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Isotropization and cosmological dimensional reduction in the cosmological models with a viscous fluid
Volume 136, number 1,2
PHYSICS LETTERS A
20 March 1989
ISOTROPIZATION AND COSMOLOGICAL DIMENSIONAL REDUCTION IN THE COSMOLOGICAL MODELS WITH A VISCOUS FLUID Wung-Hong HUANG Department ofPhysics, National Cheng Kung University, Tainan, Taiwan 700, ROC Received 2 August 1988; accepted for publication 13 January 1989 Communicated by J.P. Vigier
After extending some existing solutions ofthe Bianchi type I cosmological models with a viscous fluid to higher-dimensional theory, we use a simple algorithm to determine all Hubble functions therein. Emphasis is placed on the fact that although the shear scale function shall always decrease with cosmic time t and approach zero faster than 1-2, both the non-positive Hubble function and non-negative Hubble function can be contained in a solution of some anisotropic cosmological models with viscosity. Thus the cosmological dimensional reduction may work in higher-dimensional cosmological models with a viscous fluid.
1. Introduction To investigate the character of cosmological evolution more realistically one may take into account the dissipative processes which are caused by viscosity. Many properties arising from the introduced viscosity, such as the isotropization [1], avoidance of the initial singularity [2] and creation of matter during the evolution [3] have been found. Exact solutions of some anisotropic models with viscosity have also been obtained recently [4,5]. InthecontextsoftheKaluza—Kleintheory [6] and superstring theory [7] our universe shall start in a higher-dimensional phase with some extra spaces eventually collapsing while three others continue to expand. In higher-dimensional theories the viscosity should also arise, and one may then doubt whether the mechanism of cosmological dimensional reduction can be shown, seeing that the shearviscositywas introduced to reduce the anisotropies of the universe [1].
dimensional reduction may be found in higher-dimensional cosmological models with a viscous fluid. We also have found a simple method to determine all the Hubble functions therein. 2. Einstein’s field equation and general property The line element for a higher-dimensional Bianchi type I spacetime is 1 th2— dt2+ a2~t)clx2 ‘~‘
where 1=1, 2,..., D: The energy—momentum tensor for a viscous fluid is ~ = (a +j~)u~ u,, +j~g~~ ,pc,~, (2) h —
p~p+[(2/D)~—C]u~, A
(3) A
lCj
~
4p
The intention of this paper is to point out that although the shear scale function shall always decrease with cosmic time t and approach zero faster than 1_2, one can find solutions which contain both a non-positive Hubble function and a non-negative Hubble function in some anisotropic cosmological models with viscosity. Thus the mechanism of cosmological
I’
—
+~
+ U~U
UVA + U~U Up,~,
(4)
where a and p are the energy density and pressure, ,~the shear viscosity and C the bulk viscosity, respectively. The Einstein equations R ~ R T 5 —
=
forthe metric (1) and energy—momentum tensor (2) reduce to
0375-960 1/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
21
Volume 136, number 1,2
PHYSICS LETTERS A
D—l
2a2—a 2D W —
‘
‘
/
Using eq. (10) we can reduce eq. (13) to da2/dt 2
~
dt
2 [ /2
dt
\,
DJ
J
20 March 1989
=—2W—4~.
(15)
Since ‘i~ 0, the above equation yields the following
\
=p+(—ii—CJW—2iiH,,
(7)
inequality, dlno” ~ —2W.
(16)
where the Hubble functions H 1, the expansion scalar W and the shear scalar a are defined by
dt Using the inequality (14) we then find
~
a
2~t2. W_=>H1,
2a2m
~
H~
W2.
—
(8)
(17)
From eq. (15) we see that as long as the shear viscosity ~ or/and the expansion scalar Wdoes not approach zero faster than t 1 the shear scalar a2 shall be reduced to zero in the form of an exponential fac—
The trace part of the Einstein equation leads to
(
1 ~ W2 + 2a2 2 d W + / 1 +—) dt \ D 1 2 = [a D(p CW)]
tor, and the isotropization produced the introduced shear viscosity. Theseis results are by special cases
(9) 2 Using in eq. the (9) relation we obtain (6) to eliminate the term with a —
dW
—
.
D
(10)
of a more general theorem of Tipler [8]. However, not all models should be so. We will give in the next section some exact form of the Hubble functions to indicate arising that thefrom isotropization is viscosity. not a common property an introduced Thus, the mechanism of cosmological dimensional reduction can work in a cosmological model with a viscous fluid.
We can further obtain from the Bianchi identity + (a+p—~W)W—4,ia2=0.
(11)
~ Isotropization and dimensional reduction
(12)
We will study the five-dimensional theory with two Hubble functions h and H which correspond to those ofthree space and the extra space, respectively. Only the model with stiff matter is considered in this Let-
Eq. (6) yields d( a2! W2) dt
=
— d (ci W2)
dt
Aftersubstituting the expressions for dc/dt and d wi dt from eqs. (10) and (11) into the right-hand side of eq. (12) we finally obtain d(a2iW2)__~t~_(~e+~+ 2 [D 1 \~W 4,11 J -I (13) dt W Some general properties can now be obtained. For a fluid which satisfies the condition a—p~0eq. (10) yields the following inequality,
w>~t ‘
(14)
as ~ 0 (we consider the model with positive W). 22
ter, as we have as yet the exact solution for this case only. We note that the following solutions can be obtamed by a straightforward extension of those in refs. [4] and [5]. Models with vanishing bulk viscosity areCase already known [9]. Using eqs. (10) and (13) 1. ~=0, ?~=const. we obtam W
=
—~
a
c
2 =
e
—~‘1’
Fromeq. (18) andwiththedefinitionofWanda we then obtain
Volume 136, number 1,2
h
PHYSICS LETTERS A
1 + Ce2t2~1 H= 4t 4t
—
‘
~ 2t
3Ce2~t 2t
H=~+ 3Ce —217t 2t
(19)
2
a
=
31 ~ (1 + Ce — 3~~/2t)~1,
(20)
(21)
where C is a positive constant. From eq. (21) and with the definition of Wand am eq. (8) we then obtain
h=
4t
{ 1+
[1 + C exp ( — 3a/2t) ~
4t
4t
(24) (25)
~-~+Wh=~CW, dt
dt
+ WH= ~CW,
(26)
where the Hubble functions hand Hare those ofthree space and the extra space, respectively. Using the above equations we can find the following solutions through a straightforward extension of those in our previous paper [5]. Case 3. C= const. From eqs. (25) and (26) we
obtain dh ~
~C—h =
(-*00
{l + 3 [1 + Cexp(
—
3a/2t)
(27)
~ ~—1/2}.
(and no shear vis-
(22)
h= -~-{1—[l+Cexp(—3a/21)]”2}, 4t H=
C
The above equation can be integrated to give an exact relation between h and H. Substituting this relation into eq. (25) and integrating it we then obtain the exact solution
/2}
H= ~~_{1_3[l+Cexp(_3a/2t)]_hI2},
stiff matter with a bulk viscosity cosity) can lead to
3h(H+h)=,
where C is a positive constant. Both solutions show the property of isotropization and no dimensionalreduction solution exists now, as the shear scalar is damped out by an exponential factor. The first solution provides a positive value of h for all time. Case 2. C=0, ~=aa, where a is a constant. Using eqs. (10), (6) and (13) we obtain W= t —‘,
20 March 1989
(23)
(28)
where Cis the integration constant. This solution tells
us that the universe is isotropized by a constantbulk viscosity.
Case 4. C= Coa, where
These solutions show that the property of isOtropi-
C 0
zation is broken now and that the mechanism of cosmological dimensional reduction in which h is positive while H is negative for all time is found in the solutions of eq. (22) if C< 8. These results can be shown to be independent of the dimension ofspacetime. One can also easily check
dH dh
that the cosmological dimensional reduction still works in models with another functional form of the
as
is a constant. From eqs.
(24)— (26) we obtain — —
h(H+ h) —H h (H+ h) —
where h and H have been resealed by a factor C0. With the definition L= h+ H this equation can be written
shear viscosity, after extending the solutions in ref. [41to higher-dimensional theory.
dL dh
We next consider the models with a bulk viscosity. However, in general, we now cannot solve eq. (10) exactly as thereenters the bulk viscosity C which may be a unknown function. We therefore adopt the method introduced in our previouspaper [5]. In five
This equation can be integrated to give the general solution H—h ln (1 + H/h) = C, (30)
dimensions, the Einstein field equations containing
where C is an integration constant. We can further-
=
L (2h —1) h(L— 1)
(29)
—
23
Volume 136, number 1,2
PHYSICS LETTERS A
more express h and H in terms of the variables rand
0,
20 March 1989
Eq. (33) can lead to dln(H1—H,)
h=rcosO, H=rsinO, (31) then eq. (30) reduces to the elegant expression r=
C— ln (1 + cot 9) sin O—cos e
(32)
Plotting the above solution in the phase plane h X H we find the dynamical evolutions of the cosmological model. Depending on the integration constant C in eq. (32) we can find two stable solutions, just like those in ref. [5]. These two solutions are either driven to the infinite expansion state or to the Friedmann universe. However, we point out that when the value C is negative, we have solution with a non-negative value of h while the value of H is non-positive for all time (the final state is with h, H—~0).Thus the mechanism of cosmological dimensional reduction may work in a model with a bulk viscosity. It is noted that this fact also occurs in four-dimensional theory. Thus, contrary to the conclusion of a previous paper [5] in which we had neglected the solutions with a negative value of C, the isotropization should not be a necessary character induced by the bulk viscosity.
dln(H,—Hk) (34) dt dt This equation tells us that one can express all other — —
D —2 Hubble functions in terms of only two Hubble functions. One can see that this is a very general property and may be used to analyze many other anisotropic cosmological models. As an example, we will not describe the process ofhow to determine the three Hubble functions in the four-dimensional model with bulk viscosity which is proportional to the energy density, i.e. C= Coc. Eq. (34) yields the relation 113= (1 — C)H1 + CH2,
(35) where C is the integration constant. One can see that the H1, H2 and H3 which satisfy eq. (35) will just describe the plane intersecting through the line with H1 = H2 = 113. A cosmological solution with a fixed number C is to evolve on a fixed plane described by eq. (35). With eq. (35) we can obtain from eq. (6) a = H1 H2 + H2H3 + H3 H1 2 =(l—C)H~+2H1H2+CH2
=[(l—C)H~+aH2][H1+flH2],
(36)
4. Determination of multiple Hubble functions where a and fi can be determined as functions of C.
The methods described in the above section can only be used to find two Hubble functions at most. We will now give a simple algorithm which can be used to determine all Hubble functions. The point is that eq. (7) can lead to
d(H1H1) + (W+2~)(H1—H~)=0.
(33)
Using this relation and Einstein’s field equation (like that of eqs. (25) and (26)) we finally obtain [(1 + fl)H— 2] h
dh ~
=
H[ (1— C-F a )h —2]
(37)
where all Hubble functions have been rescaled by a factor C0, and
dt In some models with shear viscosity, as the cases 1 and 2 in section 3 and more cases in ref. [4], one may 2beand able~ to theafter explicit functional of andfind then integrating eq. forms (33) we W, a can find D —1 relations between H1 and H~(note that we have D— 1 integration constants there). Using these relations and the functional form of W we can finally determine all D Hubble functions. A constraint on these D2 —1 integration constants is given found. by the value of a 24
h=H1+flH2, H=(l—C)H1+aH2.
(38)
Eq. (37) can be easily integrated to give an expression of H relating h to H. Through eq. (38) the relation 1 to H2 is then found. Like that described in case 4 (more details can be found in ref. [5]), we can finally obtain the exact flows in the phase plane of H~x H2 and thus find the dynamical evolutions of the cosmology. We have thus found the exact solution with Hubble functions for the model discussed in three ref. [5].
Volume 136, number 1,2
PHYSICS LETTERS A
5. Discussion
The “cosmological dimensional-reduction process” was proposed by Chodos and Detweiler in 1980 [10], and there is now an extensive literature deal-
ing with more realistical higher-dimensional models containing a variety of matter fields [6,11]. However, up to our knowledge, there have been few papers concerning the viscous fluid in higher-dimensional theory [12]. This many due to the fact that the shear viscosity has been successfully introduced to smooth out the anisotropy in the universe [1], and one thus suspects that the isotropization may be a ccimmon property arising from an introduced viscosity. In this paper, we have obtained some exact
solutions of higher-dimensional models with shear viscosity or bulk viscosity and found that the isotropization is not a necessary property of the introduced viscosity.
More efforts are needed to clarify the behavior of cosmological dimensional reduction in models consisting of a variety ofmatter fields with a general form of the viscosity. The model with a compact extra
space is more natural and is worth investigating. Whether the cosmology with a viscous fluid will eventually compactify the extra space, will depend on the details of the initial condition (i.e. the value of the integration constant in cases 2 and 4) which can only be determined by the theory of quantum cosmology [13,14]. It is an interesting problem to study.
20 March 1989
References [1] C.M. Misner, Nature 214 (1967) 40; Astrophys. J. 151 (1967) 431. [2] V.A. G.L. Belinskii Murphy, Phys. Rev.Khalatnikov, D 8 (1973) 4231. [3] and I.M. Soy. Phys. JETP 42 (1976) 205. [4] A. Baneijee, S.B. Duttachoudhury and A.K. Sanyal, I. Math. Phys. 26 (1985) 3010. [5]W.H. Huang, Phys. Lett. A 129 (1988) 429. [6] M.J. Duff, (1986) 1; B.E.W. Nilsson and C.N. Pope,Phys. Rep. 130 D. Bailin and A. Love, Rep. Prog. Phys. 50 (1987)1097, and references therein. [7] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge Univ. Press, Cambridge, 1986), and references therein. [8] F. Tipler, Gen. Rd. Gray. 10 (1979) 1005; J.D. Barrow and E.J. Tipler, The anthropic cosmological principle (Oxford Univ. Press, Oxford, 1986).
[9] C.M. Misner, Astrophys. J. 151(1967) 431; R.A. Astrophys. 4 (1969) 459; Ya.B.Matzner, Zel’dovich and Space I.E.D.Sci.Novikov, Relativistic astrophysics, VoL 2 (Chicago Univ. Press, Chicago, 1983) p. 520. [10] A. Chodos and S. Detweiler, Phys. Rev. D 21(1980) 2167. [11] D. Sahdev, Phys. Lett. B 137 (1984)155; Phys. Rev. D 30 (1984) 2495; Nucl. Phys. B 209 (1982)146. P.G.O. Freund, [12] I. Waga, R.C. Falcao and R. Chanda, Phys. Rev. D 33 (1986) 1839. [13] S.W. Hawking, in: Pontificiae Accademine Scientarium Scripta Varia, VoL 48. Astrophysical cosmology (1982) 563. [14] J.B. p. Hartle and S.W. Hawking, Phys. Rev. D 28 (1983) 2960; S.W. Hawking, Nuci. Phys. B 239 (1984) 257.
25
PHYSICS LETTERS A
20 March 1989
ISOTROPIZATION AND COSMOLOGICAL DIMENSIONAL REDUCTION IN THE COSMOLOGICAL MODELS WITH A VISCOUS FLUID Wung-Hong HUANG Department ofPhysics, National Cheng Kung University, Tainan, Taiwan 700, ROC Received 2 August 1988; accepted for publication 13 January 1989 Communicated by J.P. Vigier
After extending some existing solutions ofthe Bianchi type I cosmological models with a viscous fluid to higher-dimensional theory, we use a simple algorithm to determine all Hubble functions therein. Emphasis is placed on the fact that although the shear scale function shall always decrease with cosmic time t and approach zero faster than 1-2, both the non-positive Hubble function and non-negative Hubble function can be contained in a solution of some anisotropic cosmological models with viscosity. Thus the cosmological dimensional reduction may work in higher-dimensional cosmological models with a viscous fluid.
1. Introduction To investigate the character of cosmological evolution more realistically one may take into account the dissipative processes which are caused by viscosity. Many properties arising from the introduced viscosity, such as the isotropization [1], avoidance of the initial singularity [2] and creation of matter during the evolution [3] have been found. Exact solutions of some anisotropic models with viscosity have also been obtained recently [4,5]. InthecontextsoftheKaluza—Kleintheory [6] and superstring theory [7] our universe shall start in a higher-dimensional phase with some extra spaces eventually collapsing while three others continue to expand. In higher-dimensional theories the viscosity should also arise, and one may then doubt whether the mechanism of cosmological dimensional reduction can be shown, seeing that the shearviscositywas introduced to reduce the anisotropies of the universe [1].
dimensional reduction may be found in higher-dimensional cosmological models with a viscous fluid. We also have found a simple method to determine all the Hubble functions therein. 2. Einstein’s field equation and general property The line element for a higher-dimensional Bianchi type I spacetime is 1 th2— dt2+ a2~t)clx2 ‘~‘
where 1=1, 2,..., D: The energy—momentum tensor for a viscous fluid is ~ = (a +j~)u~ u,, +j~g~~ ,pc,~, (2) h —
p~p+[(2/D)~—C]u~, A
(3) A
lCj
~
4p
The intention of this paper is to point out that although the shear scale function shall always decrease with cosmic time t and approach zero faster than 1_2, one can find solutions which contain both a non-positive Hubble function and a non-negative Hubble function in some anisotropic cosmological models with viscosity. Thus the mechanism of cosmological
I’
—
+~
+ U~U
UVA + U~U Up,~,
(4)
where a and p are the energy density and pressure, ,~the shear viscosity and C the bulk viscosity, respectively. The Einstein equations R ~ R T 5 —
=
forthe metric (1) and energy—momentum tensor (2) reduce to
0375-960 1/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
21
Volume 136, number 1,2
PHYSICS LETTERS A
D—l
2a2—a 2D W —
‘
‘
/
Using eq. (10) we can reduce eq. (13) to da2/dt 2
~
dt
2 [ /2
dt
\,
DJ
J
20 March 1989
=—2W—4~.
(15)
Since ‘i~ 0, the above equation yields the following
\
=p+(—ii—CJW—2iiH,,
(7)
inequality, dlno” ~ —2W.
(16)
where the Hubble functions H 1, the expansion scalar W and the shear scalar a are defined by
dt Using the inequality (14) we then find
~
a
2~t2. W_=>H1,
2a2m
~
H~
W2.
—
(8)
(17)
From eq. (15) we see that as long as the shear viscosity ~ or/and the expansion scalar Wdoes not approach zero faster than t 1 the shear scalar a2 shall be reduced to zero in the form of an exponential fac—
The trace part of the Einstein equation leads to
(
1 ~ W2 + 2a2 2 d W + / 1 +—) dt \ D 1 2 = [a D(p CW)]
tor, and the isotropization produced the introduced shear viscosity. Theseis results are by special cases
(9) 2 Using in eq. the (9) relation we obtain (6) to eliminate the term with a —
dW
—
.
D
(10)
of a more general theorem of Tipler [8]. However, not all models should be so. We will give in the next section some exact form of the Hubble functions to indicate arising that thefrom isotropization is viscosity. not a common property an introduced Thus, the mechanism of cosmological dimensional reduction can work in a cosmological model with a viscous fluid.
We can further obtain from the Bianchi identity + (a+p—~W)W—4,ia2=0.
(11)
~ Isotropization and dimensional reduction
(12)
We will study the five-dimensional theory with two Hubble functions h and H which correspond to those ofthree space and the extra space, respectively. Only the model with stiff matter is considered in this Let-
Eq. (6) yields d( a2! W2) dt
=
— d (ci W2)
dt
Aftersubstituting the expressions for dc/dt and d wi dt from eqs. (10) and (11) into the right-hand side of eq. (12) we finally obtain d(a2iW2)__~t~_(~e+~+ 2 [D 1 \~W 4,11 J -I (13) dt W Some general properties can now be obtained. For a fluid which satisfies the condition a—p~0eq. (10) yields the following inequality,
w>~t ‘
(14)
as ~ 0 (we consider the model with positive W). 22
ter, as we have as yet the exact solution for this case only. We note that the following solutions can be obtamed by a straightforward extension of those in refs. [4] and [5]. Models with vanishing bulk viscosity areCase already known [9]. Using eqs. (10) and (13) 1. ~=0, ?~=const. we obtam W
=
—~
a
c
2 =
e
—~‘1’
Fromeq. (18) andwiththedefinitionofWanda we then obtain
Volume 136, number 1,2
h
PHYSICS LETTERS A
1 + Ce2t2~1 H= 4t 4t
—
‘
~ 2t
3Ce2~t 2t
H=~+ 3Ce —217t 2t
(19)
2
a
=
31 ~ (1 + Ce — 3~~/2t)~1,
(20)
(21)
where C is a positive constant. From eq. (21) and with the definition of Wand am eq. (8) we then obtain
h=
4t
{ 1+
[1 + C exp ( — 3a/2t) ~
4t
4t
(24) (25)
~-~+Wh=~CW, dt
dt
+ WH= ~CW,
(26)
where the Hubble functions hand Hare those ofthree space and the extra space, respectively. Using the above equations we can find the following solutions through a straightforward extension of those in our previous paper [5]. Case 3. C= const. From eqs. (25) and (26) we
obtain dh ~
~C—h =
(-*00
{l + 3 [1 + Cexp(
—
3a/2t)
(27)
~ ~—1/2}.
(and no shear vis-
(22)
h= -~-{1—[l+Cexp(—3a/21)]”2}, 4t H=
C
The above equation can be integrated to give an exact relation between h and H. Substituting this relation into eq. (25) and integrating it we then obtain the exact solution
/2}
H= ~~_{1_3[l+Cexp(_3a/2t)]_hI2},
stiff matter with a bulk viscosity cosity) can lead to
3h(H+h)=,
where C is a positive constant. Both solutions show the property of isotropization and no dimensionalreduction solution exists now, as the shear scalar is damped out by an exponential factor. The first solution provides a positive value of h for all time. Case 2. C=0, ~=aa, where a is a constant. Using eqs. (10), (6) and (13) we obtain W= t —‘,
20 March 1989
(23)
(28)
where Cis the integration constant. This solution tells
us that the universe is isotropized by a constantbulk viscosity.
Case 4. C= Coa, where
These solutions show that the property of isOtropi-
C 0
zation is broken now and that the mechanism of cosmological dimensional reduction in which h is positive while H is negative for all time is found in the solutions of eq. (22) if C< 8. These results can be shown to be independent of the dimension ofspacetime. One can also easily check
dH dh
that the cosmological dimensional reduction still works in models with another functional form of the
as
is a constant. From eqs.
(24)— (26) we obtain — —
h(H+ h) —H h (H+ h) —
where h and H have been resealed by a factor C0. With the definition L= h+ H this equation can be written
shear viscosity, after extending the solutions in ref. [41to higher-dimensional theory.
dL dh
We next consider the models with a bulk viscosity. However, in general, we now cannot solve eq. (10) exactly as thereenters the bulk viscosity C which may be a unknown function. We therefore adopt the method introduced in our previouspaper [5]. In five
This equation can be integrated to give the general solution H—h ln (1 + H/h) = C, (30)
dimensions, the Einstein field equations containing
where C is an integration constant. We can further-
=
L (2h —1) h(L— 1)
(29)
—
23
Volume 136, number 1,2
PHYSICS LETTERS A
more express h and H in terms of the variables rand
0,
20 March 1989
Eq. (33) can lead to dln(H1—H,)
h=rcosO, H=rsinO, (31) then eq. (30) reduces to the elegant expression r=
C— ln (1 + cot 9) sin O—cos e
(32)
Plotting the above solution in the phase plane h X H we find the dynamical evolutions of the cosmological model. Depending on the integration constant C in eq. (32) we can find two stable solutions, just like those in ref. [5]. These two solutions are either driven to the infinite expansion state or to the Friedmann universe. However, we point out that when the value C is negative, we have solution with a non-negative value of h while the value of H is non-positive for all time (the final state is with h, H—~0).Thus the mechanism of cosmological dimensional reduction may work in a model with a bulk viscosity. It is noted that this fact also occurs in four-dimensional theory. Thus, contrary to the conclusion of a previous paper [5] in which we had neglected the solutions with a negative value of C, the isotropization should not be a necessary character induced by the bulk viscosity.
dln(H,—Hk) (34) dt dt This equation tells us that one can express all other — —
D —2 Hubble functions in terms of only two Hubble functions. One can see that this is a very general property and may be used to analyze many other anisotropic cosmological models. As an example, we will not describe the process ofhow to determine the three Hubble functions in the four-dimensional model with bulk viscosity which is proportional to the energy density, i.e. C= Coc. Eq. (34) yields the relation 113= (1 — C)H1 + CH2,
(35) where C is the integration constant. One can see that the H1, H2 and H3 which satisfy eq. (35) will just describe the plane intersecting through the line with H1 = H2 = 113. A cosmological solution with a fixed number C is to evolve on a fixed plane described by eq. (35). With eq. (35) we can obtain from eq. (6) a = H1 H2 + H2H3 + H3 H1 2 =(l—C)H~+2H1H2+CH2
=[(l—C)H~+aH2][H1+flH2],
(36)
4. Determination of multiple Hubble functions where a and fi can be determined as functions of C.
The methods described in the above section can only be used to find two Hubble functions at most. We will now give a simple algorithm which can be used to determine all Hubble functions. The point is that eq. (7) can lead to
d(H1H1) + (W+2~)(H1—H~)=0.
(33)
Using this relation and Einstein’s field equation (like that of eqs. (25) and (26)) we finally obtain [(1 + fl)H— 2] h
dh ~
=
H[ (1— C-F a )h —2]
(37)
where all Hubble functions have been rescaled by a factor C0, and
dt In some models with shear viscosity, as the cases 1 and 2 in section 3 and more cases in ref. [4], one may 2beand able~ to theafter explicit functional of andfind then integrating eq. forms (33) we W, a can find D —1 relations between H1 and H~(note that we have D— 1 integration constants there). Using these relations and the functional form of W we can finally determine all D Hubble functions. A constraint on these D2 —1 integration constants is given found. by the value of a 24
h=H1+flH2, H=(l—C)H1+aH2.
(38)
Eq. (37) can be easily integrated to give an expression of H relating h to H. Through eq. (38) the relation 1 to H2 is then found. Like that described in case 4 (more details can be found in ref. [5]), we can finally obtain the exact flows in the phase plane of H~x H2 and thus find the dynamical evolutions of the cosmology. We have thus found the exact solution with Hubble functions for the model discussed in three ref. [5].
Volume 136, number 1,2
PHYSICS LETTERS A
5. Discussion
The “cosmological dimensional-reduction process” was proposed by Chodos and Detweiler in 1980 [10], and there is now an extensive literature deal-
ing with more realistical higher-dimensional models containing a variety of matter fields [6,11]. However, up to our knowledge, there have been few papers concerning the viscous fluid in higher-dimensional theory [12]. This many due to the fact that the shear viscosity has been successfully introduced to smooth out the anisotropy in the universe [1], and one thus suspects that the isotropization may be a ccimmon property arising from an introduced viscosity. In this paper, we have obtained some exact
solutions of higher-dimensional models with shear viscosity or bulk viscosity and found that the isotropization is not a necessary property of the introduced viscosity.
More efforts are needed to clarify the behavior of cosmological dimensional reduction in models consisting of a variety ofmatter fields with a general form of the viscosity. The model with a compact extra
space is more natural and is worth investigating. Whether the cosmology with a viscous fluid will eventually compactify the extra space, will depend on the details of the initial condition (i.e. the value of the integration constant in cases 2 and 4) which can only be determined by the theory of quantum cosmology [13,14]. It is an interesting problem to study.
20 March 1989
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[9] C.M. Misner, Astrophys. J. 151(1967) 431; R.A. Astrophys. 4 (1969) 459; Ya.B.Matzner, Zel’dovich and Space I.E.D.Sci.Novikov, Relativistic astrophysics, VoL 2 (Chicago Univ. Press, Chicago, 1983) p. 520. [10] A. Chodos and S. Detweiler, Phys. Rev. D 21(1980) 2167. [11] D. Sahdev, Phys. Lett. B 137 (1984)155; Phys. Rev. D 30 (1984) 2495; Nucl. Phys. B 209 (1982)146. P.G.O. Freund, [12] I. Waga, R.C. Falcao and R. Chanda, Phys. Rev. D 33 (1986) 1839. [13] S.W. Hawking, in: Pontificiae Accademine Scientarium Scripta Varia, VoL 48. Astrophysical cosmology (1982) 563. [14] J.B. p. Hartle and S.W. Hawking, Phys. Rev. D 28 (1983) 2960; S.W. Hawking, Nuci. Phys. B 239 (1984) 257.
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