Astrophys Space SciDOI 10.1007/s10509-014-2083-8

L E T T E R

New exact cosmologies on the brane

Artyom V. Astashenok · Artyom V. Yurov ·Sergey V. Chervon · Evgeniy V. Shabanov ·Mohammad Sami

Received: 18 June 2014 / Accepted: 25 July 2014© Springer Science+Business Media Dordrecht 2014

Abstract We develop a method for constructing exact cos-mological solutions in brane world cosmology. New classesof cosmological solutions on Randall–Sandrum brane areobtained. The superpotential and Hubble parameter are rep-resented in quadratures. These solutions have inflationaryphases under general assumptions and also describe an exitfrom the inflationary phase without a fine tuning of the pa-rameters. Another class solutions can describe the currentphase of accelerated expansion with or without possible exitfrom it.

Keywords RS-brane · Inflation · Exact solutions

1 Introduction

Brane world scenario have been proposed more then decadeago and this scenario attracted many attention. The reasonof such attention includes the hope to describe observed ac-celerated expansion of the Universe.

To our knowledge the work by Binetruy et al. (2000a)was the first work where brane cosmology, different fromstandard Friedmann cosmology, have been proposed. It isinteresting to mention that in the first work on brane cos-mology at once it was much attention to searching of exact

A.V. Astashenok (B) · A.V. Yurov · S.V. ChervonInstitute of Physics and Technology, Baltic Federal Universityof I. Kant, Kaliningrad, Russiae-mail: artyom.art@gmail.com

S.V. Chervon · E.V. ShabanovIlya Ulyanov State Pedagogical University, Ulyanovsk, Russia

M. SamiCentre of Theoretical Physics, Jamia Millia Islamia, New Delhi,India

solutions. This situation is a diametrically opposite to that incosmological inflation theory where about a decade approx-imated solution have been under study till the works Barrow(1987), Muslimov (1990). Further development of the branecosmology scenarios was performed in works Binetruy et al.(2000b), Mukohyama (2000), Vollick (2001), Kraus (1999),Ida (2000) where few exact solutions have been found. Inthe work Mukohyama et al. (2000) it was shown the relationbetween exact solutions in Kraus (1999), Ida (2000) and Bi-netruy et al. (2000b), Mukohyama (2000). Namely it wasgiven the explicit coordinate transformation which provedthe equivalence between this two solutions, i.e. both solu-tions represent the same spacetime in different coordinatesystems.

In consideration of inflation scenario in brane cosmol-ogy it is important to analyze scalar field solutions with self-interacting potential. There are number of works devoted tothis issue. We will mention few of them where investigationof exact solutions was carried out. In the article Paul andPaul (2007) it was found exact solution at high energy limitwhen H 2 proportional to ρ2; canonical and tachyon fieldswere analyzed there. A general thick brane with a scalar fieldwas analyzed in the work Bronnikov and Meierovich (2003),where restrictions on the scalar potential was obtained andexact solutions for stepwise potentials of different shapeswas found. A general scalar field and barotropic fluid duringthe early stage of a brane-world where the Friedmann con-straint is dominated by the square of the energy density wasstudied in the work Copeland et al. (2004). In this article wepresent the method of exact solution construction and newclasses of exact solutions obtained by superpotential method(see Chervon et al. 2011 and literature cited therein). Fist ap-plication of this method for Randal–Sandrum brane cosmol-ogy was performed in the work Chervon and Sami (2009).

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Let us for a moment return to the standard inflationarymodel with a self-interacting scalar field with the poten-tial V (φ). It was proved (see Chervon and Zhuravlev 2000;Zhuravlev and Chervon 2000) that the equations of scalarcosmology in terms of superpotential W(φ) = 1

2U2(φ) +V (φ), (U(φ) = φ) look exactly as the equations in the slowroll approximation with the substitution V → W :

H 2 = 1

3M2p

W(φ) (1)

H = − 1

2M2p

φ2 (2)

3Hφ = − d

dφW(φ) (3)

where H = dadt

/a(t) = aa

is a Hubble parameter, a(t) is ascale factor, t is a cosmic time, Mp is the Plank mass.

Thus exploiting the superpotential W(φ) (or “a total en-ergy potential” in accordance with terminology in Cher-von and Zhuravlev 2000; Zhuravlev and Chervon 2000) in-stead of the potential V (φ) will considerably simplify thesystem of equation of scalar cosmology. Let us analyzethis relation from physical point of view. There are fewmethods of solutions construction for the scalar cosmologyequations (see, for example Chaadaev and Chervon 2013;Beesham et al. 2009). Searching of a scale factor and ascalar field with given potential is considered as a directtask and as a physically based approach. Generally speak-ing, it is believed that one may found the potential fromexperiment or from underlaying theory (say, from high-temperature phase transition in SU(5) grand unified theoryleading to the Coleman–Weinberg potential in “new” infla-tion). Following by this approach we note that inserting thepotential in the scalar field equation in the Hamilton–Jacobilike form (see, e.g., Chaadaev and Chervon 2013)

2M4p

(dH

dφ

)2

− 3M2pH 2 = −V (φ) (4)

one may found the solution for the Hubble parameter as thefunction on φ. After that it is possible to obtain a scalar field“velocity” from Eq. (2) and restore W(φ). The difficultiesare in solving procedure of Hamilton–Jacobi like equationfor a given potential.

From another hand one may use fine tuning of the poten-tial method (Ellis and Madsen 1991; Chervon et al. 1997)when the scale factor is given. In such a way a scalar fielddynamics for exponential, power law and other type of cos-mologies have been found in the work Ellis and Madsen(1991). These results gave possibility to analyze scalar cos-mology equations using given dynamics of scalar field inBarrow (1994) and to construct the exact solutions on thisbasis. Development of this method presented also in Zhu-ravlev and Chervon (2000) where dynamic analysis of the

scalar field leads to construction of the model admitting nat-ural exit from inflation to radiation dominated stage.

We suggest that mentioned above ideas and methods maybe applied for a scalar field cosmology on the brane.

The article is organized as follow. In Sect. 2 we presentthe model and method of exact solution construction for it.Section 3 devoted to solutions obtained for given superpo-tential. In Sect. 4 we present new exact solutions based ongiven evolution of the scalar field. In Sect. 5 we discuss vi-able cosmological models obtained in the article.

2 Cosmological models on the FRW brane: methodsof solutions construction

As an alternative to the FRW cosmology let us considerthe simplest brane model in which spacetime is homoge-neous and isotropic along three spatial dimensions, beingour 4-dimensional universe an infinitesimally thin wall, withconstant spatial curvature, embedded in a 5-dimensionalspacetime (Sahni and Shtanov 2008; Langlois 2003). In theGaussian normal coordinate system, for the brane which islocated at y = 0, one gets

ds2 = −n2dt2 + a2(t, y)γij dxidxj + dy2, (5)

where γij is the maximally 3-dimensional metric. Let t bethe proper time on the brane (y = 0), then n(t,0) = 1.Therefore, one gets the FRW metric on the brane

ds2|y=0 = −dt2 + a2(t,0)γij dxidxj . (6)

The 5-dimensional Einstein equations have the form

RAB − 1

2gABR = χ2TAB + Λ4gAB, (7)

where Λ4 is the bulk cosmological constant, χ2 =8πG(5)/c4, G(5) is the gravitational constant in 5-dimen-sional spacetime. The next step is to write the total energymomentum tensor TAB on the brane as

T AB = SA

B δ(y), (8)

with SAB = diag(−ρb,pb,pb,pb,0), where ρb and pb are

the total brane energy density and pressure, respectively.One can now calculate the components of the 5-dimen-

sional Einstein tensor which solve Einstein’s equations. Oneof the crucial issues here is to use appropriate junction con-ditions near y = 0. These reduce to the following two rela-tions:

dn

ndy |y=0+= χ2

3ρb + χ2

2pb,

da

ady |y=0+= −χ2

6ρb.

(9)

After some calculations, one obtains the following result

H 2 = χ4 ρ2b

36+ Λ4

6− k

a2+ E

a4. (10)

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This expression is valid on the brane only. Here H =a(t,0)/a(t,0) and E is an arbitrary integration constant. Theenergy conservation equation is correct, too,

ρb + 3a

a(ρb + pb) = 0. (11)

Now, let ρb = ρ +λb, where λb is the brane tension. Fur-ther we consider the fine-tuned brane with Λ4 = λ2

bχ4/6 and

the case of flat spacetime (k = 0):

a2

a2= λχ4

6

ρ

3

(1 + ρ

2λb

)+ E

a4. (12)

In what follows we will consider a single brane modelwhich mimics GR at present but differs from it at late times.We set M−2

p = 8πG = σχ4/6. For simplicity, we set E = 0(the term with E is usually called “dark radiation”). In fact,setting E �= 0 does not lead to additional solutions on a rad-ically new basis, in the framework of our approach. Equa-tion (12) can be simplified to

a2

a2= ρ

3M2p

(1 + ρ

2λb

). (13)

One can see that Eq. (13), for ρ � |λb|, differs insignifi-cantly from the FRW equation. The brane model with a pos-itive tension has been discussed in Copeland et al. (2001),Sahni et al. (2002), Sahni and Sami (2004) in the con-text of the unification of early- and late-time accelerationeras. The braneworld model with a negative tension and atime-like extra dimension can be regarded as being dualto the Randall–Sundrum model (Sahni and Shtanov 2003;Randall and Sundrum 1999; Copeland et al. 2005). Notethat, for this model, the Big Bang singularity is absent. Andthis fact does not depend upon whether or not matter violatesthe energy conditions (Ashtekar et al. 2006). This same sce-nario has also been used to construct cyclic models for theuniverse (Kanekar et al. 2001).

One can assume that in our epoch the ρ/2λ � 1 and sothere is no significant difference between the brane modeland FRW cosmology. But the universe evolution in the fu-ture or in past, for brane cosmology, can in fact differ fromsuch convenient cosmology, due to the non-linear depen-dence of the expansion rate on the energy density.

One can reduce the field equation to the slow-roll form

3HU = −W ′φ, (14)

with substitution

W = V + 1

2U2, U(φ) = φ (15)

Then Friedmann equation (13) one can write down in termsof the superpotential W

H 2 = 1

3M2p

W

(1 + W

2λb

). (16)

Considering H as positive and inserting (16) into (14)one can obtain√

3

Mp

U(φ)

√W

(1 + W

2λb

)= −W ′

φ. (17)

This is the key equation for further progress. For givenW(φ) as function of scalar field one can define the depen-dence of scalar field from time inversing the following rela-tion:

t − t0 = −√

3

Mp

∫dφ

W ′φ

√W

(1 + W

2λb

)(18)

In frames of this approach the physical potential is

V (φ) = W(φ) − M2p

6W ′2

φ

(W(φ)

(1 + W(φ)

2λb

))−1

(19)

One can also define U(φ) as function of scalar field. Inthis case the integration (17) via separation of W and φ leadsto the superpotential and Hubble parameter presentation inquadratures√

W = √2λb

× sinh

(−

√3

2λb

1

2MP

∫U(φ)dφ

), W > 0, (20)

√−W = √2λb

× cosh

(−

√3

2λb

1

2MP

∫U(φ)dφ

), W < 0,

(21)

H =√

λb

6

1

MP

sinh

(−

√3

2λb

1

MP

∫U(φ)dφ

)(22)

Note that the argument in (22) should be positive. There-fore

∫U(φ)dφ should be always negative.

The physical potential V can be obtained from the super-potential definition (15)

V (φ) = 2λb sinh2(

−√

3

2λb

1

2MP

∫U(φ)dφ

)− 1

2U(φ)2.

(23)

Thus to obtain the examples of exact solutions one cansuggest the functional dependence a scalar field φ on cosmictime t and evaluate the integral

∫U(φ)dφ:∫

U(φ)dφ =∫

U2(t)dt.

Standard calculation leads to the following formula forN(φ):

N(φ) = −√

λb

2M−2

P

×∫ √

W

(1 + W

2λb

)√(W

λb

+ 1

)2

+ 1dφ

W ′ (24)

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To understand the period with accelerating Universe ex-pansion let us calculate a

a= H + H 2. For given W(φ) we

have the following simple relation for acceleration parame-ter:

a

a= −1

6

W ′2φ

W+ W

3M2P

+ W

6λb

(W

M2P

− 1

2

W ′2φ

W(1 + W/2λb)

)(25)

The first two terms corresponds to the case of usual FRWcosmology. The last appears due to the brane tension. Com-bining (25) with (18) one can analyze the behavior a/a asfunction of time.

If we define the function U(φ) it is convenient to use theresult

a

a= − U2

2M2P

cosh

(−

√3

2λb

1

MP

∫U(φ)dφ

)

+ λb

6M2P

sinh2(

−√

3

2λb

1

MP

∫U(φ)dφ

)(26)

To investigate changing of the acceleration sign let us repre-sent (26) in the following form

6M2P

λb

a

a= Z2 − 3U2

λb

Z − 1,

Z = cosh

(−

√3

2λb

1

MP

∫U(φ)dφ

). (27)

Taking into account that Z > 1 we omit the root

Z2 = 3U2

2λb

−√

9U4

4λ2b

+ 1 (28)

as it is less then unity. For the next root

Z1 = 3U2

2λb

+√

9U4

4λ2b

+ 1 (29)

it is easy to check that Z1 > 1 for any value of U2. Thereforewe can imply that there is only one point during the evolu-tion when deceleration have been changed to acceleration.So in the framework of scalar field cosmology on the branewe have only one inflationary period. If it is early inflationthen once again as in FRW cosmology we need to solve theproblem of exit from inflation (for example using generationof the particle after inflation ends). On the other hand, if itis later inflation, we may suggest existence one more addi-tional specie at the early stage of Universe evolution whichwill be responsible for early inflation.

3 Solutions with given W(φ)

Let’s consider two simple superpotentials. The first case is

W = m2φ2

2(30)

One can easy derive the dependence t (φ). The result is

t − t0 = −√

6λb

2m2M−1

P

×[

arcsinh

(mφ

2√

λb

)+ mφ

2√

λb

√1 + m2φ2

4λb

]

For simplicity we put φ(t0) = 0. At t → ∞ scalar fieldφ → −∞. One can consider the moment t < t0 for this mo-ment φ > 0. The universe acceleration is

a

a= −m2

12+ m2φ2

6M2P

+ m2φ2

12λb

(m2φ2

2M2P

− 1

2

m2

1 + m2φ2/4λb

)(31)

For scale factor as function of scalar field we have

a = a0 exp

(± φ2

8M2P

(2 + m2φ2

4λb

))(32)

One can choose the sign − for t < 0 (i.e. φ > 0) and +for t > 0 (φ < 0). The moment t = −∞ corresponds toBig Bang singularity then universe expands so that a > 0then non-inflationary phase follows. Then universe again ex-pands with acceleration. The asymptotic of solution is

a(t) ∼ a0 exp

(λbm

2

2M2P

(t − t0)2)

. (33)

The potential of scalar field can be obtained from (19):

V (φ) = m2φ2

2− M2

P m2

3

(1 + m2φ2

4λb

)−1

. (34)

At t → ∞ this potential corresponds to free scalar field withmass m.

For case of φ − 4 superpotential

W(φ) = λφ4

4(35)

we have the following link between time and scalar field:

t − t0 = −√

3

2MP

√λ

×(

coshη − coshη0 + ln tanhη

2− ln tanh

η0

2

),

η = arcsinh

((λ

8λb

)1/2

φ2)

. (36)

The scalar field at the moment t = t0 is φ0 = (8λb

λ)1/4 ×

sinh1/2 η0 where η0 is arbitrary constant. At t → ∞ the

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scalar field φ → 0. From expression for universe acceler-ationa

a= −2λ

3φ2 + λ

12M2P

φ4

+ λ2φ6

24λb

(1

4M2P

φ2 − 2

1 + λφ4/8λb

)(37)

one can see that exit from inflation occurs. The scale factoras function of scalar field

a = a0 exp

(± φ2

8M2P

(1 + λφ4

24λb

))(38)

The negative sign corresponds to universe starting froma = 0 (φ = ∞ at t = −∞) and expanding with accelera-tion prior to some moment of time when exit from inflationoccurs.

4 Solutions with given φ(t)

To obtain exact solutions we can use the dynamics of thescalar field considered in many works. Let us start from thesimplest ones.

4.1 Logarithmic evolution of the scalar field

Let

φ = A ln(λt).

The solutions are√W = √

2λb

× sinh

(−

√3

2λb

1

2MP

(C1 − A2

t

)), W > 0,

(39)√−W = √2λb

× cosh

(−

√3

2λb

1

2MP

(C1 − A2

t

)), W < 0,

(40)

H =√

λb

6

1

MP

sinh

(−

√3

2λb

1

MP

(C1 − A2

t

)). (41)

The superpotential’s and physical potential’s presentationin terms of scalar field can be obtained using given depen-dence φ = A ln(λt):√

W = √2λb

× sinh

(−

√3

2λb

1

2MP

(C1 − A2λ exp(−φ/A)

)),

(42)√−W = √2λb

× cosh

(−

√3

2λb

1

2MP

(C1 − A2λ exp(−φ/A)

)),

(43)

Fig. 1 The potential of scalar field for logarithmic evolution of scalarfield

V (φ) = 2λb

× sinh2(

−√

3

2λb

1

2MP

(C1 − A2λ exp(−φ/A)

))

− A2λ2

2e−2φ/A. (44)

The potential V (φ) is depicted on Fig. 1. Thus we haveobtained the exact formulas for given evolution of the scalarfield. We know that for solutions under consideration wemay have only one inflection point for scalar factor a as-sociated with Eq. (29). Using the definition for Z (27) onecan obtain the equation for time corresponding to inflectionpoint:

3A2

2λbt2+

√9A4

4λ2bt

4+ 1 = cosh

(√3

2λb

A2

MP t

)(45)

It is easy to see that this equation will be true whent → ∞. We can analyze this equation numerically to findthe finite time t < ∞ which will correspond to decelerationchanges to acceleration. One can state that this time is aboutti ≈ 0.286λ

−1/2b in the units with MP = 1 and for A = 1.

Therefore we can use this approximate result to set this timeequal to beginning of (early or later) inflation. From theother hand we can analyze the transition from brane cos-mology to Friedmann one by tending the brane tension toinfinity λb → ∞. The results are:

√λb → ∞ :

√W = −√

31

2Mp

(C1 − A2

t

)

= −√3

1

2Mp

(C1 − A2λe−φ/A

), (46)

√−W = ∞, (47)

H = − 1√6Mp

(√3

2

1

Mp

(C1 − A2

t

)), (48)

V (φ) = ∞. (49)

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Here we can define the scale factor with power law—exponential behavior

a(t) = exp

(− C1t

2M2P

)t

2A2λ2

2M2P . (50)

We carefully analyzed the solution in this section. Tosimplify presentation of the next solutions let us representEqs. (20)–(23) in the following way√

W = √2λb

× sinh

(−

√3

2λb

1

2MP

[F(φ) + C1

]), W > 0,

(51)√−W = √2λb

× cosh

(−

√3

2λb

1

2MP

[F(φ) + C1

]), W < 0,

(52)

H =√

λb

6

1

MP

sinh

(−

√3

2λb

1

MP

[F

(φ(t)

) + C1])

(53)

V (φ) = 2λb sinh2(

−√

3

2λb

1

2MP

[F(φ) + C1

])

− 1

2

[U(φ)

]2. (54)

Using the general formulas (51)–(54) we will displaynew solutions by putting values for F(φ) and U(φ). Alsowe will use general formulas for transition to FriedmannUniverse (by setting λb → ∞) below√

λb → ∞ :√

W = −√

3

2Mp

(F(φ) + C1

), (55)

√−W = ∞, (56)

H = − 1

2M2p

(F

(φ(t)

) + C1), (57)

V (φ) = − 3

4MP

(F(φ) + C1

)2 − 1

2

[U(φ)

]2 (58)

4.2 Power law evolution

Let

φ = Ats, s �= 0, s �= 1/2.

The solutions are represented by functions

F(φ) = A−1/ss2

2s − 1φ2−1/s = A2s2

2s − 1t2s−1, (59)

U2(φ) = A2s2[

φ

A

] 2s−2s

. (60)

The equation for the time corresponding to inflectionpoint is

Fig. 2 The potential of scalar field for φ = At1/2

3A2s2

2λb

t2s−2 +√

9A4s4

4λb

t4s−4 + 1

= cosh

(−

√3

2λb

A2s2

(2s − 1)MP

t2s−1)

(61)

By tending the brane tension to infinity λb → ∞ we ob-tain√

λb → ∞ :√

W = −√

3

2Mp

(A2s2

2s − 1

[φ

A

] 2s−1s + C1

), (62)

√−W = ∞, (63)

H = − 1√6Mp

(√3

2

1

Mp

(C1 + A2s2

2s − 1t

2s−1s

)), (64)

V (φ) = ∞. (65)

In the case when s = 1/2 we have

F(φ) = A2

2ln

φ

A, U2(φ) = A4

4φ2(66)

The potential of scalar field in this case is depicted onFig. 2.

The equation for the time corresponding to inflectionpoint is

3A2

8tλb

+√

9A4

32t2λ2b

+ 1 = 1

2

(t−B + tB

),

B =√

3

2λb

A2

4MP

. (67)

This time is about t ≈ 4.11λ−1/2b in units MP = A = 1.

In this moment the deceleration begins.

4.3 Exponential evolution

Let

φ = Ae−λt .

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Fig. 3 The potential of scalar field for φ = A exp(−λt)

The solutions represented by functions

F(φ) = −φ2λ

2= −A2λ

2e−2λt , U2(φ) = λ2φ2 (68)

The potential of scalar field is presented on Fig. 3.The equation for the time corresponding to inflection

point is

3A2λ2

2λb

e−2λt +√

9A4λ4

4λ2b

e−4λt + 1

= cosh

(√3

2λb

A2λ

MP

e−2λt

), (69)

For inflection time we have two different solutions: (i) whenλ > 0 a → 0 at t → ∞ (and a > 0 at 0 < t < ∞); (ii) whenλ = −√

λb < 0 a = 0 at t ≈ 0.879λ−1/2b (Mp = A = 1). At

this time acceleration changes to deceleration i.e. we havethe inflation phase during finite time.

By tending the brane tension to infinity λb → ∞ we ob-tain

√λb → ∞ :

√W = −√

31

2Mp

(C1 − A2λ

2e−2λt

), (70)

√−W = ∞, (71)

H = − 1√6Mp

(√3

2

1

Mp

(C1 − A2λ

2e−2λt

)), (72)

V (φ) = ∞. (73)

The solutions above have been obtained earlier in Cher-von and Sami (2009) with a slightly different form. Here wepresented them in more suitable way and gave detailed anal-ysis. The solutions below are obtained first time and basedon the scalar field evolutions considering in cosmology Bar-row (1994), Chervon (2004). Investigation of cosmologicalparameters for such evolution of scalar field was performedin the work Chervon and Fomin (2008).

Fig. 4 The scalar field potential for φ = A ln(tanh(λt))

Table 1 The duration ofaccelerated expansion for modelφ = A ln(tanh(λt)) for various λ

λ/λ1/2

0.8 0.38 < tλ1/2b < 0.53

0.6 0.315 < tλ1/2b < 0.382

0.4 0.295 < tλ1/2b < 1.2

0.2 0.286 < tλ1/2b < 2.31

4.4 New classes of solutions

4.4.1 φ = A ln(tanh(λt))

The solution is represented by functions

F(φ) = A2λ(2 cosh(φ/A)

) = A2λ(tanh(λt) + coth(λt)

),

(74)

U2(φ) = A2λ2(1 − exp(2φA

))2

exp(2φA

)(75)

The potential of scalar field is depicted on Fig. 4.The inflection point for the scale factor a(t) can be ob-

tained as a solution of the following equation:

3A2λ2

2λb cosh2(λt) sinh2(λt)+

√9A4λ4

4λ2b cosh4(λt) sinh4(λt)

+ 1

= cosh

(√3

2λb

A2λ

Mp

(tanh(λt) + coth(λt)

))(76)

This equation has solutions at λ < 0.82λ1/2b (for A =

MP = 1). The accelerated expansion begins at t = t1 andends at t = t2. In Table 1 the duration of inflation stage aregiven for various λ.

4.4.2 φ = A ln(tan(λt))

The solution is represented by formulas

F(φ) = A2λ(2 sinh(φ/A)

) = A2λ(tan(λt) − cot(λt)

), (77)

U2(φ) = A2λ2(1 + exp(2φA

))2

exp(2φA

)(78)

Astrophys Space Sci

Fig. 5 The scalar field potential for φ = A ln(tan(λt))

Table 2 The duration ofaccelerated expansion for modelφ = A ln(tan(λt)) for various λ

λ/λ1/2

0.6 0.265 < tλ1/2b < 2.355

0.8 0.251 < tλ1/2b < 1.712

1.0 0.239 < tλ1/2b < 1.335

1.2 0.225 < tλ1/2b < 1.083

1.4 0.215 < tλ1/2b < 0.907

On Fig. 5 one can see the dependence of potential fromscalar field.

The time for inflection point can be found as a solutionof equation:

3A2λ2

2λb cos2(λt) sin2(λt)+

√9A4λ4

4λ2b cos4(λt) sin4(λt)

+ 1

= cosh

(√3

2λb

A2λ

Mp

(tan(λt) − cot(λt)

))(79)

We have the same situation as in previous case. For vari-ous values of λ we have two roots corresponding to momentof beginning of inflation and to moment of exit from infla-tion. In Table 2 these results are given for various valuesof λ.

4.4.3 φ = A/ sinh(λt)

The solution is described by the functions

F(φ) = A2λ coth(λt)1

3

[φ2

A2− 1

], (80)

U2(φ) = λ2

A2φ4

(1 +

(A

φ

)2)(81)

The potential of scalar field is depicted on Fig. 6.The inflection point can be found from corresponding

equation

3A2λ2 cosh2(λt)

2λb sinh4(λt)+

√9A4λ4 cosh4(λt)

4λ2b sinh8(λt)

+ 1

Fig. 6 The scalar field potential for φ = A/ sinh(λt)

Fig. 7 The scalar field potential for φ = A arctan(exp(λt))

Table 3 The duration of accelerated expansion for model φ =A/ sinh(λt) for various λ (A = MP = 1)

λ/λ1/2

0.6 0.665 < tλ1/2b < 3.5

0.8 0.545 < tλ1/2b < 2.63

1.0 0.468 < tλ1/2b < 2.11

1.2 0.414 < tλ1/2b < 1.765

1.4 0.374 < tλ1/2b < 1.52

= cosh

(√3

2λb

A2λ cosh(λt)

3Mp

[1

sinh2(λt)− 1

])(82)

Once again for various values of λ we have phase of ac-celerated expansion during finite time. In Table 3 one cansee results for various values of λ.

4.4.4 φ = A arctan(exp(λt))

The solution is represented by functions

F(φ) = −A2λ cos2(φ/A)

2= − A2λ

2(1 + exp(2λt)), (83)

U2(φ) = A2λ2 tan2(φA

)

(1 + tan2(φA

)). (84)

The potential of scalar field is presented on Fig. 7.

Astrophys Space Sci

Fig. 8 The scalar field potential for φ = A sin−1(λt)

From the equation

3A2λ2 exp(2λt)

2λb(1 + exp(2λt))2+

√9A4λ4 exp(4λt)

4λ2b(1 + exp(2λt))4

+ 1

= cosh

(√3

2λb

A2λ

2Mp(1 + exp(2λt))

)(85)

one can find that for λ > 0 a > 0 (0 < t < ∞) and for λ < 0we have the moment when deceleration begins. For examplefor λ = −λ

1/2b we have t ≈ 0.093λ

−1/2b .

4.4.5 φ = A sin−1(λt)

The solution is described by formulas

F(φ) = A2λ cot(λt)1

3

[1 − 1

sin2(λt)

], (86)

U2(φ) = λ2

A2φ4

(1 −

(A

φ

)2). (87)

On Fig. 8 one can see the potential of scalar field.The inflection point can be found from equation

3A2λ2 cos2(λt)

2λb sin4(λt)+

√9A4λ4 cos4(λt)

4λ2b sin8(λt)

+ 1

= cosh

(√3

2λb

A2λ cot(λt)

3Mp

[1 − 1

sin2(λt)

]). (88)

For λ = λ1/2b we have the phase of accelerated expansion

for t > 0.4λ−1/2b .

5 Discussions

In this paper we have discussed a simple method of con-struction of exact solutions for cosmological equations onRS brane. Despite simplicity, the method allows for acquire-ment of solutions characterized by interesting properties.These solutions have inflationary phases under quite generalassumptions. This is an indication that inflationary regimeseems a common occurrence in cosmology not requiring anyspecial initial conditions. We have the following cases:

(i) Accelerated expansion begins at some moment of timeand lasts forever (logarithmic evolution of scalar field,φ ∼ sin−1(λt)). In principle current observed accelera-tion can be described by this case.

(ii) Acceleration take place before t < tf when decelera-tion starts (power and exponential evolution of scalarfield or φ ∼ arctan(exp(λt))). These solutions may cor-respond to inflation in beginning of cosmological evo-lution and exit from inflation phase.

(iii) Acceleration occurs during interval ti < t < tf (φ ∼

ln(tanh(λt)), φ ∼ ln(tan(λt)), φ ∼ 1/ sinh(λt)). Theinterpretation of such solutions is obvious: we have thepossible exit from inflation or current accelerated ex-pansion or initial inflation rolling to later slow inflation(with further exit from it).

Therefore these models can not only describe inflationbut also describe an exit from the inflationary phase withouta fine tuning of the parameters. This fact can be seen as ev-idence of the existence of a realistic model that contains aninflationary phase in the early universe stage and a currentphase of accelerated expansion.

Acknowledgements This work is supported in part by project 14-02-31100 (RFBR, Russia) (AVA).

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