Bianchi Type I Cosmology with Scalar and Spinor ...

Bianchi Type I Cosmology with Scalar and Spinor Tachyon

Alireza Sepehri, Anirudh Pradhan & Somayyeh Shoorvazi

International Journal of Theoretical Physics ISSN 0020-7748 Int J Theor Phys DOI 10.1007/s10773-014-2354-8

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Author's personal copy Int J Theor Phys DOI 10.1007/s10773-014-2354-8

Bianchi Type I Cosmology with Scalar and Spinor Tachyon Alireza Sepehri · Anirudh Pradhan · Somayyeh Shoorvazi

Received: 5 June 2014 / Accepted: 25 September 2014 © Springer Science+Business Media New York 2014

Abstract Recently, some authors propose a scenario where inflation is driven by scalar and spinor fields. In this scenario, the universe undergoes anisotropic inflationary expansion due to a preferred direction dominated by spinor. The main question arises that what is the origin of spinor inflation in 4D Universe? To answer to this question , we propose a new model in super string theory that allows to take into account the spinor string tachyon in additional to scalar one which stretch between branes and antibranes. In this model, the inflation of 4D Universe is controlled by the spinor type of tachyon and evolves from nonphantom phase to phantom one and consequently, phantom-dominated era of the universe accelerates and ends up in big-rip singularity. We test our model against WMAP and Planck data and obtain the ripping time. We find that the finite time that Big Rip singularity occurs is t = 65(Gyr). According to our calculations, N 60 case leads to ns 0.96, where N and ns are the number e-folds and the spectral index respectively. Thus our results are consistent with experimental data and thus the spinor inflationary model works. Keywords Cosmology · Scalar and spinor field · Inflationary model

A. Sepehri () Faculty of Physics, Shahid Bahonar University, P.O. Box 76175 Kerman, Iran e-mail: [email protected]; [email protected] A. Pradhan Department of Mathematics, G. L. A. University, Mathura 281 406, Uttar Pradesh, India e-mail: [email protected]; [email protected] S. Shoorvazi Department of Physics, Islamic Azad University, Neyshabur Branch, Neyshabur, Iran e-mail: [email protected]

Author's personal copy Int J Theor Phys

1 Introduction One main problem of the inflation theory is how to attach the universe to the end of the inflation epoch. An interesting solution of this problem is the study of inflation in the context of spinor inflation scenario [1]. In this model, the expansion of universe is controlled by the spinor field. Until now, the spinor inflationary Universe has been discussed in many papers [2–7]. For example, some authors considered a system of interacting spinor and scalar fields in a gravitational field given by a Bianchi type-I cosmological model. They showed that within the framework of the simplest model of interacting spinor and scalar fields, the Cosmological constant plays very important role [1]. In another research, selfconsistent solutions to the nonlinear spinor field equations in General Relativity has been studied for the case of Bianchi type-I (B-I) space-time. It has been shown that, for some special type of nonlinearity, the model provides regular solution, but this singularity-free solutions are attained at the cost of broken dominant energy condition in Hawking-Penrose theorem [2]. In another paper, the role of inhomogeneity in the evolution of spinor was studied. In this consideration, authors obtained the oscillatory mode of expansion of the universe for some special choice of spinor field nonlinearity [3]. Also, in similar paper, the evolution of the volume scale of universe, energy density and corresponding Hubble constant has been derived [4]. In one scenario, the effect of magneto-fluid on the interacting scalar and spinor fields in Bianchi type I universe was investigated [5]. In another one, different characteristic of matter influencing the evolution of the Universe has been simulated by means of a nonlinear spinor field [6]. In one investigation, it was explained that the specific behavior of spinor field in curve space-time with the exception of FRW model almost always gives rise to non-trivial non-diagonal components of the energy-momentum tensor. This non-triviality of non-diagonal components of the energy-momentum tensor imposes some severe restrictions either on the spinor field or on the metric functions [7]. These models are consistent with experimental data [8–12]. Now, the question arises that what is the origin of spinor inflationary model in 4D universe? We answer to this question in brane-antibrane system. In this system, it might be reasonable to ignore the tachyon in the ultraviolet where the branes and antibranes are well separated and the tachyon is small, it is likely to condense and acquire large values in the infrared where the branes meet. In this condition, inflation of universe is influenced by tachyon potential between branes and evolves from non phase to one. The tachyonic energy density grows without bound and becomes infinite in finite time, dominates all other forms of energy, such that the gravitational repulsion and finally brings our brief epoch of universe to an end. When the universe approaches this future singularity; the Milky Way, solar system, Earth, and ultimately the molecules, atoms, nuclei, and nucleons of which we are composed will be ripped by the infinite energy in a Big Rip [13]. Clearly, the scalar string tachyon couldn’t produce this type of inflation and we are forced to introduce spinor string tachyon as the main cause of spinor inflation in four dimensional universe. So far, many investigators used higher spin tachyons to explain different phenomena [14–23]. For example, some authors consider a Fermi gas of free tachyons in the frames of statistical thermodynamics and mechanics of continuous medium [17]. They claim that apart from any possible theoretical speculation, there are the physical principles to test this theory [18]. Finally, the first set of OPERA data suggested that the muon neutrino could propagate at a speed v larger than of light [19]. According to this experiment, many authors have considered the possibility that neutrinos may be Dirac tachyons [20–23]. Motivated by these researches, we propose a new model which includes both scalar and spinor string tachyons. We calculate the effective action for tachyonic dynamics in this system and show that all


cosmological events that happen in our universe is due to the spinor string tachyon in extra dimension. The outline of the paper is as the following. In Section 2, we construct spinor inflationary model of Universe in super string theory and show that all cosmological parameters depend on the separation distance between two branes. Also, in section, we consider Big Rip singularity and calculate the relation between ripping time and separation between branes in extra dimension. In Section 3, we test our model against the observational data from Planck and WMAP collaborations and obtain the ripping time. The last Section 4 is devoted to summary and conclusion.

2 The Spinor Inflationary Model of Universe in Super String Theory In this section, we propose a new model that allows to take into account the spinor string tachyon in additional to scalar one which stretch between D4-branes and anti-D4-branes. We will show that scalar and spinor string tachyons can control the expansion of four dimensional Universe. Also, we will enter the effects of this singularity on the results of the derivation of scale factors, Hubble parameter and other important parameters in spinor model of cosmology. We will show that these parameters depend on the ripping time and are influenced by separation distance of branes in extra dimension. In our model, equation of state parameter in 4D Universe may change due to flowing energy from extra dimension and decrease from higher values of −1 (non-phase) to lower values (one). The D4/D4 model [24–27] is formulated by placing D4-branes into the D4-brane background with the following metric: ds 2 =

U R

3 R 32 dU 2 2 ημν dx μ dx ν + f (U )(dx 4)2 + + U 2 d24 , U f (U )

eφ = gs

U R

3 4

, F4 =

U3 2πNc ε4 , f (U ) = 1 − k3 , V4 U

(1)

where ημν = diag(−1, +1, +1, +1) and Uk is a constant parameter of the solution. R is 2 ´ 5α related to the 5 − d Yang-Mills coupling by R 3 = λ4π . Also, d4 , ε4 and V4 = 8π3 are the line element, the volume form and the volume of a unit S 4 respectively. This metric has a singularity at U = Uk in the U − x4 subspace which can be avoided only if x4 has a specific 3 1 2 periodicity. The radius of the circle in the x4 direction is Rk = 23 ( R Uk ) . Now consider a set of Nf D4-D4-brane pairs in the above background, placed at points xL4 = l/2 and xR4 = −l/2 respectively on the circle so that the separation between the brane and antibrane is l. So far, the effective action for the case of a single brane-anti-brane pair with scalar string tachyon has been proposed in [28]. A generalization of this action to the super string case when Dirac and spinor tachyons are also present is known as: S = −

d 9σ

d 2θ

d 2 θ¯ V (H, l)e−iφ

M Aab(i) = gab + (Y ∗ )M a(i) Yb(i) +

− det AL + − det AR ,

1

(H )(H )∗ + (H )∗ (H ) gab , 2

(2)


where H = T (x) +

√

i i μ 2θ α α + θ 2 + √ θ 2 θ¯ α˙ γ 5 γ μ ∂μ α + √ θ 2 θ¯ α˙ γαα˙ ∂μ α αα˙ 2 2

1 +i θ γ μ θ¯ ∂μ T − θ 2 θ¯ 2 T , 4 H∗ = T ∗ +

√

i ¯ α˙ + θ¯ 2 + √i θ¯ 2 ∂μ ¯ α˙ θ α γ 5 γ μ ¯ α˙ θ α γ μ 2θ¯ α˙ + √ θ¯ 2 ∂μ αα˙ αα˙ 2 2

1 −i(θ γ μ θ¯ )∂μ T ∗ − θ 2 θ¯ 2 T ∗ , 4 Y M = XM +

√

i i μ 2θ α ψαM + θ 2 B M + √ θ 2 θ¯ α˙ γ 5 γ μ ∂μ ψ M,α + √ θ 2 θ¯ α˙ γαα˙ ∂μ ψ M,α αα˙ 2 2

1 +i(θ γ μ θ¯ )∂μ XM − θ 2 θ¯ 2 XM , 4 V (H, l) = gs V (H ) Q,

(3)

, i = L, R and we have used the fact that the background does not depend on x4 . The indices a, b run over the world-volume directions of the branes while the indices M, N run over the background ten-dimensional space-time directions. In above equation, the super tachyon H unifies the bosonic (T ) and spinor ( ) tachyon with an another bosonic tachyon () [29, 30]. Also, the super string Y unifies the bosonic (X) and fermionic (ψ) string with another bosonic string (B). Also, Dμ is the covariant derivative of spinor field: Dμ =

∂ − μ ∂x μ

Dμ =

∂ + μ , ∂x μ

(4)

γ are the Dirac matrices in curve space-time and obey the following algebra γ μ γ ν + γ ν γ μ = 2g μν ,

(5)

and are connected with the flat space-time Dirac matrices γ in the following way: γ μ = eμa γa ,

gμν = eμa eνb ηab ,

(6)

where ηab = diag(1, −1, −1, −1) and eμa is a set of tetrad 4-vectors. The spinor affine connection matrices are uniquely determined up to an additive multiple of the unit matrix by the equation: ∂γν ρ − μν γρ − μ γν + γν μ = 0, ∂x μ

(7)

with the solution: μ =

1 1 ρ γ a γ ν ∂μ eμa − γρ γ ϑ μν , 4 4

(8)


By using the convention 2π α´ = 1 and after some algebra, the action given by (2) takes the form − det AL + − det AR , S = − d 9 γ V (H, l)e−iφ 1

(∂a T )(∂b T )∗ + (∂a T )∗ (∂b T ) (Ai )ab = gab + (1 + T 2 )∂a XiM ∂b XiM + 2 1¯ 1 ¯ ψ¯ iM ψiM ((1 + γ 5 )γ μ ∂μ ψiM − ∂μ ) − gab −ψ¯ iM (1 + γ 5 )γ μ ab ab 2 2 M ∗ +(B ∗ )M (9) i Bi gab + ()() gab . For simplicity, we assume that all fields and tachyons are functions of (U ) only, V (H ) ∼ V (T )V ( )V () and ψ ∼ B ∼ l. Let us set all the derivatives with respect to U of tachyons and xi4 to zero. Moreover, we choose xL4 = l/2 and xR4 = −l/2 so that the separation between the brane and antibrane is l. In this case, the action given by (9) reduces to − 3 4 U 4 (10) U4 DL,H + DR,H , S = −V4 d x dU V (H ) R where DL,H = DR,H ≡ DH and − 3 3 2 U 2 l (U )2 −1 U + |T (U )|2 + |T (U )|2 l(U )2 + f (U ) DH = f (U ) R R 4 ¯ ) (1 + γ 5 )γ 0 (U )l(U ) − (U ¯ ) (U )l(U )4 + 2 , − (U (11) where a ‘prime’ denotes a derivative with respect to its argument U . The equations of motion obtained from this action are: 13 13 U4 V (T ) U4 2 2 T (U )l(U ) + DH − T (U ) √ T (U ) = √ V (T ) DH DH

13

U4 (U ) √ DH

13

U 4 f (U ) √ DH 4

U R

13 U4 V ( )

4 ¯ (U )l(U ) + = √ DH − (U ) , V ( ) DH

13

U4 () √ DH

3 2

l (U )

V ( ) f (U ) ¯ l(U ) − V ( ) 4 3

(12)

13

U 4 = √ DH U R

13

U4 =√ DH

3 2

V () DH − ()2 , V ()

T (U )2 l(U ) −

V (T ) f (U ) V (T ) 4

V () f (U ) l (U ) (U ) + V () 4

(13)

U R

(14)

U R

3 2

3 2

l (U )T (U ) +

l (U ) (U )

. (15)

Note that the ‘prime’ on V denotes a derivative with respect to its argument tachyon and not a derivative with respect to U . To solve these equations, we should determine explicit form of V . There are several proposals for V which satisfy these requirements [31], although no rigorous derivation exists. In view of this, in the following analysis we will avoid using any specific expression for V , except when needed for explicit numerical calculations. It will, however, be necessary for us to specify the asymptotic form of the potential for


large tachyon. We will assume that in our parametrization this behavior is given by V (H ), V (H ) = eDH where DH is given in (11). On the hand, a potential satisfying this property, in addition to the properties listed above is [32, 33] τ4 V (H ) = (16) √ cosh π DH where τ4 is the D4-brane tension. Using this potential, we can solve (12)–(15). The solutions are: l(U ) = l0 + l1 U + .... 1 T ∼ + .... l 1 ∼ 2 + .... (l) 1 (17) ∼ + .... l where l√ 0 = πRk , is maximum separation between the brane and antibrane and l1 = 2 4 U f , where f = f (U ). The effective potential for the tachyon can be obtained R U k 0 0 0 3 k 0 from the action (10) and it is given by Veff (H ) ∼ √ ¯ ) (1 + γ 5 )γ 0 (U )l(U ) sech π |T (U )|2 + |T (U )|2 l(U )2 − (U ¯ ) (U )l(U )4 + 2 − (U

¯ ) 1 + γ 5 γ 0 (U )l(U ) − (U ¯ ) (U )l(U )4 + 2 → |T (U )|2 + |T (U )|2 l(U )2 − (U √ 2 2 2 2 Veff (l) ∼ sech π − 1+ 2 − 4. (l)2 (l)4 (l) (l)

(18)

This equation shows that when the branes-antibranes become close to each other, the interaction potential increases and at higher energies there exists more channels for string tachyon production in this system. Consequently, the effect of string tachyon on the universe inflation becomes systematically more effective. Also, the effective potential depends on the scalar and spinor tachyons and consequently all evolutions in cosmological observations are due to these types of tachyon between branes in extra dimension. From this point of view, our model is consistent with spinor inflation theory [1]. Let us now discuss the spinor inflationary model of Universe in super string theory. For this, we need to compute the contribution of the flavour brane-antibrane system to the four-dimensional Universe energy momentum tensor. The energy momentum tensor is obtained from it by calculating its functional derivative with respect to the background 2 δS ten-dimensional metric gMN . The precise relation is T MN = √−detg δgMN [34]. We get: Tiab = −V (H )

Ti44

U R

U = −V (H ) R

− 3 4 ¯ ) 1 + γ5 γ0 DH g ab − (U − 3 4

f

−1 U −3/2 √

R

DH

ab

(U )l(U ) , a, b = U,

¯ + () + f T l − l 2 2

4

2

U R

3 2

l 2 4

.

(19)


The conservation law of energy-momentum relates the tensors calculated in brane-antibrane system with ones associated with the four dimensional universe with the following equation: T μν = √

2 δgMN δS δgMN = TMN , δg δg δgμν − det g MN μν

(20)

where T μν is the energy-momentum tensor of 4D Universe in ten dimensional space-time with the metric of the form: ds 2 = dsU2 ni,1 + dsU2 ni,2 + b 2 dw2 − dτ1 − dτ2 .

(21)

Here dsU2 ni1 = dsU2 ni2 = −dt 2 + A(t)2 dx 2 + B(t)2 dy 2 + C(t)2 dz2 ,

(22)

where w is the extra space-like coordinates perpendicular to two Universes, τ1 and τ2 are the extra time-like coordinates and scale factors A, B, C and b are assumed to be functions of time only. In this model, we introduce two four dimensional Universe that interact with each other and form a binary system. To obtain the energy-momentum tensor in this system, we use of the Einstein’s field equation in presence of fluid flow that reads as: 1 Rij − gij R = k T¯ij . 2

(23)

In many recent works, it has been shown that the spinor field possesses non-diagonal components of the energy momentum tensor [7, 35]. The main question arises that what is the reason for non-triviality of non-diagonal components of the energy-momentum tensor? We can answer to this question in string theory. The following (24) shows that the specific behavior of energy momentum tensor associated with spinor tachyon in brane-antibrane system gives rise to non-trivial non-diagonal components of the energy-momentum tensor in four dimensional universe. We have: T

ij

= TiN

δgNj = TiN cos(θNj ), δgij

(24)

Setting the solution of this equation with the line element given by (21) in the conservation law of energy-momentum tensor given in (24) and employing (19) yield: k T¯11 = KT11 + KT1i cos(θ1i ) = −V (H )

+

i

=

U R

− 3 4

¯ ) (1 + γ 5 )γ 0 DH (1 − (U (U )l(U ))), 11

U V (H ) R

− 3 4 ¯ ) (1 + γ 5 )γ 0 cos(θ1i ) (U )l(U ))), DH ( (U

b¨ 2C¨ 2B˙ C˙ 2b˙ B˙ 2b˙ C˙ 2B¨ + + + + + B C b CB bB bC

1i


k T¯22 = KT22 + KT2i cos(θ2i ) = − 3 4 U ¯ ) (1 + γ 5 )γ 0 DH (1 − (U (U )l(U ))), −V (H ) 22 R − 3 4 U ¯ ) (1 + γ 5 )γ 0 cos(θ2i ) (U )l(U ))), + V (H ) DH ( (U 2i R i

2C¨ b¨ 2A˙ C˙ 2b˙ A˙ 2b˙ C˙ 2A¨ + + + + + A C b CA bA bC k T¯33 = KT33 + KT3i cos(θ3i ) = − 3 4 U ¯ ) (1 + γ 5 )γ 0 −V (H ) DH (1 − (U (U )l(U ))), 33 R − 3 4 U ¯ ) (1 + γ 5 )γ 0 cos(θ3i ) (U )l(U ))), + V (H ) DH ( (U 3i R =

i

2B¨ b¨ 2B˙ A˙ 2b˙ B˙ 2b˙ A˙ 2A¨ + + + + + A B b AB bB bA k T¯44 = KT44 + KT4i cos(θ4i ) = − 3 4 U ¯ ) (1 + γ 5 )γ 0 cos(θ4i ) (U )l(U ))), V (H ) DH ( (U + 4i R i

− 3 −1 U −3/2 3/2 2 4 f U U l R 2 2 4 2 ¯ −V (H ) √ + () + f T l − l , R R 4 DH =

=

2B¨ 2C¨ 2A˙ C˙ 2A˙ B˙ 2A˙ C˙ 2A¨ + + + + + A B C CA AB AC

10 10 i k T¯10 = KT10 + KT10 cos(θ10i ) =

V (H ) +

i

=

U R

− 3 4

00

V (H )

¯ ) (1 + γ 5 )γ 0 DH (1 − (U (U )l(U )))

3 U −4 R

¯ ) (1 + γ 5 )γ 0 cos(θ0i ) (U )l(U ))) DH ( (U 0i

2C˙ B˙ 2C˙ A˙ 2A˙ b˙ 2B˙ b˙ 2C˙ b˙ 2A˙ B˙ + + + + + , (25) AB CB CA Ab Bb Cb is the angle between momentums in anisotropic universe and brane-antibrane

where θij system. The non diagonal components of higher-dimensional stress-energy tensor have the following relations with pressure and density: 1

p1 = KT 1 = KT11 + KT1i cos(θ1i ), 2

p2 = KT 2 = KT22 + KT2i cos(θ2i ), 3

p3 = KT 3 = KT33 + KT3i cos(θ3i ),

Author's personal copy Int J Theor Phys 4

p = KT 4 = KT44 + KT4i cos(θ4i ), 10

10 i ρ = KT 10 = KT10 + KT10 cos(θ10i ),

(26)

where p¯ is the pressure in the extra space-like dimension. In above equation, we are allowing the pressure in the extra dimension to be different, in general, from the pressure in the 3D space. Hence, this stress-energy tensor describes a homogeneous, anisotropic perfect fluid in ten dimensions. Equation (26) shows that non-triviality of non-diagonal components of the energy-momentum tensor imposes some severe restrictions on the pressures and density.By adopting the metric ansatzs in (1) and (21), the conservation law in (20) and (25) and the perfect fluid stress-energy tensor given by (26), the field equations are of the form: 2C¨ b¨ 2B˙ C˙ 2b˙ B˙ 2b˙ C˙ 2B¨ + + + + + = B C b CB bB bC − 3 4 U ¯ ) (1 + γ 5 )γ 0 −V (H ) DH 1 − (U (U )l(U ) 11 R 3 U −4 ¯ ) (1 + γ 5 )γ 0 cos(θ1i ) (U )l(U ) , V (H ) DH (U + 1i R

−p1 =

(27)

i

b¨ 2C¨ 2A˙ C˙ 2b˙ A˙ 2b˙ C˙ 2A¨ + + + + + = A C b CA bA bC − 3 4 U ¯ ) (1 + γ 5 )γ 0 −V (H ) DH 1 − (U (U )l(U ) , 22 R 3 U −4 ¯ ) (1 + γ 5 )γ 0 cos(θ2i ) (U )l(U ) , V (H ) DH (U + 2i R

−p2 =

(28)

i

b¨ 2B¨ 2B˙ A˙ 2b˙ B˙ 2b˙ A˙ 2A¨ + + + + + = A B b AB bB bA − 3 4 U ¯ ) (1 + γ 5 )γ 0 −V (H ) DH 1 − (U (U )l(U ) 33 R 3 U −4 ¯ ) (1 + γ 5 )γ 0 cos(θ3i ) (U )l(U ) , V (H ) DH (U + 3i R

−p3 =

(29)

i

2B¨ 2C¨ 2A˙ C˙ 2A˙ B˙ 2A˙ C˙ 2A¨ + + + + + = A B C CA AB AC − 3 −1 U −3/2 3 2 4 f U U 2 l ¯ 4 + ()2 + f −V (H ) T 2 l 2 − l √R R R 4 DH −p¯ =

+

i

ρ=

U V (H ) R

− 3 4

¯ ) (1 + γ 5 )γ 0 cos(θ4i ) (U )l(U ) , DH (U 4i

2C˙ B˙ 2C˙ A˙ 2A˙ b˙ 2B˙ b˙ 2C˙ b˙ 2A˙ B˙ + + + + + = AB CB CA Ab Bb Cb

(30)


U V (H ) R

− 3 4

¯ ) (1 + γ 5 )γ 0 DH 1 − (U (U )l(U )) 00

3 U −4 ¯ ) (1 + γ 5 )γ 0 cos(θ0i ) (U )l(U ) , V (H ) DH (U + 0i R

(31)

i

where we set the higher-dimensional coupling constant equal to one, k = 1. The above equations help us to explain all properties of the current Universe in terms of evolutions in brane-antibrane system. These equations constraint the pressure and density in our Universe to tachyon fields and expresses that any increase or decrease in these parameters is due to tachyon potential in extra dimension. By taking derivative from the left side of (31) and employing (27)–(31), we obtain: 2A˙ 2B˙ 2C˙ b˙ ¯ = ρ. ˙ (−ρ − p1 ) + (−ρ − p2 ) + (−ρ − p3 ) + (−ρ − p) A B C b

(32)

˙ B˙ C˙ b˙ ¯ By defining H1 = A A , H2 = B , H3 = C and H = b as Hubble parameters in 4D Universe and extra dimension, we can rewrite (31) and (32) in following manner:

2(H1H2 + H1 H3 + H2 H3 + H1 H¯ + H2 H¯ + H3 H¯ ) = ρ.

(33)

ρ˙ + 2H1 (ρ + p1 ) + 2H2 (ρ + p2 ) + 2H3 (ρ + p3 ) + 2H¯ (ρ + p) ¯ = 0.

(34)

Equation (34) is, in fact, conservation equation in ten dimensional space-time. This equation is analogous to standard 4D FRW cosmology with H¯ = 0, H1 = H2 = H3 and dsU2 ni2 = 0. Relating the 3D and higher-dimensional pressures to the density (p1 = ω1 ρ, p2 = ω2 ρ, p3 = ω3 ρ, p¯ = ωρ) ¯ and employing (33) in (34) lead to following equation: 2H˙ 1 (H2 + H3 + H¯ ) + 2H˙ 2 (H1 + H3 + H¯ ) + 2H˙ 3 (H1 + H2 + H¯ ) + 2H˙¯ (H1 + H2 + H3 ) + 2(H1 H2 + H1 H3 + H2 H3 + H1 H¯ + H2 H¯ + H3 H¯ ) ×

2H1 (1 + ω1 ) + 2H2 (1 + ω2 ) + 2H3 (1 + ω3 ) + 2H¯ (1 + ω) ¯ = 0,

(35)

where ω and ω¯ are equation of state parameters in 4D Universes and extra dimension respectively. Notice that these parameters can in general be time-dependent. For simplicity and in agreement with Refs. [36, 37], we assume: ¯ 1. H2 = m2 H1 , H3 = m3 H1 , H¯ = mH

(36)

Also, substituting (36) in to (35) yields: H˙ 1 (2m1 + 2m2 + 2m3 + 2m ¯ + 2m2 m3 + 2m2 m ¯ + 2mm ¯ 3) + 4(H1 )2 (m2 + m3 + m ¯ + m2 m3 + m2 m ¯ + mm ¯ 3 ) ((1 + ω1 )+ ¯ + ω)) ¯ = 0, m2 (1 + ω2 ) + m3 (1 + ω3 ) + m(1

(37)

Solving above equation and performing some algebra, we find a consistent solution for the Hubble parameters and scale factors in 4D Universes and extra dimension : H1 =

F , trip − t

A(t) = (trip − t)F ,


F =

(2m1 + 2m2 + 2m3 + 2m ¯ + 2m2 m3 + 2m2 m ¯ + 2mm ¯ 3) . ((1 + m)(m ¯ + m ) + m ¯ + m m ) ((1 + ω ) + m (1 + ω ) + m (1 + ω3 ) + m(1 ¯ + ω)) ¯ 2 3 2 3 1 2 2 3 (38)

H2 =

V , trip − t

B(t) = (trip − t)V , V = m2 F. X , H3 = trip − t C(t) = (trip − t)X , X = m3 F. G , H¯ = trip − t

(39)

(40)

b(t) = (trip − t)G , G = mF. ¯ (41) The (38), (39) and (41) show that F is smaller than zero and 3D scale factor and Hubble parameters at a certain time (t = ts ) are infinity. So far, researchers have believed that if 3D equation of state parameter is lower than negative one, dark energy is phantom energy and at Big Rip singularity, ingredients of which universe is made will be ripped; whereas our calculations indicate that occurrence of this singularity is related to all four ω, ¯ ω1 , ω2 and ω3 . Substituting these equations in (27)–(31), we obtain: l = trip − t 1/2

3 Uk (trip − t), θ12 = θ34 = θ14 = θ23 = 0 2 R 3/2 2F (F − 1) + 2V (V − 1) + 2X(X − 1) + 2V X + 2F X + 2F V

θ13 = θ24 ∼ =

T ==

2F (F − 1) + 2V (V − 1) + 2X(X − 1) + 2V X + 2F X + 2F V trip − t

(42)

The (42) provides some interesting results which can be used to explain the reasons for occurrence of Big Rip singularity in present era of Universe. According to these calculations, when two branes are located at a large distance from each other (l = l0 and t=0), tachyon field is almost zero, while approaching the two together, the value of this field increases and tends to infinity at t = trip . In this situation, brane-antibrane system disappears and consequently one singularity happens in our four dimensional Universe. Another interesting point that comes out from this equation is that time of this singularity is proportional to the initial distance between two branes.

3 Considering the Signature of Spinor Tachyon in Observational Data In previous section, we have proposed a model that allows to consider the spinor inflation of Universe in super string theory. Here, we examine the correctness of our model with observation data and obtain some important parameters like ripping time. Until now, eight


possible asymptotic solutions for cosmological dynamics have been proposed [38]. Three of these solutions have non-inflationary scale factor and another three ones of solutions give de Sitter, intermediate and power-low inflationary expansions. Finally, two cases of these solutions have asymptotic expansion with scale factor (a = a0 exp(A[lnt]λ ). This version of inflation is named logamediate inflation [39]. Our model lets us to study-tachyon inflationary model in the scenarios of logamediate inflation with λ = 1. In this condition, the model is converted to power-law inflation. Using (38) and (39), the number of e-folds may be found: 1 (H1 + H2 + H3 ) 3

t trip − t0 N= H dt = |Z| ln trip − t t0 Z =F +V +X

H =

(43)

where t0 denotes the beginning time of inflation epoch. In Fig. 1, we present the number of e-folds N for intermediate scenario as a function of the t where t is the age of Universe. In this plot, we choose Z = −260, t0 = 0 and trip = 65(Gyr). We find that N = 60 leads to tuniverse = 13.5(Gyr). This result is compatible with with both Planck and WMAP7 data [8–12]. It is clear that the number of efolds N is much larger for older Universe. This is because, as the age of Universe increases, the distance between branes becomes smaller and the tachyon field acquires bigger values. Finally, we compare our model with the scalar spectral index [40]: ns − 1 = −6ε + 2η

(44)

1200 1000

N

800 600 400 200 0 0

10

20

30

40

50

60

70

TIME Fig. 1 The number of e-folds N in spinor inflation scenario as a function of t for Z= −260, t0 = 0 and trip = 65(Gyr)


0.970 0.968

ns

0.966 0.964 0.962 0.960 0

10

20

30

40

50

60

70

TIME Fig. 2 The Scalar spectral index ns in spinor inflation scenario as a function of t for Z = −260, t0 = 0 and trip = 65(Gyr)

Here k is co-moving wavenumber and ε and η are slow-roll parameters of the inflation which are given by: ε=−

1 dlnH H dt

η=−

H¨ . H H˙

(45)

In Fig. 2, we show the Scalar spectral index ns for scenario as a function of the age of Universe. In this plot, we choose Z = −260 and t0 = 0. Comparing this figure with Fig. 1, we find that N 60 case leads to ns 0.96. This standard case is consistent with both Planck and WMAP7 data [8–12]. At this point, the finite time that Big Rip singularity occurs is trip = 45(Gyr).

4 Summary and Discussion In this research, we write a new effective action in super string theory which includes all three types of scalar and spinor tachyons. We show that the expansion of universe is controlled by spinor string tachyon. We construct spinor inflation in super string theory and show that all cosmological parameters can be given in terms of the time that the two branes approached together in extra dimensions. According to our results, when the distance between branes increases, the Number of e-fold and the spectral index increases and tends to infinity at Big Rip singularity. This is because, as the separation distance between branes decreases, the interaction potential increases and at higher energies there exists more channels for tachyon production in brane-antibrane system; consequently, the effect of string


radiation from extra dimension on the universe inflation becomes systematically more effective. We find that N 60 case leads to ns 0.96 which are consistent with experimental data. At this point, the finite time that Big Rip singularity occurs is trip = 65(Gyr). Acknowledgments Authors thank Ali Mohammadi and B. Saha for their helpful motivations. One of the authors (A. Pradhan) would like to thank the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India for providing facility & support during a visit where part of this work was done. The financial support by University Grants Commission, New Delhi, India under the research grant (Project F.No. 41-899/2012(SR)) is gratefully acknowledged. The authors also thank the anonymous referees for their constructive comments.

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