luis p. chimento and martın g. richarte- k-essence and extended tachyons in brane-worlds

8/3/2019 Luis P. Chimento and Martn G. Richarte- k-essence and extended tachyons in brane-worlds

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arXiv:1002.0369v1

[astro-ph.CO]2Feb2010

k-essence and extended tachyons in brane-worlds

Luis P. Chimento 1,2 and Martn G. Richarte 2

1 Department of Theoretical Physics, University of the Basque Country, P.O. Box 644, 48080Bilbao, Spain and IKERBASQUE, the Basque Foundation for Science, 48011, Bilbao, Spain.2 Departamento de Fsica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos

Aires, Ciudad Universitaria, Pabellon I, 1428 Buenos Aires, ArgentinaE-mail: [email protected], [email protected]

Abstract. We study a k-essence field evolving linearly with the cosmic time and the atypicalk-essence model on a homogeneous and isotropic flat 3-brane. We show that the k-field is drivenby an inverse quadratic polynomial potential. The solutions represent expanding, contractingor bouncing universes with a finite time span and some of them end in a big crunch or a bigrip. Besides, by selecting the extended tachyonic kinetic functions we analyze the high and lowenergy limits of our model, obtaining the nearly power law solution. We introduce a tachyonfield with negative energy density and show that the universe evolves between two singularities.

1. Introduction

Recent progresses in Superstring/M-theory have offered new perspectives to understand thecosmological evolution of the universe [1], [2]. In these theories our Universe is conceived as a3-brane embedded in a higher dimensional spacetime usually referred as the bulk [3]. Amongmany interesting models coming from this new scenarios, there are two possibilities that seemto resolve different problems in the standard cosmology, the RS [2] and DGP [4] scenarios.

Several works have been investigated in cosmological scenarios with non-canonical kineticterm usually known as k-essence models [5]-[10]. In particular, some efforts in the framework ofk-essence have been directed toward model building using power law solutions which preserve[6], [11],[12], [13] or violate the weak-energy condition [14]. In addition, a k-essence model witha divergent sound speed, called atypical k-essence, was carefully analyzed in Refs. [8], [15]. Inthe latter case it showed that the model fix the form of the Lagrangian for k-essence matter.Then it would be interesting to study brane-worlds models supported by k-essence.

Recently, power law solutions on a 3-dimensional brane coupled to the tachyonic field wereobtained [16] by using a well known algorithm developed in [11]. Further the high and low energylimits for the tachyonic potential turns out to be V = V0

1 and V = V02 respectively.

Here, we present a model with k-essence localized on the Friedmann-Robertson-Walker(FRW) brane and obtain the scale factor and potential for a k-field evolving linearly with thetime. Also, we examine the divergent sound speed model and apply these results to the extendedtachyon field cases. Finally the conclusions are stated.
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2. k-essence in brane-worlds

We will explore the evolution of a universe filled with a k-essence field in a flat FRW spacetime.

Using the perfect fluid analogy, the energy density and pressure are given by

= V()[F 2xFx], p = L = V()F(x). (1)

where F(x) is a function of the kinetic energy x = 2, Fx = dF/ dx and V() is a positivepotential. We shall focus on cosmological branes with the induced metric g = diag[1, a2ji],where a is the scale factor. Then the modified Einstein equations on the brane are [17]

3H2 = +3

22, (2)

(Fx + 2xFxx) + 3HFx +V

2V(F 2xFx) = 0, (3)

with H = a/a the Hubble expansion rate and d/d. Also, by writing the equation of statefor the k-essence as p = ( 1), the barotropic index reads = 2xFx/(F 2xFx). Thequadratic term (2) modifies the standard cosmology dominating at the early times, H /,whereas in the limit we recover standard relativity H .

For x = x0 = const and Fx(x0) = 0, we get = 0. Hence, from Eqs. (2) and (3) the energydensity and the expansion rate become constants and the brane exhibits a de Sitter phase.

Other kind of solutions can be found by re-writing the Eqs. ( 2)-(3)

3H2 = V +3

22V2,

+

V

V+ 3H = 0, (4)

with = /V = F 2xFx and assuming the constraint = 0 = const. It is satisfied for anykinetic function F when i. the k-field evolves linearly with the cosmological time, = 0t withx0 = 20 = const or ii. for the kinetic function fulfilling F 2xFx = 0 [6], [8], [13], hence

F = 0 + x, (5)

where is an arbitrary constant. It is associated with a divergent sound velocity and with theextended tachyon model considered in [6]. The function F generates a divergent sound speedmodel called atypical k-essence [8], [15].

i. In this case the barotropic index = 0 = const and one finds the potential V = V0a30 ,

after integrating the conservation equation (4b). For this potential the Eq. (4a) reads

y2 = 320

0y +

320

2

, (6)

with y = a30 and 0 = V00. Its general solution is given by

a() = (30)1/30

2

4

1/30

, (7)

where = 0t. This solution has singularities at s = 0 and/or s = 4/. For 0 > 0, we havefour expanding universes, two of them with 0 > 0 evolve from an initial singularity at

s= 0

or s = 4/. The scale factor begins as a ( s)1/30 in the high energy regime and ends asa 2/30 in the low energy regime. The remaining two universes with 0 < 0 end in a final


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big rip at s = 0 or s = 4/ having the final behavior (s )1/30 . For 0 < 0, the solutionshave an extremum at e = 2/ where ae = (30/2)1/30 . They represent two universes with afinite time span, one of them begins from a singularity, reaches a maximum value ae, and endsin a big crunch at s = 4/. The other begins from a singularity at s = 0, reaches a minimumat ae, where it bounces, and ends in a final big rip at s = 4/. In addition we have the timereversal of these solutions.

In turn, by combining (t) with a(t) and V = V0a30 , we obtain the following potential

V =V0

30

20

2

420

00

1

. (8)

In the early universe quadratic contributions in become important and the potentialapproaches V 1 when 0. It seems to be the counterpart of the solution forquintessence cosmology driven by an inverse square potential [18], [19]. In the limit

, the

potential V 2 and we recover the power law solution [11], [12] [13].The Eq. (8) shows that the brane correction shifts the inverse square potential to the inverse

linear one at high energy. It has a minimum at e = 20/0 or a maximum at e = 20/0.Finally, by associating the brane equations to an effective fluid description, we find a linearbarotropic equation of state in the low energy limit. However, in the high energy regime the

k-essence source becomes a modified Chaplygin gas, Pef (20 1)ef + 01/2ef /

3.

ii. By combining the Eq. (5) and its associated barotropic index = x/0 with the Eq.

(4b) we find H = 0V/(3V). So by using a = 0V() into the Eq. (2) one gets the potential

V =1

3

402

1

. (9)

The potential on the brane (9) generalizes the inverse square one of the Friedmann cosmologyshifting the divergence to = 40/. So, for asymptotic power-law expansions the lineark-field model driven by a nearly inverse square potential and the atypical model are isomorphic.

2.1. Extended Tachyons in brane-worlds

Recently, it was proposed that the tachyon Lagrangian could be extended in such a way to allowthe barotropic index any value [6] generating phantom and complementary tachyons in additionto the ordinary one [6], [20]. This extended tachyons generalize the Chaplygin gas introducingthe phantom and complementary Chaplygin gases. The kinetic functions for these tachyons are

Ft = (1 2rt )1/2r, 0 < t = 2rt < 1 (10)Fph = (1 +

2rph)

1/2r, < ph = 2rph < 0 (11)Fc = (2rc 1)1/2r, 1 < c = 2rc < (12)

where r is a real parameter. The scale factor and potential for the three sets of extended tachyonfields are given by Eqs. (7) and (8) with 0 =

2r0t , 0 = 2r0ph, 0 = 2r0c and 0 = (1 2r0t )1/2,

0 = (1 + 2r0ph)

1/2, 0 = (2r0c 1)1/2 respectively.The ordinary and complementary tachyons lead to expanding scenarios while the former has

an accelerated expansion for 2r0t < 2/3. A universe dominated by a phantom tachyon ends in

a big rip at s = 0 or s = 4/ where the scale factor blows up as a 1/32r0ph and the

potential diverges as V

1

ph. The ordinary and the complementary tachyons satisfy the weak

energy condition 0 and +p 0, and the null energy condition +p 0 while the phantomtachyon violates both conditions.


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As Eq. (2) is quadratic in the energy density, we introduce a tachyon with negative energydensity by making F F in Eqs. (10)-(12) and keeping V > 0. The universe evolves betweentwo singularities having a finite time span ts = 4/|0|. The scale factor has a maximum forordinary and complementary tachyons, ending in a big crunch or has a minimum where thephantom tachyon bounces before it blows up in a big rip.

3. Summary

We have found solutions for 3-dimensional cosmological brane with k-essence and extendedtachyons, whose metric corrpesponds to a spatially flat FRW spacetime, when the ratio /V isconstrained to be constant for linear k-field or for the atypical k-essence model. For 0 > 0 wehave obtained power law behavior at the initial and final stages. However, in the case 0 < 0 theuniverse bounces, has a finite time span and ends in a big crunch (0 > 0) or a big rip (0 < 0).The quadratic brane correction has shifted the inverse square potential to the inverse linearone at high energy, so that the k-field is driven by an inverse quadratic polynomial potential.Finally, we have analyzed the extended tachyons with negative energy density and showed thatthe scale factor evolves between two singularities having a finite time span which depends onthe brane tension and the barotropic index. For the ordinary and complementary tachyons thescale factor ends in a big crunch while for the phantom one it bounces and ends in a big rip.

Acknowledgments

LPC thanks the University of Buenos Aires for the partial support of this work during itsdifferent stages under Project X044, and the Consejo Nacional de Investigaciones Cientficas yTecnicas under Project PIP 114-200801-00328. MGR is supported by the Consejo Nacional deInvestigaciones Cientficas y Tecnicas (CONICET).

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luis p. chimento and martın g. richarte- k-essence and extended tachyons in brane-worlds

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