Alexandre Barreira

Nonlinear structure formation in the cubic Galileon gravity model

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 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 ~ 20% with respect to the standard ΛCDM model today. The modified expansion rate accounts for ~ 15% of this enhancement, while the fifth force is responsible for only ~ 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 (k0.1h Mpc−1), the fifth force leads to only a modest increase (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 (l50), the predicted integrated Sachs-Wolfe effect is in tension with Planck/WMAP results.

Linear Perturbations in Galileon Gravity Models

We discussed in the previous chapter that, before undergoing dedicated studies of nonlinear structure formation in modified gravity models, one should first learn about their goodness-of-fit to the data from the CMB, SNIa and BAO.

Spherical collapse in Galileon gravity: fifth force solutions, halo mass function and halo bias

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 the standard value at late times (z1), 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 z1 feel an overall weaker gravity, which suppresses halo formation. However, the matter density in the Quartic model is higher than in standard ΛCDM, 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 ΛCDM. Dark matter halos are also less biased than in ΛCDM, 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.

Simulating the quartic Galileon gravity model on adaptively refined meshes

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.

Nonlinear Structure Formation in Nonlocal Gravity

We now turn our attention to large scale structure formation in nonlocal gravity models. In these models, the modifications to gravity arise via the addition of nonlocal terms (i.e. which depend on more than one point in spacetime) to the Einstein field equations. These terms typically involve the inverse of the d’Alembertian operator, \(\Box ^{-1}\), acting on curvature tensors.

The Observational Status of Galileon Gravity After Planck

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 ΛCDM, 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 ΛCDM 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 ΛCDM, 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.

Modified Gravity N-body Code Comparison Project

Self-consistent 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 N-body codes for the f (R), Dvali-Gabadadze-Porrati and Symmetron models, starting from the same initial conditions. We find that the fractional deviation of the matter power spectrum from Lambda cold dark matter agrees to better than 1 per cent up to k ˜ 5-10 h Mpc-1 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 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.

Weak lensing by voids in modified lensing potentials

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 ≈ 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.

Halo model and halo properties in Galileon gravity cosmologies

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.

Parameter space in Galileon gravity models

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 nine-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 cosmic microwave background data rather than the cosmic microwave background 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 Wilkinson microwave anisotropy probe 9-year results better than the standard Λ-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 2σ in the Galileon model compared with ΛCDM. We highlight other potential constraints of the Galileon model using galaxy clustering and weak lensing measurements.