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Unification of Dark fields and Inflation To cite this article: J D García-Aguilar and A Pérez-Lorenzana 2011 J. Phys.: Conf. Ser. 315 012009

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VI International Workshop on the Dark side of the Universe (DSU 2010) Journal of Physics: Conference Series 315 (2011) 012009

IOP Publishing doi:10.1088/1742-6596/315/1/012009

Uni cation of Dark elds and In ation. 1

2

J. D. Garc a-Aguilar , A. P erez-Lorenzana

1; 2 Departamento de Fsica, Centro de Investigacion y de Estudios Avanzados del I.P.N., Apdo. Post. 14-740 07000 Mexico D.F., Mexico. E-mail: 1 [email protected], 2 [email protected] We present a partial uni ed model for in ation, dark matter and dark energy, based on a global SU (2)  U (1) symmetry, where the corresponding cosmological scalar elds that source such components of the Universe appear as dierent degrees of freedom of fundamental representations. The protecting symmetry allows to build successful potentials to describe both the early and late eras of accelerated expansion of the Universe, and admit for the interpretation of Dark energy as quintessence or phantom matter. The model also provides a limit where the elds have a dual behavior. Abstract.

1. Introduction

Measurements of the microwave cosmic background spectrum by the Cosmic Background Explorer (COBE) and most recently by the Wilkinson Microwave Anisotropy Probe (WMAP), and the observations of distant type Ia supernovae and galaxy cluster measurements, point towards the unavoidable necessity of dark components of the Universe. Both in the form of dark matter (DM) and dark energy (DE). Apart from the cosmological constant (), one of the favorite candidates for DE are scalar elds [1], for which acceleration from the Friedmann equation is easy to achieve by choosing an appropriate potential energy and tuning model parameters. For all such models the equation of state P = ! leads to ! > 1. A simple way to realize this scenario is to introduce a scalar eld ' minimally coupled to gravity, called quintessence, which gives pressure and energy density 1 2 (1) P = 2 '_ V ('); 1 '_ 2 + V ('); (2)  = 2 such that ! becomes a dynamical parameter which in general varies through time and space. In contrast, a cosmological constant is static, with a xed energy density,  = , and pressure, P = , such that ! = 1. Scalar elds had also been proposed as being dark matter. We postulate that all cosmology scalar elds could be interrelated by symmetries [2], and here we present a model that exempli es the realization of such an idea. In this context we start by considering a model with four real scalar elds, which latter on we will identify as sources for hybrid in ation, dark matter and dark energy. Thus, in order to clearly identify the natural symmetries of the system, we look upon the Lagrangian density Published under licence by IOP Publishing Ltd

1

VI International Workshop on the Dark side of the Universe (DSU 2010) Journal of Physics: Conference Series 315 (2011) 012009

without potential term

L = 12 @ 'i@'i

IOP Publishing doi:10.1088/1742-6596/315/1/012009

= 1; : : : ; 4: We notice that this Lagrangian density can be expressed as L = 12 @T
@  = 21 ij @i@ j ; where we have written the eld components as i

0

=

1 '1 B '2 C B C @ '3 A ; '4

(3)

and the background metric contraction is implicit. As it is easy to see from this expression, the symmetry of the Lagrangian corresponds to the group of symmetry transformations on eld space that preserve the Euclidean metric. That is the symmetry group O(4), consisting of all four by four orthogonal real matrices. Unfortunately given any two arbitrary quadruplet representations,  and , there is only one invariant bilinear form under O(4) transformations, T I = ij i j . This indicates, that if one chooses O (4) as a fundamental symmetry, the only possible potential one can admit for the system is a general function of T  = ij i j . That hardly builds any cosmological scenario. However the reduced subgroups SO(4) and SU (4) have extra invariants. Additional invariants are found for smaller groups. We shall only use the other associated invariants of a particular subgroup to build the potential of our uni ed theory. This way we assume that only part of the total symmetry of the kinetic term is actually fundamental for the theory. 2. Evolution equations of Universe

The dynamics of the universe whit the Robertson-Walker metric is described by the Einstein's equations. Let us consider an ideal perfect uid as the source of the energy momentum tensor  T = Diag ( ; p; p; p). The Einstein's equations are [1] H2

=

_ H

=

 2 a_ a

= 1 2  aK2 ; 3MP l 1 ( + P ) + K ; 2 a2

2MP l

where H is the Hubble parameter,  and P denote the total energy density and pressure of all the species present in the universe at a given epoch. 3. A simple model

As a working example, let is consider a complex scalar eld system, where the kinetic term becomes 1 T = @  @  ; 2

and thus the largest symmetry becomes O (4; C). 2

VI International Workshop on the Dark side of the Universe (DSU 2010) Journal of Physics: Conference Series 315 (2011) 012009

Next, we consider the subgroup represented as 0 ei=2

G

=

U

(1)  SU (2) 

IOP Publishing doi:10.1088/1742-6596/315/1/012009

SU

(4), whose elements can be 1

0

0 0 i=2 B 0 C e 0 0 C G=B 0 0 @ 0 0 cos ( =2) i sin ( =2) A ; 0 0 i sin (0 =2) cos (0 =2) for which the four scalars correspond to the representations 1 (1; 1); 2 (1; 1) and (3 ; 4 ) (2; 0). So the G invariants given in terms of component elds, are = j3 j2 + j4 j2 ; (1 )ij i j = 3 4 + 4 3 ; out of which we may build the simple potential 1 M 2 2
+ m2 2 + g2 2 2 U = 2 1 2 1 2 4 2 2 +M 2 j3 j2 + j4 j2 + 2 (3 4 + 4 3 ) : (4) 21 ;

22 ;

ij i j

Thus we get two decoupled subsystems (V1 (1 ; 2 ) and V2 (3 ; 4 )). 4. Cosmology of the two subsystems

The potential V1 (1 ; 2 ) corresponds to the simple model due to Linde [3] for hybrid in ation. When in ation ends, the dynamics of potential V2 (3 ; 4 ) begins. Now, the potential V2 (3 ; 4 ) can be rewritten in the form 3 ; 4 V2 = = y M2 ;






M2 2

2 M2



3 4






(5) (6)

where M2 is the mass matrix of the potential and  = 3 ; 4 . Diagonalizing M2 we nd the squared masses of eld '+ and ' mass eigenstates, which are given by M 2  2 . If 2 = M 2 M 2 , then the potential in the eigenmode system is expressed as 2 2 V2 ('+ ; ' ) = 2M 2 j'+ j + M 2 j' j : This potential is steeper along '+ , with an almost at direction along ' , when M 2  M 2 (see Fig. 1). If initial conditions are such that '3 and '4 are about M , then '+ shall roll down the potential towards the local minimum at V2 (0; ' ) then, dynamics of '+ ends and the dynamics of ' begins, which is a quintessential in ation stage, for which we rst need to take M 2 of the order of the observed cosmological constant. We can take ' as the quintessence eld and '+ as the eld of Dark Matter due to it is high mass which makes it a superheavy particle (a candidate of dark matter [4]). Due to small value of '+ mass (in comparison of MP l ), Dark Matter and Dark Energy elds don't contribute to the value of Hubble's constant at the early Universe. 5. Summary

In this work we explore the uni cation of cosmological elds in the same way as for particle physics. We show, with an speci c example, that it is possible to build adequate cosmological potentials that are invariant under a global symmetry, which may explain the currently dynamics of the Universe. 3

VI International Workshop on the Dark side of the Universe (DSU 2010) Journal of Physics: Conference Series 315 (2011) 012009

Figure 1.

IOP Publishing doi:10.1088/1742-6596/315/1/012009

Potential V2 ('+ ; ' ) with M 2  M 2 .

Acknowledgments

Work supported in part by Conacyt, Mexico, under grant number 54576. References

[1] E. J. Copeland, M. Sami, S. Tsujikawa, arXiv:hep-th/0603057v3 (2006) [2] A. Perez-Lorenzana, M. Montesinos and T. Matos, Phys. Rev. D 77, 063507 (2008); J. D. Garca Aguilar, MSc.Thesis, CINVESTAV (2009). [3] Andrei Linde, Phys. Lett B 259 38 (1991) [4] G. Bertone, D. Hooper and J. Silk arXiv:hep-ph/0404175v2 (2004)

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