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Inflation in universes with a massive scalar field
Volume 163B, number 5,6
PHYSICS LETTERS
28 November 1985
INFLATION IN UNIVERSES WITH A MASSIVE SCALAR FIELD Tsvl P I R A N 1 and R u t h M. W I L L I A M S 2
Institutefor TheoretwalPhysws, Umversltyof Cahforma, Santa Barbara, CA 93106, USA Recewed 2 August 1985
We &scuss the evolutmn of toy umverses coupled to a masswe scalar field Under most mmal condmons such a umverse enters an mflatmnary phase from wluch ~t emerges naturally When the umverse ~s amsotrop~c or mhomogeneous, the amsotropy or mhomogenelty freezes dunng the mflaUonary phase
In vaew of the recent great interest in inflationary cosmological models it Is instructwe to study the evolution of a umverse containing a massive scalar field. We consider first the evolution of a homogeneous unwerse. Later, using a recently developed 3 + 1 Regge calculus formalism [ 1 ] we discuss the evolutaon o f a homogeneous but anasotropac model and o f an mhomogeneous and anasotropic model. We find that under most untial condations a universe coupled to a masswe scalar field enters an inflationary phase from wluch it emerges naturally [ 1]* 1. When the umverse Is anisotroplc or inhomogeneous, the anasotropy or irthomogeneaty freezes durmg the mflataonary phase. The evolution equations for a Fnedmann universe coupled to a masswe homogeneous scalar field are H2 = (8~r/3mp2) (½ m 2 0 2 + ~a d2) + k/R 2 ,
(1)
and + 3H4+m20=
O,
(2)
where 0 as the scalar field, mats mass, mp Planck's mass, H = t~/R and • corresponds to a tune denvative i Permanent address The Racah Institute for Physics, The Hebrew Umverslty, Jerusalem, Israel and The Institute for Advanced Study, Pnnceton, N J, USA 2 Permanent address Gtrton College and Department of Apphed Mathematics and Theorettcal Physics, Cambridge, England *l This result was found independently by Belmskl, Gnshchuk, Khalatnikov and Zel'doweh [ 2 ] 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The trajectories of the solutions [neglecting the curvature term m eq. (1)] on the dunenslonless ¢, plane are shown m fig. la, for a variety of inlml condxtions (the curvature term decreases rapadly when the umverse expands and, as we show later, its mclusaon does not change the reported results). Using the slow rolling eonsaderations of the inflationary models (see e.g. ref. [3]) one can observe that the umverse approaches a De Sitter phase when the rates of change o f 0 and d are small compared with H
~/H~ "~ d/HO ~ m2 /3H 2 '~ 1 .
(3)
Thas occurs along the two horizontal lines at ~ ~ +~ X X / ~ m m p m fig. la. Under these condations the pressure o f the scalar field, p = ~(d2 _ m202) Is negative and it supplies the energy for the exponenml expansion. During the mtlataonary phase p ~, -E#(Ee~ = ~m202 + ½q~2) as can be seen m fig. lb. The Hubble constant dunng the De Sitter phase is
iI 2 ~ 41rm202/3m2p .
(4)
Eq (3) sets a lunit on the mltlal amphtude of the scalar field.
02 ~" m2/4rr .
(5)
A stronger lmaat on 0 can be obtained from the fact that the universe has to expand at least 64 e-fold during the inflationary phase [4]. This leads to
02 ~" 8m2/Ir .
(6)
The conditaon that perturbataons should not grow too 331
Volume 163B, number 5,6
PHYSICS LETTERS
28 November 1985 P
2O i
'
'
'
1
10 c-a
i
-10
-20
I -4
-2
0
4
2
J
0
10
20
E
Ftg. 1 (a) TraJectories of the homogeneous universe on the normalized (0/V~'~')mp, ~ / ~ m p m ) plane Most of the trajectories lead to the mflatmnaty phase m which ~ ~ 1/3 The internal ettcle corresponds to osctllatlons of the scalar field and "dusthke" behavaour at late tune. There is an unphclt phase transition on the lines 4~= ±~, since the pressure Is negative when 02 < ~2. (b) TraJectories of the homogeneous umverse on the (E0, p) plane. The line p ~ - E 0 corresponds to the inflationary phase. much during the inflationary phase [5] leads to a hmlt on m, the mass of the scalar field. m < 10 -5 m p .
(7)
During the mltlal phase the scalar field, and therefore H decrease gradually. Eventually ¢~is small enough and the umverse goes out of the De Sitter phase. This happens without an apparent phase transmon or a change m the scalar potentml o(~) = ~ rn 252 However as the amphtude of the scalar field decreases the pressure of the scalar field becomes positive again. After the De Sitter phase the universe enters a massive dust-like phase with H ,~ m. The scalar field oscillates and using the WKB approxlmatmn we obtain ~b~ ~0 R - 3 / 2 e lint .
(8)
With this approxmaation H satisfies H 2 ~ 41rm2¢2o/3R 3 .
(9)
This behavmur is described m fig. la by the slowly shrinking circular curves (E~ ~ R - 3 ) near the ongm 332
which correspond to the post.inflationary phase. The most amazing feature of the scalar field cosmology (see fig. la) is that if the initial energy density, E~, of the scalar field is large enough the configuration evolves into a De Sitter phase, even when the initial ~ is large. The lmtlal trajectory is ~ ~ c o n s t a n t until the mflanonary phase begins. Therefore to have a long enough inflationary phase the inflationary condmon [eq. (3)] must be satisfied initially Only m a very lmaited range of initial (¢, ~) phase space, the size of which vanishes rapidly as the energy densxty of the scalar field increases, the configuration evolves directly towards the masswe dust-like configuration. As the universe evolves towards the low ~ region, an imphclt phase transition occurs with the pressure becoming negatwe and a De Sitter inflationary phase develops. We conclude thus that the inflationary phase is generic m a masswe scalar field universe. However the actual length of the inflation will depend on the mitml amplitude and phase of the field, with expansion by 64 e-fold reqmrmg that eq. (6) be satisfied.
Volume 163B, number 5,6
PHYSICS LETrERS
To consider the effect of deviations from isotropy and homogeneity we have constructed a toy model of a universe built out of five tetrahedra forming the surface of a four stmplex. The evolution of tlus umverse is described m term of the edge lengths of these tetrahedra using a 3 + 1 Regge calculus formalism that we have developed recently [ 1]. Altogether there are 10 edges. To construct a homogeneous but amsotrop~c model we choose two sets of five equal edges Ia and l b (see fig. 2a). Our dynamical variables are then the geometric mean of the edge lengths, R = (la lb)l/2, which is the scale length of the umverse, and the ratio 00 = (la/lb), wluch measures the amsotropy. The scalar field, $, is def'med on the vertices with all its values being equal. The equations for this toy unwerse are: H 2 + ] H ~ e t ' / a - 662/~/12002~, + (4/R 20)(8 l'Vt'~ + 8 2/Vr~)
= (41r/3m 2) (~2 + m 2~2),
82
=
2/r
28 November 1985
--
[2 - 260 2 + 004\ arccosl~ - - -
/
~, ~ (4 - 002) !
-2arccos~ 2 (3602-2) /2) (12b) ~o(400 - 1) 1/2 (4 - 602) 1 ' and ' denotes a 00 denvatwe. An immediate feature of these equations is that asymptotically, i.e. for large R , ~3 = o3 = 0 is an approximate solution of eq. (10c). Asymptotically the amsotropy will freeze at a given value (depending on the initial conditions). This does not change even after the universe goes out of the mflaUonary phase provided that R is large enough. When cb vanishes eqs. (10a), (10b) reduce to eqs. (1)(2) (up to the curvature term which is now small). H and the scalar field follow the same pattern described m fig. la. A typical evolution of this anisotropic universe is shown in fig. 3, in which the De Sxtter phase, with an almost constant H, and
(10a)
a
the scalar field evolution equataon: \
~+ (3/-/+ o3ot'/ot)~ + m2~b = 0 ,
(10b)
;.-
and an evolution equation for the anisotropy factor:
/\
..
/<
-.;\ i
- \
\
/ \',.
\
~(00o{/3ot + [3/4 w a ' ) + CoH( o.,,ot'/ ot + 3/3/400ot') ~ _ + Co2(a"00130t + 318a00 + 3'18a'00 2 -- 3/4a'00 2)
\
',
)
/- - - f / ~
= ( 3 [ R 2o~') (8 lX/r~ - 8 2/~¢/-~)
- (200/R 2a) (6 lX/~ + 6 2/x/-~) •
(1 0c)
b
Eqs. (10a), (10b) reduce to eqs. (1), (2) (up to a different numerical factor in the curvature term) and eq. (1 0c) becomes an identity when the anisotropy vanishes. The functions ~t, ~ and the deficit angles [6] 8a and 8 b are functions of 00 only: ot = (-00 3 + 200 + 200-1 - oo-3)1/2 ,
(1 la)
/3 = (3603 + 260 + 200-1 + 300-3)/~,,
(1 lb)
81
= 2 n - arccos(2004--2e 2- + 1 \ 4602 - 1 I - 2 arccos((4w 2 \
00(3 - 26o2) _ 1-~i-/~~ _--~o2)1/2],
Ftg. 2 Five tetrahodra, which compose the surface of a four umplex. (a) A homogeneous but amsotropic conf'~mratlon
with all vertices equal and two dfffexent edge lengths (sohd (12a)
lines and dashed lines). (b) A conf'~zratmn with one pecubar vertex, and two types of edge lengths (sohd lines and dashed lines).
333
Volume 163B, number 5,6
PHYSICS LETTERS
28 November 1985
2
1
I
-1
,
I
0
I 2
,
I
J
I 4
I
L
I
i 6
I
,
J
i 8
,
,
t
l i0
L
,
L
I 12 t
Fig 3 Hubble's constant (sohd hne), the anlsotropy parameter (dashed line), and normahzed ~ (dashed-dotted line) durmg a typical evolution of an amsotroplc but homogeneous toy umverse with a scalar field. The initial values were chosen so that the umverse wdl get into (at t ~ 1) and out of (at t ~ 9 5) an inflationary phase.
,\,.......................................................................
0
-1
i
i
I 2
i
i
t
I 4
i
t
i
I 6
i
i
i
I 8
i
i
i
I 10
i
J
12 t
Fig 4 Hubble's constant (solid line), the amsotropy parameter (dashed line), the scalar field ratio (dotted-dashed line) and the normahzed average ~ (short dashed hne) during a typical evolution of the mhomogeneous toy umverse with a scalar field The m;tlal values were chosen so that the umverse wKl get mto (at t ~ 1) and out of (at t ~ 8) an inflatlonaxy phase. 334
Volume 163B, number 5,6
PHYSCIS LETTERS
the freezing of the imhomogenelty are apparent. We have not neglected the curvature term in solvmg these equations; however the solution is not affected by it at all. As a second toy model we have considered, again, a unwerse built out of five tetrahedra. However, now one vertex is singled out so that its scalar field and the edges connecting it are different from all other scalar fields and all other edges (see fig. 2b) Clearly, this model is mhomogeneous and amsotropic. It is described by two edge lengths (or a length scale and an anlsotropy parameter) and by values of the scalar field on the two types of vertices and, of course, their derwarives. The structure of the evolution equations is remarkably similar to the structure of eqs. (10), but is somewhat more complicated. The mare new feature is an addltlon of a spatial derivative term m the scalar field equation. However, hke the curvature term this has a 1/R 2 dependence and its effect vanishes as the universe expands exponentmlly. The full set of equations will be written elsewhere [1]. The evolution of this unwerse resembles the previous cases. Again a De Sitter phase appears naturally under most initial conditions. As the radius of the universe increases the curvature terms (and the spatial gradient terms) become unimportant and the mhomogeneity freezes both m the different edge length and the values of the scalar field A typical evolution of this sort is shown in fig. 4
28 November 1985
The amsotropy and mhomogenelty that appear in these toy models is very coarse. In other words we consider here only a large scale mhomogenelty or a large scale amsotropy. This Is due to the small number of edge lengths involved. The observation that these become frozen does not mean that the same will be true for small scale deviations from homogenelty. These may be better studied using perturbation calculations or much more detailed Regge calculus models. It Is a pleasure to acknowledge discussions with S. Bludman. We are grateful to the ITP at Santa Barbara and to Meudon Observatory (TP) where some of this work was done This research was supported m part by the National Science Foundation under Grant No. PHY84-07219.
References [1] T Ptran, and R M Wflhams,prepnnt (1985), R M Williams, lecture given at Fourth Marcel Grossmann meeting (Rome, June 1985) [2] V A Behnskl, lecture given at Fourth Marcel Grossmann meeting (Rome, June 1985). [3] R H Brandenberger,Rev Mod Phys 57 (1985)1 [4] S W Hawking, DAMTPpreprmt (Sept 1984). [5] V.A Rubakov, M V Sazhm,and A V Veryaskm,Phys Lett l15B (1982) 189 [6] R M Wflhams, Gen. Rel Gray. 17 (1985) 559.
335
PHYSICS LETTERS
28 November 1985
INFLATION IN UNIVERSES WITH A MASSIVE SCALAR FIELD Tsvl P I R A N 1 and R u t h M. W I L L I A M S 2
Institutefor TheoretwalPhysws, Umversltyof Cahforma, Santa Barbara, CA 93106, USA Recewed 2 August 1985
We &scuss the evolutmn of toy umverses coupled to a masswe scalar field Under most mmal condmons such a umverse enters an mflatmnary phase from wluch ~t emerges naturally When the umverse ~s amsotrop~c or mhomogeneous, the amsotropy or mhomogenelty freezes dunng the mflaUonary phase
In vaew of the recent great interest in inflationary cosmological models it Is instructwe to study the evolution of a umverse containing a massive scalar field. We consider first the evolution of a homogeneous unwerse. Later, using a recently developed 3 + 1 Regge calculus formalism [ 1 ] we discuss the evolutaon o f a homogeneous but anasotropac model and o f an mhomogeneous and anasotropic model. We find that under most untial condations a universe coupled to a masswe scalar field enters an inflationary phase from wluch it emerges naturally [ 1]* 1. When the umverse Is anisotroplc or inhomogeneous, the anasotropy or irthomogeneaty freezes durmg the mflataonary phase. The evolution equations for a Fnedmann universe coupled to a masswe homogeneous scalar field are H2 = (8~r/3mp2) (½ m 2 0 2 + ~a d2) + k/R 2 ,
(1)
and + 3H4+m20=
O,
(2)
where 0 as the scalar field, mats mass, mp Planck's mass, H = t~/R and • corresponds to a tune denvative i Permanent address The Racah Institute for Physics, The Hebrew Umverslty, Jerusalem, Israel and The Institute for Advanced Study, Pnnceton, N J, USA 2 Permanent address Gtrton College and Department of Apphed Mathematics and Theorettcal Physics, Cambridge, England *l This result was found independently by Belmskl, Gnshchuk, Khalatnikov and Zel'doweh [ 2 ] 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The trajectories of the solutions [neglecting the curvature term m eq. (1)] on the dunenslonless ¢, plane are shown m fig. la, for a variety of inlml condxtions (the curvature term decreases rapadly when the umverse expands and, as we show later, its mclusaon does not change the reported results). Using the slow rolling eonsaderations of the inflationary models (see e.g. ref. [3]) one can observe that the umverse approaches a De Sitter phase when the rates of change o f 0 and d are small compared with H
~/H~ "~ d/HO ~ m2 /3H 2 '~ 1 .
(3)
Thas occurs along the two horizontal lines at ~ ~ +~ X X / ~ m m p m fig. la. Under these condations the pressure o f the scalar field, p = ~(d2 _ m202) Is negative and it supplies the energy for the exponenml expansion. During the mtlataonary phase p ~, -E#(Ee~ = ~m202 + ½q~2) as can be seen m fig. lb. The Hubble constant dunng the De Sitter phase is
iI 2 ~ 41rm202/3m2p .
(4)
Eq (3) sets a lunit on the mltlal amphtude of the scalar field.
02 ~" m2/4rr .
(5)
A stronger lmaat on 0 can be obtained from the fact that the universe has to expand at least 64 e-fold during the inflationary phase [4]. This leads to
02 ~" 8m2/Ir .
(6)
The conditaon that perturbataons should not grow too 331
Volume 163B, number 5,6
PHYSICS LETTERS
28 November 1985 P
2O i
'
'
'
1
10 c-a
i
-10
-20
I -4
-2
0
4
2
J
0
10
20
E
Ftg. 1 (a) TraJectories of the homogeneous universe on the normalized (0/V~'~')mp, ~ / ~ m p m ) plane Most of the trajectories lead to the mflatmnaty phase m which ~ ~ 1/3 The internal ettcle corresponds to osctllatlons of the scalar field and "dusthke" behavaour at late tune. There is an unphclt phase transition on the lines 4~= ±~, since the pressure Is negative when 02 < ~2. (b) TraJectories of the homogeneous umverse on the (E0, p) plane. The line p ~ - E 0 corresponds to the inflationary phase. much during the inflationary phase [5] leads to a hmlt on m, the mass of the scalar field. m < 10 -5 m p .
(7)
During the mltlal phase the scalar field, and therefore H decrease gradually. Eventually ¢~is small enough and the umverse goes out of the De Sitter phase. This happens without an apparent phase transmon or a change m the scalar potentml o(~) = ~ rn 252 However as the amphtude of the scalar field decreases the pressure of the scalar field becomes positive again. After the De Sitter phase the universe enters a massive dust-like phase with H ,~ m. The scalar field oscillates and using the WKB approxlmatmn we obtain ~b~ ~0 R - 3 / 2 e lint .
(8)
With this approxmaation H satisfies H 2 ~ 41rm2¢2o/3R 3 .
(9)
This behavmur is described m fig. la by the slowly shrinking circular curves (E~ ~ R - 3 ) near the ongm 332
which correspond to the post.inflationary phase. The most amazing feature of the scalar field cosmology (see fig. la) is that if the initial energy density, E~, of the scalar field is large enough the configuration evolves into a De Sitter phase, even when the initial ~ is large. The lmtlal trajectory is ~ ~ c o n s t a n t until the mflanonary phase begins. Therefore to have a long enough inflationary phase the inflationary condmon [eq. (3)] must be satisfied initially Only m a very lmaited range of initial (¢, ~) phase space, the size of which vanishes rapidly as the energy densxty of the scalar field increases, the configuration evolves directly towards the masswe dust-like configuration. As the universe evolves towards the low ~ region, an imphclt phase transition occurs with the pressure becoming negatwe and a De Sitter inflationary phase develops. We conclude thus that the inflationary phase is generic m a masswe scalar field universe. However the actual length of the inflation will depend on the mitml amplitude and phase of the field, with expansion by 64 e-fold reqmrmg that eq. (6) be satisfied.
Volume 163B, number 5,6
PHYSICS LETrERS
To consider the effect of deviations from isotropy and homogeneity we have constructed a toy model of a universe built out of five tetrahedra forming the surface of a four stmplex. The evolution of tlus umverse is described m term of the edge lengths of these tetrahedra using a 3 + 1 Regge calculus formalism that we have developed recently [ 1]. Altogether there are 10 edges. To construct a homogeneous but amsotrop~c model we choose two sets of five equal edges Ia and l b (see fig. 2a). Our dynamical variables are then the geometric mean of the edge lengths, R = (la lb)l/2, which is the scale length of the umverse, and the ratio 00 = (la/lb), wluch measures the amsotropy. The scalar field, $, is def'med on the vertices with all its values being equal. The equations for this toy unwerse are: H 2 + ] H ~ e t ' / a - 662/~/12002~, + (4/R 20)(8 l'Vt'~ + 8 2/Vr~)
= (41r/3m 2) (~2 + m 2~2),
82
=
2/r
28 November 1985
--
[2 - 260 2 + 004\ arccosl~ - - -
/
~, ~ (4 - 002) !
-2arccos~ 2 (3602-2) /2) (12b) ~o(400 - 1) 1/2 (4 - 602) 1 ' and ' denotes a 00 denvatwe. An immediate feature of these equations is that asymptotically, i.e. for large R , ~3 = o3 = 0 is an approximate solution of eq. (10c). Asymptotically the amsotropy will freeze at a given value (depending on the initial conditions). This does not change even after the universe goes out of the mflaUonary phase provided that R is large enough. When cb vanishes eqs. (10a), (10b) reduce to eqs. (1)(2) (up to the curvature term which is now small). H and the scalar field follow the same pattern described m fig. la. A typical evolution of this anisotropic universe is shown in fig. 3, in which the De Sxtter phase, with an almost constant H, and
(10a)
a
the scalar field evolution equataon: \
~+ (3/-/+ o3ot'/ot)~ + m2~b = 0 ,
(10b)
;.-
and an evolution equation for the anisotropy factor:
/\
..
/<
-.;\ i
- \
\
/ \',.
\
~(00o{/3ot + [3/4 w a ' ) + CoH( o.,,ot'/ ot + 3/3/400ot') ~ _ + Co2(a"00130t + 318a00 + 3'18a'00 2 -- 3/4a'00 2)
\
',
)
/- - - f / ~
= ( 3 [ R 2o~') (8 lX/r~ - 8 2/~¢/-~)
- (200/R 2a) (6 lX/~ + 6 2/x/-~) •
(1 0c)
b
Eqs. (10a), (10b) reduce to eqs. (1), (2) (up to a different numerical factor in the curvature term) and eq. (1 0c) becomes an identity when the anisotropy vanishes. The functions ~t, ~ and the deficit angles [6] 8a and 8 b are functions of 00 only: ot = (-00 3 + 200 + 200-1 - oo-3)1/2 ,
(1 la)
/3 = (3603 + 260 + 200-1 + 300-3)/~,,
(1 lb)
81
= 2 n - arccos(2004--2e 2- + 1 \ 4602 - 1 I - 2 arccos((4w 2 \
00(3 - 26o2) _ 1-~i-/~~ _--~o2)1/2],
Ftg. 2 Five tetrahodra, which compose the surface of a four umplex. (a) A homogeneous but amsotropic conf'~mratlon
with all vertices equal and two dfffexent edge lengths (sohd (12a)
lines and dashed lines). (b) A conf'~zratmn with one pecubar vertex, and two types of edge lengths (sohd lines and dashed lines).
333
Volume 163B, number 5,6
PHYSICS LETTERS
28 November 1985
2
1
I
-1
,
I
0
I 2
,
I
J
I 4
I
L
I
i 6
I
,
J
i 8
,
,
t
l i0
L
,
L
I 12 t
Fig 3 Hubble's constant (sohd hne), the anlsotropy parameter (dashed line), and normahzed ~ (dashed-dotted line) durmg a typical evolution of an amsotroplc but homogeneous toy umverse with a scalar field. The initial values were chosen so that the umverse wdl get into (at t ~ 1) and out of (at t ~ 9 5) an inflationary phase.
,\,.......................................................................
0
-1
i
i
I 2
i
i
t
I 4
i
t
i
I 6
i
i
i
I 8
i
i
i
I 10
i
J
12 t
Fig 4 Hubble's constant (solid line), the amsotropy parameter (dashed line), the scalar field ratio (dotted-dashed line) and the normahzed average ~ (short dashed hne) during a typical evolution of the mhomogeneous toy umverse with a scalar field The m;tlal values were chosen so that the umverse wKl get mto (at t ~ 1) and out of (at t ~ 8) an inflatlonaxy phase. 334
Volume 163B, number 5,6
PHYSCIS LETTERS
the freezing of the imhomogenelty are apparent. We have not neglected the curvature term in solvmg these equations; however the solution is not affected by it at all. As a second toy model we have considered, again, a unwerse built out of five tetrahedra. However, now one vertex is singled out so that its scalar field and the edges connecting it are different from all other scalar fields and all other edges (see fig. 2b) Clearly, this model is mhomogeneous and amsotropic. It is described by two edge lengths (or a length scale and an anlsotropy parameter) and by values of the scalar field on the two types of vertices and, of course, their derwarives. The structure of the evolution equations is remarkably similar to the structure of eqs. (10), but is somewhat more complicated. The mare new feature is an addltlon of a spatial derivative term m the scalar field equation. However, hke the curvature term this has a 1/R 2 dependence and its effect vanishes as the universe expands exponentmlly. The full set of equations will be written elsewhere [1]. The evolution of this unwerse resembles the previous cases. Again a De Sitter phase appears naturally under most initial conditions. As the radius of the universe increases the curvature terms (and the spatial gradient terms) become unimportant and the mhomogeneity freezes both m the different edge length and the values of the scalar field A typical evolution of this sort is shown in fig. 4
28 November 1985
The amsotropy and mhomogenelty that appear in these toy models is very coarse. In other words we consider here only a large scale mhomogenelty or a large scale amsotropy. This Is due to the small number of edge lengths involved. The observation that these become frozen does not mean that the same will be true for small scale deviations from homogenelty. These may be better studied using perturbation calculations or much more detailed Regge calculus models. It Is a pleasure to acknowledge discussions with S. Bludman. We are grateful to the ITP at Santa Barbara and to Meudon Observatory (TP) where some of this work was done This research was supported m part by the National Science Foundation under Grant No. PHY84-07219.
References [1] T Ptran, and R M Wflhams,prepnnt (1985), R M Williams, lecture given at Fourth Marcel Grossmann meeting (Rome, June 1985) [2] V A Behnskl, lecture given at Fourth Marcel Grossmann meeting (Rome, June 1985). [3] R H Brandenberger,Rev Mod Phys 57 (1985)1 [4] S W Hawking, DAMTPpreprmt (Sept 1984). [5] V.A Rubakov, M V Sazhm,and A V Veryaskm,Phys Lett l15B (1982) 189 [6] R M Wflhams, Gen. Rel Gray. 17 (1985) 559.
335