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Chaotic inflation in supergravity
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
CHAOTIC INFLATION IN SUPERGRAVITY A.S. GONCHAROV and A.D. LINDE
Lebedev Physical Institute, Moscow 11 7924, USSR Received 7 December 1983
A new realization of the chaotic inflation scenario is suggested in the context ofN = 1 supergravity.
One of the most interesting realizations of the inflationary universe scenario [ 1 - 3 ] is based on the investigation of the cosmological consequences of N = 1 supergravity coupled to matter [ 4 - 8 ] . This version of the inflationary universe scenario became especially attractive after the primordial monopole problem in this scenario was solved [ 6 - 8 ] and a consistent realization of this scenario was suggested [8], based on the idea of chaotic inflation [3]. However, this scenario still was to some extent incomplete. For realization of this scenario some supplementary chiral superfield q~ (different from the Polonyi field [9]) was introduced [4], and the effective potential V(z, z*) of the first (scalar) component z of this superfield was investigated [10]
V(z, z*) = exp(~zz*)(2 Igz - lz*gl2 - 3 [gl2) ,
(1)
where gz =- dg[dz, g(z) is the superpotential,
g(z) = u 3 f ( z ) .
(2)
Here/a is some mass parameter, f(z) is some arbitrary dimensionless function of the field z. The system of units in which Mp/x/-~ = 1 is used, whereMp ~ 1019 GeV is the Planck mass [ 10]. In refs. [4,5] the superpotential was represented in the following form: m
f(z) =
Xn_ zn ,
(3)
n=l n and it was assumed that )to t> O, kl > O. The effective potential V@) with respect to the real part ~0 of the field z was written as follows,
V@) =/.t6(a +/3~02 - '7~03 + 6tp4 + ...),
(4)
where c~,/3, 7, 6 ... are some functions of Xn, which can be chosen in such a way that the minimum of V@) is at ~0= ~00 = 1, and V@0) = 0. The last condition is necessary in order to have zero cosmological constant at the minimum of V@). In refs. [6,8] another effective potential was used, V(~0) = 3U6(1 - o~2~02+ ¼o~4tp4).
(5)
At present there is almost no idea what would be a preferable choice for the superpotential g(z), and therefore it is not quite clear which effective potential is more natural. However, there exists a possible constraint on the function g(z). In the theories under consideration the mass of the gravitino mG is proportional to g@0), and is assumed to be very small, mG "~ 102 GeV [11], or m G ~ 10 -16 in the units used above ( M p / x / ~ = 1). This is necessary in order to obtain a comparatively simple solution to the gauge hierarchy problem in the context of supergravity [12]. In such theories the value o f g ( z ) in the minimum of V(z, z*) should be almost precisely zero. In refs. [6,8] it was not assumed that g(~00) = 0, since the theories with m G ~ 102 GeV lead to some cosmological difficulties [13]. One can overcome these difficulties in the inflationary universe scenario [14,5], but only in a rather restricted class of theories [15]. Nevertheless it would be important to understand whether it is possible to realize the inflationary universe scenario in supergravity under the condition g(~00) = 0, as was assumed in refs. [4,5]. To make our investigation as simple as possible we introduce the function 27
Volume 139B, number 1,2 ~O(z) = exp(¼z 2) f(z),
PHYSICS LETTERS (6)
and rewrite the effective potential V(z, z*) (1) as follows:
V(z, z*) = U6 e x p [ - ~ ( z - z*) 2] 1 * × (21 ~bz + g(z - z) ff 12 - 3 1~b12),
(7)
where ~z -= d~/dz. At the real axis z = ~ the effective potential has a very simple form, V(~0) = U6(2~02 - 3 ~ 2 ) .
(8)
Now let us note, that i f ( 0 ) = f(0), ff~o(0)= f~(0), and that from the condition f(~00) = 0 follows that ff (~o0) = 0. Let us consider according to refs. [4,5] the superpotentialf(z) = F-,m=o(Xn/n)z n (3) with k 0/> 0, ~1 ]> 0. In such theories ~k(0) = X0/> 0, ff~o(0) = X1 > 0. It can be easily understood, that in this case the condition ~O(~00)= 0 can be satisfed only if ~O(~) becomes zero at some point ~ between ~o = 0 and ~o = ~00, in which ~ ( ~ 4: 0. From eq. (8) it follows that V(~---)< 0, which means that the point ~ = ~o0, in which ff(~0) = 0, is not the absolute minimum of V(~o) in the theories considered in refs. [4,5], and that a deeper minim u m with V ( ~ < 0 lies somewhere between ¢ = 0 and ~0= ~o0 [16]. This result is a strengthened version of the theorem, which has been proven independently in ref. [ 17]. The authors of ref. [17] have concluded, that the inflationary universe scenario in supergravity can be realized only in the theories, in which V(~00) = 0, g(~00) 4: 0, just as in the model suggested in refs. [6,8]. This would mean that inflation in supergravity is incompatible with the abovementioned possibility to solve the gauge hierarchy problem [12]. This general conclusion of the authors of ref. [ 17] (as distinct from the theorem discussed above) was based on the investigation of the high-temperature corrections to V(z, z*) (1). However, as was shown by one of the present authors in ref. [8], the only possibility to implement the inflationary universe scenario in the context of supergravity is connected with the chaotic inflation scenario [3], which is not based on the study of high-temperature effects. The main aim of the present paper is to implement the chaotic inflation scenario in the context of supergravity in such a way as to have both V(¢) = 0 and g(~0) = 0 in the absolute minimum of the effective po28
3 May 1984
tential. We will not try to propose here a most natural superpotential, since at present we have no criterion for making a choice between the different possible functions g(z). We would like just to show, that the superpotentials of the necessary type do actually exist, and to reveal some basic features of the chaotic inflation scenario in supergravity [8]. As an example of an appropriate superpotential let us consider the function ~k(z) = th ~"sh ~',
~"= ~ + i7/,
(9)
where ~"= V ~ ( z - tp0),
(10)
and ~00 is some real number, see below. It is seen, that #(z) = 0 at z -- ~00. The effective potential in the new variables is V(~', ~'*) = 3p 6 exp[~(Im ~-)2] X { [ ~ - + 1(~-* _ ~-) ~[2 _ 1~[2}.
(ll)
With the superpotential (9) the effective potential (1 1) at the real axis is given by V(~) = 3/~6(* 2 -- ~2) = 9/.t6(1 - 2/3 ch2~ - 1/3 ch4~).
(12)
This potential has a minimum at ~ = 0 (~0 = ~00) with V(~00) = 0. At large If[ the effective potential grows and asymptotically approaches 9# 6 . In the complex plane V(~', ~'*) is positive semidefinite, V(~', ~'*) = 0 only at ~"= iTrn, n = 0, +1, +2, .... Due to the presence of the factor exp [~(Im D 2] in (11), the potential V(~', ~'*) is exponentially large everywhere outside the narrow region near the real axis. The only exceptions are exponentially narrow holes ( o f the width of order e x p ( - ~ r r 2 n 2) near the above-mentioned points ~"= inn, n = +1, +2 . . . . . This potential is ideally fitted for the realization of the chaotic inflation scenario [3]. According to this scenario the universe initially was filled with some chaotic spatial distribution of the non-equilibrium field z. During the rolling down of the field z to the minimum of the effective potential some of the domains of the universe, Idled with a sufficiently large field z, exponentially grow and acquire a size exceeding the size of the observable part of the universe [3]. In a recent modification of this scenario [18] the large
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
field z appears after the quantum creation of the universe. The chaotic inflation scenario can be implemented in the theory with the effective potential ( 9 ) - ( 1 2 ) as follows. Almost independently of the initial value of the field z in the universe, this field exponentially rapidly rolls down to the real axis, ~"= ~ = Vc~(~o - ~P0). After this process the main part of the universe contains some real field ~o, which typically is very large, I~o - ¢ 0 1 >> 1. (The probability to roll down to the holes near ~"= inn, n = -+1, +2 .... or directly to the minimum at ~"= 0 is negligibly small.) Then the field ~0 exponentially slowly rolls down along the real axis to the minimum of V(~o) at ~p = tp0. During this process the domains of the universe filled with a sufficiently homogeneous field ~0(homogeneous at a scale l k H -1 [3]) expand according to the Einstein equation
with a probability, which is almost ~-independent, at least for sufficiently large ~. Therefore a considerable part of the universe (presumably, the main part of the universe) initially was in a state with [~ - ~0[ > 2. The regions of the universe with [~ - ~0[ k 2 exponentially expand and acquire a size exceeding the size of the observable part of the universe: l ~ 1028 cm [3]. Now let us investigate the spectrum of inhomogeneities, which arise after the inflation. As was shown in ref. [ 19], these inhomogeneities arise from quantum fluctuations of the field % which at the beginning of the process of their amplification at the point ~o= ~, have the momentum
H = d Z / d t = 3 - 1 / 2 ( V + ½~2)1/2,
t~2 = K2 exp(-2zXZ,) ~ 3p 6 exp(-2zXZ.),
(13)
where Z is the logarithm of the scale factor a(t), Z = In a. The evolution of the field ~ois governed by equation
+ 3H~ = -dV/d~o.
(14)
In the region [~o - ~ooI> 1, which is of main interest for us, the field ~p rolls down very slowly, and the terms ~ 2 in (13) and ~ in (14) can be neglected [3]. In this region the potential V(~0) is very flat and can be approximated by the asymptotic formula V(~o) = 9/a6 [1 - } e x p ( - ~ l ~ o - tP01)] .
(15)
During the rolling of the field ¢ to the minimum of V(~o) the scale factor a(t) = a(~o(t)) grows as follows:
ln[a(~oo)/a(~o)] = Z(~o0) - Z(~0)
f
~o
~ d~o=f
x
~oo
9 exp(v~l~o - ~P01) •
(16)
From eq. (16) follows that the universe expands more than e 70 times (which is necessary for the realization of the inflationary universe scenario) if the value of 1~0- ~P0I initially was sufficiently large, [~o - ~o0l/> 2. According to the chaotic inflation scenario, in a theory with the effective potential (15) any initial value of ~o can exist in a given domain of the universe
K2 ~ H 2 ~ 3/16 .
(17)
At the end of inflation this momentum decreases, (18)
where ~ Z . = Z(~o0) -Z(~o,). According to ref. [19], the spectrum of density perturbations after inflation is given by
8p (K)/p = (2rr3) - 1/2H2/l~[l~=so . = (3#3/rrV c~) ~ Z , 0.3/13 ln(3~6/K2).
(19)
The galaxies arise due to the amplification of fluctuations with zSZ, ~ 50, hence
6p/p ~ 30/~ 3 .
(20)
This means, that 6pip ~ 10 - 4 i f # 3 "" 3 X 10 -6, i.e. /.t ~ 10 -2, which seems quite reasonable. In these theories, unlike all other theories considered up to now, we do not need any other small parameters to obtain a sufficiently large inflation and the small value of density perturbations 6pip ~ 10 -4. As for the parameter ~o0, it determines the rate of decay of particles ~0 to other particles, F~o ~/a3~o 2 [5,8], and with an appropriate choice of this parameter one can get an adequate temperature TR of the universe after reheating, which is important for the solution of the gravitino problem in this scenario [15]. In particular, at ~00 = 1 in our scenario T R ~ 1010-1011 GeV, and the decrease (increase) of ~00 leads to a decrease (increase) of TR proportional to ~00. A discussion of the gravitino problem and of the baryon asymmetry generation in our scenario is contained in ref. [15]. As was shown above, the observable part of the uni29
Volume 139B, number 1,2
PHYSICS LETTERS
verse in our scenario was formed when the field ~ was in the region I¢ - ~001~< 2. Therefore this scenario can be realized not only in the theory with the superpotential (9), but in any theory, in which ff (z) is approximately given by (9) in the region Iz - ¢01 ~ 2. Thus, we have found a rather wide class of superpotentials, which have all the properties necessary for the realization of the chaotic inflation scenario in supergravity [8]. An attractive feature of the proposed scenario is that it does not contain any small parameters except for the parameter/2 ~ 10 -2. In this respect our scenario differs from all previous versions of the inflationary universe scenario suggested so far. However, the most important feature of this scenario is that g(~0) = 0 and V(~0) = 0 at the absolute m i n i m u m of V(¢), which is necessary in order to solve the gauge hierarchy problem in supergravity.
References [1] A.H. Guth, Phys. Rev. D23 (1981) 347. [2] A.D. Linde, Phys. Lett. 108B (1982) 389; 114B (1982) 431; l16B (1982) 335,340; A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [3] A.D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 38 (1983) 149; Phys. Lett. 129B (1983) 177. [4] D.V. Nanopoulos, K.A. Olive, M. Srednicki and K. Tamvakis, Phys. Lett. 123B (1983) 41.
30
3 May 1984
[5] D.V. Nanopoulos, K.A. Olive and M. Srednicki, Phys. Lett. 127B (1983) 30. [6] A.D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 37 (1983) 606. [7] A.D. Linde, Phys. Lett. 131B (1983) 330. [8] A.D. Linde, Phys. Lett. 132B (1983) 317. [9] J. Polonyi, Budapest preprint KFKI-1977-93 (1977). [10] E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. GirardeUo and P. van Nieuwenhuizen, Nucl. Phys. B147 (1979) 105. [11] J. Ellis and D.V. Nanopoulos, Phys. Lett. l16B (1982) 133. [12] L.E. Ib~fiez, Phys. Lett. 118B (1982) 73; R. Barbieri, S. Ferrara and C.A. Savoy, Phys. Lett. l19B (1982) 343; P. Nath, R. Arnowitt and A.H. Chamseddine, Phys. Rev. Lett. 49 (1982) 970. [13] S. Weinberg, Phys. Rev. Lett. 48 (1982) 1303. [14] J. Ellis, A.D. Linde and D.V. Nanopoulos, Phys. Lett. 118B (1982) 59. [15] M.Yu. Khlopov and A.D. Linde, Phys. Lett. 138B (1984) 265. [16] A.S. Goncharov and A.D. Linde, submitted to Zh. Eksp. Teor. Fiz. [17] B.A. Ovrut and P.J. Steinhardt, Phys. Lett. 133B (1983) 161. [18] A.D. Linde, Quantum creation of inflationary universe, submitted to Phys. Lett. [191 S.W. Hawking, Phys. Lett. l15B (1982) 295; A.A. Starobinsky, Phys. Lett. l17B (1982) 175; A.H. Guth and S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110; J.M. Bardeen, P.J. Steinhardt and M.S. Turner, Phys. Rev. D28 (1983) 679.
PHYSICS LETTERS
3 May 1984
CHAOTIC INFLATION IN SUPERGRAVITY A.S. GONCHAROV and A.D. LINDE
Lebedev Physical Institute, Moscow 11 7924, USSR Received 7 December 1983
A new realization of the chaotic inflation scenario is suggested in the context ofN = 1 supergravity.
One of the most interesting realizations of the inflationary universe scenario [ 1 - 3 ] is based on the investigation of the cosmological consequences of N = 1 supergravity coupled to matter [ 4 - 8 ] . This version of the inflationary universe scenario became especially attractive after the primordial monopole problem in this scenario was solved [ 6 - 8 ] and a consistent realization of this scenario was suggested [8], based on the idea of chaotic inflation [3]. However, this scenario still was to some extent incomplete. For realization of this scenario some supplementary chiral superfield q~ (different from the Polonyi field [9]) was introduced [4], and the effective potential V(z, z*) of the first (scalar) component z of this superfield was investigated [10]
V(z, z*) = exp(~zz*)(2 Igz - lz*gl2 - 3 [gl2) ,
(1)
where gz =- dg[dz, g(z) is the superpotential,
g(z) = u 3 f ( z ) .
(2)
Here/a is some mass parameter, f(z) is some arbitrary dimensionless function of the field z. The system of units in which Mp/x/-~ = 1 is used, whereMp ~ 1019 GeV is the Planck mass [ 10]. In refs. [4,5] the superpotential was represented in the following form: m
f(z) =
Xn_ zn ,
(3)
n=l n and it was assumed that )to t> O, kl > O. The effective potential V@) with respect to the real part ~0 of the field z was written as follows,
V@) =/.t6(a +/3~02 - '7~03 + 6tp4 + ...),
(4)
where c~,/3, 7, 6 ... are some functions of Xn, which can be chosen in such a way that the minimum of V@) is at ~0= ~00 = 1, and V@0) = 0. The last condition is necessary in order to have zero cosmological constant at the minimum of V@). In refs. [6,8] another effective potential was used, V(~0) = 3U6(1 - o~2~02+ ¼o~4tp4).
(5)
At present there is almost no idea what would be a preferable choice for the superpotential g(z), and therefore it is not quite clear which effective potential is more natural. However, there exists a possible constraint on the function g(z). In the theories under consideration the mass of the gravitino mG is proportional to g@0), and is assumed to be very small, mG "~ 102 GeV [11], or m G ~ 10 -16 in the units used above ( M p / x / ~ = 1). This is necessary in order to obtain a comparatively simple solution to the gauge hierarchy problem in the context of supergravity [12]. In such theories the value o f g ( z ) in the minimum of V(z, z*) should be almost precisely zero. In refs. [6,8] it was not assumed that g(~00) = 0, since the theories with m G ~ 102 GeV lead to some cosmological difficulties [13]. One can overcome these difficulties in the inflationary universe scenario [14,5], but only in a rather restricted class of theories [15]. Nevertheless it would be important to understand whether it is possible to realize the inflationary universe scenario in supergravity under the condition g(~00) = 0, as was assumed in refs. [4,5]. To make our investigation as simple as possible we introduce the function 27
Volume 139B, number 1,2 ~O(z) = exp(¼z 2) f(z),
PHYSICS LETTERS (6)
and rewrite the effective potential V(z, z*) (1) as follows:
V(z, z*) = U6 e x p [ - ~ ( z - z*) 2] 1 * × (21 ~bz + g(z - z) ff 12 - 3 1~b12),
(7)
where ~z -= d~/dz. At the real axis z = ~ the effective potential has a very simple form, V(~0) = U6(2~02 - 3 ~ 2 ) .
(8)
Now let us note, that i f ( 0 ) = f(0), ff~o(0)= f~(0), and that from the condition f(~00) = 0 follows that ff (~o0) = 0. Let us consider according to refs. [4,5] the superpotentialf(z) = F-,m=o(Xn/n)z n (3) with k 0/> 0, ~1 ]> 0. In such theories ~k(0) = X0/> 0, ff~o(0) = X1 > 0. It can be easily understood, that in this case the condition ~O(~00)= 0 can be satisfed only if ~O(~) becomes zero at some point ~ between ~o = 0 and ~o = ~00, in which ~ ( ~ 4: 0. From eq. (8) it follows that V(~---)< 0, which means that the point ~ = ~o0, in which ff(~0) = 0, is not the absolute minimum of V(~o) in the theories considered in refs. [4,5], and that a deeper minim u m with V ( ~ < 0 lies somewhere between ¢ = 0 and ~0= ~o0 [16]. This result is a strengthened version of the theorem, which has been proven independently in ref. [ 17]. The authors of ref. [17] have concluded, that the inflationary universe scenario in supergravity can be realized only in the theories, in which V(~00) = 0, g(~00) 4: 0, just as in the model suggested in refs. [6,8]. This would mean that inflation in supergravity is incompatible with the abovementioned possibility to solve the gauge hierarchy problem [12]. This general conclusion of the authors of ref. [ 17] (as distinct from the theorem discussed above) was based on the investigation of the high-temperature corrections to V(z, z*) (1). However, as was shown by one of the present authors in ref. [8], the only possibility to implement the inflationary universe scenario in the context of supergravity is connected with the chaotic inflation scenario [3], which is not based on the study of high-temperature effects. The main aim of the present paper is to implement the chaotic inflation scenario in the context of supergravity in such a way as to have both V(¢) = 0 and g(~0) = 0 in the absolute minimum of the effective po28
3 May 1984
tential. We will not try to propose here a most natural superpotential, since at present we have no criterion for making a choice between the different possible functions g(z). We would like just to show, that the superpotentials of the necessary type do actually exist, and to reveal some basic features of the chaotic inflation scenario in supergravity [8]. As an example of an appropriate superpotential let us consider the function ~k(z) = th ~"sh ~',
~"= ~ + i7/,
(9)
where ~"= V ~ ( z - tp0),
(10)
and ~00 is some real number, see below. It is seen, that #(z) = 0 at z -- ~00. The effective potential in the new variables is V(~', ~'*) = 3p 6 exp[~(Im ~-)2] X { [ ~ - + 1(~-* _ ~-) ~[2 _ 1~[2}.
(ll)
With the superpotential (9) the effective potential (1 1) at the real axis is given by V(~) = 3/~6(* 2 -- ~2) = 9/.t6(1 - 2/3 ch2~ - 1/3 ch4~).
(12)
This potential has a minimum at ~ = 0 (~0 = ~00) with V(~00) = 0. At large If[ the effective potential grows and asymptotically approaches 9# 6 . In the complex plane V(~', ~'*) is positive semidefinite, V(~', ~'*) = 0 only at ~"= iTrn, n = 0, +1, +2, .... Due to the presence of the factor exp [~(Im D 2] in (11), the potential V(~', ~'*) is exponentially large everywhere outside the narrow region near the real axis. The only exceptions are exponentially narrow holes ( o f the width of order e x p ( - ~ r r 2 n 2) near the above-mentioned points ~"= inn, n = +1, +2 . . . . . This potential is ideally fitted for the realization of the chaotic inflation scenario [3]. According to this scenario the universe initially was filled with some chaotic spatial distribution of the non-equilibrium field z. During the rolling down of the field z to the minimum of the effective potential some of the domains of the universe, Idled with a sufficiently large field z, exponentially grow and acquire a size exceeding the size of the observable part of the universe [3]. In a recent modification of this scenario [18] the large
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
field z appears after the quantum creation of the universe. The chaotic inflation scenario can be implemented in the theory with the effective potential ( 9 ) - ( 1 2 ) as follows. Almost independently of the initial value of the field z in the universe, this field exponentially rapidly rolls down to the real axis, ~"= ~ = Vc~(~o - ~P0). After this process the main part of the universe contains some real field ~o, which typically is very large, I~o - ¢ 0 1 >> 1. (The probability to roll down to the holes near ~"= inn, n = -+1, +2 .... or directly to the minimum at ~"= 0 is negligibly small.) Then the field ~0 exponentially slowly rolls down along the real axis to the minimum of V(~o) at ~p = tp0. During this process the domains of the universe filled with a sufficiently homogeneous field ~0(homogeneous at a scale l k H -1 [3]) expand according to the Einstein equation
with a probability, which is almost ~-independent, at least for sufficiently large ~. Therefore a considerable part of the universe (presumably, the main part of the universe) initially was in a state with [~ - ~0[ > 2. The regions of the universe with [~ - ~0[ k 2 exponentially expand and acquire a size exceeding the size of the observable part of the universe: l ~ 1028 cm [3]. Now let us investigate the spectrum of inhomogeneities, which arise after the inflation. As was shown in ref. [ 19], these inhomogeneities arise from quantum fluctuations of the field % which at the beginning of the process of their amplification at the point ~o= ~, have the momentum
H = d Z / d t = 3 - 1 / 2 ( V + ½~2)1/2,
t~2 = K2 exp(-2zXZ,) ~ 3p 6 exp(-2zXZ.),
(13)
where Z is the logarithm of the scale factor a(t), Z = In a. The evolution of the field ~ois governed by equation
+ 3H~ = -dV/d~o.
(14)
In the region [~o - ~ooI> 1, which is of main interest for us, the field ~p rolls down very slowly, and the terms ~ 2 in (13) and ~ in (14) can be neglected [3]. In this region the potential V(~0) is very flat and can be approximated by the asymptotic formula V(~o) = 9/a6 [1 - } e x p ( - ~ l ~ o - tP01)] .
(15)
During the rolling of the field ¢ to the minimum of V(~o) the scale factor a(t) = a(~o(t)) grows as follows:
ln[a(~oo)/a(~o)] = Z(~o0) - Z(~0)
f
~o
~ d~o=f
x
~oo
9 exp(v~l~o - ~P01) •
(16)
From eq. (16) follows that the universe expands more than e 70 times (which is necessary for the realization of the inflationary universe scenario) if the value of 1~0- ~P0I initially was sufficiently large, [~o - ~o0l/> 2. According to the chaotic inflation scenario, in a theory with the effective potential (15) any initial value of ~o can exist in a given domain of the universe
K2 ~ H 2 ~ 3/16 .
(17)
At the end of inflation this momentum decreases, (18)
where ~ Z . = Z(~o0) -Z(~o,). According to ref. [19], the spectrum of density perturbations after inflation is given by
8p (K)/p = (2rr3) - 1/2H2/l~[l~=so . = (3#3/rrV c~) ~ Z , 0.3/13 ln(3~6/K2).
(19)
The galaxies arise due to the amplification of fluctuations with zSZ, ~ 50, hence
6p/p ~ 30/~ 3 .
(20)
This means, that 6pip ~ 10 - 4 i f # 3 "" 3 X 10 -6, i.e. /.t ~ 10 -2, which seems quite reasonable. In these theories, unlike all other theories considered up to now, we do not need any other small parameters to obtain a sufficiently large inflation and the small value of density perturbations 6pip ~ 10 -4. As for the parameter ~o0, it determines the rate of decay of particles ~0 to other particles, F~o ~/a3~o 2 [5,8], and with an appropriate choice of this parameter one can get an adequate temperature TR of the universe after reheating, which is important for the solution of the gravitino problem in this scenario [15]. In particular, at ~00 = 1 in our scenario T R ~ 1010-1011 GeV, and the decrease (increase) of ~00 leads to a decrease (increase) of TR proportional to ~00. A discussion of the gravitino problem and of the baryon asymmetry generation in our scenario is contained in ref. [15]. As was shown above, the observable part of the uni29
Volume 139B, number 1,2
PHYSICS LETTERS
verse in our scenario was formed when the field ~ was in the region I¢ - ~001~< 2. Therefore this scenario can be realized not only in the theory with the superpotential (9), but in any theory, in which ff (z) is approximately given by (9) in the region Iz - ¢01 ~ 2. Thus, we have found a rather wide class of superpotentials, which have all the properties necessary for the realization of the chaotic inflation scenario in supergravity [8]. An attractive feature of the proposed scenario is that it does not contain any small parameters except for the parameter/2 ~ 10 -2. In this respect our scenario differs from all previous versions of the inflationary universe scenario suggested so far. However, the most important feature of this scenario is that g(~0) = 0 and V(~0) = 0 at the absolute m i n i m u m of V(¢), which is necessary in order to solve the gauge hierarchy problem in supergravity.
References [1] A.H. Guth, Phys. Rev. D23 (1981) 347. [2] A.D. Linde, Phys. Lett. 108B (1982) 389; 114B (1982) 431; l16B (1982) 335,340; A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [3] A.D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 38 (1983) 149; Phys. Lett. 129B (1983) 177. [4] D.V. Nanopoulos, K.A. Olive, M. Srednicki and K. Tamvakis, Phys. Lett. 123B (1983) 41.
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