- Email: [email protected]
Creating a world in the laboratory
Volume 256, number 3,4
PHYSICS LETTERS B
14 March 1991
Creating a world in the laboratory D a l i a S. G o l d w i r t h 1
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel and Centerfor Astrophysics, Cambridge. MA 02138, USA 2 Received 1 August 1990
We study the possibility of getting a deflationary phase followed by an inflationary phase at late stages of the collapse of the universe or a collapse to a black hole. We find that a deflationary phase during collapse is extremely unstable and it is very unlikely to reach it. This introduces a natural asymmetry between the expanding phase and the collapsing phase of a closed universe.
At the very late stages o f a collapse to a black hole or a collapse o f a closed universe the m a t t e r temperature and density are similar to those that were present in the very early universe. This suggests that the evolution at this stage will resemble that o f the early universe, in a time reversed order. It is especially interesting if cosmological inflation [ 1 ] can take place in time reversed order. Such an evolution will lead to an exponential decrease, which we call deflation, that will turn into an exponential expansion, by a mechanism which will be explained later. This solution will give rise to a baby universe inside the black hole instead of the singularity. The appealing idea to get rid o f the singularity which inevitably arises in a collapse process in classical general relativity [2 ] led to the ad hoc suggestion [ 3 ] that a cosmological constant appears when the s p a c e - t i m e curvature reaches a maximal value o f 1/121, and this gives rise to a de Sitter phase at the state o f deflation. The only physical way known today to obtain such an equation o f state is via the cosmological inflation mechanism. In the following we study the possibility that a deflation phase will a p p e a r during gravitational collapse. During inflation [ 1 ] the energy density is d o m i nated by the potentia' t e r m of a scalar field, 0, which acts as a cosmological constant, giving rise to a de Sitter phase. At the end o f inflation [4] ~ oscillates around its true m i n i m u m , decaying into fermions and E-mail address: Dalia@hujivms. 2 Address after September 1990. 3 54
radiation through its coupling to them and reheating the universe. During the collapse we reach a stage in which the conditions are similar to those at the end o f the cosmological inflation and it is natural to ask whether inflation could take place in reverse. The first question is whether hot fermions can generate a field? The next question is whether the ~ field will follow the slow rolling solution and enter a deflation stage, or will it follow another trajectory? We study the collapse o f a spherically s y m m e t r i c homogeneous sphere o f size R. Since we are concerned only with the interior region that collapses we ignore the surface effects. Clearly one needs a shell with non-vanishing energy density to match between the homogeneous interior and the Schwarzschild exterior. F o r full discussion o f the surface effects a n d the matching between the inside metric and the outside Schwarzschild metric see ref. [5]. Einstein equations in such a case are
R_H R
' H=_N/~ ~
1
R2 ,
(1)
where H is the " H u b b l e constant", p is the energy density and the dot denotes a derivative with respect to the c o m o v i n g time coordinate. We use units in which c = 8 7 r G = 1. O u r m a t t e r source is a radiation field, which we represent by a massless scalar field ~,, coupled to a massive scalar field ~ with a potential V(~, iff) = V(~) +g021p,2. This representation o f the r a d i a t i o n field highly simplifies the calculation and
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 256, number 3,4
PHYSICS LETTERS B
enables us to couple between the scalar field and the radiation field. The energy density and pressure are 02
~2
k2~t2
P = T + -2- + ~ 5 -
+ V(0, ~u)
(2a)
and
~2
~2
P= T + 2
k2qy2 3R 2
V(O"q/) '
(2b)
where k2q/2/R 2 represents an "effective" gradient term (for details see ref. [6] ). From eqs. (2a) and {2b) it follows that on average ~ imitates a radiation ~p~,. The massive scalar field 0 satisfies field, i.e. P~,= the evolution equation 6"+3H¢)+ d V ( 0 ' ~') = 0 ,
(3a)
dO and the radiation field ~u satisfies the evolution equation
k2~ dV{O, qJ) = 0 . (fi+ 3tI~,+ ~ T + dg~
(3b)
During inflation [ 1 ] the energy density, p, is dominated by the potential term, V(O), and O must change slowly in time so that eq. (3a) becomes (what is known as the slow rolling evolution equation) 1 dV b~ -- -3H dO "
(4)
In this case we have an effective cosmological constant leading to an exponential increase of the scale factor of universe, R. The same condition that leads to inflation must hold in order to get deflation. In particular V(0) must dominate the energy density. However, from eq. (2a) it follows that both the gradient term and the kinetic terms grow during a collapse. It is essential therefore that these terms will be exponentially small at the beginning of deflation. The essence of this work is the question: can one reach such conditions? Once the deflationary phase is reached R will decrease quasiexponentially (at the rate of H). During this period the curvature term, 1/R 2, increases [see eq. ( 1 ) ], while p is almost a constant. [HI decreases until p= 3 / R 2 when the solution reaches a turning point at which H changes sign and the deflation turns into inflation. In general collapse can turn to expansion if
14 March 1991
the energy density increases with R slower than 1/ R 2. This can happen for equations of state p = cep with c~< - ~ . The only "natural" matter source that can have such an equation of state is a massive scalar field when p ~ V(0). From these arguments we see that the only way to turn a collapse into an expansion is to have a massive scalar field as the dominant matter field and to have a mechanism in which p is dominated by the scalar field potential for long enough so that the turning point is reached before the kinetic terms or the gradient term start to dominate. We solve eqs. ( 1 ), (2a) and ( 3 ) numerically using R u n g e - K u t t a methods. We consider two different scenarios. In the first case we begin with a radiation field and a scalar field. We expect that the radiation field will be damped through its coupling to the scalar field, 0, and we look for the possibility that the energy density will be eventually dominated by the potential term of the 0 field. To start such a process the initial conditions must be such that H is small compared to the derivative of V (so that it will not dominate the evolution of 0 and q/), i.e. dV 3H0< ~,
dV 3H~9< ~ ,
(5a)
and the change in H must be small compared to the change in the derivatives of V, which is known as the WKB approximation: d2l/
9 H 2< - d02 '
d2V
9H 2 < - d~2 '
(5b)
Such conditions lead to an oscillating solution with slow increase in the amplitude due to the collapse. Eventually the radiation field, q/, decays into the scalar field, 0. We start with the initial massive scalar field, 0i, at the m i n i m u m of its potential, V(0), with a small 0i. We choose the initial ~ui and ~i so that the WKB approximation for eqs. (3a) and (3b) is valid, i.e., for which conditions (5a) and (5b) are satisfied. Fig. 1 shows trajectories o f the solution in the (G ~) phase plane, with 0i = 1 and 0i = 0, for different initial ~'~and ¢q = 0 . In this calculation we used a Coleman-Weinberg ( C W ) type of potential (with 2 = 10 -4, or= 1 ),
V(O)'----/~04In~5-0"5 + 2
(6)
355
Volume 256, number 3,4
PHYSICS LETTERS B
o, t . . . .
/ /
00L 'l; .-~
o
-.o.12 / -.01K 0
/
,/
/
I .5
,
,
,
If I i
,
I
,
1.5
Fig. 1. Trajectories in the (0, ~) plane with ~ = 1 and ~i =0 for different initial % and ~i =0. The number on the curves correspond to the values of ~q. (This calculation was done with k= 5 and CW potential. ) We see the same type of behavior for other potentials which correspond to both "chaotic" and " n e w " inflation. In all the cases the radiation field is converted to the scalar field but the solution diverges exponentially from the slow rolling solution, i.e. away from the deflationary phase. At first these results might be puzzling since inflation is an attractor in the (~, 0) phase plane [ 71. The main difference between collapse and expansion is the sign of H. During expansion H is positive leading to an exponential decay of 0 [see eq. (3a) with a positive H] which makes the slow rolling phase an attractor. During a collapse H is (at least initially) negative leading to an exponential increase of 0 and ~ [see eqs. (3a) and (3b) with a negative H ] . Hence, even a very small initial kinetic term causes the solution to diverge exponentially from the deflationary solution. F r o m these arguments it follows that, unlike the cosmological inflation which is generic [7], deflation cannot be generic in a gravitational collapse. We have shown numerically that even if we chose special initial conditions, where H does not d o m i n a t e the evolution initially and without kinetic terms, the solution departs from the deflationary solution. We turn now to consider a more artificial situation 356
14 March 1991
in which the massive scalar is the only matter source, and we ask whether there are initial conditions that will lead to deflation. From the physical point of view we assume that some u n k n o w n mechanism provides us with initial (~i, 0i ) and we ask what happens next. Because of the previous arguments it is clear that not all initial conditions that lead to cosmological inflation will lead to deflation. Fig. 2 shows trajectories in the (0, 0) phase plane for different initial 0i and 0~ for CW type of potential [see eq. (6) ] with 2 = 10-4 a n d a = 1. It is important to notice that this phase space is different from the (¢, 0) phase space commonly used to discuss inflation [ 7 ] in the sense that trajectories can intersect each other. This happens for two reasons: First the curvature term in eq. ( 1 ) adds R dependence to eq. (2a), the phase space is though three dimensional and not two dimensional. The second reason is that H can have either + or - sign, so that trajectories leaving a point with a given 0, 0 and
.01
.005
•"e.
0
-.005
-.01
-1
-.5
0
.5
1.5
Fig. 2. Trajectories in the (0, 0) plane. The dotted line is the slow rolling solution, eq. (4), with H<0, i.e., deflation. The dashdotted line is the slow rolling solution with H> 0, i.e. inflation. The dashed lines are trajectories which reach the inflationary phase. The two long dashed lines begin at ~i=0.5 and in one case q)i=-l.5966687e-3, while in the other case 0i= 1.596668e- 3. Changingthe last two digits of~i causes the solution to divert from the slowroiling solution (the solid lines that begin at ~i= 0.5 ). The short dashed line begins at Oi= 10-4 and 0i= 10 -8, changing slightly the initial conditions to ~ = 10 2 causes the solution to divert (solid line). -
Volume 256, number 3,4
PHYSICS LETTERS B
R can follow two different directions, one corresponding to collapse and the other to expansion. In this phase space we have two special curves: the inflationary slow rolling solution and the deflationary slow rolling solution (for details see fig. 2). Most of the trajectories do not reach the deflation phase. However, a tiny part of the phase space leads to deflation followed by inflation. Only when the field is initially on the slow rolling trajectory for deflation [eq. (4) for deflation] does it continue to follow it. Eventually it reaches a turning point and it proceeds to seek the inflationary slow rolling curve. Since the deflationary solution is unstable a small deviation from it causes the solution to diverge from it and trajectories which have slightly different initial values never reach the inflationary solution. The solution is considerably unstable, changing ¢i from 1 . 5 9 6 6 6 8 e - 3 to 1 . 5 9 6 6 6 9 / 7 e - 3 causes the solution to diverge (see fig. 2 the trajectories that start at ¢i = 0.5 ). When we use a power law ("chaotic" inflation) potential we cannot find any initial conditions that lead to inflation. Even when the initial conditions are on the slow rolling phase the trajectories diverge. Figs. 3a and 3b show the evolution of the scale factor R as a function of H and as a function of the time, for the case where we get deflation followed by inflation. The amount of inflation depends significantly on the initial values of¢i and ~i. Changing by one the eighth digit of ~ from the one used in fig. 3 reduces the amount of inflation by 130 e-folds. 150
' ' ' J ' ' ' I ' ' ' I ')J
14 March 1991
The question which is left open is whether there is any physical mechanism which will "create" a scalar field which will be the dominant matter field, and whether this mechanism will ever favor those conditions which lead to deflation. Even if by some miracle the field is in those very specific conditions it is very doubtful that it will be stable there. First there are the surface effects which we have neglected but which are there and obviously will perturb the solution which is unstable, away from inflation. Moreover, we have assumed homogeneity. Adding inhomogeneity, which is the more natural case, will again perturb the solution in the same way. This work does not contradict the results by Farhi and Guth [8] because their main assumptions are different from ours. Farhi and Guth studied an expanding spherical de Sitter bubble separated by a thin wall from the outside region where the geometry is Schwarzschild. They found that in such a case the expanding bubble must be associated with an initial singularity which is spacelike. They conclude that the need for initial spacelike singularity would preclude the creation of a child universe from a black hole which is produced in the laboratory. In our work we show that it is practically impossible to obtain the configuration that they consider, i.e., it is impossible to find an expanding de Sitter bubble inside a black hole. As was mentioned before in this work we have neglected the surface effects. This means that the results 150
(a)
' l '
(b)
100
~3scillating
lO0 0
-1
50
50
0
0 0
20000
40000
time
60000
r"t,, -.005
Deflation ,,It[ill 0 H
J .005
Fig. 3. The evolution of the scale factor R in the case where we get inflation (this corresponds to one of the long dashed line solutions of fig. 2. ) Initially there is a short period at which ]HI grows until p becomes dominated by V(~). Once p ~ const. × R decreases quasiexponentially at the rate of H and since the curvature term I / R 2 is not negligible, [H] decreases. At t = 1831 p is equal to 3 / R 2 and the evolution of R reaches a turning point and R begins to increase exponentially. We stop the calculation when ~ reaches the oscillating region around a (the end of the inflationary phase) and from now on the regular Friedmann solution holds. (a) R as a function of time. (b) R as a function of H.
357
Volume 256, number 3,4
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are true also in the case o f the collapse o f the w h o l e universe. W h a t this w o r k shows is t h a t the c o n t r a c t ing phase o f a closed u n i v e r s e is d i f f e r e n t f r o m the t i m e reversal o f the e x p a n d i n g phase. Inflation is q u i t e generic in the b e g i n n i n g o f the e x p a n d i n g phase; h o w e v e r , it is a l m o s t hopeless to get it at the e n d o f the c o n t r a c t i n g phase. T h i s p r o d u c e s a classical a s y m m e t r y b e t w e e n the e x p a n d i n g p h a s e a n d the collapsing phase o f a closed universe. T h i s a s y m m e t r y has a statistical n a t u r e since the u n i v e r s e can deflate d u r i n g the collapse b u t o n l y u n d e r e x t r e m e l y special initial c o n d i t i o n s . M o s t c o n d i t i o n s will n o t l e a d to deflation. T h e reverse is true d u r i n g the e x p a n s i o n . T h i s a s y m m e t r y m i g h t s h a d e s o m e light on the direction o f the t h e r m o d y n a m i c a r r o w o f time. It is a p l e a s u r e to a c k n o w l e d g e helpful d i s c u s s i o n s with N. Deruelle, J. Katz, T. P i r a n a n d J.W. York. T h i s research was s u p p o r t e d by a grant f r o m the U S Israel B i n a t i o n a l Science F o u n d a t i o n to t h e H e b r e w University.
358
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References [ 1 ] For recent reviews of inflation see A.D. Linde, Rep. Prog. Phys. 47 (1984) 925; in: Proc. XXII Intern. Conf. on High energy physics (Leipzig, 1984) p. 125; Inflation and quantum cosmology (Academic Press, Boston, 1990 ); M.S. Turner, in: Proc. Carg6se School of Fundamental physics and cosmology, eds. J. Adouze and J. Tran Thanh Van (Editions Fronti6res, Gif-sur-Yvette, France, 1985 ). [2] See e.g.S.W. Hawking and G.F.R. Ellis, The large scale structure of space-tim e ( Cambridge U.P., Cambridge, 1973 ). [3] V.P. Frolov, M.A. Markov and V.F. Mukhanov, Phys. Lett. B216 (1989) 272; Phys. Rev. D 41 (1990) 383. [4] A. Albrecht, P.J. Steinhardt, M.S. Turner and F. Wilczek, Phys. Rev. Lett. 48 (1982) 1437. [ 5 ] K. Sato, M. Sasaki, H. Kodama and K. Maeda, Prog. Theor. Phys. 65 ( 1981 ) 1443; 66 ( 1981 ) 2052; K. Sato, Prog. Theor. Phys. 66 ( 1981 ) 2287; H. Kodama, M. Sasaki and K. Sato, Prog. Theor. Phys. 68 (1982) 1979; K. Maeda, K. Sato, M. Sasaki and H. Kodama, Phys. Len. B 108 (1982) 98; K. Sato, H. Kodama, M. Sasaki and K. Maeda, Phys. Len. B 108 (1982) 103. [6] T. Piran, Phys. Lett. B 181 (1986) 238. [7] V.A. Belinsky, L.P. Grishchuk, 1.M. Khalatnikov and Ya.B. Zel'dovich, Phys. Len. B 155 (1985) 232; A.D. Linde, Phys. Len. B 162 (1985) 281; T. Piran and R.M. Williams, Phys. Lett. B 163 (1985) 331; V.A. Belinsky, H. lshihara, I.M. Khalatnikov and H. Sato, Prog. Theor. Phys. 79 (1988) 676. [8] E. Farhi and A.H. Guth, Phys. Len. B 183 (1987) 149.
PHYSICS LETTERS B
14 March 1991
Creating a world in the laboratory D a l i a S. G o l d w i r t h 1
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel and Centerfor Astrophysics, Cambridge. MA 02138, USA 2 Received 1 August 1990
We study the possibility of getting a deflationary phase followed by an inflationary phase at late stages of the collapse of the universe or a collapse to a black hole. We find that a deflationary phase during collapse is extremely unstable and it is very unlikely to reach it. This introduces a natural asymmetry between the expanding phase and the collapsing phase of a closed universe.
At the very late stages o f a collapse to a black hole or a collapse o f a closed universe the m a t t e r temperature and density are similar to those that were present in the very early universe. This suggests that the evolution at this stage will resemble that o f the early universe, in a time reversed order. It is especially interesting if cosmological inflation [ 1 ] can take place in time reversed order. Such an evolution will lead to an exponential decrease, which we call deflation, that will turn into an exponential expansion, by a mechanism which will be explained later. This solution will give rise to a baby universe inside the black hole instead of the singularity. The appealing idea to get rid o f the singularity which inevitably arises in a collapse process in classical general relativity [2 ] led to the ad hoc suggestion [ 3 ] that a cosmological constant appears when the s p a c e - t i m e curvature reaches a maximal value o f 1/121, and this gives rise to a de Sitter phase at the state o f deflation. The only physical way known today to obtain such an equation o f state is via the cosmological inflation mechanism. In the following we study the possibility that a deflation phase will a p p e a r during gravitational collapse. During inflation [ 1 ] the energy density is d o m i nated by the potentia' t e r m of a scalar field, 0, which acts as a cosmological constant, giving rise to a de Sitter phase. At the end o f inflation [4] ~ oscillates around its true m i n i m u m , decaying into fermions and E-mail address: Dalia@hujivms. 2 Address after September 1990. 3 54
radiation through its coupling to them and reheating the universe. During the collapse we reach a stage in which the conditions are similar to those at the end o f the cosmological inflation and it is natural to ask whether inflation could take place in reverse. The first question is whether hot fermions can generate a field? The next question is whether the ~ field will follow the slow rolling solution and enter a deflation stage, or will it follow another trajectory? We study the collapse o f a spherically s y m m e t r i c homogeneous sphere o f size R. Since we are concerned only with the interior region that collapses we ignore the surface effects. Clearly one needs a shell with non-vanishing energy density to match between the homogeneous interior and the Schwarzschild exterior. F o r full discussion o f the surface effects a n d the matching between the inside metric and the outside Schwarzschild metric see ref. [5]. Einstein equations in such a case are
R_H R
' H=_N/~ ~
1
R2 ,
(1)
where H is the " H u b b l e constant", p is the energy density and the dot denotes a derivative with respect to the c o m o v i n g time coordinate. We use units in which c = 8 7 r G = 1. O u r m a t t e r source is a radiation field, which we represent by a massless scalar field ~,, coupled to a massive scalar field ~ with a potential V(~, iff) = V(~) +g021p,2. This representation o f the r a d i a t i o n field highly simplifies the calculation and
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 256, number 3,4
PHYSICS LETTERS B
enables us to couple between the scalar field and the radiation field. The energy density and pressure are 02
~2
k2~t2
P = T + -2- + ~ 5 -
+ V(0, ~u)
(2a)
and
~2
~2
P= T + 2
k2qy2 3R 2
V(O"q/) '
(2b)
where k2q/2/R 2 represents an "effective" gradient term (for details see ref. [6] ). From eqs. (2a) and {2b) it follows that on average ~ imitates a radiation ~p~,. The massive scalar field 0 satisfies field, i.e. P~,= the evolution equation 6"+3H¢)+ d V ( 0 ' ~') = 0 ,
(3a)
dO and the radiation field ~u satisfies the evolution equation
k2~ dV{O, qJ) = 0 . (fi+ 3tI~,+ ~ T + dg~
(3b)
During inflation [ 1 ] the energy density, p, is dominated by the potential term, V(O), and O must change slowly in time so that eq. (3a) becomes (what is known as the slow rolling evolution equation) 1 dV b~ -- -3H dO "
(4)
In this case we have an effective cosmological constant leading to an exponential increase of the scale factor of universe, R. The same condition that leads to inflation must hold in order to get deflation. In particular V(0) must dominate the energy density. However, from eq. (2a) it follows that both the gradient term and the kinetic terms grow during a collapse. It is essential therefore that these terms will be exponentially small at the beginning of deflation. The essence of this work is the question: can one reach such conditions? Once the deflationary phase is reached R will decrease quasiexponentially (at the rate of H). During this period the curvature term, 1/R 2, increases [see eq. ( 1 ) ], while p is almost a constant. [HI decreases until p= 3 / R 2 when the solution reaches a turning point at which H changes sign and the deflation turns into inflation. In general collapse can turn to expansion if
14 March 1991
the energy density increases with R slower than 1/ R 2. This can happen for equations of state p = cep with c~< - ~ . The only "natural" matter source that can have such an equation of state is a massive scalar field when p ~ V(0). From these arguments we see that the only way to turn a collapse into an expansion is to have a massive scalar field as the dominant matter field and to have a mechanism in which p is dominated by the scalar field potential for long enough so that the turning point is reached before the kinetic terms or the gradient term start to dominate. We solve eqs. ( 1 ), (2a) and ( 3 ) numerically using R u n g e - K u t t a methods. We consider two different scenarios. In the first case we begin with a radiation field and a scalar field. We expect that the radiation field will be damped through its coupling to the scalar field, 0, and we look for the possibility that the energy density will be eventually dominated by the potential term of the 0 field. To start such a process the initial conditions must be such that H is small compared to the derivative of V (so that it will not dominate the evolution of 0 and q/), i.e. dV 3H0< ~,
dV 3H~9< ~ ,
(5a)
and the change in H must be small compared to the change in the derivatives of V, which is known as the WKB approximation: d2l/
9 H 2< - d02 '
d2V
9H 2 < - d~2 '
(5b)
Such conditions lead to an oscillating solution with slow increase in the amplitude due to the collapse. Eventually the radiation field, q/, decays into the scalar field, 0. We start with the initial massive scalar field, 0i, at the m i n i m u m of its potential, V(0), with a small 0i. We choose the initial ~ui and ~i so that the WKB approximation for eqs. (3a) and (3b) is valid, i.e., for which conditions (5a) and (5b) are satisfied. Fig. 1 shows trajectories o f the solution in the (G ~) phase plane, with 0i = 1 and 0i = 0, for different initial ~'~and ¢q = 0 . In this calculation we used a Coleman-Weinberg ( C W ) type of potential (with 2 = 10 -4, or= 1 ),
V(O)'----/~04In~5-0"5 + 2
(6)
355
Volume 256, number 3,4
PHYSICS LETTERS B
o, t . . . .
/ /
00L 'l; .-~
o
-.o.12 / -.01K 0
/
,/
/
I .5
,
,
,
If I i
,
I
,
1.5
Fig. 1. Trajectories in the (0, ~) plane with ~ = 1 and ~i =0 for different initial % and ~i =0. The number on the curves correspond to the values of ~q. (This calculation was done with k= 5 and CW potential. ) We see the same type of behavior for other potentials which correspond to both "chaotic" and " n e w " inflation. In all the cases the radiation field is converted to the scalar field but the solution diverges exponentially from the slow rolling solution, i.e. away from the deflationary phase. At first these results might be puzzling since inflation is an attractor in the (~, 0) phase plane [ 71. The main difference between collapse and expansion is the sign of H. During expansion H is positive leading to an exponential decay of 0 [see eq. (3a) with a positive H] which makes the slow rolling phase an attractor. During a collapse H is (at least initially) negative leading to an exponential increase of 0 and ~ [see eqs. (3a) and (3b) with a negative H ] . Hence, even a very small initial kinetic term causes the solution to diverge exponentially from the deflationary solution. F r o m these arguments it follows that, unlike the cosmological inflation which is generic [7], deflation cannot be generic in a gravitational collapse. We have shown numerically that even if we chose special initial conditions, where H does not d o m i n a t e the evolution initially and without kinetic terms, the solution departs from the deflationary solution. We turn now to consider a more artificial situation 356
14 March 1991
in which the massive scalar is the only matter source, and we ask whether there are initial conditions that will lead to deflation. From the physical point of view we assume that some u n k n o w n mechanism provides us with initial (~i, 0i ) and we ask what happens next. Because of the previous arguments it is clear that not all initial conditions that lead to cosmological inflation will lead to deflation. Fig. 2 shows trajectories in the (0, 0) phase plane for different initial 0i and 0~ for CW type of potential [see eq. (6) ] with 2 = 10-4 a n d a = 1. It is important to notice that this phase space is different from the (¢, 0) phase space commonly used to discuss inflation [ 7 ] in the sense that trajectories can intersect each other. This happens for two reasons: First the curvature term in eq. ( 1 ) adds R dependence to eq. (2a), the phase space is though three dimensional and not two dimensional. The second reason is that H can have either + or - sign, so that trajectories leaving a point with a given 0, 0 and
.01
.005
•"e.
0
-.005
-.01
-1
-.5
0
.5
1.5
Fig. 2. Trajectories in the (0, 0) plane. The dotted line is the slow rolling solution, eq. (4), with H<0, i.e., deflation. The dashdotted line is the slow rolling solution with H> 0, i.e. inflation. The dashed lines are trajectories which reach the inflationary phase. The two long dashed lines begin at ~i=0.5 and in one case q)i=-l.5966687e-3, while in the other case 0i= 1.596668e- 3. Changingthe last two digits of~i causes the solution to divert from the slowroiling solution (the solid lines that begin at ~i= 0.5 ). The short dashed line begins at Oi= 10-4 and 0i= 10 -8, changing slightly the initial conditions to ~ = 10 2 causes the solution to divert (solid line). -
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R can follow two different directions, one corresponding to collapse and the other to expansion. In this phase space we have two special curves: the inflationary slow rolling solution and the deflationary slow rolling solution (for details see fig. 2). Most of the trajectories do not reach the deflation phase. However, a tiny part of the phase space leads to deflation followed by inflation. Only when the field is initially on the slow rolling trajectory for deflation [eq. (4) for deflation] does it continue to follow it. Eventually it reaches a turning point and it proceeds to seek the inflationary slow rolling curve. Since the deflationary solution is unstable a small deviation from it causes the solution to diverge from it and trajectories which have slightly different initial values never reach the inflationary solution. The solution is considerably unstable, changing ¢i from 1 . 5 9 6 6 6 8 e - 3 to 1 . 5 9 6 6 6 9 / 7 e - 3 causes the solution to diverge (see fig. 2 the trajectories that start at ¢i = 0.5 ). When we use a power law ("chaotic" inflation) potential we cannot find any initial conditions that lead to inflation. Even when the initial conditions are on the slow rolling phase the trajectories diverge. Figs. 3a and 3b show the evolution of the scale factor R as a function of H and as a function of the time, for the case where we get deflation followed by inflation. The amount of inflation depends significantly on the initial values of¢i and ~i. Changing by one the eighth digit of ~ from the one used in fig. 3 reduces the amount of inflation by 130 e-folds. 150
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The question which is left open is whether there is any physical mechanism which will "create" a scalar field which will be the dominant matter field, and whether this mechanism will ever favor those conditions which lead to deflation. Even if by some miracle the field is in those very specific conditions it is very doubtful that it will be stable there. First there are the surface effects which we have neglected but which are there and obviously will perturb the solution which is unstable, away from inflation. Moreover, we have assumed homogeneity. Adding inhomogeneity, which is the more natural case, will again perturb the solution in the same way. This work does not contradict the results by Farhi and Guth [8] because their main assumptions are different from ours. Farhi and Guth studied an expanding spherical de Sitter bubble separated by a thin wall from the outside region where the geometry is Schwarzschild. They found that in such a case the expanding bubble must be associated with an initial singularity which is spacelike. They conclude that the need for initial spacelike singularity would preclude the creation of a child universe from a black hole which is produced in the laboratory. In our work we show that it is practically impossible to obtain the configuration that they consider, i.e., it is impossible to find an expanding de Sitter bubble inside a black hole. As was mentioned before in this work we have neglected the surface effects. This means that the results 150
(a)
' l '
(b)
100
~3scillating
lO0 0
-1
50
50
0
0 0
20000
40000
time
60000
r"t,, -.005
Deflation ,,It[ill 0 H
J .005
Fig. 3. The evolution of the scale factor R in the case where we get inflation (this corresponds to one of the long dashed line solutions of fig. 2. ) Initially there is a short period at which ]HI grows until p becomes dominated by V(~). Once p ~ const. × R decreases quasiexponentially at the rate of H and since the curvature term I / R 2 is not negligible, [H] decreases. At t = 1831 p is equal to 3 / R 2 and the evolution of R reaches a turning point and R begins to increase exponentially. We stop the calculation when ~ reaches the oscillating region around a (the end of the inflationary phase) and from now on the regular Friedmann solution holds. (a) R as a function of time. (b) R as a function of H.
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are true also in the case o f the collapse o f the w h o l e universe. W h a t this w o r k shows is t h a t the c o n t r a c t ing phase o f a closed u n i v e r s e is d i f f e r e n t f r o m the t i m e reversal o f the e x p a n d i n g phase. Inflation is q u i t e generic in the b e g i n n i n g o f the e x p a n d i n g phase; h o w e v e r , it is a l m o s t hopeless to get it at the e n d o f the c o n t r a c t i n g phase. T h i s p r o d u c e s a classical a s y m m e t r y b e t w e e n the e x p a n d i n g p h a s e a n d the collapsing phase o f a closed universe. T h i s a s y m m e t r y has a statistical n a t u r e since the u n i v e r s e can deflate d u r i n g the collapse b u t o n l y u n d e r e x t r e m e l y special initial c o n d i t i o n s . M o s t c o n d i t i o n s will n o t l e a d to deflation. T h e reverse is true d u r i n g the e x p a n s i o n . T h i s a s y m m e t r y m i g h t s h a d e s o m e light on the direction o f the t h e r m o d y n a m i c a r r o w o f time. It is a p l e a s u r e to a c k n o w l e d g e helpful d i s c u s s i o n s with N. Deruelle, J. Katz, T. P i r a n a n d J.W. York. T h i s research was s u p p o r t e d by a grant f r o m the U S Israel B i n a t i o n a l Science F o u n d a t i o n to t h e H e b r e w University.
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References [ 1 ] For recent reviews of inflation see A.D. Linde, Rep. Prog. Phys. 47 (1984) 925; in: Proc. XXII Intern. Conf. on High energy physics (Leipzig, 1984) p. 125; Inflation and quantum cosmology (Academic Press, Boston, 1990 ); M.S. Turner, in: Proc. Carg6se School of Fundamental physics and cosmology, eds. J. Adouze and J. Tran Thanh Van (Editions Fronti6res, Gif-sur-Yvette, France, 1985 ). [2] See e.g.S.W. Hawking and G.F.R. Ellis, The large scale structure of space-tim e ( Cambridge U.P., Cambridge, 1973 ). [3] V.P. Frolov, M.A. Markov and V.F. Mukhanov, Phys. Lett. B216 (1989) 272; Phys. Rev. D 41 (1990) 383. [4] A. Albrecht, P.J. Steinhardt, M.S. Turner and F. Wilczek, Phys. Rev. Lett. 48 (1982) 1437. [ 5 ] K. Sato, M. Sasaki, H. Kodama and K. Maeda, Prog. Theor. Phys. 65 ( 1981 ) 1443; 66 ( 1981 ) 2052; K. Sato, Prog. Theor. Phys. 66 ( 1981 ) 2287; H. Kodama, M. Sasaki and K. Sato, Prog. Theor. Phys. 68 (1982) 1979; K. Maeda, K. Sato, M. Sasaki and H. Kodama, Phys. Len. B 108 (1982) 98; K. Sato, H. Kodama, M. Sasaki and K. Maeda, Phys. Len. B 108 (1982) 103. [6] T. Piran, Phys. Lett. B 181 (1986) 238. [7] V.A. Belinsky, L.P. Grishchuk, 1.M. Khalatnikov and Ya.B. Zel'dovich, Phys. Len. B 155 (1985) 232; A.D. Linde, Phys. Len. B 162 (1985) 281; T. Piran and R.M. Williams, Phys. Lett. B 163 (1985) 331; V.A. Belinsky, H. lshihara, I.M. Khalatnikov and H. Sato, Prog. Theor. Phys. 79 (1988) 676. [8] E. Farhi and A.H. Guth, Phys. Len. B 183 (1987) 149.