fulltext

Torsion and quadratic bulk viscosity in a homogeneous

isotropic universe

C. WOLF

Department of Physics, North Adams State College - North Adams, MA 01247, USA (ricevuto il 2 Aprile 1996; approvato il 15 Gennaio 1997)

Summary. — By considering the competitive factors of spin-generated torsion and quadratic bulk viscosity in a Robertson-Walker universe we obtain the first integrals of the equation of motion for a wide range of equations of state. All of the cosmologies have an evolution that is retarded by the effects of torsion and bulk viscosity either accelerates or decelerates the expansion depending on the magnitude of the bulk viscosity coefficient.

PACS 98.80 – Cosmology.

1. – Introduction

Ever since the rebirth of interest in cosmology with the great innovation brought about by inflationary cosmology [1-4], numerous factors have been studied in a cosmological setting that have a microphysical origin rooted in quantum physics. Bulk viscosity though a phenomenological concept has been interpreted as representing the conversion of massive string modes to massless modes as well as representing the dissapative effects brought about by particle creation [5]. In somewhat the same direction, Gurovich and Starobinsky have pointed out that a bulk viscosity coefficient proportional to the curvature squared is a representation of particle creation by vacuum polarization in a background gravitational field [6]. Padmanabhan and Chitre have shown that bulk viscosity in a flat universe leads to inflation, they emphasize that bulk viscosity can represent the viscous drag on a superconducting string on a magnetic field, the drag brought about by monopole-monopole interactions and the viscous drag due to photon and neutrino viscosity [7]. In two previous notes we have shown that inflation results from cosmologies admitting energy-dependent and curvature-dependent bulk viscosity [8, 9]. Another factor that has profound effects on early-universe evolution is spin-generated torsion [10]. Torsion is a consequence of three considerations in physical theory, the first is the result of including spin into the structure of space-time, the second is a result of considering a gauge theory of gravity, and the third is a consequence of considering a unified theory of electromagnetism and gravity [11]. The simplest torsion theory, the Einstein-Cartan theory, allows the metric

to be calculated from the energy-momentum tensor of matter, and the torsion is calculated from the spin of matter [12, 13]. Numerous results of spin-generated torsion theories have appeared in a cosmological setting including the elimination of cosmological singularities [14], the halting of the unimpeded collapse of a dust sphere [15], the stabilization of a black hole against radiative decay [16], the spectral splitting of microwaves in a magnetic field [17] and the rotation of the plane of polarization of EM waves in a magnetic field [18]. In early universe evolution Demianski et al. [19] and Gasperini [20] have shown that spin-generated torsion leads to solutions that are inflationary in character. Numerous authors have found acceptable

cosmological solutions of spin torsion theories within the context of the

Einstein-Cartan theory [21-24] and spin torsion theories have also proven to leave specific signatures in the spectrum of elementary particles [25]. In ref. [24] we have shown that when spin-generated torsion is embedded in a Robertson-Walker cosmology it is equivalent to replacing, P KP2 (2pG/C2) S2_{, and e Ke2 (2pG/C}2) S2

(

S2

4 a( 1 /2 ) SmnSmnb 4 average of spin density squared

)

.

In the present note we include both factors, quadratic bulk viscosity [26] and spin-generated torsion [24] in the evolution of a Robertson-Walker cosmology. The cosmological equations yield an exact first integral for a wide range of equations of state and the character of the solutions is dependent on the magnitude of the bulk viscosity coefficient. It is hoped that the subtle interplay between bulk viscous effects and spin-generated effects will be further investigated by students of the early universe to find out exactly how these factors effect inflationary solutions and the development of primordial perturbations that provide seeds to large-scale structure.

2. – Torsion and quadratic bulk viscosity in a homogeneous isotropic universe

We begin our analysis by considering a Robertson-Walker line element

( dS)2_{4 dt}2_{2 R}2

g

dr 2 1 2Kr2 1 r 2 ( du)2_{1 r}2sin2u( df)2

h

(2.1)

(

K 4061, R(t) 4scale factor

)

. For the Ricci component we have

R004 3 RO R , Rij4

y

RO R 1 2

u

R n R

v

2 1 2 K R2

z

gij. (2.2)

The energy-momentum tensor of a perfect fluid is

Tmn4 (P 1 e) UmUn2 gmnP .

(2.3)

To take into account the effect of quadratic bulk viscosity [26] and torsion with spherically symmetric distributed spins [24] we replace

e Ke2 2 pG C2 S

2 (2.4)

and P KP2 2 pG C2 S 2 2 b

u

R n R

v

2 (2.5)

(here b 4coefficient of quadratic bulk viscosity). For the Einstein equations we have (k 48pG/C4₎

Rmn4 2k

k

Tmn2

1 2Tgmn

l

. (2.6)

The components of eq. (2.3) with the replacement in eq. (2.4) and eq. (2.5) are

T_{004 e 2} 2 pG C2 S 2_{, T} ij4 2

u

P 2 2 pG C2 S 2 2 b

u

R n R

v

2

v

gij (2.7) with T 4Tmngmnfe 2 2 pG C2 S 2 2 3

u

P 2 2 pG C2 S 2 2 b

u

R n R

v

2

v

. The Einstein equations become

3 R n n R 4 2k

y

e 2 2 pG C2 S 2 2 1 2

u

e 23P1 4 pG C2 1 3 b

u

R n R

v

2

v

z

, (2.8) R n n R 1 2

u

R n R

v

2 1 2 K R2 4 2k

C

`

`

`

D

2P 1 2 pG C2 S 2 2 b

u

R n R

v

2 21 2

u

e 23P1 4 pG C2 S 2 1 3 b

u

R n R

v

2

v

E

`

`

`

F

. (2.9)

We also consider the equation of state

P 4ge .

(2.10)

Inserting eq. (2.10) into eq. (2.8) and eq. (2.9) and solving eq. (2.8) for e and substituting into eq. (2.9) we obtain

RO R

g

4 3 g 11

h

1 2

u

R n R

v

2 2

u

R n R

v

2

g

2 kb 3 g 11

h

1 2 K R2 1 4 pGkS2 0(g 21) C2_R6 ( 3 g 11) 4 0 . (2.11) Here we substitute S2

4 S02/R6(S04 total spin in given three-volume). We now reduce eq. (2.11) by the following substitution: P 4 R

n , R n n 4 P( dP/dR), giving dP dR 1 1 R

g

3 g 11 2 2 bk 2

h

P 42

y

K( 3 g 11) 2 R 1 KGkS2 0(g 21) C2R5

z

P 21_. (2.12)

Equation (2.12) is the Bernoulli equation with

P2 4 Z . (2.13)

Substituting eq. (2.13) into eq. (2.12) gives dZ dR 1 1 R

(

( 3 g 11)2bk

)

Z 42

y

K( 3 g 11) R 1 2 pGkS02 C2_R5 (g 21)

z

. (2.14)

We now consider the following cases of eq. (2.14).

Case I. Radiation or highly relativistic particles, g 41/3.

a) _{g 4} 1 3 , 2 2bkD0 . (2.15) Equation (2.14) becomes dZ dR 1 1 R( 2 2bk) Z42

g

2 K R 2 4 pGkS02 C2_R5

h

, (2.16)

the integrating factor of eq. (2.16) is

e

g

2 2bk R

h

dR 4 R2 2bk. Equation (2.16) integrates to ZR2 2bk_{4 2}2 KR 2 2bk 2 2kb 2 4 pGkS02 3 C2_R2 1bk_{( 2 1bk)} 1 C1 or (R n )2 4 2 2 K 2 2kb 2 4 pGkS2 0 3 C2 ( 2 1kb) R4 1 C1 R2 2bk , (2.17) C14 integration constant.

We see from eq. (2.17) that quadratic bulk viscosity accelerates the expansion and the torsion slows the evolution of R.

b) _{2 2kb40.} Equation (2.16) becomes dZ dR 4 2 2 K R 1 4 pGkS2 0 3 C2_R5 . (2.18) Equation (2.18) integrates to Z 422K ln R2 4 pGkS 2 0 12 C2_R4 1 C1 or (R n )2 4 ln R22 K2 pGkS 2 0 3 C2R4 1 C1. (2.19)

Equation (2.19) gives an evolutionary rate of R

n

that is diminished by torsion and a cancellation of the term 2 (R

n

/R)2 due to bulk viscosity leads to the anomalous dependence of ln R22 K_{for (R}n

)2_.

c) _{g 4} 1

3 , 2 2bkE0 .

In this case eq. (2.16) becomes dZ dR 2 1 R(bk 22) Z42 2 K R 1 4 pGkS2 0 3 C2R5 (2.20)

using the integrating factor

m 4e2(bk 2 2 ) ln R 4 R2(bk 2 2 ), eq. (2.20) integrates to (R n )2₄ 2 K bk 22 2 4 bGkS2 0 3 C2_{(kb 12) R}4 1 C1R bk 22_. (2.21)

Equation (2.21) gives an evolution of R diminished by torsion but increased by bulk viscosity because of the term C Q Rbk 22_{. Also the first term 2 K/(bk 22) generates an}

anti-intuitive evolution for (Rn)2

by increasing the evolution for K D0 and decreasing it for K E0.

Case II. g 40 (dust).

Equation (2.14) becomes in this case dZ dR 1 1 R( 1 2bk) Z42

g

K R 2 2 pGkS2 0 C2_R5

h

. (2.22) a) g 40, 12bkD0.

Equation (2.22) has an integrating factor

e( 1 2bk) ln R_{4 R}( 1 2bk)_. (2.23)

Using eq. (2.23), eq. (2.22) integrates to (R n )2 4 2 K 1 2bk 2 2 pGkS2 0 C2_{( 3 1kb) R}4 1 C1 R1 2bk . (2.24)

Equation (2.24) gives an evolutionary rate diminished by torsion, diminished by bulk viscosity in the first term and enhanced by bulk viscosity in the last term.

b) g 40, 12bk40. Equation (2.22) integrates to Z 42K ln R2 pGkS 2 0 2 C2R4 1 C1

or (R n )2 4 ln R2K2 pGkS 2 0 2 C2R4 1 C1. (2.25)

Here eq. (2.25) gives an evolutionary rate for R

n

diminished by torsion with the anomalous dependence of ln, R2K _{generated by the cancellation of the kinetic term} 2(R

n

/R) by bulk viscosity.

c) g 40, 12bkE0.

In this case eq. (2.22) becomes dZ dR 2 1 R(bk 21) 42

g

K R 2 2 pGkS2 0 C2_R5

h

; (2.26)

the integrating factor of eq. (2.26) is

e2(bk 2 1 ) ln R

4 R2(bk 2 1 ). (2.27)

Using eq. (2.27), eq. (2.26) integrates to (Rn)2 4 K bk 21 2 2 pGkS02 C2 (kb 13) R4 1 C1R bk 21_. (2.28)

Equation (2.28) again gives an evolutionary rate diminished by torsion and enhanced by bulk viscosity in the last term with the anti-intuitive dependence on the curvature in the first term due to bulk viscosity.

Case III. Stiff matter g 41.

In this case, eq. (2.14) becomes dZ dR 1 1 R( 4 2kb) Z42 4 K R . (2.29)

We note here that torsion does not effect the solution because of the term containing (g 21) in the last term of eq. (2.14).

a) g 41, 42bkD0.

In this case the integrating factor in eq. (2.29) becomes

e



_{( 4 2bk)}dR

R

4 R4 2bk; (2.30)

multiplying eq. (2.29) by eq. (2.30) and integrating gives (R n )2_{4 2} 4 K 4 2kb 1 C1 R4 2kb . (2.31)

Here bulk viscosity enhances the expansion rate in the second term and decreases it in the first term.

b) g 41, 42bk40.

In this case eq. (2.29) integrates to (R

n

)2

4 ln R24 K1 C1. (2.32)

Again we get the anomalous term ln R24 K _{because of the cancellation of the kinetic} term due to quadratic bulk viscosity, the cosmology has a maximum radius for K D0.

c) g 41, 42kbE0.

In this case eq. (2.29) becomes dZ dR 2 1 R(kb 24) 42 4 K R ; (2.33)

the integrating factor of eq. (2.33) is

e2(kb 2 4 ) ln R

4 R2(kb 2 4 ). (2.34)

Using eq. (2.34), eq. (2.33) integrates to

(R n )2 4 4 K bk 24 1 C1R (bk 24)_.

Again we get enhancement of the evolution of R due to the second term and the anti-intuitive term due in the curvature in the first term generates an ever expanding cosmology for position K.

Case IV. Decompressive Matter [27] P 4ge, g42n. a) n E1/3, 123n2bkD0. Equation (2.14) becomes dZ dR 1 1 R( 1 23n2bk) Z42

y

K( 1 23n) R 2 2 pGS02( 1 1n) k C2_R5

z

; (2.35)

the integrating factor of eq. (2.35) is

e

( 1 23n2bk)



dR

R

4 R1 23n2bk. (2.36)

Using eq. (2.36), eq. (2.35) integrates to

(R n )2 4 2 K( 1 23n) 1 23n2bk 2 2 pGS2 0k C2 ( 3 13n1bk) R4 1 C1 R1 23n2bk . (2.37)

Here eq. (2.37) gives a diminished evolutionary rate due to torsion, and an enhanced rate of evolution due to the last term and diminished rate due to the first term.

b) n E1/3, (123n2kb) 40. Equation (2.35) becomes dZ dR 4 2

y

K( 1 23n) R 2 2 pGS2 0( 1 1n) k C2_R5

z

. (2.38) Equation (2.38) integrates to (Rn)2 4 ln R2K( 1 2 3 n)2 pGkS 2 0( 1 1n) 2 C2_R4 1 C1. (2.39)

In eq. (2.39) we get a diminished evolutionary rate due to torsion and the anomalous term due to the cancellation of the kinetic term 2(R

n

/R)2_{by bulk viscosity.}

c) n E1/3, 123n2kbE0.

Here eq. (2.35) becomes dZ dR 2 1 R(bk 13n21) 42 K( 1 23n) R 1 2 pGS2 0k( 1 1n) C2R5 ; (2.40)

the integrating factor of eq. (2.40) is

e

2(bk 1 3 n 2 1 )



dR

R

4 R2(bk 1 3 n 2 1 ). (2.41)

Using eq. (2.41) to integrate eq. (2.40), we obtain

(Rn_{) 4} K( 1 23n) bk 13n21 2 2 pGS02k C2 (bk 13n13) 1 C1R bk 13n21_. (2.42)

Again, eq. (2.42) gives a diminished rate of evolution due to torsion, an enhanced rate due to bulk viscosity and the anti-intuitive curvature contribution due to bulk viscosity in the first term.

d) n D1/3, 123n2n2kbE0.

In this case eq. (2.35) gives dZ dR 2 1 R(bk 13n21) Z4 K( 3 n 21) R 1 2 pGk2 0( 1 1n) C2_R5 . (2.43)

Here the integrating factor is

e2(bk 1 3 n 2 1 ) ln R

4 R2(bk 1 3 n 2 1 ). (2.44)

Using eq. (2.44), eq. (2.43) integrates to (Rn)2 4 2 K( 3 n 21) (bk 13n21) 2 2 pGS2 0k( 1 1n) C2 (bk 13n13) R4 1 C1R bk 13n21_.

We see from eq. (2.45) that again torsion diminishes the evolutionary rate, while bulk viscosity increases it in the last term and diminishes the evolutionary rate (K D0) in the first term.

3. – Conclusion

The above results demonstrate that torsion diminishes the rate of evolution of the scale factor and quadratic bulk viscosity has a tendency to enhance the rate of evolution in the term that depends on the arbitrary constant of integration. We also get the anti-intuitive dependence on the curvature constant K when the bulk viscosity coefficient is large, when the torsion term is of the same order of magnitude as the bulk viscosity term containing C1 we expect a delaying effect in the evolution of the cosmology, such a period might be related to the period ordinarily required to percolate the true vacuum in extended theories of inflation [4] wherein the effective Hubble constant R

n

/R becomes small enough so that an accelerated expansion does not create too inhomogeneous a universe. To obtain solutions for R(t) all of the above solutions for R

n

(t) would have to be integrated numerically since the integrals are to complicated to evaluate in terms of elementary functions. It would also be interesting to see how the two factors (torsion and bulk viscosity) would effect the evolution of the multi-dimensional cosmologies and whether or not this would favor compactification of the extra dimensions [28, 29].

* * *

I would like to thank the Physics Departments at Williams College and Harvard University for the use of their facilities.

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