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Stochastic non de sitter inflation
0083--6656/93$24.00 © 1993 PergamonPress Ltd
V/stas/n Astronomy,Vol. 37, pp. 481--485, 1993 Printed in Great Britain. All rights reserved.
STOCHASTIC NON DE SITTER INFLATION Boris L. Spokoiny Department of Physics, Kyoto University, Kyoto 606, Japan We generalize the stochastic approach to quasi-power-law inflationary Universes,obtain the corresponding Lmagevin and Fokker-Plaack equations for the scalar fidd driving inflation and find stationary solutions to the above FP equation.
The stochastic approach to quasi-de Sitter Universes (Starobinsky (1986), Sasaki et a1.(1988), etc.) appeared to be very fruitful and so it seems reasonable to extend it to a wider class of the Universes. In the chaotic inflationary model it is supposed that the evolution of the Universe is driven by some scalar field @ with the potential satisfying the condition(lo#V(qb)) ~ << M~ 1, which provides the slow rolling down of the scalar field along the potential and quasi-exponential evolution of the Universe.In this note we generalize the stochastic approach to the models which do not satisfy this restriction .We weaken this restriction and suppose that
(loaV(@))" << M~ 2.
(1)
So our models include quasi-exponential potentials IogV(c~) = A~/M +/ogl)(~) with a slowly varying/ogl)(#). Such potentials give rise to the power-law evolution of the Universe which generally is not of the quasi-de Sitter form.For the quasi-de Sitter Universe it is enough to consider only the perturbations of the scalar field and the account of the gravitational perturbations gives a correction of the order of ]-I/H ~ which is small.For the power-law inflationary Universe this correction is not small,this is why the normalization of the scalar perturbations obtained by previous authors (e.g.,Ortolan,Lucchin & Matarrese (1988)) who did not take into account this effect is not correct.Other authors (Salopek and Bond (1991)) in fact simply generalized the Fokker-Planck equation derived for the de Sitter Universe to the pure power-law inflationary Universe by introducing a new constant (a diffusion coefficient) which was not calculated by them.So we think that the detailed investigation of the models satisfying (1) is important. First we consider the theory of scalar perturbations in the model with a pure exponential potential: 1 2 1 2 1 L = - ~ M R + ~(0~¢) - ~V0 exp (A~b/M) (2) The equations for the homogeneous background have a power-law solution (Lucchin & Mat arrese (1985)) a(t) = t n+t,
~ = ~(f) = ~o - ~ - ~ M
log t,
(3)
where n = 2/A 2 - 1. Sohtion (3)is an attractor sohtion of the system (2) (see Halliwell (1987), Yokoyama & Ma~cla (1988)). In the longitudinal gauge we may represent the action for the scalar perturbations (of the scalar field and metric) in the form
482
B. L Spokoiny
s:
f
- (,<, -
+ o,,/o -
+
(4),
where k is a wave vectord/ is a conformal time, Xk = h k / ~° , ht is a perturbation of metric. Eq.(4) may be obtained from the paper by Makino and Sasaki (1991) and it gives us a correct normalization of the perturbation Xk (as for one degree of freedom) and is very important. Substituting the solution (3) into eq.(4) and varying the action (4) with respect to Xi we get an equation which can be easily solved in Bessel functions. The solution that has a proper asymptotic behavior at - k y >> 1 and the usual normalization for one degree of freedom is of the form:
Xk = -
-/--/-/(~)½ + ~,t - k '71'
!, 2V-'l 2 2kM
(5)
The perturbation of the scalar field is given by 6@t = --2M2(~'l'/~'a)xk -- 2 M 2 ( x ' k i a ) and at - k , / < < 1 we find $@k=
i H,1+~ ~f~,s_-..!~
(6)
where H . is defined by the equation for the ttubble "constant":
H(,/) = (I +
1
n)H,(-H,rl)-,
F ( n ) = 2~V~(1 + ~) co,~r(½
-
~)
(7)
For the de Sitter space F = F(oo) = 1 The perturbation of the metric at -kT/<< 1 is
iF(n) Hi, +~ hk = - 2Vrff + 1 M k ] + ~ "
(8)
Now we want to work in the synchronous coordinate system: ds 2 = dt 2 - aS(t, r-')dl 2, so we should make a transformation of time t = t + T(t, r-), T = f dth, then
~¢,(k, t) = ~¢~ + CTk,
~,(k, 0 = hk + #r~.
(9)
The above transformation of time is correct for long wave perturbations with -kT/ << 1. Now we are ready to apply the stochastic approach to the power-law evolution of the Universe.
sto
ac non de Sitter
Inflation
483
As usual we represent any operator F as a sum of a long wave part f(t, ~ (with k! = ~ < ell) and a short wave part (with ~ > ell) (Starobinsky (1986)):
F
,-3+ f
d3k
0(k
-
,aH)L(k, t)e
(10)
where the perturbation is(k, t) under the integral is given in a synchronous coordinate system and satisfies the equations of motion for perturbations.We write down five equations similar to eq.(10) where we substitute triples (~b;~(t, ~; ~,(k, t)), (~; v(t, r~; ~,(k, t)), (¢; wit, ~; ~,(k, t)), (a; a(t, r-3;&,(k, t)), (&; H(t, ~ ; ~,(k, t)) into eq.(10) instead of the triple ( f ; f(t, r-');.~(k, 4)). Here a(t, ~ = log a(t, ~, The equation for the scalar field and the ~ -Einstein equation after the substitution of the expressions (10) into them may be expressed in the form So + $1 + S~ = 0, where So is the contribution of the long wave part only, $1 is linear in the short wave part and $2 is at least quadratic in the short wave part.We find that $1 = 0 since the short wave part perturbations satisfy the exact equations of motion for perturbations by definition.We neglect higher order contributions,so we put $2 = 0 and we get So = 0 and the above equations read
w + 3Hv + V' = O, H 2 = (v212 + V(~))/3M ~. (11) After introducing a new variable 0 = v/H we obtain from (10)-(11) after some calculations d~ 1 . d3k = 0 + -~e(aH) f .-:=--~8(k - eaH)(te i~, d--~ J
(12)
)~2 V~ dad--~'O = -(3 - ~--~)(t~ + M2-~) + X,
(13)
where ~k' = ¢, - ( ~ / H ) a ,
= 6~t - (~///)ak
(14)
is a gauge invariant quantity and after the substitution of (3),(6) and (8) into the expression for X we find that X = 0 ! This is a very important result since it shows that we have chosen good variables (t~ and a) . Of course we take only the leading terms in a series in (-k)?) to obtain (6) and (8).However the next terms in the series for the Bessel function (5) have an extra smallness of the order of maz((k)l) 2, (-k)?)l+~) which after the substitution into the expression for X will give us a factor of the order of 81 oc max(e 2, ~el+Z,) due to 8(k -¢aH). So we may neglect this contribution to X if e is sufficiently small. If we take into account the difference between a real quasi-power law inflationary background and the pure power law background (3) that was used for the calculation of the dynamics of perturbations we will have a contribution to X which is suppressed by the factor 8~ = O ( ~logV ) " which corresponds to the factor ~rn~ in the usual quasi-de Sitter model (M.Sasaki et ai.(1988)). So we put X = O.
484
B. L Spokoiny
A general solution to the eq.(13) (with X = 0) contains two modes. As usual we consider only the slowly varying mode
0 = -M2(IogV) '
(15)
which is an asymptotically leading mode. We remind that the solution (3) is n o r a general solution ,but only an attractor solution and in this sense it corresponds to (15). After the substitution of (15) into (12) one finds a Langevin equation
d~o/da = - M 2 V ' I V + q(a)
(16)
where ~/(a) is a white noise with the correlation function
(,7(a),7(a'))
= 200(9)
Y_~2)
M2[V'~2
1 - ~ T ~ vo, t a~,
1
_ a'),
(17)
-~-t'Ir J
where n - n(~o) = 2(
~V ~ )2 _ I,
1 I ( n + 2 ) 2 F(n) 2 2Do(~o) = 121r2 e~ (n + I) ~ (I + ~)2+~-"
(18)
From the Langevin eq.(16) we obtain a Fokker-Planck equation
OP
, 0 V'
I_..~ O~[A(~o)V½ 0_OA(~o)V½ P],
(19)
where M2 r V' ~2
A(~o)2 = D0(~o) 1 - -~-t~rj, , . M rye2 1 - --g-t-p-j
(20)
As usual the Fokker-Planck equation (19) has two stationary solutions,one of which const " 4 1 1 P0(~o) = A ( - ~ 2 ~ exPIM j/ =7-~,2 A(~o) d(7)] corresponds to the "quantum creation" of the Universe since it has the Hawking-Moss instanton type form and the other is approximately P(~o) m joV/M2V ~ and describes the "classical creation" of the Universe,i.e.,the evolution from the singularity (Starobinsky(1986)).
Stochastic non de Sitter Inflation
485
REFERENCES Halliwell J. (1987)Scalar fields in cosmology with an exponential potential Phys.Leti. B185,341344. Luechin F.gz Matarrese S.(1985) Power-law inflation Phys.Rev.D32,1316-1322. Makino N.g~ Sasaki M.(1991)The density perturbation in the chaotic inflation with non-minimal eoupling.Progr. Theor.Ph~ts. 86,103-118. Ortolan A.,Lucchin F.~ Matarrese S.(1988) Non-Gaussian perturbations from inflationary dynamics. Phys.Rev.D38,465-471. Salopek D.S.& Bond J.R.(1991)Stochastic inflation and nonlinear gravity. 1005-1031.
Phys.Rev. D43,
Sasaki M.,Nambu Y.Sz Nakao K.(1988)Classical behavior of a scMar field in the inflationary universe.Nucl.Phys.B308,868-884. Starobinsky,A.A.(1986) Stochastic de Sitter (inflationary) stage in the early Universe. Lecture Noles in Physics.246,107-126 (Proceedings of the seminar on Field Theory, Quanlum Gravii~l , and Strings,Meudon and Paris VI,France 1984/85,edited by H.J.de Vega and N.Sanchez) Yokoyama J.~ Mazda K.(1988) On the dynamics of the power law inflation due to exponentiM potentiM.Ph~is.J~eff. B207,31-35.
V/stas/n Astronomy,Vol. 37, pp. 481--485, 1993 Printed in Great Britain. All rights reserved.
STOCHASTIC NON DE SITTER INFLATION Boris L. Spokoiny Department of Physics, Kyoto University, Kyoto 606, Japan We generalize the stochastic approach to quasi-power-law inflationary Universes,obtain the corresponding Lmagevin and Fokker-Plaack equations for the scalar fidd driving inflation and find stationary solutions to the above FP equation.
The stochastic approach to quasi-de Sitter Universes (Starobinsky (1986), Sasaki et a1.(1988), etc.) appeared to be very fruitful and so it seems reasonable to extend it to a wider class of the Universes. In the chaotic inflationary model it is supposed that the evolution of the Universe is driven by some scalar field @ with the potential satisfying the condition(lo#V(qb)) ~ << M~ 1, which provides the slow rolling down of the scalar field along the potential and quasi-exponential evolution of the Universe.In this note we generalize the stochastic approach to the models which do not satisfy this restriction .We weaken this restriction and suppose that
(loaV(@))" << M~ 2.
(1)
So our models include quasi-exponential potentials IogV(c~) = A~/M +/ogl)(~) with a slowly varying/ogl)(#). Such potentials give rise to the power-law evolution of the Universe which generally is not of the quasi-de Sitter form.For the quasi-de Sitter Universe it is enough to consider only the perturbations of the scalar field and the account of the gravitational perturbations gives a correction of the order of ]-I/H ~ which is small.For the power-law inflationary Universe this correction is not small,this is why the normalization of the scalar perturbations obtained by previous authors (e.g.,Ortolan,Lucchin & Matarrese (1988)) who did not take into account this effect is not correct.Other authors (Salopek and Bond (1991)) in fact simply generalized the Fokker-Planck equation derived for the de Sitter Universe to the pure power-law inflationary Universe by introducing a new constant (a diffusion coefficient) which was not calculated by them.So we think that the detailed investigation of the models satisfying (1) is important. First we consider the theory of scalar perturbations in the model with a pure exponential potential: 1 2 1 2 1 L = - ~ M R + ~(0~¢) - ~V0 exp (A~b/M) (2) The equations for the homogeneous background have a power-law solution (Lucchin & Mat arrese (1985)) a(t) = t n+t,
~ = ~(f) = ~o - ~ - ~ M
log t,
(3)
where n = 2/A 2 - 1. Sohtion (3)is an attractor sohtion of the system (2) (see Halliwell (1987), Yokoyama & Ma~cla (1988)). In the longitudinal gauge we may represent the action for the scalar perturbations (of the scalar field and metric) in the form
482
B. L Spokoiny
s:
f
- (,<, -
+ o,,/o -
+
(4),
where k is a wave vectord/ is a conformal time, Xk = h k / ~° , ht is a perturbation of metric. Eq.(4) may be obtained from the paper by Makino and Sasaki (1991) and it gives us a correct normalization of the perturbation Xk (as for one degree of freedom) and is very important. Substituting the solution (3) into eq.(4) and varying the action (4) with respect to Xi we get an equation which can be easily solved in Bessel functions. The solution that has a proper asymptotic behavior at - k y >> 1 and the usual normalization for one degree of freedom is of the form:
Xk = -
-/--/-/(~)½ + ~,t - k '71'
!, 2V-'l 2 2kM
(5)
The perturbation of the scalar field is given by 6@t = --2M2(~'l'/~'a)xk -- 2 M 2 ( x ' k i a ) and at - k , / < < 1 we find $@k=
i H,1+~ ~f~,s_-..!~
(6)
where H . is defined by the equation for the ttubble "constant":
H(,/) = (I +
1
n)H,(-H,rl)-,
F ( n ) = 2~V~(1 + ~) co,~r(½
-
~)
(7)
For the de Sitter space F = F(oo) = 1 The perturbation of the metric at -kT/<< 1 is
iF(n) Hi, +~ hk = - 2Vrff + 1 M k ] + ~ "
(8)
Now we want to work in the synchronous coordinate system: ds 2 = dt 2 - aS(t, r-')dl 2, so we should make a transformation of time t = t + T(t, r-), T = f dth, then
~¢,(k, t) = ~¢~ + CTk,
~,(k, 0 = hk + #r~.
(9)
The above transformation of time is correct for long wave perturbations with -kT/ << 1. Now we are ready to apply the stochastic approach to the power-law evolution of the Universe.
sto
ac non de Sitter
Inflation
483
As usual we represent any operator F as a sum of a long wave part f(t, ~ (with k! = ~ < ell) and a short wave part (with ~ > ell) (Starobinsky (1986)):
F
,-3+ f
d3k
0(k
-
,aH)L(k, t)e
(10)
where the perturbation is(k, t) under the integral is given in a synchronous coordinate system and satisfies the equations of motion for perturbations.We write down five equations similar to eq.(10) where we substitute triples (~b;~(t, ~; ~,(k, t)), (~; v(t, r~; ~,(k, t)), (¢; wit, ~; ~,(k, t)), (a; a(t, r-3;&,(k, t)), (&; H(t, ~ ; ~,(k, t)) into eq.(10) instead of the triple ( f ; f(t, r-');.~(k, 4)). Here a(t, ~ = log a(t, ~, The equation for the scalar field and the ~ -Einstein equation after the substitution of the expressions (10) into them may be expressed in the form So + $1 + S~ = 0, where So is the contribution of the long wave part only, $1 is linear in the short wave part and $2 is at least quadratic in the short wave part.We find that $1 = 0 since the short wave part perturbations satisfy the exact equations of motion for perturbations by definition.We neglect higher order contributions,so we put $2 = 0 and we get So = 0 and the above equations read
w + 3Hv + V' = O, H 2 = (v212 + V(~))/3M ~. (11) After introducing a new variable 0 = v/H we obtain from (10)-(11) after some calculations d~ 1 . d3k = 0 + -~e(aH) f .-:=--~8(k - eaH)(te i~, d--~ J
(12)
)~2 V~ dad--~'O = -(3 - ~--~)(t~ + M2-~) + X,
(13)
where ~k' = ¢, - ( ~ / H ) a ,
= 6~t - (~///)ak
(14)
is a gauge invariant quantity and after the substitution of (3),(6) and (8) into the expression for X we find that X = 0 ! This is a very important result since it shows that we have chosen good variables (t~ and a) . Of course we take only the leading terms in a series in (-k)?) to obtain (6) and (8).However the next terms in the series for the Bessel function (5) have an extra smallness of the order of maz((k)l) 2, (-k)?)l+~) which after the substitution into the expression for X will give us a factor of the order of 81 oc max(e 2, ~el+Z,) due to 8(k -¢aH). So we may neglect this contribution to X if e is sufficiently small. If we take into account the difference between a real quasi-power law inflationary background and the pure power law background (3) that was used for the calculation of the dynamics of perturbations we will have a contribution to X which is suppressed by the factor 8~ = O ( ~logV ) " which corresponds to the factor ~rn~ in the usual quasi-de Sitter model (M.Sasaki et ai.(1988)). So we put X = O.
484
B. L Spokoiny
A general solution to the eq.(13) (with X = 0) contains two modes. As usual we consider only the slowly varying mode
0 = -M2(IogV) '
(15)
which is an asymptotically leading mode. We remind that the solution (3) is n o r a general solution ,but only an attractor solution and in this sense it corresponds to (15). After the substitution of (15) into (12) one finds a Langevin equation
d~o/da = - M 2 V ' I V + q(a)
(16)
where ~/(a) is a white noise with the correlation function
(,7(a),7(a'))
= 200(9)
Y_~2)
M2[V'~2
1 - ~ T ~ vo, t a~,
1
_ a'),
(17)
-~-t'Ir J
where n - n(~o) = 2(
~V ~ )2 _ I,
1 I ( n + 2 ) 2 F(n) 2 2Do(~o) = 121r2 e~ (n + I) ~ (I + ~)2+~-"
(18)
From the Langevin eq.(16) we obtain a Fokker-Planck equation
OP
, 0 V'
I_..~ O~[A(~o)V½ 0_OA(~o)V½ P],
(19)
where M2 r V' ~2
A(~o)2 = D0(~o) 1 - -~-t~rj, , . M rye2 1 - --g-t-p-j
(20)
As usual the Fokker-Planck equation (19) has two stationary solutions,one of which const " 4 1 1 P0(~o) = A ( - ~ 2 ~ exPIM j/ =7-~,2 A(~o) d(7)] corresponds to the "quantum creation" of the Universe since it has the Hawking-Moss instanton type form and the other is approximately P(~o) m joV/M2V ~ and describes the "classical creation" of the Universe,i.e.,the evolution from the singularity (Starobinsky(1986)).
Stochastic non de Sitter Inflation
485
REFERENCES Halliwell J. (1987)Scalar fields in cosmology with an exponential potential Phys.Leti. B185,341344. Luechin F.gz Matarrese S.(1985) Power-law inflation Phys.Rev.D32,1316-1322. Makino N.g~ Sasaki M.(1991)The density perturbation in the chaotic inflation with non-minimal eoupling.Progr. Theor.Ph~ts. 86,103-118. Ortolan A.,Lucchin F.~ Matarrese S.(1988) Non-Gaussian perturbations from inflationary dynamics. Phys.Rev.D38,465-471. Salopek D.S.& Bond J.R.(1991)Stochastic inflation and nonlinear gravity. 1005-1031.
Phys.Rev. D43,
Sasaki M.,Nambu Y.Sz Nakao K.(1988)Classical behavior of a scMar field in the inflationary universe.Nucl.Phys.B308,868-884. Starobinsky,A.A.(1986) Stochastic de Sitter (inflationary) stage in the early Universe. Lecture Noles in Physics.246,107-126 (Proceedings of the seminar on Field Theory, Quanlum Gravii~l , and Strings,Meudon and Paris VI,France 1984/85,edited by H.J.de Vega and N.Sanchez) Yokoyama J.~ Mazda K.(1988) On the dynamics of the power law inflation due to exponentiM potentiM.Ph~is.J~eff. B207,31-35.