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Initial conditions for new inflation
Volume 243, number 1,2
PHYSICS LETTERS B
21 June 1990
Initial conditions for new inflation D a l i a S. G o l d w i r t h Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Received 19 February 1990
It is well known that an initial kinetic term does not prevent the onset of "chaotic" inflation. In this paper we extend this investigation to "new" inflation scenarios. We find that in the case of "new" inflation as in the case of "chaotic" inflation there exists a curve that attracts most of the solutions. However, unlike the "chaotic" inflation case where the attractor is the inflationary solution, in the "new" inflation case the attractor is the slow rolling solution and only a small part of it corresponds to inflation. So unlike the "chaotic" case where the inflationary stage is generic, in the "new" inflation case only very specific initial conditions lead to inflation.
During inflation the energy density o f the universe must be d o m i n a t e d by the potential t e r m o f the scalar field [ 1 ]. In recent years several groups [2 ] have investigated the influence o f an initial kinetic t e r m ( o f the scalar field) on the onset o f " c h a o t i c " inflation. They f o u n d that such an initial kinetic t e r m decays rapidly a n d it does not prevent the onset o f inflation. In this work, we extend this study to " n e w " inflation scenarios. We follow the classical evolution o f a F r i e d m a n n universe coupled to a scalar field in an a t t e m p t to find which initial conditions will lead to inflation. We use an explicit numerical c o m p u t a t i o n o f the phase space trajectories. We also o b t a i n analytical a p p r o x i m a t i o n s to the trajectories o f the scalar field in different regions and show that our numerical results agree to a very high degree with these approximations. In the following we use units in which c = 8~zG= 1. We solve numerically, using R u n g e - K u t t a methods, the homogeneous Einstein equations coupled to a massive scalar field: ~=H,
/¢
H 2 = ~ 2 + 1V(q0)- ~--~,
(I)
where R is the scale factor, H is the H u b b l e "constant" and x = - 1, 0, 1 for open, fiat or closed uniElectronic mail: Dalia@hujivms
verse. The scalar field equation
satisfies the evolution
• + 3 H ~ + V' ( q b ) = 0 ,
(2)
where • is the derivative with respect to t i m e a n d ' the derivative with respect to ~ . We define/7--- tb a n d eq. (2) becomes two first order differential equations: dO H ~-= ,
(3a)
d// - - 3 H / 7 - V' ( @ ) . dt
(3b)
We use two t y p i c a l ' p o t e n t i a l s for the scalar field: a quartic potential o f the form V ( ~ ) = 2 ( q~ - a 2 ) 2 and a C o l e m a n - W e i n b e r g ( C W ) type o f potential
V(~) = 2 ~ 4
[ l n ( O 2 / a 2) - ½] + ½,~.a4.
The results presented in the following are with the C W potential. The general features o f the b e h a v i o r o f the trajectories do not change when we use a quartic type potential. We choose different initial conditions ~o, in the range I ~o I < a, and/70 in the range [/7ol < x/~ tr and we follow the evolution until the field begins to oscillate a r o u n d the true v a c u u m at + a. Fig. 1 displays the trajectories in the (x, y ) plane, where x a n d y are dimensionless coordinates,
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
41
Volume 243, number 1,2
PHYSICS LETTERS B
1
.5 0
--.~
-1
-1
-.5
0
.5
X Fig. 1. Trajectories in the (x, y) plane. The dashed lines are p = ½/72+ V(~) =const. lines [from p = V(0) t o p = 2 V ( 0 ) ].
21 June 1990
only a narrow stripe of trajectories reach this region it follows that only a very small part of the phase space leads to inflation, i.e. only very specific initial @oand Ho will result in inflation. This behavior differs from the behavior of such trajectories with "chaotic" inflation type potential, in which inflation takes place everywhere along the curve that attracts most of the trajectories. The behavior of the trajectories can be also analyzed analytically. To understand the evolution of the scalar field we define three regions (see fig. 3) in the ((P, H) plane. Region (I) where V(~) < 1//2, region (II) where V ( ~ ) > ½ H 2 and region (III) where 9//2< V" (4,) which is the oscillating region of the scalar field. Curve F describes the slow rolling solution where the evolution equation for the scalar field can be approximated by V'
17=~
1
(~)
3H
'
t/~ v / ~ .
(5)
Region O on curve F corresponds to the region which leads to inflation of more then a few e-folds, i.e. the region wheref~ H d t > N (Nis the number of e-folds, and N>~ 60 for sufficient inflation). According to Steinhardt and Turner [ 3 ] this can be approximated by
05
0
3H 2
-05
I v"(,~O I
>N,
H ~ x / ] V(~b),
(6)
where ~bb is the value of the field at the point where -.1
-.05
0
.05
1
X
1 &lliJ I
Fig. 2. Enlargement of the region around the origin of fig. 1. The × corresponds to points that have N (number of e-folds) of 21 600, 100, 60, 10 and 1. It can be seen that Ndrops very quickly, and only the first three curves around zero lead to sufficient inflation.
2y, t=471
dx r/, y = d--~.
(4)
We see that there is a curve that attracts most of the trajectories (we will see later that this curve corresponds to the slow rolling solution). However, an examination of fig. 2 (which depicts an enlargement of the central region of fig. 1 ) reveals that only a very small part of this curve corresponds to inflation. Since 42
--~ III-~
III
-.5
-1
I I I
II
.5
0 --
• ==, H=
I I I I I I I I I I'1
/
~tl,, -i
-.5
0
.5
1
X Fig. 3. Schematic picture of the different regions in the (x, y) phase plane.
Volume 243, number 1,2
PHYSICSLETTERSB
it reaches curve F and ~ is the value of the field at the end of the slow rolling phase, i.e. when it enters region (III). In "new" inflation if ~ # 0 initially then if the field's velocity H has the same sign as ~, i.e. it is in the direction leading away from ~ = 0, the field will be driven down the potential hill without ever reaching the inflationary phase. So we will focus on the situation where the velocity is in the direction leading towards zero. In this case it follows from eq. (2) that I~1 will decrease as long as the damping force 3HH is larger than V' (qb) which tends to drive • away from 0. The crucial question is where does V' ( ~ ) balance 3HH? To see this we derive approximate solutions to the motion of the field in each of the regions described in fig. 3. This approximation does not depend on the specific form of the potential so it is valid for both "chaotic" and "new" inflation. In region (I), eq. (3b) can be approximated by [/~ -v/~H 2 ,
(7)
from which it follows that
H=
/70 1 + (t-To) x/~ I/7ol
(8a)
and • = g3o- x/~-]l n ( - ~ ) ,
where ~ and H~ are the values of • and H a t tg, the time when the field entered this region. [In eq. ( 1 lb) we have assumed that V remains almost constant. When the field enters region (II), ½H2= V(~) and in this stage the factor 3 in eq. (1 lb) should be replaced by 6. This factor changes quickly to 3 as the field moves into the region. ] The evolution inside region (II) continues with I~1 decreasing until 1I' ( ~ ) = 3HH. At this stage the trajectory reaches curve F [see eq. ( 11b) ] with @b=tP~4 ~
• ;+ ~
H~= x / 2 V ( ~ ) .
(9)
When the field is in region (II) H is approximately ~ - V and the evolution equation (3b) for H can be approximated by /'/~ - 3 v / ~ - ~ ) H ,
(t-t~) ]
(lla)
and
H$-H ~ = ¢ ] + ~ ,
(12)
H~
< • ~6o) ,
(13)
where ~ b(6°) is calculated from eq. (6) with N = 60. If the field begins in region (I), it follows from eqs. (9), (13) and the remark about the factor 3 in eq. ( I I b ) that the universe will inflate if ~ ~ I/xf3, i.e. the universe will enter the inflationary phase if it starts with g3o, N ~ [ ~ 2 2 +in (
/~o
~]
(14,
(10)
so that H=H~ exp[-~)
H~
(where we have neglected Hb since it is exponentially smaller then H~ ). The evolution continues along the curve F until it reaches regions (III) where it oscillates. In the "chaotic" inflation case ~b> 4 is sufficient for inflation (with N > 60). Hence most initial conditions are suitable for entering inflation. In the "new" inflation scenario in order to get sufficient inflation the trajectory must reach region O, where both tp and H a r e very close to zero. We can now trace the solution backwards from O to find which initial conditions lead to inflation. Inside region (II) such initial conditions must satisfy
(8b)
where Do and ~o are the values of H and • at the time 70 when this phase begins. When ½H2= V(~) the field will enter region (II) with the initial conditions
21 June 1990
(llb)
One can easily verify from figs. 1 and 2 that these approximations are in very good agreement with the numerical results. According to the standard picture of the "new" inflation scenario [ 1 ] the scalar field will be localized around ~ g 0 at the end of the phase transition. Our calculations show that it also must be at rest there. It 43
Volume 243, number 1,2
PHYSICS LETTERS B
has been shown [4] that the • field does not have enough time to thermalize during the phase transition so that it cannot be localized near zero everywhere. We find, from the classical evolution of the field equations, that only a very a small part of the ( ~ , / 7 ) phase space will lead to the inflating region near the origin. It is a pleasure to acknowledge helpful discussions with R.H. Brandenberger and T. Piran. This research was supported by a grant from the US-Israel Binational Science Foundation to the Hebrew University.
44
21 June 1990
References [ 1 ] For a recent review of inflation see M.S. Turner, in: Proc. Carg~se School of Fundamental physics and cosmology, eds. J. Adouze and J. Tran Thanh Van (Editions Fronti~res, Gifsur-Yvette, France, 1985 ). [2 ] V.A. Belinsky, L.P. Grishchuk, I.M. Khalatnikov and Ya.B. Zel'dovich, Phys. Lett. B 155 ( 1985 ) 232; A.D. Linde, Phys. Lett. B 162 (1985) 281; T. Piran and R.M. Williams, Phys. Lett. B 163 (1985) 331; T. Piran, Phys. Lett. B 181 (1986) 238; V.A. Belinsky, H. Ishihara, I.M. Khalatnikov and H. Sato, Prog. Theor. Phys. 79 (1988) 676. [3] P.J. Steinhardt and M.S. Turner, Phys. Rev. D 29 (1984) 2162. [4] A.D. Linde, Phys. Lett. B 132 (1983) 317; B 162 (1985) 281; G.F. Mazenko, W.G. Unruh and R.M. Wald, Phys. Rev. D 31 (1985) 273.
PHYSICS LETTERS B
21 June 1990
Initial conditions for new inflation D a l i a S. G o l d w i r t h Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Received 19 February 1990
It is well known that an initial kinetic term does not prevent the onset of "chaotic" inflation. In this paper we extend this investigation to "new" inflation scenarios. We find that in the case of "new" inflation as in the case of "chaotic" inflation there exists a curve that attracts most of the solutions. However, unlike the "chaotic" inflation case where the attractor is the inflationary solution, in the "new" inflation case the attractor is the slow rolling solution and only a small part of it corresponds to inflation. So unlike the "chaotic" case where the inflationary stage is generic, in the "new" inflation case only very specific initial conditions lead to inflation.
During inflation the energy density o f the universe must be d o m i n a t e d by the potential t e r m o f the scalar field [ 1 ]. In recent years several groups [2 ] have investigated the influence o f an initial kinetic t e r m ( o f the scalar field) on the onset o f " c h a o t i c " inflation. They f o u n d that such an initial kinetic t e r m decays rapidly a n d it does not prevent the onset o f inflation. In this work, we extend this study to " n e w " inflation scenarios. We follow the classical evolution o f a F r i e d m a n n universe coupled to a scalar field in an a t t e m p t to find which initial conditions will lead to inflation. We use an explicit numerical c o m p u t a t i o n o f the phase space trajectories. We also o b t a i n analytical a p p r o x i m a t i o n s to the trajectories o f the scalar field in different regions and show that our numerical results agree to a very high degree with these approximations. In the following we use units in which c = 8~zG= 1. We solve numerically, using R u n g e - K u t t a methods, the homogeneous Einstein equations coupled to a massive scalar field: ~=H,
/¢
H 2 = ~ 2 + 1V(q0)- ~--~,
(I)
where R is the scale factor, H is the H u b b l e "constant" and x = - 1, 0, 1 for open, fiat or closed uniElectronic mail: Dalia@hujivms
verse. The scalar field equation
satisfies the evolution
• + 3 H ~ + V' ( q b ) = 0 ,
(2)
where • is the derivative with respect to t i m e a n d ' the derivative with respect to ~ . We define/7--- tb a n d eq. (2) becomes two first order differential equations: dO H ~-= ,
(3a)
d// - - 3 H / 7 - V' ( @ ) . dt
(3b)
We use two t y p i c a l ' p o t e n t i a l s for the scalar field: a quartic potential o f the form V ( ~ ) = 2 ( q~ - a 2 ) 2 and a C o l e m a n - W e i n b e r g ( C W ) type o f potential
V(~) = 2 ~ 4
[ l n ( O 2 / a 2) - ½] + ½,~.a4.
The results presented in the following are with the C W potential. The general features o f the b e h a v i o r o f the trajectories do not change when we use a quartic type potential. We choose different initial conditions ~o, in the range I ~o I < a, and/70 in the range [/7ol < x/~ tr and we follow the evolution until the field begins to oscillate a r o u n d the true v a c u u m at + a. Fig. 1 displays the trajectories in the (x, y ) plane, where x a n d y are dimensionless coordinates,
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
41
Volume 243, number 1,2
PHYSICS LETTERS B
1
.5 0
--.~
-1
-1
-.5
0
.5
X Fig. 1. Trajectories in the (x, y) plane. The dashed lines are p = ½/72+ V(~) =const. lines [from p = V(0) t o p = 2 V ( 0 ) ].
21 June 1990
only a narrow stripe of trajectories reach this region it follows that only a very small part of the phase space leads to inflation, i.e. only very specific initial @oand Ho will result in inflation. This behavior differs from the behavior of such trajectories with "chaotic" inflation type potential, in which inflation takes place everywhere along the curve that attracts most of the trajectories. The behavior of the trajectories can be also analyzed analytically. To understand the evolution of the scalar field we define three regions (see fig. 3) in the ((P, H) plane. Region (I) where V(~) < 1//2, region (II) where V ( ~ ) > ½ H 2 and region (III) where 9//2< V" (4,) which is the oscillating region of the scalar field. Curve F describes the slow rolling solution where the evolution equation for the scalar field can be approximated by V'
17=~
1
(~)
3H
'
t/~ v / ~ .
(5)
Region O on curve F corresponds to the region which leads to inflation of more then a few e-folds, i.e. the region wheref~ H d t > N (Nis the number of e-folds, and N>~ 60 for sufficient inflation). According to Steinhardt and Turner [ 3 ] this can be approximated by
05
0
3H 2
-05
I v"(,~O I
>N,
H ~ x / ] V(~b),
(6)
where ~bb is the value of the field at the point where -.1
-.05
0
.05
1
X
1 &lliJ I
Fig. 2. Enlargement of the region around the origin of fig. 1. The × corresponds to points that have N (number of e-folds) of 21 600, 100, 60, 10 and 1. It can be seen that Ndrops very quickly, and only the first three curves around zero lead to sufficient inflation.
2y, t=471
dx r/, y = d--~.
(4)
We see that there is a curve that attracts most of the trajectories (we will see later that this curve corresponds to the slow rolling solution). However, an examination of fig. 2 (which depicts an enlargement of the central region of fig. 1 ) reveals that only a very small part of this curve corresponds to inflation. Since 42
--~ III-~
III
-.5
-1
I I I
II
.5
0 --
• ==, H=
I I I I I I I I I I'1
/
~tl,, -i
-.5
0
.5
1
X Fig. 3. Schematic picture of the different regions in the (x, y) phase plane.
Volume 243, number 1,2
PHYSICSLETTERSB
it reaches curve F and ~ is the value of the field at the end of the slow rolling phase, i.e. when it enters region (III). In "new" inflation if ~ # 0 initially then if the field's velocity H has the same sign as ~, i.e. it is in the direction leading away from ~ = 0, the field will be driven down the potential hill without ever reaching the inflationary phase. So we will focus on the situation where the velocity is in the direction leading towards zero. In this case it follows from eq. (2) that I~1 will decrease as long as the damping force 3HH is larger than V' (qb) which tends to drive • away from 0. The crucial question is where does V' ( ~ ) balance 3HH? To see this we derive approximate solutions to the motion of the field in each of the regions described in fig. 3. This approximation does not depend on the specific form of the potential so it is valid for both "chaotic" and "new" inflation. In region (I), eq. (3b) can be approximated by [/~ -v/~H 2 ,
(7)
from which it follows that
H=
/70 1 + (t-To) x/~ I/7ol
(8a)
and • = g3o- x/~-]l n ( - ~ ) ,
where ~ and H~ are the values of • and H a t tg, the time when the field entered this region. [In eq. ( 1 lb) we have assumed that V remains almost constant. When the field enters region (II), ½H2= V(~) and in this stage the factor 3 in eq. (1 lb) should be replaced by 6. This factor changes quickly to 3 as the field moves into the region. ] The evolution inside region (II) continues with I~1 decreasing until 1I' ( ~ ) = 3HH. At this stage the trajectory reaches curve F [see eq. ( 11b) ] with @b=tP~4 ~
• ;+ ~
H~= x / 2 V ( ~ ) .
(9)
When the field is in region (II) H is approximately ~ - V and the evolution equation (3b) for H can be approximated by /'/~ - 3 v / ~ - ~ ) H ,
(t-t~) ]
(lla)
and
H$-H ~ = ¢ ] + ~ ,
(12)
H~
< • ~6o) ,
(13)
where ~ b(6°) is calculated from eq. (6) with N = 60. If the field begins in region (I), it follows from eqs. (9), (13) and the remark about the factor 3 in eq. ( I I b ) that the universe will inflate if ~ ~ I/xf3, i.e. the universe will enter the inflationary phase if it starts with g3o, N ~ [ ~ 2 2 +in (
/~o
~]
(14,
(10)
so that H=H~ exp[-~)
H~
(where we have neglected Hb since it is exponentially smaller then H~ ). The evolution continues along the curve F until it reaches regions (III) where it oscillates. In the "chaotic" inflation case ~b> 4 is sufficient for inflation (with N > 60). Hence most initial conditions are suitable for entering inflation. In the "new" inflation scenario in order to get sufficient inflation the trajectory must reach region O, where both tp and H a r e very close to zero. We can now trace the solution backwards from O to find which initial conditions lead to inflation. Inside region (II) such initial conditions must satisfy
(8b)
where Do and ~o are the values of H and • at the time 70 when this phase begins. When ½H2= V(~) the field will enter region (II) with the initial conditions
21 June 1990
(llb)
One can easily verify from figs. 1 and 2 that these approximations are in very good agreement with the numerical results. According to the standard picture of the "new" inflation scenario [ 1 ] the scalar field will be localized around ~ g 0 at the end of the phase transition. Our calculations show that it also must be at rest there. It 43
Volume 243, number 1,2
PHYSICS LETTERS B
has been shown [4] that the • field does not have enough time to thermalize during the phase transition so that it cannot be localized near zero everywhere. We find, from the classical evolution of the field equations, that only a very a small part of the ( ~ , / 7 ) phase space will lead to the inflating region near the origin. It is a pleasure to acknowledge helpful discussions with R.H. Brandenberger and T. Piran. This research was supported by a grant from the US-Israel Binational Science Foundation to the Hebrew University.
44
21 June 1990
References [ 1 ] For a recent review of inflation see M.S. Turner, in: Proc. Carg~se School of Fundamental physics and cosmology, eds. J. Adouze and J. Tran Thanh Van (Editions Fronti~res, Gifsur-Yvette, France, 1985 ). [2 ] V.A. Belinsky, L.P. Grishchuk, I.M. Khalatnikov and Ya.B. Zel'dovich, Phys. Lett. B 155 ( 1985 ) 232; A.D. Linde, Phys. Lett. B 162 (1985) 281; T. Piran and R.M. Williams, Phys. Lett. B 163 (1985) 331; T. Piran, Phys. Lett. B 181 (1986) 238; V.A. Belinsky, H. Ishihara, I.M. Khalatnikov and H. Sato, Prog. Theor. Phys. 79 (1988) 676. [3] P.J. Steinhardt and M.S. Turner, Phys. Rev. D 29 (1984) 2162. [4] A.D. Linde, Phys. Lett. B 132 (1983) 317; B 162 (1985) 281; G.F. Mazenko, W.G. Unruh and R.M. Wald, Phys. Rev. D 31 (1985) 273.