Utilisation de l'IBM Blue Gene/P de l'IDRIS :
Preheating metric perturbations after inflation

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We propose to study the numerical evolution of the full set of Einstein equations for an expanding universe, in a 3-dimensional spatial lattice with N points and size L per spatial dimension, and the source term given by the matter Lagrangian of two coupled scalar fields. The scalar potential is that of a hybrid model of inflation, and the evolution will start soon after the end of inflation. Writing the equations in the so called BSSN (Baumgarte-Sahpiro-Shibata-Nakamura) formalism in General Relativity, the system to be integrated is given by a set of 21xN³ first order partial differential equations. The code used is a parallel program in C, that makes use of the SUNDIALS (Suite of Nonlinear Differential Algebraic equation Solvers) library and the FFTW (Fast Fourier Transform) library.

The inflationary paradigm remains extremely successful in solving the horizon and flatness problems of the standard cosmology, and at the same time in explaining the origin of structure in the observable Universe. The implementation of the inflationary mechanism usually requires the presence of at least one scalar field, the inflaton. After inflation, one should recover the radiation dominated (RD) epoch in standard cosmology. During the first stages of the transition from inflation to RD, non-perturbative effects may dominate, giving rise to the parametric amplification of field and metric fluctuations, a process known as « preheating ». Preheating of field fluctuations has been extensively studied in the recent literature; as well as that of fields plus scalar metric perturbatons, and fields plus tensor metric perturbations. The latter could lead to a stochastic background of primordial gravity waves that might be detected by future gravity waves experiments.
The aim of the project is to study the evolution of the coupled system field-gravity during preheating integrating the full set of Einstein equations with matter sources provided by the fields, and including scalar, tensors and vector metric perturbations. Tensor modes are sourced by field and scalar metric perturbations (and eventually by vectors). Previous studies only include field perturbations in the source term for tensors. Therefore, the question is whether the predictions obtained in previous studies, including only fields perturbations as the source for tensors, are modified when including the evolution of all metric modes.

During the 2009 campaign, we have been working with a development account in Babel, where the time initially allocated has been used to perform some tests, change the program to render it more efficient, and perform the strong scaling test and the the extension curve in order to pass to production mode, which has been recently granted in October 2009.
All the variables integrated present damped oscillatory behavior in time, with a typical average value for the frequency of order (1) in program units, and a damping factor given by the expansion rate H. The initial value of the latter, H₀, is one of the problem parameters. We have performed a test of strong scaling type by running a small lattice with N=32 and; H₀=0.003, integrating until t_p=100. The reference value for the number of processors was NP=64. In Fig. 1 we have plotted the acceleration ersus the no. of processors (LHS), and the efficiency (RHS), defined as:
                                       Acc(N_P)= tref/t(NP),  Eff(NP)=Acc(NP)/(NP/Nref).


With the time allocated left, we have run an example with N=64, L=10π and; H₀=0.006. In Fig. 2 we show the results for the evolution of the energy density in GW, and their power spectrum versus comoving momentum at different times.
We have compared the case when including all kind of metric perturbations with that usually studied in the literature where only the fluctuations of the field and tensors are kept. The aim now is to run lattices with N=128 and N=256, for different values of the lattice size L, and the damping parameter H₀. By varying L and N we can check the sensitivity to both the infrared cutoff kmin=2π/L and the ultravioltet cutoff kmax= N (2π/L) of the lattice. For the physical problem at hand we need kmin < 0.4, and Δk no larger than 0.2, in order to check where the shift of the peak of the GW spectrum due to the effect of the scalar and vector metric perturbations.

Figure 2: LHS plot: energy density for primordial GW produced after inflation, integrating the full set of Einstein equations (solid lines), and including only tensor perturbations (dashed lines). RHS plot: power spectrum for primordial gravity waves at different times. The system has been integrated in a lattice with N=64 sites per spatial dimension and size L=10π; g is the value of the coupling between the 2 scalar fields in hybrid inflation.