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The anisotropic Bianchi V cosmological model (*)

A. HARVEY(**) (***)

Department of Physics, Queens College of the City University of New York Kissena Boulevard, Flushing, NY 11367-0904, USA

(ricevuto il 30 Luglio 1996; approvato il 25 Marzo 1997)

Summary. — One of the more important sub-sets of cosmological models is the generalization of the isotropic, negative-curvature, Friedmann-Robertson-Walker models to the anisotropic Bianchi V class. This class is the subject of continuing studies, the most recent of which are by Lorenz, Nayak and Bhuyan, and Fiser et al. In none of these cases has the cosmological constant been used. Motivated by the fact that a) if the cosmological constant exists it plays a major role in the evolution of the universe and b) recent determinations of the Hubble parameter imply an age for the universe substantially smaller than that for mature stars, we have re-examined the Bianchi V model with cosmological constant included for both electromagnetic radiation and pressureless dust sources. We find that current astronomical data and the isotropy of the cosmic background microwave radiation do not rule out such a model. PACS 04.20 – Classical general relativity.

1. – Introduction

The first published material on a Bianchi V model was by Heckmann and Schucking [1]. They applied the Einstein equations with cosmological constant to the case of a synchronous, diagonal metric and pressureless dust source. A number of results were stated without details or discussion. Somewhat later an explicit vacuum solution in non-synchronous form for the equations without cosmological constant was given by Joseph [2]. Since then there have been many studies of Bianchi V symmetric models, the most recent being those of Lorenz [3] who discusses tilted models with stiff matter and electromagnetic fields, Nayak and Bhuyan [4] who discuss perfect-fluid sources and null geodesics, and Fiser et al. [5] who discuss perfect fluids with various

(*) The author of this paper has agreed to not receive the proofs for correction. (**) Professor Emeritus, City University of New York.

(***) E-mail: harveyHscires.acf.nyu.edu

equations of state. None of these authors include the effects of a cosmological constant. The recent announcements [6] of new determinations of the Hubble parameter which imply an age for the universe of half that of the most mature stars which can be observed suggest that a re-examination of the Bianchi V models with the cosmological constant is desirable.

We consider models with radiation gas or pressureless dust sources both with (for the sake of completeness and comparison) and without cosmological constant. Some comments on the vacuum model are made. We determine the conditions under which analogs of the Friedmann equation for these models exist. For all these cases we determine, modulo evaluation of an elliptic or hyperelliptic integral, solutions in synchronous form. The behavior of all models in the limit of large t is identical and depends only on the cosmological constant: if L 40 the space-time becomes expanding and flat; if L D0 the space-time expands, but does not become flat; if LE0 all models terminate in a big crunch with a total lifetime of 13 3109 years. The behavior of solutions in the vicinity of the initial singularity in all cases is identical; they are smooth and curvature-dominated and do not depend on the presence or absence of a cosmological constant. At the present time the effect of the cosmological constant is not yet apparent and the space-time is not distinguishable from an expanding, flat space-time. Several new solutions are exhibited. And, it is demonstrated that a Bianchi V cosmological model is compatible with the isotropic cosmic microwave back-ground radiation (CMBR).

We start with a reconstruction of the generalized Friedmann equation of Heckmann and Schucking in the course of which we obtain an integral of the Einstein equations which facilitates construction of solutions. A synchronous vacuum metric is derived which, modulo a coordinate transformation, is the same as Joseph’s solution. We explore the cases of pressureless dust and radiation gas source both with and without the cosmological constant. For the sake of completeness we discuss briefly some of the previous work and we provide several new exact solutions. (For the most part the studies cited earlier [3-5] utilize the formalism introduced by Misner [7] for the study of homogeneous cosmologies. The calculations, results, and discussion we present are more conveniently made in a synchronous coordinate basis. The sign conventions of Misner, Thorne and Wheeler [8] are used. R. _{denotes dROdt and c41.)}

2. – The metric and field equations

The general Bianchi V metric may be diagonalized to ds2

4 2dt21 R2dx2

1 e22 x(S2_dy2

1 U2dz2_{) ,}

(1)

where R, S, and U are functions of t provided the Einstein tensor (or, equivalently, the energy-momentum tensor) is diagonal [9]. Note that, with ds having the dimensions of a length, 2dt is similarly dimensioned (setting c41 does not change its dimensions). The factor e22 x_{requires that x be dimensionless, from which it follows that R has the}

dimensions of a length and S, y, and z are dimensionless. (It is a remarkable fact that this metric, just as the k 421 Friedmann-Robertson-Walker models, is of isotropic

constant negative curvature [10].) The concomitant components of the Einstein tensor are G0 04 2 S.U. SU 2 U.R. UR 2 R.S. SR 1 3 R2 , (2a) G0 14 12 R. R 2 S. S 2 U. U 4 R 2_G1 0, (2b) G1 14 2 S n n S 2 U n n U 2 S.U. SU 1 1 R2 , (2c) G2 24 2 U n n U 2 R n n R 2 U.R. UR 1 1 R2 , (2d) G334 2 R n n R 2 S n n S 2 R.S. RS 1 1 R2 . (2e)

For the several cases to be discussed, i.e. vacuum, radiation gas, and pressureless dust, each in a comoving coordinate system, the stress-energy tensor is Tm

n4 0 for T01

or T1

0. This assumes that in the case of radiation the coordinate system is taken in the

frame in which the radiation is isotropic and for dust that in which the 4-velocity is orthogonal to the hypersurfaces of transitivity. This leads, in all cases, to an integral of the system, SU 4R2 _{where an integration constant has been absorbed. The simple}

choice S 4U4R imposes rotational symmetry and yields the metric ds2

4 2dt21 R2dx2

1 R2e22 x_{( dy}2

1 dz2) . (3)

This is just a form of the constant-negative-curvature Friedmann-Robertson-Walker metric [11]:

ds2

4 2dt21 R2[ dx21 sinh2x( du21 sin2u df2) ] . (38)

A slightly more general case, a tilted LRS model with cosmological constant and pressureless dust, was discussed by Farnsworth [12]. In such models neither T01 nor

T1

0normally vanishes. Under these circumstances the more general LRS metric

ds2

4 2dt21 R2dx21 S2e22 x_{( dy}2

1 dz2) (4)

was admitted. (Tilted models lie outside the scope of this paper and will not be discussed further.)

The next simplest choice is the replacement S KRS and UKRS21_{. This change}

entails no loss of generality and the metric becomes that of Heckmann and Schucking [1]: ds2

4 2dt21 R2dx2

1 R2e22 x_(S2_dy2

1 S22dz2_{) .}

The Einstein equations are G004 S.2 S2 2 3 R.2 R2 1 3 R2 4 gT 0 0, (6a) G1 14 22 R n n R 2 R.2 R2 2 S.2 S2 1 1 R2 4 gT 1 1, (6b) G224 22 R n n R 2 R.2 R2 1 S n n S 2 2 S.2 S2 1 3 R.S. RS 1 1 R2 4 gT 2 2, (6c) G3 34 22 R n n R 2 R.2 R2 2 S n n S 2 3 R.S. RS 1 1 R2 4 gT 3 3, (6d) where Tm

n for the dust and radiation cases are, respectively, (2r, 0, 0, 0) and

(2r, rO3, rO3, rO3) and g48pG. If Lc0 the term 2da

bL is added to the

right-hand side of each equation.

For these Einstein equations and a stress-energy tensor for which any 2 components Ti

i (i 41, 2, 3) are equal (and this will include the cases in which we are

interested and most other cases which are not tilted models) there exists an integral of the system obtained by subtracting one of the corresponding Einstein equations from the other. The result is, modulo a numerical factor,

S n n S 2 S.2 S2 1 3 R.S. RS 4 0 . (7)

The first 2 terms combine to become (S._{OS) and the resulting equation is simply} (S./S)Q S./S 1 3 R . R 4 0 (8)

with the immediate solution

S.

S 4

k R3 .

(9a)

The choice of the positive root is irrelevant. The form of the metric is essentially unchanged by the substitution S KS21_{. Such a substitution would change the sign of}

the root. Hence, the choice of sign is arbitrary and without effect on results. In some cases it is more convenient to use eq. (9a) in the form

d log S dR 4 k R3_R. . (9b)

substitution into eq. (6a) results in k2 R6 2 3 R.2 R2 1 3 R2 4 gT 0 0. (10a)

This result is clearly also valid for the case of a non-vanishing cosmological constant. If it is present, eq. (10a) becomes

k2 R6 2 3 R.2 R2 1 3 R2 4 gT 0 02 L . (10b) Inasmuch as T0

04 2 r for both radiation and pressureless dust, it is convenient to

rewrite both in the form R.2 4 1 1 grR 2 3 1 a4 R4 1 LR2 3 , (10c) where a4_{4 k}2_{O3 .}

This equation is not as simple as it appears. Firstly, it must be augmented by either eqs. (9a) or (9b) for a complete picture of the evolution of a particular model. Secondly, there seems to be no sensible way of delimiting the constant of integration a. Lastly, the density, r, varies with time (or R). This problem may be eliminated by noting that for both disordered radiation and pressureless dust a constant is provided by the covariant divergence Tmn

; n. These, respectively, are rR44 M and rR34 M . The two

cases may be subsumed in R.2 4 1 1 m 3 Re 1 a4 R4 1 LR2 3 , (11)

where m 48pGM and e42 for radiation and 1 for pressureless dust. This is the generalized Friedmann equation [1] and is central to our work.

The 2 lowest-order scalars of the Riemann tensor which do not vanish identically are occasionally useful to determine whether or not a singularity is a true singularity or an artifact of the coordinate system being used. These are Rab

gdRgdab and Rab gdRgdlmRlmabwhich evaluate to K24 16 k2 R6

u

R.2 R2 1 k2 3 R6 2 1 R2

v

, (12a) K24 16 a4 R6

g

gr 1L1 6 a4 R6

h

, (12b) K34 32 k4 R12

u

R.2 R2 2 k2 9 R6 2 1 R2

v

, (12c) K34 96 a8 R12

g

gr 1L1 2 a4 R6

h

, (12d)

where gr 48pGMOR4

for radiation and 8 pGMOR3_{for dust. Equations (12a) and (12c),}

before simplification, contain terms in S

n n

OS , S .

OS , and R .

OR . Equation (7) may be used to eliminate S

n n

OS and then (9a) to eliminate S .

OS . Lastly, eq. (11) is used to eliminate R._{OR . Other scalars do not provide any additional useful information.}

3. – The vacuum solution

Equation (11) for the vacuum case reduces to dR dt 4

k

a4 1 R41 LR6/3 R2 . (13)

This may be used to eliminate R. in eq. (9b). The result is d log S dR 4 k3 a2 R

k

a4 1 R41 LR6/3 . (14)

If L 40 this reduces to a pseudo-elliptic integral [13] which evaluates to S 4n

u

k

a 4 1 R42 a2

k

a4 1 R41 a2

v

k3/4 , (15)

where n is a constant of integration which may be absorbed. With minor manipulation, the metric, written in terms solely of R, is

ds2 4 2dt21 R2dx21 R2( 1 2k3)e22 x_{[ (}

_k

_a4 1 R42 a2)k3dy21 (16) 1 (

k

a4 1 R41 a2)k3_dz2_{] .}

R, of course, is given in terms of t implicitly, but inasmuch as it is a monotonic function of t it is suitable as a time-coordinate. To this end eq. (13) may be used again. The result is ds2 4 2 R 4 a4_{1 R}4dR 2 1 R2dx2 1 R2( 1 2k3)e22 x_{[ (}

_k

_a4 1 R42 a2)k3_dy2 1 (17) 1 (

k

a4_{1 R}4_{1 a}2)k3_dz2_{] .}

This is readily converted to Joseph’s metric [2], viz ds2

4 2a2[ sinh ( 2 t)( dt2

2 dx2) 2e22 x_{sinh ( 2 t)}

₍

_tanhk3_{(t) dy}2

1 tanh2k3(t) dz2

₎

_{] ,}

(18)

by the simple transformation R2

4 a2sinh ( 2 t). For the choice a 41 this is just Joseph’s metric.

The solution of eq. (13) is 1

2F(f , 1 /k2) 2E(f, 1/k2) 2

sn u dn u

1 2cn u 4 t 2 t0, (19)

where F and E are the (incomplete) elliptic integrals of the first and second kinds, respectively; sn u , dn u , and cn u are Jacobian elliptic functions [13]. The upper and lower limits of integration are, respectively, am u24 p and am u14 cos21f(R22 1 ) O

(R2

1 1 ). With R 4 0 at t04 0 the result is

20.846 2 1

2 F(f , 1 /k2) 1E(f, 1/k2) 1

R

k

1 1R4

1 1R2 4 t .

(20)

A number of subsequent integrals are also elliptic integrals and their evaluation follows a similar pattern.

If the cosmological constant does not vanish eqs. (13) and (14) are hyperelliptic integrals [13]. In their limiting forms these will be discussed below.

4. – Radiation gas

If it is assumed that the initial singularity is a hot big bang, then to investigate the metric behavior in its vicinity a reasonable choice for a source would be a radiation gas with the stress-energy tensor indicated earlier. Equation (11) with e 42 becomes

R.2_{4 1 1} m 3 R2 1 a4 R4 1 LR2 3 . (21)

This leads to a hyperelliptic integral. The cognate to eq. (14) is d log S dR 4 a2k3 R

k

a4 1 mR2/3 1R4 1 LR6/3 . (22)

If L 40 this reduces to a pseudo-elliptic integral with the result S 4

u

k

R 4 1 mR21 1 2

k

mR2_{1 1}

k

R4_{1 mR}2_{1 1 1}

k

mR2_{1 1}

v

(k3/4 )kmR2 1 1 (23)

and the metric may be written:

ds2 4 2 R 4_dR2 a4 1 mR2/3 1R4 1 R 2_dx2 1 (24) 1R2e22 x

u

k

R 4 1 mR21 1 2

k

mR2_{1 1}

k

R4_{1 mR}2_{1 1 1}

k

mR2_{1 1}

v

(k3/2 )kmR21 1 dy2 1 1R2e22 x

u

k

R 4 1 mR21 1 1

k

mR2 1 1

k

R4 1 mR21 1 2

k

mR2 1 1

v

(k3/2 )kmR21 1 dz2_.

5. – Pressureless dust

The treatment for pressureless dust parallels very closely that for radiation gas. With the stress-energy tensor for pressureless dust indicated earlier eq. (11) reduces to

R.2 4 1 1 m 3 R 1 a4 R4 1 LR2 3 , (25)

where m has precisely the same definition as before. This, like the vacuum case, eq. (12), leads to a hyperelliptic integral for R, implicitly, in terms of t. The parallel to eq. (22) is

d log S dR 4 a2 k3 R

k

a4 1 mR2/3 1R4 1 LR6/3 . (26)

If L 40 this leads to a pseudo-elliptic integral which evaluates to S 4

u

k

R 4 1 mR31 1 2

k

mR3 1 1

k

R4 1 mR31 1 1

k

mR3 1 1

v

(k3/4 )kmR3_{1 1} . (27)

The metric then is ds2 4 2 R 4_dR2 a4 1 mR3/3 1R4 1 R 2_dx2 1 (28) 1R2e22 x

u

k

R 4 1 mR31 1 2

k

mR3 1 1

k

R4 1 mR31 1 1

k

mR3 1 1

v

(k3/2 )kmR31 1 dy2 1 1R2e22 x

u

k

R 4 1 mR31 1 1

k

mR3 1 1

k

R4 1 mR31 1 2

k

mR3 1 1

v

(k3/2 )kmR31 1 dz2_. 6. – Limiting forms

The generalized Friedmann equation, R.2 4 1 1 m 2 Re 1 a4 R4 1 LR2 3 (11)

(where e 41, 2 for dust and radiation, respectively), is the starting point for discussing the behavior of various models in the limits of either t K0 or tKQ.

6._{1. t K0. – In the limit tK0 this reduces to R}2

dR 4a2_{dt independently of source}

or presence of a cosmological constant of whatever sign. All such models are independent of L. With discard of a non-essential integration constant R 4 (3a2_t)1 O3

and use of eq. (9b) yields S 4Rk3

4 ( 3 a2t)2k3_{. The corresponding metric is}

ds2

4 2dt21 ( 3 a2t)2 /3_dx2

1 e22 x[ ( 3 a2_t)2 /3( 1 1k3)_dy2

1 ( 3 a2t)2 /3( 1 2k3)_dz2_{] ,}

(29)

where numerical constants have been absorbed in y and z. As might be expected, the constants K2and K3blow up.

6._{2. t KQ. – For the limit tKQ the existence of a source and, if it exists, its nature} are irrelevant. However, the presence or absence of a cosmological constant and, if present, its sign make important differences. If L 40 eq. (11) reduces to dR4dt. Then, eq. (9b) provides S 4e2a2k3O(2t2₎

, so that S K1. The resultant metric is simply that for an expanding flat space-time

ds2_{4 2dt}2_{1 t}2dx2_{1 e}22 x_t2

dy2_{1 e}22 x_t2

dz2 (30)

in which the hypersurfaces of transitivity are 3-pseudospheres.

6._{3. t KQ; LD0. – If LD0 then eq. (11) reduces to R}. _{4 R}k_{LO3. Thus, R4e}kLO3t. Despite the difference in the dependence of R on t, it still turns out that S K1. The resulting metric is ds2 4 2dt21 e2kL/3 tdx2 1 e2(kL/3 t 2x)( dy2 1 dz2) . (31)

The space-time is not flat nor it not become so asymptotically. The curvature invariants K2and K3take on the asymptotic values ( 4 O27) L2and ( 4 O243) L3.

6._{4. t KQ; LE0. – If LE0 then eq. (11) reduces to} R.2

4 1 2 LR

2

3 (118)

with the immediate solution

R 4k3 /L sin (kL/3 t) (32)

which implies an ultimate big crunch. The concomitant solution for S is

S 4

g

tan kL/3 t 2

h

a2L 4k3 e 2a 2_{Lcos (kL/3 t)} 2k3 sin2(L/3 t) _. (33) The metric is ds2_{4 2dt}2₁ 3 Lsin 2 (kL/3 t) dx2₁ (34) 1 3 Lsin 2₍ kL/3 t) e22 x

g

_tan kL/3 t 2

h

a2_L 4k3 e 2a 2_{Lcos (kL/3 t)} 2k3 sin2_{(L/3 t)} dy2 1 1 3 Lsin 2₍ kL/3 t) e22 x

g

_tan kL/3 t 2

h

2a 2_L 4k3 e a2_{Lcos (kL/3 t)} 2k3 sin2_{(L/3 t)} dz2_.

7. – Comments

At the present time the (visible) matter density may be estimated to be 10231_{gm cm}21 _{and the average peculiar velocity of a galaxy is roughly 300 km s}21_.

From these data a simple calculation shows that rmB 3 3 106pm3 c22. Also, the mass

density equivalent of black-body background radiation at 2.74 K is about 4.7 3 10234_{gm cm}23_{. Consequently, the background radiation and dynamical galactic}

pressure may be neglected and the relevant metric is eq. (28). A simple order-of-magnitude calculation shows that this model is indistinguishable from the flat, expanding space-time noted earlier, i.e. R Bt and SB1. If Lc0 the flatness will not persist and the space-time will ultimately become curved.

Studies based on the galactic redshift-magnitude [14] provide the limits: 22310255

G L G12310255_cm22_{. This implies, for L E0 models, a time interval from initial}

singu-larity to recollapse of about 13 3109_{years. This is scarcely large enough to solve the age}

problem. For models with L D0 there is no such problem; they expand indefinitely. There remains the question of whether a Bianchi V cosmological model with positive cosmological constant is viable. The answer is yes. Present astronomical data do not indi-cate the presence of enough mass to prevent the continued expansion of the universe. There is no observational data which would rule out a Bianchi V, LD0, cosmological model.

* * *

A number of conversations with E. SCHUCKING were very enlightening. Pithy

comments by M. A. H. MACCALLUM who also indicated how eq. (38) can be obtained

from eq. (3) were extremely useful. I am very grateful to both gentlemen. Most of the calculations were made with the general-relativity packages SHEEP and STENSOR of I. FRICK and L. HO¨RNFELDT, respectively. Gratitude must be expressed for the installation of these packages on my PC by J. SKEA. Support by New York University in

the form of computer time is deeply appreciated.

R E F E R E N C E S

[1] SCHUCKING E. and HECKMANN O., Proceedings of the Solvay Conference (Institut International de Physique Solvay, Bruxelles) 1958, p. 157; see also WITTEN L. (Editor), Gravitation: An Introduction to Current Research (John Wiley & Sons, New York) 1962, p. 445.

[2] JOSEPHV., Proc. Cambridge Philos. Soc., 62 (1966) 87. [3] LORENZD., Gen. Relativ. Gravit., 13 (1981) 795.

[4] NAYAKB. K. and BHUYANG. B., Gen. Relativ. Gravit., 18 (1986) 79. [5] FISERK., ROSQUISTK. and UGGLAC., Gen. Relativ. Gravit., 24 (1992) 679. [6] See, e.g., JACOBYJ. H., Nature, 371 (1994) 741.

[7] See, e.g., RYANM. P. and SHEPLEYL. C., Homogeneous Relativistic Cosmologies (Princeton University Press, Princeton) 1975, Chapt. 9 or Chapt. 30 of [8].

[8] MISNER C. W., THORNE K. S. and WHEELER J. A., Gravitation (W. H. Freeman and Company, San Francisco) 1971.

[9] MACCALLUMM. A. H., Phys. Lett. A, 40 (1972) 385. [10] HARVEYA., J. Math. Phys., 21 (1980) 870.

[11] MACCALLUMM. A. H., private communication. [12] FARNSWORTHD. L., J. Math. Phys., 8 (1967) 2315.

[13] BYRD P. F. and FRIEDMANN M. D., Handbook of Elliptic Integrals, 2nd revised edition (Springer-Verlag, New York) 1971, pp. 142-143.