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The isotropization of low-density cosmological models
Volume 12A, number 2
PHYSICS LETTERS
25 June1979
THE ISOTROPIZATION OF LOW-DENSITY COSMOLOGICAL MODELS N. CADERNI ’ Institute of Astronomy, University of Cambridge, UK
and R. FABBRI Istituto de Fisica Superiore, University of Florence, Italy Received 24 April 1979
We analyse the evolution of Bianchi type-VIIh cosmological models during the lepton era and the subsequent adiabatic epochs. We find that these models cannot generally attain to the lowdensity regime with a reasonable anisotropy level because of a curvature-anisotropy coupling. The primordial damping due to neutrino viscosity, although very effective for the entropy production, cannot help to match to isotropy requirement at the present epoch.
According to the “philosophy” of chaotic cosmology dissipative processes play an important role in the early stages of the cosmological expansion, when primordial irregularities are damped out and the universe becomes smooth and isotropic at large scales [ 1,2]. Since large amounts of radiation entropy are produced during the damping process, one hopes to establish a link between the expansion isotropy and the large specific entropy observed today. We carried out extensive investigations to check whether neutrino viscosity [l] can provide the suitable damping mechanism in the most general of homogeneous anisotropic models, i.e. the Bianchi spaces with anisotropic curvature tensor: we analyzed in previous papers [3,4] the evolution of spaces of types VIII and IX, and found that a large entropy enhancement (up to N 1012 for suitable initial conditions) can occur in such spaces, but the anisotropic part of curvature usuaIly prevents the Hubble expansion from being sufficiently isotropic today. This contrasts with results found for low-density spaces with isotropic curvature tensor [S] . Therefore we are compelled to extend the analysis to type VII, which is as generic as type VIII or IX and, being the largest generalization of the open
Friedmann model, is moreover favoured by the possibility of providing low density models [6]. Collins and Hawking [7] have shown that in models of type VII, the expansion anisotropy does not become arbitrarily small at times I + m; however, Doroshkevich et al. [8] state that such models admit a “Milne stage” where (a) the matter density is much smaller than the critical density, and (b) the expansi6n anisotropy remains constant and small. Therefore, without any concern for the ultimate fate of the universe * l, here we consider the question, whether these models can dissipate large expansion anisotropies in the lepton era and become consistent with observational isotropy constraints [lo]. To answer this question we consider the type VII,, spaces with metric ds2 = - dt2 t R:(cos C#J s1, - sin 9 S22)2 t R&sin $J52, + cos r#~ Q2)2 + R$ !2$ , where the Q are the canonical frame 1-forms and Rk are the cosmic scale factors: the Einstein equations read *r The scatology of an open universe is discussed by Barrow
1 E.S.A. International Fellow.
and Tipler [9 1.
71
Volume 72A, number 2
dHk
PHYSICS LETTERS
25 June 1979
a 2
I
B
rL~L~ nR.R.~2 1 /
x ilk ~ [\ R. + ~(e
—
—
—
p) + ~
1 \ R.R~/ J (flI?~\2
Ii
+ 2~5(H— Hk),
A
-2
2
dØ = 2a(2H
3
—
H1
R1
—
H2) (h—
R2 — ~
-20
‘
where the Hubble parameters H=-~-
=~-~H
describe the expansion rate; also, R * is the curvature of 3-space, and p the energy density and pressure of the cosmological fluid, ~v and i~,the volume and shear viscosity coefficients [11], a and ~k are dimensionless constants related to the geometrical structure of space [12]. Eqs. (1) can be integrated numerically to give the entropy amplification parameter ~ defined in ref. [13] and the shear parameters ~
=
)
-ao
-30,.
-so -~
—~
log H11,
2
~ H , 1 Li .s~I1~k_H\ k
Fig. 1. The shear parameters at the end of the lepton era (sub. script 1) versus the initial Hubble parameter H~.The curves are
I,~\2IR —
—
I
WI I
~) ~
1 2
R, \2 I 2’—1 )~ 1’J I,~Ofl ) ~
,
~j
which describe the anisotropy of the Hubble expansion and the rotation of the canonical frame axes, respectively. In agreement with the results found for types VIII and IX curvature turns out to have little direct influence on the entropy production, and entropy enhancements as large as E 1012 are allowed in the lepton era. The results are similar to those of refs. [3,4] and we do not repeat them here. As far as the evolution of shear is concerned, we find that typically most of the shear resides in AH, since a strongly anisotropic curiature tensor gives AR ~ AH. These parameters as found at the end of the lepton era are shown in fig. 1. The -~
72
labelled by the solution class (A or B) *defmed in ref. [13] and . correspondmg values of log Rjn/Hml. assume here aby= the 1O~, but the results depend weakly on thisWeparameter.
2I
I we have a mod-
results show that, unless IR*1H erate anisotropy ~
-~
<
(The subscript f denotes the end of the lepton era.) However, this fact does not imply an acceptable anisotropy at the present epoch, since a curvature—anisotropy coupling makes the expansion anisotropy to increase at large times. In the more general cases, where the curvature tensor is strongly anisotropic, we have (if n is such that R ‘x tn)
Volume 72A, number 2
PHYSICS LETTERS
‘~~)
8/~~i) n \2/R* \2
AH
3a2(_~ \R — 2 R1
-
A
/
both a sensibl~curvature and a large dissipation in the lepton era tui~nout to attain a quasi-isotropic Milne
‘
—2
AR
(3\
H’
25 June 1979
“ /
2 I increases by several orders of magThe ratio IR*/H nitude during the lepton era (see fig. 2); moreover it is expected to increase by 14 orders of magnitude during the subsequent evolution, so that only a very small curvature at the beginning gives an acceptable anisotropy (A~2~ I 0~for open spaces),The expansion anisotropy must become large whenever the models approach the stage R* ——H2. This result is of particular importance because the universe can switch to the Milne epoch only when the condition R * —6H2 is realized. (The analytical results of ref. [8] are not correct, since they do not take into account the anisotropy enhancement which accompanies the passage to the low-density regime.) We emphasize that none of the models which exitibit
epoch. As a n~atterof fact we can have high dissipation E ~ 1 only if initially AH ~ I; this condition makes the curvature tensor to become rapidly anisotropic evenanisotropy if it was strictly the beginning. the of the isotropic curvatureattensor reacts on Then the expansion anisotropy in agreement with eq. (3). To check more strongly the impossibility of reaching a quasi-isotropic Mime epoch, we also posed “mitial” conditions at redshifts z = 10—1O~with very small expansion anisotropies and relatiVely large curiatures (conditions inconsistent with the evolution of our models in the lepton era). We found thatAH increases sharply, rapidly adjusting itself to the regime
4 0
~J
.
‘~
________________________________________________________________________
o
—10
,‘~
“
,
~
I,’
/
, ‘~~‘
—!_~~~
I
‘A 0
I
——
Jog
log Hi,,
Fig. 2. The curvature at the end of the lepton era R~versusthe initial Hubble parameterHin. For the curve labelling see the caption to fig. I.
Fig. 3. The comparative evolution of shear and curvature in the
adiabatic regime (z <2 x 1O~).In this model we assume a = i04, (R*/H2)f~~ _10b0, and (R~,,is 0fR~)f = 3 X 10—6. The behaviour shown in the figure is typical for models with slightly anisotropic curvature.
73
Volume 72A, number 2
PHYSICS LETTERS
(3). Even in the case where the isotropic part of the curvature dominates the anisotropic part, namely IR~0I ~“ IR11~50!,the expansion anisotropy increases rapidly, although eq. (3) does not hold any more (see fig. 3). We finally tested the likelthood of an isotropic Milne epoch at the present epoch in the following 2 and AH ~ manner: 1 at z 10 we artificially posed R* —6H and checked whether the isotropic configuration had a lifetime at least of order 1010 yr. In this case we got A 1 at z 3 with the rapid (i.e. less than io~ yrs) destruction of the Milne epoch and its substitution by a curvature dominated regime. We conclude that either (i) the universe is remarkably flat, IR *1H2 I ~ 1, and has a quasi-critical energy density e 3H2, or (ii) it is allowed to have a low matter density but it is very special in other respects, i.e., it must be quasi-Friedmann since the beginning or belong to a non-generic type (for instance type V). A strong dissipation is possible in the first stages of the expansion, so that the large specific entropy observed today is certainly not surprising, but this is not sufficient to explain the isotropy of the universe at the present epoch: we must anyhow resort to rather special models.
74
25 June 1979
References [11C.W. Misner, Phys. Rev. Lett. 9 (1967) 533; Astrophys. [2] [3] [4]
J. 158 (1968) 431. M.J. Rees, Phys. Lett.Phys. 28 (1972) 1669.(1978) 19. N. Caderni and R.Rev. Fabbri, Lett. 67A N. Caderni and R. Fabbri, Nuovo Cimento, to be published. N. Caderni and R. Fabbri, Phys. Lett. 68A (1978) 144.
[5] [6] J.R. Gott, i.E. Gunn, D.N. Schramm and B.M. Tinsley, Astrophys. J. 194 (1974) 543. [7] Soc. C.B. Collins and S.W. Not. R. Astron. 262 (1973) 307; Hawking, Astrophys.Mon. J. 180 (1973) 317. [8] A.G. Doroshkevich, V.N. Lukash and I.D. Novikov, Soy. Phys. JETP 37 (1973) 739; See also I.D. Novikov, in: Confrontation of cosmological theories with observational data, ed. M.S. Longair (Reidel, Dordrecht, 1974). [9] J.D. Barrow and F.J. Tipler, Nature 276 (1978) 453. [10] G.F. Smoot, M.V. Gorenstein and R.A. Muller, Phys. Rev. Lett. 39 (1977) 898. [11] N. Caderni, R. Fabbri, L.J. van den Horn and Th. J. Siskens, Phys. Lett. 66A (1978) 251. [12] These constants are listed in L. Landau and E. Lifshitz, Theorie des Champs (Mir, Moscow, 1970) p. 477. [13] N. Cadernj and R. Fabbri, Nuovo Cimento 44B (1978)
228.
PHYSICS LETTERS
25 June1979
THE ISOTROPIZATION OF LOW-DENSITY COSMOLOGICAL MODELS N. CADERNI ’ Institute of Astronomy, University of Cambridge, UK
and R. FABBRI Istituto de Fisica Superiore, University of Florence, Italy Received 24 April 1979
We analyse the evolution of Bianchi type-VIIh cosmological models during the lepton era and the subsequent adiabatic epochs. We find that these models cannot generally attain to the lowdensity regime with a reasonable anisotropy level because of a curvature-anisotropy coupling. The primordial damping due to neutrino viscosity, although very effective for the entropy production, cannot help to match to isotropy requirement at the present epoch.
According to the “philosophy” of chaotic cosmology dissipative processes play an important role in the early stages of the cosmological expansion, when primordial irregularities are damped out and the universe becomes smooth and isotropic at large scales [ 1,2]. Since large amounts of radiation entropy are produced during the damping process, one hopes to establish a link between the expansion isotropy and the large specific entropy observed today. We carried out extensive investigations to check whether neutrino viscosity [l] can provide the suitable damping mechanism in the most general of homogeneous anisotropic models, i.e. the Bianchi spaces with anisotropic curvature tensor: we analyzed in previous papers [3,4] the evolution of spaces of types VIII and IX, and found that a large entropy enhancement (up to N 1012 for suitable initial conditions) can occur in such spaces, but the anisotropic part of curvature usuaIly prevents the Hubble expansion from being sufficiently isotropic today. This contrasts with results found for low-density spaces with isotropic curvature tensor [S] . Therefore we are compelled to extend the analysis to type VII, which is as generic as type VIII or IX and, being the largest generalization of the open
Friedmann model, is moreover favoured by the possibility of providing low density models [6]. Collins and Hawking [7] have shown that in models of type VII, the expansion anisotropy does not become arbitrarily small at times I + m; however, Doroshkevich et al. [8] state that such models admit a “Milne stage” where (a) the matter density is much smaller than the critical density, and (b) the expansi6n anisotropy remains constant and small. Therefore, without any concern for the ultimate fate of the universe * l, here we consider the question, whether these models can dissipate large expansion anisotropies in the lepton era and become consistent with observational isotropy constraints [lo]. To answer this question we consider the type VII,, spaces with metric ds2 = - dt2 t R:(cos C#J s1, - sin 9 S22)2 t R&sin $J52, + cos r#~ Q2)2 + R$ !2$ , where the Q are the canonical frame 1-forms and Rk are the cosmic scale factors: the Einstein equations read *r The scatology of an open universe is discussed by Barrow
1 E.S.A. International Fellow.
and Tipler [9 1.
71
Volume 72A, number 2
dHk
PHYSICS LETTERS
25 June 1979
a 2
I
B
rL~L~ nR.R.~2 1 /
x ilk ~ [\ R. + ~(e
—
—
—
p) + ~
1 \ R.R~/ J (flI?~\2
Ii
+ 2~5(H— Hk),
A
-2
2
dØ = 2a(2H
3
—
H1
R1
—
H2) (h—
R2 — ~
-20
‘
where the Hubble parameters H=-~-
=~-~H
describe the expansion rate; also, R * is the curvature of 3-space, and p the energy density and pressure of the cosmological fluid, ~v and i~,the volume and shear viscosity coefficients [11], a and ~k are dimensionless constants related to the geometrical structure of space [12]. Eqs. (1) can be integrated numerically to give the entropy amplification parameter ~ defined in ref. [13] and the shear parameters ~
=
)
-ao
-30,.
-so -~
—~
log H11,
2
~ H , 1 Li .s~I1~k_H\ k
Fig. 1. The shear parameters at the end of the lepton era (sub. script 1) versus the initial Hubble parameter H~.The curves are
I,~\2IR —
—
I
WI I
~) ~
1 2
R, \2 I 2’—1 )~ 1’J I,~Ofl ) ~
,
~j
which describe the anisotropy of the Hubble expansion and the rotation of the canonical frame axes, respectively. In agreement with the results found for types VIII and IX curvature turns out to have little direct influence on the entropy production, and entropy enhancements as large as E 1012 are allowed in the lepton era. The results are similar to those of refs. [3,4] and we do not repeat them here. As far as the evolution of shear is concerned, we find that typically most of the shear resides in AH, since a strongly anisotropic curiature tensor gives AR ~ AH. These parameters as found at the end of the lepton era are shown in fig. 1. The -~
72
labelled by the solution class (A or B) *defmed in ref. [13] and . correspondmg values of log Rjn/Hml. assume here aby= the 1O~, but the results depend weakly on thisWeparameter.
2I
I we have a mod-
results show that, unless IR*1H erate anisotropy ~
-~
<
(The subscript f denotes the end of the lepton era.) However, this fact does not imply an acceptable anisotropy at the present epoch, since a curvature—anisotropy coupling makes the expansion anisotropy to increase at large times. In the more general cases, where the curvature tensor is strongly anisotropic, we have (if n is such that R ‘x tn)
Volume 72A, number 2
PHYSICS LETTERS
‘~~)
8/~~i) n \2/R* \2
AH
3a2(_~ \R — 2 R1
-
A
/
both a sensibl~curvature and a large dissipation in the lepton era tui~nout to attain a quasi-isotropic Milne
‘
—2
AR
(3\
H’
25 June 1979
“ /
2 I increases by several orders of magThe ratio IR*/H nitude during the lepton era (see fig. 2); moreover it is expected to increase by 14 orders of magnitude during the subsequent evolution, so that only a very small curvature at the beginning gives an acceptable anisotropy (A~2~ I 0~for open spaces),The expansion anisotropy must become large whenever the models approach the stage R* ——H2. This result is of particular importance because the universe can switch to the Milne epoch only when the condition R * —6H2 is realized. (The analytical results of ref. [8] are not correct, since they do not take into account the anisotropy enhancement which accompanies the passage to the low-density regime.) We emphasize that none of the models which exitibit
epoch. As a n~atterof fact we can have high dissipation E ~ 1 only if initially AH ~ I; this condition makes the curvature tensor to become rapidly anisotropic evenanisotropy if it was strictly the beginning. the of the isotropic curvatureattensor reacts on Then the expansion anisotropy in agreement with eq. (3). To check more strongly the impossibility of reaching a quasi-isotropic Mime epoch, we also posed “mitial” conditions at redshifts z = 10—1O~with very small expansion anisotropies and relatiVely large curiatures (conditions inconsistent with the evolution of our models in the lepton era). We found thatAH increases sharply, rapidly adjusting itself to the regime
4 0
~J
.
‘~
________________________________________________________________________
o
—10
,‘~
“
,
~
I,’
/
, ‘~~‘
—!_~~~
I
‘A 0
I
——
Jog
log Hi,,
Fig. 2. The curvature at the end of the lepton era R~versusthe initial Hubble parameterHin. For the curve labelling see the caption to fig. I.
Fig. 3. The comparative evolution of shear and curvature in the
adiabatic regime (z <2 x 1O~).In this model we assume a = i04, (R*/H2)f~~ _10b0, and (R~,,is 0fR~)f = 3 X 10—6. The behaviour shown in the figure is typical for models with slightly anisotropic curvature.
73
Volume 72A, number 2
PHYSICS LETTERS
(3). Even in the case where the isotropic part of the curvature dominates the anisotropic part, namely IR~0I ~“ IR11~50!,the expansion anisotropy increases rapidly, although eq. (3) does not hold any more (see fig. 3). We finally tested the likelthood of an isotropic Milne epoch at the present epoch in the following 2 and AH ~ manner: 1 at z 10 we artificially posed R* —6H and checked whether the isotropic configuration had a lifetime at least of order 1010 yr. In this case we got A 1 at z 3 with the rapid (i.e. less than io~ yrs) destruction of the Milne epoch and its substitution by a curvature dominated regime. We conclude that either (i) the universe is remarkably flat, IR *1H2 I ~ 1, and has a quasi-critical energy density e 3H2, or (ii) it is allowed to have a low matter density but it is very special in other respects, i.e., it must be quasi-Friedmann since the beginning or belong to a non-generic type (for instance type V). A strong dissipation is possible in the first stages of the expansion, so that the large specific entropy observed today is certainly not surprising, but this is not sufficient to explain the isotropy of the universe at the present epoch: we must anyhow resort to rather special models.
74
25 June 1979
References [11C.W. Misner, Phys. Rev. Lett. 9 (1967) 533; Astrophys. [2] [3] [4]
J. 158 (1968) 431. M.J. Rees, Phys. Lett.Phys. 28 (1972) 1669.(1978) 19. N. Caderni and R.Rev. Fabbri, Lett. 67A N. Caderni and R. Fabbri, Nuovo Cimento, to be published. N. Caderni and R. Fabbri, Phys. Lett. 68A (1978) 144.
[5] [6] J.R. Gott, i.E. Gunn, D.N. Schramm and B.M. Tinsley, Astrophys. J. 194 (1974) 543. [7] Soc. C.B. Collins and S.W. Not. R. Astron. 262 (1973) 307; Hawking, Astrophys.Mon. J. 180 (1973) 317. [8] A.G. Doroshkevich, V.N. Lukash and I.D. Novikov, Soy. Phys. JETP 37 (1973) 739; See also I.D. Novikov, in: Confrontation of cosmological theories with observational data, ed. M.S. Longair (Reidel, Dordrecht, 1974). [9] J.D. Barrow and F.J. Tipler, Nature 276 (1978) 453. [10] G.F. Smoot, M.V. Gorenstein and R.A. Muller, Phys. Rev. Lett. 39 (1977) 898. [11] N. Caderni, R. Fabbri, L.J. van den Horn and Th. J. Siskens, Phys. Lett. 66A (1978) 251. [12] These constants are listed in L. Landau and E. Lifshitz, Theorie des Champs (Mir, Moscow, 1970) p. 477. [13] N. Cadernj and R. Fabbri, Nuovo Cimento 44B (1978)
228.