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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 than decay, at late times. This leaves clear signatures in the low-$l$ CMB power spectrum through the modified integrated Sachs-Wolfe effect, strongly enhances the linear growth of matter density perturbations and makes distinctive predictions for other cosmological signals such as weak lensing and the power spectrum of density fluctuations. The quasi-static approximation is shown to work quite well from small to the near-horizon scales. We demonstrate that Galileon models display a rich phenomenology due to the large parameter space and the sensitive dependence of the model predictions on the Galileon parameters. Our results show that some Galileon models are already ruled out by present data and that future higher significance galaxy clustering, ISW and lensing measurements will place strong constraints on Galileon gravity., Comment: version 2. Minor changes to figures and text, and typo corrected in Eq. (A7). References added. Published in PRD

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 compared to standard gravity, making it essential to use the full CMB data rather than the CMB distance priors. We demonstrate that meaningful constraints are only possible in the Galileon parameter space after taking advantage of a scaling degeneracy. We find that the Galileon model can fit the WMAP 9-year results better than the standard \Lambda-Cold Dark Matter model, but gives a slightly worse fit overall once lower redshift distance measurements are included. The best-fitting cosmological parameters (e.g. matter density, scalar spectral index, fluctuation amplitude) can differ by more than 2\sigma\ in the Galileon model compared with \Lambda CDM. We highlight other potential constraints of the Galileon model using galaxy clustering and weak lensing measurements., Comment: 21 pages, 7 figures and 2 tables v2. Presentational changes in some figures. References added. Matches version published in PRD

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 $\Lambda$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., Comment: 16 pages, 5 figures and 2 tables; The latest version corrects small typos in equations that do not affect the results

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 (BAO) data. Our result shows that the Vainshtein mechanism in this model is very efficient in suppressing the spatial variations of the scalar field. However, the time variation of the effective Newtonian constant caused by the curvature coupling of the Galileon field cannot be suppressed by the Vainshtein mechanism. This leads to a significant weakening of the strength of gravity in high-density regions at late times, and therefore a weaker matter clustering on small scales. We also find that without the Vainshtein mechanism the model would have behaved in a completely different way, which shows the crucial role played by nonlinearities in modified gravity theories and the importance of performing self-consistent N-body simulations for these theories., Comment: 22.15 pages, 4 figures; minor corrections, version accepted for publication in JCAP

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 late times ($z \lesssim 1$), becoming larger if the density is low, but smaller if the density is high. This shows that the Vainshtein mechanism at high densities is not enough to screen all of the modifications of gravity. This makes halos that collapse at $z \lesssim 1$ feel an overall weaker gravity, which suppresses halo formation. However, the matter density in the Quartic model is higher than in standard $\Lambda$CDM, which boosts structure formation and dominates over the effect of the weaker gravity. In the Quartic model there is a significant overabundance of high-mass halos relative to $\Lambda$CDM. Dark matter halos are also less biased than in $\Lambda$CDM, with the difference increasing appreciably with halo mass. However, our results suggest that the bias may not be small enough to fully reconcile the predicted matter power spectrum with LRG clustering data., Comment: 20 pages, 5 figures, 2 tables; v2: references added and minor changes to the text

The velocity field of dark matter and galaxies reflects the continued action of gravity throughout cosmic history. We show that the low-order moments of the pairwise velocity distribution, $v_{12}$, are a powerful diagnostic of the laws of gravity on cosmological scales. In particular, the projected line-of-sight galaxy pairwise velocity dispersion, $\sigma_{12}(r)$, is very sensitive to the presence of modified gravity. Using a set of high-resolution N-body simulations we compute the pairwise velocity distribution and its projected line-of-sight dispersion for a class of modified gravity theories: the chameleon \fR gravity and Galileon gravity (cubic and quartic). The velocities of dark matter halos with a wide range of masses would exhibit deviations from General Relativity at the $(5-10)\sigma$ level. We examine strategies for detecting these deviations in galaxy redshift and peculiar velocity surveys. If detected, this signature would be a "smoking gun" for modified gravity., Comment: 5 pages, 2 figures, accepted for publication in Physical Review Letters. Minor changes to text, some references added. The Fig. 1 now presents results for HOD LRG galaxy mock catalogues, the comments in the text changed accordingly

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 mildly nonlinear regime, but captures reasonably well the effects of the Vainshtein screening mechanism on small scales. In the Cubic model, the screening mechanism hides essentially all of the effects of the fifth force inside haloes. In the case of the Quartic model, the screening mechanism leaves behind residual modifications to gravity, which make the effective gravitational strength time-varying and smaller than the standard value. Compared to normal gravity, this causes a deficiency of massive haloes and leads to a weaker matter clustering on small scales. For both models, we show that there are realistic halo occupation distributions of Luminous Red Galaxies that can match both the observed large-scale clustering amplitude and the number density of these galaxies., Comment: 17 pages, 7 figures and 5 tables; v2 contains a HOD analysis of LRGs (Fig.5)

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 $\Lambda{\rm CDM}$, the Galileon model constraints are compatible with local determinations of the Hubble parameter and predict nonzero neutrino masses at over $5\sigma$ significance. We also identify that the low-$l$ part of the CMB lensing spectrum may be able to distinguish between $\Lambda{\rm CDM}$ and Galileon models. In the Cubic model, the lensing potential deepens at late times on sub-horizon scales, which is at odds with the current observational suggestion of a positive ISW effect. Compared to $\Lambda$CDM, the Quartic and Quintic models predict less ISW power in the low-$l$ region of the CMB temperature spectrum, and as such are slightly preferred by the Planck data. We illustrate that residual local modifications to gravity in the Quartic and Quintic models may render the Cubic model as the only branch of Galileon gravity that passes Solar System tests., Comment: 25 pages, 11 figures, 2 tables. Comments welcomed; v2 contains an extra appendix discussing the role of tensor perturbations

We study the nonlinear growth of structure in nonlocal gravity models with the aid of N-body simulation and the spherical collapse and halo models. We focus on a model in which the inverse-squared of the d'Alembertian operator acts on the Ricci scalar in the action. For fixed cosmological parameters, this model differs from $\Lambda{\rm CDM}$ by having a lower late-time expansion rate and an enhanced and time-dependent gravitational strength ($\sim 6\%$ larger today). Compared to $\Lambda{\rm CDM}$ today, in the nonlocal model, massive haloes are slightly more abundant (by $\sim 10\%$ at $M \sim 10^{14} M_{\odot}/h$) and concentrated ($\approx 8\%$ enhancement over a range of mass scales), but their linear bias remains almost unchanged. We find that the Sheth-Tormen formalism describes the mass function and halo bias very well, with little need for recalibration of free parameters. The fitting of the halo concentrations is however essential to ensure the good performance of the halo model on small scales. For $k \gtrsim 1 h/{\rm Mpc}$, the amplitude of the nonlinear matter and velocity divergence power spectra exhibits a modest enhancement of $\sim 12\%$ to $15\%$, compared to $\Lambda{\rm CDM}$ today. This suggests that this model might only be distinguishable from $\Lambda{\rm CDM}$ by future observational missions. We point out that the absence of a screening mechanism may lead to tensions with Solar System tests due to local time variations of the gravitational strength, although this is subject to assumptions about the local time evolution of background averaged quantities., Comment: 19 pages, 8 figures, 5 tables

We present a comprehensive derivation of linear perturbation equations for different matter species, including photons, baryons, cold dark matter, scalar fields, massless and massive neutrinos, in the presence of a generic conformal coupling. Starting from the Lagrangians, we show how the conformal transformation affects the dynamics. In particular, we discuss how to incorporate consistently the scalar coupling in the equations of the Boltzmann hierarchy for massive neutrinos and the subsequent fluid approximations. We use the recently proposed K-mouflage model as an example to demonstrate the numerical implementation of our linear perturbation equations. K-mouflage is a new mechanism to suppress the fifth force between matter particles induced by the scalar coupling, but in the linear regime the fifth force is unsuppressed and can change the clustering of different matter species in different ways. We show how the CMB, lensing potential and matter power spectra are affected by the fifth force, and find ranges of K-mouflage parameters whose effects could be seen observationally. We also find that the scalar coupling can have the nontrivial effect of shifting the amplitude of the power spectra of the lensing potential and density fluctuations in opposite directions, although both probe the overall clustering of matter. This paper can serve as a reference for those who work on generic coupled scalar field cosmology, or those who are interested in the cosmological behaviour of the K-mouflage model., Comment: 22 pages, 4 figures

We show that Solar System tests can place very strong constraints on K-mouflage models of gravity, which are coupled scalar field models with nontrivial kinetic terms that screen the fifth force in regions of large gravitational acceleration. In particular, the bounds on the anomalous perihelion of the Moon imposes stringent restrictions on the K-mouflage Lagrangian density, which can be met when the contributions of higher-order operators in the static regime are sufficiently small. The bound on the rate of change of the gravitational strength in the Solar System constrains the coupling strength $\beta$ to be smaller than $0.1$. These two bounds impose tighter constraints than the results from the Cassini satellite and Big Bang Nucleosynthesis. Despite the Solar System restrictions, we show that it is possible to construct viable models with interesting cosmological predictions. In particular, relative to $\Lambda$-CDM, such models predict percent-level deviations for the clustering of matter and the number density of dark matter haloes. This makes these models predictive and testable by forthcoming observational missions., Comment: 15 pages

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 estimates in the modified gravity models are, within the errorbars, the same as in $\Lambda$CDM. This result demonstrates that, for the Nonlocal model, the modifications to gravity are too weak at the cluster redshifts, and for the Galileon model, the screening mechanism is very efficient inside the cluster radius. However, at distances $\sim \left[2-20\right] {\rm Mpc}/h$ from the cluster center, we find that the surrounding force profiles are enhanced by $\sim20-40\%$ in the Cubic Galileon model. This has an impact on dynamical mass estimates, which means that tests of gravity based on comparisons between lensing and dynamical masses can also be applied to the Cubic Galileon model., Comment: 21 pages, 9 figures, 2 tables. Comments welcome

We study lensing by voids in Cubic Galileon and Nonlocal gravity cosmologies, which are examples of theories of gravity that modify the lensing potential. We find voids in the dark matter and halo density fields of N-body simulations and compute their lensing signal analytically from the void density profiles, which we show are well fit by a simple analytical formula. In the Cubic Galileon model, the modifications to gravity inside voids are not screened and they approximately double the size of the lensing effects compared to GR. The difference is largely determined by the direct effects of the fifth force on lensing and less so by the modified density profiles. For this model, we also discuss the subtle impact on the force and lensing calculations caused by the screening effects of haloes that exist in and around voids. In the Nonlocal model, the impact of the modified density profiles and the direct modifications to lensing are comparable, but they boost the lensing signal by only $\approx 10\%$, compared with that of GR. Overall, our results suggest that lensing by voids is a promising tool to test models of gravity that modify lensing., Comment: 19 pages, 8 figures. v2 matches the version published in JCAP

Self-consistent ${\it N}$-body simulations of modified gravity models are a key ingredient to obtain rigorous constraints on deviations from General Relativity using large-scale structure observations. This paper provides the first detailed comparison of the results of different ${\it N}$-body codes for the $f(R)$, DGP, and Symmetron models, starting from the same initial conditions. We find that the fractional deviation of the matter power spectrum from $\Lambda$CDM agrees to better than $1\%$ up to $k \sim 5-10~h/{\rm Mpc}$ between the different codes. These codes are thus able to meet the stringent accuracy requirements of upcoming observational surveys. All codes are also in good agreement in their results for the velocity divergence power spectrum, halo abundances and halo profiles. We also test the quasi-static limit, which is employed in most modified gravity ${\it N}$-body codes, for the Symmetron model for which the most significant non-static effects among the models considered are expected. We conclude that this limit is a very good approximation for all of the observables considered here., Comment: 31 pages, 16 figures. Version to appear in MNRAS

We introduce and demonstrate the power of a method to speed up current iterative techniques for N-body modified gravity simulations. Our method is based on the observation that the accuracy of the final result is not compromised if the calculation of the fifth force becomes less accurate, but substantially faster, in high-density regions where it is weak due to screening. We focus on the nDGP model which employs Vainshtein screening, and test our method by running AMR simulations in which the solutions on the finer levels of the mesh (high density) are not obtained iteratively, but instead interpolated from coarser levels. We show that the impact this has on the matter power spectrum is below $1\%$ for $k < 5h/{\rm Mpc}$ at $z = 0$, and even smaller at higher redshift. The impact on halo properties is also small ($\lesssim 3\%$ for abundance, profiles, mass; and $\lesssim 0.05\%$ for positions and velocities). The method can boost the performance of modified gravity simulations by more than a factor of 10, which allows them to be pushed to resolution levels that were previously hard to achieve., Comment: 15 pages, 8 figures, 1 table. Comments welcomed