Quintom fields from chiral anisotropic cosmology J. Socorro1S. P erez-Pay an2Rafael Hern andez-Jim enez3 Abraham Espinoza-Garc ıa2and Luis Rey D ıaz-Barr on2

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Quintom ﬁelds from chiral anisotropic cosmology

J. Socorro,1, ∗S. P´erez-Pay´an,2, †Rafael Hern´andez-Jim´enez,3, ‡

Abraham Espinoza-Garc´ıa,2, §and Luis Rey D´ıaz-Barr´on2, ¶

1Departamento de F´ısica, DCeI, Universidad de Guanajuato-Campus Le´on, C.P. 37150, Le´on, Guanajuato, M´exico

2Unidad Profesional Interdisciplinaria de Ingenier´ıa,

Campus Guanajuato del Instituto Polit´ecnico Nacional.

Av. Mineral de Valenciana #200, Col. Fraccionamiento Industrial Puerto Interior,

C.P. 36275, Silao de la Victoria, Guanajuato, M´exico.

3Departamento de F´ısica, Centro Universitario de Ciencias Exactas e Ingenier´ıa, Universidad de Guadalajara.

Av. Revoluci´on 1500, Colonia Ol´ımpica C.P. 44430, Guadalajara, Jalisco, M´exico.

In this paper we present an analysis of a chiral anisotropic cosmological scenario from the per-

spective of quintom ﬁelds. In this setup quintessence and phantom ﬁelds interact in a non-standard

(chiral) way within an anisotropic Bianchi type I background. We present our examination from

two fronts: classical and quantum approaches. In the classical program we ﬁnd analytical solutions

given by a particular choice of the emerged relevant parameters. Remarkably, we present an expla-

nation of the “big-bang” singularity by means of a “big-bounce”. Moreover, isotropization is in fact

reached as the time evolves. On the quantum counterpart the Wheeler-DeWitt equation is analyt-

ically solved for various instances given by the same parameter space from the classical study, and

we also include the factor ordering Q. Having solutions in this scheme we compute the probability

density, which is in eﬀect damped as the volume function and the scalar ﬁelds evolve; and it also

tends towards a ﬂat FLRW framework when the factor ordering constant Q ≪0. This result might

indicate that for a ﬁxed set of parameters, the anisotropies quantum-mechanically vanish for very

small values of the parameter Q. Finally, classical and quantum solutions reduce to their ﬂat FLRW

counterparts when the anisotropies vanish.

PACS numbers:

I. INTRODUCTION

The rather small deviation from isotropy observed in the cosmic microwave background (CMB) radiation [1] makes

it plausible that at very early times the universe was indeed anisotropic, therefore prompting the introduction of

anisotropic cosmological models to describe the evolution of the universe near the initial singularity [2, 3]. The

Bianchi type I model is a natural choice for such a background given that its isotropic limit is the spatially ﬂat

Friedmann-Robertson-Lamaˆıtre-Walker (FRLW) model (see, e.g., [4]). Indeed, the Bianchi type I model has been

recently considered to explain the aforementioned tiny variations in the CMB by a number of researchers [5–8].

On the other hand, the multi-ﬁeld cosmology paradigm has proven to be an eﬀective framework to account (in a

single model) for several important characteristics/ingredients of the universe, e.g., early acceleration (inﬂation) [9–

25], dark matter [26–28], late acceleration [29–46]. With respect to early and late acceleration, the crossing of the

phantom divide line is a most wanted feature in scalar ﬁeld cosmology; it has been shown that this crossing cannot

be achieved by considering a single scalar ﬁeld/ﬂuid (unless stability is not demanded) [47]. The standard quintom

scenario [48] considers two scalar ﬁelds, a quintessence and a phantom, in order to realize such crossing in a simple

way. As a byproduct, quintom ﬁelds allow (in particular cases) the avoidance of the initial singularity by means of a

bounce [49] (see also the review [47]). In the conventional quintom scenario (and in ordinary multi-ﬁeld cosmology)

the scalar ﬁelds interact in the following way:

Lϕ=δabgµν ∇µϕa∇νϕb+V(ϕa, ϕb).(1)

An incarnation of multi-ﬁeld cosmology is the so called chiral-cosmology [50], in which the scalar ﬁelds deﬁne an

“internal space” with a certain metric component mab. They also interact in a non-standard manner within their

∗Electronic address: socorro@ﬁsica.ugto.mx

†Electronic address: saperezp@ipn.mx

‡Electronic address: rafaelhernandezjmz@gmail.com

§Electronic address: aespinoza@ipn.mx

¶Electronic address: lrdiaz@ipn.mx

arXiv:2210.01186v2 [gr-qc] 13 Jun 2023

kinetic terms, their couplings are governed by the metric mab (in short, we will replace δab →mab in (1)). This metric

can be seen as arising from casting a non-minimally coupled multi-scalar-tensor theory as General Relativity (i.e., in

going from the Jordan frame to the Einstein frame) [51]. Non-minimal couplings are indeed required when considering

the quantization of scalar ﬁelds in curved backgrounds [52], the use of non-canonical ﬁelds in (eﬀective) descriptions

of the early universe in Einstein’s general relativity is therefore theoretically consistent with standard quantum ﬁeld

theory.

In the present investigation we consider a Bianchi type I framework within a generalized quintom scenario, in

which the scalar ﬁelds deﬁne a chiral space with a certain metric mab (so that the ﬁelds are not canonical). The

exact classical solutions are obtained, then with them we show that the initial singularity is avoided by means of a

bounce. Moreover, exact quantum solutions will show that the wave function of the universe is damped with respect

to the average scale factor. Similar conclusions were made in the corresponding isotropic case [53]. In the remaining

part of this introduction we proceed to describe the generalities of the chiral cosmology which we will be employing.

We consider the following simple case of two scalar ﬁelds, a quintessence ﬁeld ϕ1and phantom ﬁeld ϕ2(with their

corresponding potential terms) within the chiral cosmology paradigm [50, 53–59]

L=√−gR+mab ξab(ϕc,gµν ) + C(ϕc),(2)

where R is the Ricci scalar, ξab(ϕc,gµν ) = −1

2gµν ∇µϕa∇νϕbthe kinetic energy, and C(ϕc) = V(ϕa, ϕb) the corre-

sponding scalar ﬁeld potential, with mab a 2×2 constant matrix; we consider the particular form mab =1 m12

m12 −1.

Thus, the Einstein-Klein-Gordon ﬁeld equations are

Gαβ =−1

2mab ∇αϕa∇βϕb−1

2gαβ gµν ∇µϕa∇νϕb+1

2gαβ C(ϕc),(3)

mcb∇ν∇νϕb−∂C(ϕc)

∂ϕc

= 0 ,(4)

where a, b, c = 1,2. From (3) we read oﬀ the energy-momentum tensor for the scalar ﬁelds (ϕ1, ϕ2), as

8πGTαβ (ϕ1, ϕ2) = −1

2mab ∇αϕa∇βϕb−1

2gαβ gµν ∇µϕa∇νϕb+1

2gαβ V(ϕ1, ϕ2).(5)

and considering the analogy with a barotropic perfect ﬂuid for the scalar ﬁelds,

Tαβ (ϕc) = (ρ+ P)uα(ϕc)uβ(ϕc) + P gαβ ,(6)

we have that the pressure P and the energy density ρof the scalar ﬁelds are

P(ϕc) = 1

2mab ξab −1

2C(ϕc), ρ(ϕc) = 1

2mab ξab +1

2C(ϕc),(7)

the four-velocity becomes uαuβ=∇αϕa∇βϕb

2ξab .

We will employ the scalar potential term C(ϕc) = V1(ϕ1)+V2(ϕ2) = V01e−λ1ϕ1+ V02e−λ2ϕ2(with λ1,λ2non-

negative) and the line element to be considered for this two-ﬁeld cosmological model will be that of the anisotropic

Bianchi type I model, which in Misner’s parameterization is given by

ds2=−N2dt2+ e2Ω+2β++2√3β−dx2+ e2Ω+2β+−2√3β−dy2+ e2Ω−4β+dz2,(8)

where the scale factors are A = eΩ+β++√3β−,B=eΩ+β+−√3β−,C=eΩ−2β+and (β+, β−) are the anisotropic param-

eters. Also (Ω, β+, β−) are scalar functions depending on time, and N = N(t) is the lapse function. Plugging in (8)

into (2) we obtain the following Lagrangian density (we eliminate the second time derivatives, previously)

L= e3Ω 









6˙

Ω2

N−6˙

β+2

N−6˙

β−2

N−(˙

ϕ2)2

2N −m12

N˙

ϕ1˙

ϕ2+(˙

ϕ2)2

2N + N (V1(ϕ1)+V2(ϕ2))









.(9)

Henceforth we will be utilizing the Lagrangian density (9) as starting point for our study. The document is organized

as follows. Section II is devoted to set up the classical scheme via the Hamiltonian formalism, and to obtain exact

classical solutions for several cases. In section III the Wheeler-DeWiit equation is constructed considering a semi-

general factor ordering, and exact quantum solutions are presented for various cases as well. Final remarks are stated

in section IV.

II. CLASSICAL SCHEME

In this section we present the classical solutions via the Hamiltonian formalism. We start with the momenta

Πq=∂L/∂ ˙q (with qi= Ω, β+, β−, ϕ1, ϕ2), which are calculated in the usual way, yielding

ΠΩ=12

Ne3Ω ˙

Ω,

Π+=−12

Ne3Ω ˙

β+,

Π−=−12

Ne3Ω ˙

β−,

Πϕ1=1

Ne3Ω(−˙

ϕ1−m12 ˙

ϕ2),

Πϕ2=1

Ne3Ω(˙

ϕ2−m12 ˙

ϕ1),

Ω = N

12e−3ΩΠΩ,

β+=−N

12e−3ΩΠ+,

β−=−N

12e−3ΩΠ−,

ϕ1=−e−3Ω N

1 + (m12)2Πϕ1+ m12Πϕ2,

ϕ2= e−3Ω N

1 + (m12)2m12Πϕ1+ Πϕ2,

(10)

then the Lagrangian density (9) is rewritten in a canonical way, i.e. Lcanonical = Πi˙qi−NH, so we arrive at the

Hamiltonian density

H=e−3Ω

24 (Π2

Ω−Π2

+−Π2

−−12Π2

ϕ1

△−12 m12 Πϕ1Πϕ2

△+ 12Π2

ϕ2

△−24V01 e6Ω−λ1ϕ1−24V02 e6Ω−λ2ϕ2),(11)

where △= 1 + (m12)2. We now consider the canonical transformation (Ω, ϕ1, ϕ2, β+, β−)↔(ξ1, ξ2, ξ3, ξ4, ξ5)

ξ1= 6Ω −λ1ϕ1,

ξ2= 6Ω −λ2ϕ2,

ξ3= 6Ω + λ1ϕ1+λ2ϕ2,

ξ4=β+,

ξ5=β−,

←→

Ω = ξ1+ξ2+ξ3

18 ,

ϕ1=−2ξ1+ξ2+ξ3

3λ1

ϕ2=ξ1−2ξ2+ξ3

3λ2

β+=ξ4,

β−=ξ5,

(12)

with the new conjugate momenta (P1,P2,P3,P4,P5) given by

ΠΩ= 6P1+ 6P2+ 6P3,

Πϕ1=λ1(−P1+ P3),

Πϕ2=λ2(−P2+ P3),(13)

Π+= P4,

Π−= P5.

Therefore the Hamiltonian density, in the gauge N = 24e3Ω, becomes

H= 12 (3 −Λ1) P2

1+ 12 (3 + Λ2) P2

2+ 12 (3 −2Λ12 + Λ2−Λ1) P2

+ 24 [(3 + Λ1+ Λ12) P1+ (3 + Λ12 −Λ2) P2] P3

+ 24 (3 −Λ12) P1P2−P2

4−P2

5−24 V01eξ1+ V02eξ2,(14)

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QuintomfieldsfromchiralanisotropiccosmologyJ.Socorro,1,∗S.P´erez-Pay´an,2,†RafaelHern´andez-Jim´enez,3,‡AbrahamEspinoza-Garc´ıa,2,§andLuisReyD´ıaz-Barr´on2,¶1DepartamentodeF´ısica,DCeI,UniversidaddeGuanajuato-CampusLe´on,C.P.37150,Le´on,Guanajuato,M´exico2UnidadProfesionalInterdisciplinariadeIngenie...

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Quintom fields from chiral anisotropic cosmology J. Socorro1S. P erez-Pay an2Rafael Hern andez-Jim enez3 Abraham Espinoza-Garc ıa2and Luis Rey D ıaz-Barr on2

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