## N-body simulation of DGP model July 1, 2009

Posted by keithkchan in Cosmology, Journal club.

I am very happy to have Kwan Chuen Chan from Center for Particles Physics and Cosmology, New York University to talk about their new paper. He is a grad student at NYU, working with Roman Scoccimarro. His office is in Room 538 CCPP.
Keith

Thank Keith for inviting me to blog about our recent paper. In this post I will briefly talk about the paper that Roman Scoccimarro and I just uploaded to the arXiv. I will keep it brief and elementary, so for more details, please refer to the original paper arXiv:0906.4548.

Here is the abstract

Large-Scale Structure in Brane-Induced Gravity II. Numerical Simulations
Authors: K. C. Chan, Roman Scoccimarro
(Submitted on 24 Jun 2009)
Abstract: We use N-body simulations to study the nonlinear structure formation in brane-induced gravity, developing a new method that requires alternate use of Fast Fourier Transforms and relaxation. This enables us to compute the nonlinear matter power spectrum and bispectrum, the halo mass function, and the halo bias. From the simulation results, we confirm the expectations based on analytic arguments that the Vainshtein mechanism does operate as anticipated, with the density power spectrum approaching that of standard gravity within a modified background evolution in the nonlinear regime. The transition is very broad and there is no well defined Vainshtein scale, but roughly this corresponds to k_*= 2 h/Mpc at redshift z=1 and k_*=1 h/Mpc$at z=0. We checked that while extrinsic curvature fluctuations go nonlinear, and the dynamics of the brane-bending mode$C$receives important nonlinear corrections, this mode does get suppressed compared to density perturbations, effectively decoupling from the standard gravity sector. At the same time, there is no violation of the weak field limit for metric perturbations associated with$C\$. We find good agreement between our measurements and the predictions for the nonlinear power spectrum presented in paper I, that rely on a renormalization of the linear spectrum due to nonlinearities in the modified gravity sector. A similar prediction for the mass function shows the right trends but we were unable to test this accurately due to lack of simulation volume and mass resolution. Our simulations also confirm the induced change in the bispectrum configuration dependence predicted in paper I.

DGP model is an extra-dimension model, which has one co-dimension, and ordinary matter lives on the 3-brane. The graviton propagator is modified in the infrared. One of the interesting properties of this model is that it exhibits self-accelerating solution. The hope was that the recent observed cosmic acceleration may be due to modification of gravity rather than the mysterious dark energy. However, both theoretically and observationally, this model is proved to be unfavorable. However, this model has inspired a bunch of more sophisticated models such as degravitation, galleon. One of the serious problem in modification of gravity is that it induces new degrees of freedom. The theory can usually be approximated as a scalar-tensor theory. But any scalar degree of freedom is likely to be highly constrained by current solar system experiments. There are two nice ways have been put forward to evade this kind of constraints. One of them is chameleon mechanism, which has been realized in $f(R)$ gravity. The other mechanism is called Vainshtein effect, which is incorporated in DGP and some massive gravity models. The scalar degree of freedom becomes strongly coupling and frozen because of the derivative self-interactions. The theory effectively becomes GR.

In this paper, using numerical simulations, we study this type of brane-induced gravity in the nonlinear regime, in particular the Vainshtein effect. We compute the cosmological observables: the power spectrum, bispectrum, mass function, and bias, which give us the signatures of the DGP model, and help us to differentiate modified gravity model from dark energy. In the companion paper arXiv:0906.4545 by Scoccimarro, the model is studied by perturbative calculations. Some of the results are checked against the numerical results in this work.

The method we used is the N-body simulation, which is largely similar to the standard gravity one. However, in GR, the field equation in the subhorizon, non-relativistic regime is just the Poisson equation, now we need to solve a fully nonlinear partial differential equation. Let me write down the equations although I am not attempting to explain it in details
$\bar{\nabla}^2 \phi - \frac{1}{\eta} \sqrt{ - \bar{\nabla}^2 } \phi + \frac{1}{2 \eta} \bar{\nabla}^2 C + \frac{ 3 \eta^2 - 5 \eta + 1 }{2 \eta^2 (2 \eta -1) } \sqrt{ - \bar{\nabla}^2 } C = \frac{3}{2} \frac{\eta -1 }{\eta} \delta$
$(\bar{\nabla}^2 C)^2 + \alpha \bar{\nabla}^2 C - (\bar{ \nabla}_{ij} C)^2 + \frac{ 3 \beta (\eta -1) }{2 \eta-1 } \sqrt{ - \bar{\nabla}^2 } C = \frac{ 3( \eta -1 ) } {\eta } ( 1- \beta \bar{\nabla}^{-1} ) \delta,$
The first equation is analogous to the Poisson equation, but now we have one more field C, whose equation of motion is given by the second one. The nonlocal term like $\sqrt{ - \bar{\nabla}^2 } C$ can be easily handled in the Fourier space. The real headache comes from the nonlinear derivative terms $(\bar{\nabla}^2 C)^2$ and $(\bar{ \nabla}_{ij} C)^2$. One of the major achievement in this paper is that we developed a convergent method to solve this set of equations consistently. It involves alternate use of relaxation and Fast Fourier transform (so we call it FFT-relaxation method). Although that is a main result of the paper, I am not going to talk about it in details so as not to get too technical and dry. But interested readers are welcome to read the original paper.

Let me get to the results. As I have mentioned, from the simulations we have measured the power spectrum, bispectrum, mass function and bias. Here I only show the power spectrum.

In the first figure we show the power spectrum from three different models, which are the fully nonlinear DGP model (nlDGP), linearized DGP model (lDGP) and the GR with the same expansion history as the DGP model (GRH), which essentially is the GR limit. In order to see the difference more clearly, we have shown the ratios of power spectrum from various models, $P_{\rm nlDGP} / P_{\rm lDGP}$ and $P_{\rm GRH} / P_{\rm nl DGP}$ in the lower figure. In the large scales (small k), the full nonlinear DGP model reduces to the linear one. More interestingly in the nonlinear regime (large k limit), the fully nonlinear DGP model approaches the GR with the same expansion history. This demonstrates that Vainshtein effect drives the model towards GR limit in the large k regime. The transition is broad and the limit is not yet fully attained in the range shown here.

OK, let me summarize some of the main results here. We have developed a convergent algorithm, FFT-relaxation method, to solve the fully nonlinear field equations in the DGP model. This enables to compute the observables like the power spectrum in the DGP model using numerical simulations. We have demonstrated the Vainshtein effect, and the Vainshtein radius at $z =0$ is about 1 h/Mpc. For more details, please refer to our original paper arXiv:0906.4548.