Attractors in dark energy models with Born-Infeld scalar field

Attractors in dark energy models with Born-Infeld scalar field

Ronaldo Carlotto Batista; Luís Raul Weber Abramo; Thiago dos Santos Pereira

Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo, Brazil

ABSTRACT

We derive, in the large scale limit, analytical solutions for a dark energy model described by the Born-Infeld field plus a perfect fluid, both for homogeneous background and first order perturbations. These analytical results are compared with numerical solutions in a model with radiation, pressureless matter and the field. We investigate the non-adiabatic perturbed pressure associated with this attractor and whether it corresponds to isocurvature contribution.

I. INTRODUCTION

Recent observations [1] indicate that expansion of the universe is accelerated. Models with cosmological constant L and canonical scalar fields [2] have been proposed to explain this acceleration. Another class of scalar field theory is the Born-Infeld theory [3]:

We use a inverse power-law potential V(j) = M^4+aj^-a, where, for dark energy models, we must have 0 < a < 2 [4]. The equation of motion of such field is:

which, at homogeneous level becomes

In this case, the density and pressure are:

and w_j = p_j/r_j.

The background evolution is given by the Friedmann equation, with 8pG = 1,

the equation (3) plus the equations of continuity for radiation and matter.

II. ATTRACTORS

The equation (3) may assume the solution = 0. When a perfect fluid with equation of state w_f = g_f-1 dominates the background evolution, we have H = and the field grows linearly in time [4]:

where A = for a < 2/g_f, and A = 1 for a > 2/g_f.

We evaluate first order scalar perturbations in the newtonian-longitudinal gauge [5], with no anisotropic stress:

In this case, the first order perturbations of the field obey the equation:

In the large scale limit k/aH 1, assuming = 0, H = , and the background solution (6), the field perturbation has the following attractor solution:

where c₁ is a constant. Neglecting the decaying mode, we have

In this case, the field evolves as a perfect fluid with constant equation of state, given by w_j = A²-1.

When the attractor regime is reached at matter domination, we have g_f = 1, then always a < 2/g_f and A = . But, when the attractor regime is reached at radiation domination, we have g_f = 4/3, then we may have a < 2/g_f = 1.5, thus A = , or a > 2/g_f = 1.5 and A = 1. In the latter case, the field perturbations can grow indefinitely when the transition between radiation domination and matter domination is going on. This behaviour is strongly dependent on the initial conditions for the field perturbations. Hence, we call models with a < 1.5 "stable" and with a > 1.5 "unstable".

III. ISOCURVATURE PERTURBATION

We want to investigate, in the large scale limit, whether the Born-Infeld field generates isocurvature perturbations. The function [6]

where the prime denotes the derivative with respect to N = ln(a/a₀), has the following equation of motion [5, 6]

where dp_nad is the total nonadiabatic pressure. From (12), we see that z can only vary in large scales if the nonadiabatic pressure is nonnegligible.

To verify if the Born-Infeld field generates isocurvature modes we set adiabatic conditions between matter and radiation, d_mi = 0.75d_ri, and evaluate the functions dj/j,F and z for a = 1 (stable case - Figure 1) and a = 1.51 (unstable case - Figure 2) .

To illustrate the dependence of the evolution of the field perturbations on its initial conditions and how this can generate isocurvature modes, we show in Figure 3, for three different sets of initial conditions, the evolution of z function, for three different values of a.

IV. CONCLUSIONS

For dark energy models, the isocurvature modes tend to be very small. This occurs because to approach w_j = -1 today, we need a near zero, what decreases the isocurvature modes. On the other hand, since we have a > 1.5, the isocurvature modes can be very significant. But, in this case, we do not have good models for dark energy, once we have, in the matter dominated era, w_j = -0.25 at least, what does not allow values of w_j close to -1 today. For this case, we have a model closer to a dark matter model.

Acknowledgments

This work was supported by CNPq and FAPESP.

References

[1] A. Riess et al, Astron. J. 116, 1009 (1998); C. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003)

[2] B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988); C. Wetterich, Nucl. Phys. B302, 645 (1988).

[3] T. Padmanabhan, Phys. Rev.D 66, 021301 (2002); J. Bagla, H. Jassal, and T. Padmanabhan Phys. Rev. D 67, 063504 (2003); L. R. Abramo, F. Finelli and Thiago S. Pereira, Phys. Rev. D 70, 063517 (2004).

[4] L. R. Abramo and F. Finelli, Phys. Lett. B575, 165 (2003).

[5] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Phys. Rept. 215, 203 (1992).

[6] J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. D28, 679 (1983).

Received on 17 October, 2005

Publication Dates

• Publication in this collection
  16 Jan 2006
• Date of issue
  Dec 2005

History

• Received
  17 Oct 2005