from accelerating phase to phantom phase in the future.
We use N-body simulation to study the structure formation in the Cubic Galileon Gravity model where along with the usual kinetic and potential term we also have a higher derivative self-interaction term. We find that the large scale structure provides a unique constraining power for this model. The matter power spectrum, halo mass function, galaxy-galaxy weak lensing signal, marked density power spectrum as well as count in cell are measured. The simulations show that there are less massive halos in the Cubic Galileon Gravity model than corresponding $Λ$CDM model and the marked density power s…
Galileons are higher-derivative theories of a real scalar which nevertheless admit second order equations of motion. They have interesting applications as dark energy models and in early universe cosmology, and have been conjectured to arise as descriptions of brane dynamics in string theory. In the present paper, we study the bosonic sector of globally N=1 supersymmetric extensions of the cubic Galileon Lagrangian in detail. Supersymmetry requires that the Galileon scalar now becomes paired with a second real scalar field. We prove that the presence of this second scalar causes the equations …
We use the latest CMB data from Planck, together with BAO measurements, to constrain the full parameter space of Galileon gravity. We constrain separately the three main branches of the theory known as the Cubic, Quartic and Quintic models, and find that all yield a very good fit to these data. Unlike in $Λ{\rm CDM}$, the Galileon model constraints are compatible with local determinations of the Hubble parameter and predict nonzero neutrino masses at over $5σ$ significance. We also identify that the low-$l$ part of the CMB lensing spectrum may be able to distinguish between $Λ{\rm CDM}$ and Ga…
In this paper we obtain the bispectrum of dark matter density perturbations in the frame of covariant cubic Galileon theories. This result is obtained by means of a semi-analytic approach to second-order perturbations in Galileon cosmologies, assuming Gaussian initial conditions. In particular, we show that, even in the presence of large deviations of the linear growth-rate w.r.t. the ΛCDM one, at the bispectrum level such deviations are reduced to a few percent.
We re-explore the Bi- and Multi-Galileon models with trivial asymptotic conditions at infinity and show that propagation of superluminal fluctuations is a common and unavoidable feature of these theories, unlike previously claimed in the literature. We show that all Multi-Galileon theories containing a Cubic Galileon term exhibit superluminalities at large distances from a point source, and that even if the Cubic Galileon is not present one can always find sensible matter distributions in which there are superluminal modes at large distances. In the Bi-Galileon case we explicitly show that the…
We use recent measurements of the expansion history of the universe to place constraints on the parameter space of cubic Galileon models, in particular we concentrate on those models which contain the simplest Galileon term plus a linear potential. This gives strong constraints on the Lagrangian of these models. Most dynamical terms in the Galileon Lagrangian are constraint to be small and the acceleration is effectively provided by a constant term in the scalar potential, thus reducing, effectively, to a LCDM model for current acceleration. The effective equation of state is indistinguishable…
In this paper, we study the effects of general relativistic corrections on the observed galaxy power spectrum in thawing class of cubic Galileon model with linear potential that preserves the shift symmetry. In this scenario, the observed galaxy power spectrum differs from the standard matter power spectrum mainly due to redshift space distortion (RSD) factor and relativistic effects. The RSD term enhances the matter power spectrum both at larger and smaller scales whereas the relativistic terms further enhance the matter power spectrum only at larger scales. In comparison with $Λ$CDM, the obs…
We present for the first time an explicit exposition of quantum corrections within the cubic Galileon theory including the effect of quantum gravity, in a background- and gauge-invariant manner, employing the field-reparametrisation approach of the covariant effective action at 1-loop. We show that the consideration of gravitational effects in combination with the non-linear derivative structure of the theory reveals new interactions at the perturbative level, which manifest themselves as higher-operators in the associated effective action, which' relevance is controlled by appropriate ratios …
We investigate the nonlinear growth of large-scale structure in the generalised Proca theory, in which a self-interacting massive vector field plays the role of driving the acceleration of the cosmic expansion. Focusing to the Proca Lagrangian at cubic order -- the cubic vector Galileon model -- we derive the simplified equations for gravity as well as the longitudinal and transverse modes of the vector field under the weak-field and quasi-static approximations, and implement them in a modified version of the ECOSMOG N-body code. Our simulations incorporate the Vainshtein screening effect, whi…
We show that, in the presence of massive neutrinos, the Galileon gravity model provides a very good fit to the current CMB temperature, CMB lensing and BAO data. This model, which we dub $ν \rm{Galileon}$, when assuming its stable attractor background solution, contains the same set of free parameters as $Λ\rm{CDM}$, although it leads to different expansion dynamics and nontrivial gravitational interactions. The data provide compelling evidence ($\gtrsim 6σ$) for nonzero neutrino masses, with $Σm_ν\gtrsim 0.4\ {\rm eV}$ at the $2σ$ level. Upcoming precision terrestrial measurements of the abso…
N-body Simulations of Large-Scale Structure in the Generalized Cubic Covariant Galileon Model
Instituto de Astrofisíca e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa（里斯本大学科学学院天体物理与空间科学研究所）；Dipartimento di Fisica “E. Pancini”, Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学E. Pancini物理系）；INFN Sezione di Napoli, Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学意大利国家核物理研究所那不勒斯分部）；Institute for Computational Cosmology, Department of Physics, Durham University（杜伦大学物理学院计算宇宙学研究所）
We consider a disformal coupling between Standard Model matter and a cubic Galileon scalar sector, assumed to be a relict of some other physics that solves the cosmological constant problem rather than a solution in its own right. This allows the energy density carried by the Galileon scalar to be sufficiently small that it evades stringent constraints from the integrated Sachs-Wolfe effect, which otherwise rules out the cubic Galileon theory. Although the model with disformal coupling does not exhibit screening, we show there is a `screening-like' phenomenon in which the energy density carrie…
We revisit the cosmological dynamics of the cubic Galileon model in light of the recently proposed model-independent analyses of the Pantheon supernova data. At the background level, it is shown to be compatible with data and preferred over standard quintessence models. Furthermore, the model is shown to be consistent with the trans-Planckian censorship conjecture (as well as other Swampland conjectures). It is shown that for the given parametrization, the model fails to satisfy the bounds on the reconstructed growth index derived from the Pantheon data set at the level of linear~perturbations…
We investigate the generalized cubic covariant Galileon model, a kinetically driven dark energy model within the Horndeski class of theories. The model extends the cubic covariant Galileon by including power laws of the field derivatives in the K-essence and cubic terms which still allow for tracker solutions. We study the shape of the viable parameter space by enforcing stability conditions which include the absence of ghost, gradient and tachyon instabilities and the avoidance of strong coupling at early time. We study here the relevant effects of the modifications induced by the model on so…
We study relativistic stars in Hordenski theories that evade the gravitational wave constraints and exhibit the Vainshtein mechanism, focusing on a model based on the cubic Galileon Lagrangian. We derive the scalar field profile for static spherically symmetric objects in asymptotically de Sitter space-time with a linear time dependence. The exterior solution matches to the black hole solution found in the literature. Due to the Vainshtein mechanism, the stellar structure is indistinguishable from that of General Relativity with the same central density as long as the radius of the star is sho…
The generalised Proca (GP) theory is a modified gravity model in which the acceleration of the cosmic expansion rate can be explained by self interactions of a cosmological vector field. In this paper we study a particular sub-class of the GP theory, with up to cubic order Lagrangian, known as the cubic vector Galileon (cvG) model. This model is similar to the cubic scalar Galileon (csG) in many aspects, including a fifth force and the Vainshtein screening mechanism, but with the additional flexibility that the strength of the fifth force depends on an extra parameter -- interpolating between …
We investigate the performance of semi-analytical modelling of large-scale structure in Galileon gravity cosmologies using results from N-body simulations. We focus on the Cubic and Quartic Galileon models that provide a reasonable fit to CMB, SNIa and BAO data. We demonstrate that the Sheth-Tormen mass function and linear halo bias can be calibrated to provide a very good fit to our simulation results. We also find that the halo concentration-mass relation is well fitted by a power law. The nonlinear matter power spectrum computed in the halo model approach is found to be inaccurate in the mi…
We analyse the speed of gravitational waves in coupled Galileon models with an equation of state $ω_ϕ=-1$ now and a ghost-free Minkowski limit. We find that the gravitational waves propagate much faster than the speed of light unless these models are small perturbations of cubic Galileons and the Galileon energy density is sub-dominant to a dominant cosmological constant. In this case, the binary pulsar bounds on the speed of gravitational waves can be satisfied and the equation of state can be close to -1 when the coupling to matter and the coefficient of the cubic term of the Galileon Lagran…
The galileon model was recently proposed to locally describe a class of modified gravity theories, including the braneworld DGP model. We discuss spontaneous symmetry breaking of the self-accelerating branch in a multi-galileon theory with internal global symmetries. We show a modified version of Goldstone's theorem is applicable to the symmetry breaking pattern and discuss its implications. We also derive the Hamiltonian of a general multi-galileon theory and discuss its implications.
Theories of modified gravity attempt to reconcile physics at the largest and the smallest scales by explaining the accelerated expansion of our universe without introducing the cosmological constant. One class of such theories, known as Galileon theories, predict lensing potentials of spherically symmetric bodies, such as dark matter halos, to receive a feature-like modification at the 5% level. With the advent of next-generation photometric surveys, such modifications can serve as novel probes of modified gravity. Assuming an LSST-like fiducial dataset, we produce halo-shear power spectra for…
We consider the covariant galileon gravity taking into account the third order and fourth order scalar field Lagrangians L_3(π) and L_4(π) consisting of three and four $π$'s with four and five derivatives acting on them respectively. The background dynamical equations are set up for the system under consideration and the stability of the self accelerating solution is demonstrated in general setting. We extended this study to the general case of the fifth order theory. For spherically symmetric static background, we spell out conditions for suppression of fifth force effects mediated by the gal…
In this paper we investigate the decoupling limit of a particular class of multi-gravity theories, i.e. of theories of interacting spin-2 fields. We explicitly compute the interactions of helicity-0 modes in this limit, showing that they take on the form of multi-Galileons and dual forms. In the process we extend the recently discovered Galileon dualities, deriving a set of new multi-Galileon dualities. These are also intrinsically connected to healthy, but higher-derivative, multi-scalar field theories akin to `beyond Horndeski' models.
We study spherical collapse in the Quartic and Quintic Covariant Galileon gravity models within the framework of the excursion set formalism. We derive the nonlinear spherically symmetric equations in the quasi-static and weak-field limits, focusing on model parameters that fit current CMB, SNIa and BAO data. We demonstrate that the equations of the Quintic model do not admit physical solutions of the fifth force in high density regions, which prevents the study of structure formation in this model. For the Quartic model, we show that the effective gravitational strength deviates from unity at…
We consider a galileon field coupled to gravity. The standard no-hair theorems do not apply, because of the galileon's peculiar derivative interactions. We prove that, nonetheless, static spherically symmetric black holes cannot sustain non-trivial galileon profiles. Our theorem holds regardless of whether there are non-minimal couplings between the galileon and gravity of the covariant galileon type.
We determine the concentration-mass relation of 19 X-ray selected galaxy clusters from the CLASH survey in theories of gravity that directly modify the lensing potential. We model the clusters as NFW haloes and fit their lensing signal, in the Cubic Galileon and Nonlocal gravity models, to the lensing convergence profiles of the clusters. We discuss a number of important issues that need to be taken into account, associated with the use of nonparametric and parametric lensing methods, as well as assumptions about the background cosmology. Our results show that the concentration and mass estima…
Galileon gravity offers a robust gravitational theory for explaining cosmic acceleration, having a rich phenomenology of testable behaviors. We explore three classes of Galileon models -- standard uncoupled, and linearly or derivatively coupled to matter -- investigating the expansion history with particular attention to early time and late time attractors, as well as the linear perturbations. From the relativistic and nonrelativistic Poisson equations we calculate the generalizations of the gravitational strength (Newton's constant), deriving its early and late time behavior. By scanning thro…
We present the first constraints on the full parameter space of the Galileon modified gravity model, considering both the cosmological parameters and the coefficients which specify the additional terms in the Lagrangian due to the Galileon field, which we call the Galileon parameters. We use the latest cosmic microwave background measurements, along with distance measurements from supernovae and baryonic acoustic oscillations, performing a Monte Carlo Markov Chain exploration of the 9-dimensional parameter space. The integrated Sachs-Wolfe signal can be very different in Galileon models compar…
We review the different frameworks in which Galileon scalar fields have been seen to emerge such an in DGP, New Massive Gravity and Ghost-free massive Gravity and emphasize their relation with the Lovelock invariant in braneworld models. The existence of a non-renormalization theorem for Galileon scalar fields makes them especially attractive candidates for inflation as well as for late-time acceleration. In particular we review the self-accelerating and degravitating branches of solutions present in Galileon models when arising from Massive Gravity and discuss their phenomenology.
Certain scalar fields with higher derivative interactions and novel classical and quantum mechanical properties - the Galileons - can be naturally covariantized by coupling to nonlinear massive gravity in such a way that their symmetries and number of degrees of freedom are unchanged. We study the propagating degrees of freedom in these models around cosmologically interesting backgrounds. We identify the conditions necessary for such a theory to remain ghost free, and consider when tachyonic instabilities can be avoided. We show that on the self-accelerating branch of solutions, the kinetic t…
Galileon gravity is a robust theoretical alternative to general relativity with a cosmological constant for explaining cosmic acceleration, with interesting properties such as having second order field equations and a shift symmetry. While either its predictions for the cosmic expansion or growth histories can approach standard ΛCDM, we demonstrate the incompatibility of both doing so simultaneously. Already current observational constraints can severely disfavor an entire class of Galileon gravity models that do not couple directly to matter, ruling them out as an alternative to ΛCDM.
We present a perturbative treatment of nonlinear galaxy clustering in the context of the cubic Galileon modified gravity model, in terms of 2nd order Lagrangian Perturbation theory and an extension of ellipsoidal collapse that includes Vainshtein screening. We numerically implement such prescriptions in the approximate PINOCCHIO code, and use it to generate realisations of the matter density field and halo catalogues with different prescriptions for ellipsoidal collapse. We investigate the impact of three different approximations in the computation of collapse times on the halo mass function, …
We calculate the power emitted in scalar modes for a binary systems, including binary pulsars, with a conformal coupling to the most general Galileon effective field theory by considering perturbations around a static, spherical background. While this method is effective for calculating the power in the cubic Galileon case, here we find that if the quartic or quintic Galileon dominate, for realistic pulsar systems the classical perturbative expansion about spherically symmetric backgrounds breaks down (although the quantum effective theory is well-defined). The basic reason is that the equatio…
In this paper we examine the cosmological consequences of fourth order Galileon gravity. We carry out detailed investigations of the underlying dynamics and demonstrate the stability of one de Sitter phase. The stable de Sitter phase contains a Galileon field $π$ which is an increasing function of time (\dotπ>0). Using the required suppression of the fifth force, supernovae, BAO and CMB data, we constrain parameters of the model. We find that the $π$ matter coupling parameter $β$ is constrained to small numerical values such that $β$<0.02. We also show that the parameters of the third and four…
We study the cosmology of Galileon modified gravity models in the linear perturbation regime. We derive the fully covariant and gauge invariant perturbed field equations using two different methods, which give consistent results, and solve them using a modified version of the {\tt CAMB} code. We find that, in addition to modifying the background expansion history and therefore shifting the positions of the acoustic peaks in the cosmic microwave background (CMB) power spectrum, the Galileon field can cluster strongly from early times, and causes the Weyl gravitational potential to grow, rather …
A scalar-tensor theory of gravity can be made not only to account for the current cosmic acceleration, but also to satisfy solar-system and laboratory constraints, by introducing a non-linear derivative interaction for the scalar field. Such an additional scalar degree of freedom is called "Galileon". The basic idea is inspired by the DGP braneworld, but one can construct a ghost-free model that admits a self-accelerating solution. We perform a fully relativistic analysis of linear perturbations in Galileon cosmology. Although the Galileon model can mimic the background evolution of standard $…
We compare observational data of growth rate with the prediction by Galileon theory. For the same value of the energy density parameter $Ω_{m,0}$, the growth rate in Galileon models is enhanced compared with the $Λ$CDM case, due to the enhancement of Newton's constant. The smaller $Ω_{m,0}$ is, the more suppressed growth rate is. Hence the best fit value of $Ω_{m,0}$ in the Galileon model is 0.16 from only the growth rate data, which is considerably smaller than such value obtained from observations of supernovae Ia, the cosmic microwave background and baryon acoustic oscillations. This result…
In this paper we study the evolution of a spherical matter overdensity in the context of the recently introduced Galileon field theory. Our analysis considers the complete covariant Lagrangian in four dimensions. This theory is composed by a potential and a standard kinetic term, a cubic kinetic term and two additional terms that include the coupling between the Galileon and the metric, to preserve the original properties of Galileons also in curved space-times. Here we extend previous studies, which considered both the quintessence and the cubic terms, by focussing on the role of the last two…
We develop a numerical algorithm to solve the high-order nonlinear derivative-coupling equation associated with the quartic Galileon model, and implement it in a modified version of the RAMSES N-body code to study the effect of the Galileon field on the large-scale matter clustering. The algorithm is tested for several matter field configurations with different symmetries, and works very well. This enables us to perform the first simulations for a quartic Galileon model which provides a good fit to the cosmic microwave background (CMB) anisotropy, supernovae and baryonic acoustic oscillations …
We studied the cosmological constraints on the Galileon gravity obtained from observational data of the growth rate of matter density perturbations, the supernovae Ia (SN Ia), the cosmic microwave background (CMB), and baryon acoustic oscillations (BAO). For the same value of the energy density parameter of matter $Ω_{m,0}$, the growth rate $f$ in Galileon models is enhanced, relative to the $Λ$CDM case, because of an increase in Newton's constant. The smaller $Ω_{m,0}$ is, the more growth rate is suppressed. Therefore, the best fit value of $Ω_{m,0}$ in the Galileon model, based only the grow…
Galileon models are a class of effective field theories that have recently received much attention. They arise in the decoupling limit of theories of massive gravity, and in some cases they have been treated in their own right as scalar field theories with a specific nonlinearly realized global symmetry (Galilean transformation). It is well known that in the presence of a source, these Galileon theories admit superluminal propagating solutions, implying that as quantum field theories they must admit a different notion of causality than standard local Lorentz invariant theories. We show that in…
We use N-body simulations to study the statistics of massive halos and redshift space distortions for theories with a standard ΛCDM expansion history and a galileon-type scalar field. The extra scalar field increases the gravitational force, leading to enhanced structure formation. We compare our measurements of the real space matter power spectrum and halo properties with fitting formula for estimating these quantities analytically. We find that a model for power spectrum, halo mass-function and halo bias, derived from ΛCDM simulations can fit the results from our simulations of modified grav…
We consider background dynamics of generalized Galileon theories in the context of inflation, where gravity and inflaton are non-minimally coupled to each other. In the inflaton oscillation regime, the Hubble parameter and energy density oscillate violently in many cases, in contrast to the Einstein gravity with minimally coupled inflaton. However, we find that there is an adiabatic invariant in the inflaton oscillation regime in any generalized Galileon theory. This adiabatic invariant is useful in estimating the expansion law of the universe and also the particle production rate due to the o…
In the context of a cubic Galileon model in which the Vainshtein mechanism suppresses the scalar field interactions with matter, we study low-density stars with slow rotation and static relativistic stars. We develop an expansion scheme to find approximated solutions inside the Vainshtein radius, and show that deviations from General Relativity (GR), while considering rotation, are also suppressed by the Vainshtein mechanism. In a quadratic coupling model, in which the scalarisation effect can significantly enhance deviations from GR in normal scalar tensor gravity, the Galileon term successfu…
We present a detailed study of the collapse of a spherical matter overdensity and the non-linear growth of large scale structures in the Galileon ghost condensate (GGC) model. This model is an extension of the cubic covariant Galileon (G3) which includes a field derivative of type $(\nabla_μϕ\nabla^μϕ)^2$ in the Lagrangian. We find that the cubic term activates the modifications in the main physical quantities whose time evolution is then strongly affected by the additional term. Indeed, the GGC model shows largely mitigated effects in the linearised critical density contrast, non-linear effec…
The Galileon model is a modified gravity model that can explain the late-time accelerated expansion of the Universe. In a previous work, we derived experimental constraints on the Galileon model with no explicit coupling to matter and showed that this model agrees with the most recent cosmological data. In the context of braneworld constructions or massive gravity, the Galileon model exhibits a disformal coupling to matter, which we study in this paper. After comparing our constraints on the uncoupled model with recent studies, we extend the analysis framework to the disformally coupled Galile…
How deep can a cosmic void be? Voids-informed theoretical bounds in Galileon gravity
Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）；INFN Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；ICTP, International Centre for Theoretical Physics（国际理论物理中心）；Center for Computational Astrophysics, Flatiron Institute（平坦研究所计算天体物理学中心）；Università degli Studi di Torino（都灵大学）；INFN-Sezione di Torino（意大利国家核物理研究所都灵分部）；INAF-Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino（意大利国家天体物理研究所都灵天文台）
Cosmological models with Galileon gravity are an alternative to the standard $Λ{\rm CDM}$ paradigm with testable predictions at the level of its self-accelerating solutions for the expansion history, as well as large-scale structure formation. Here, we place constraints on the full parameter space of these models using data from the cosmic microwave background (CMB) (including lensing), baryonic acoustic oscillations (BAO) and the Integrated Sachs-Wolfe (ISW) effect. We pay special attention to the ISW effect for which we use the cross-spectra, $C_\ell^{\rm T g}$, of CMB temperature maps and f…
