#### Stochastic inflation with quantum and thermal noise

Eur. Phys. J. C
Stochastic inflation with quantum and thermal noise
Z. Haba 0
0 Institute of Theoretical Physics, University of Wroclaw , 50-204 Wrocław , Poland
We add a thermal noise to Starobinsky equation of slow roll inflation. We calculate the number of e-folds of the stochastic system. The power spectrum and the spectral index are evaluated from the fluctuations of the e-folds using an expansion in the quantum and thermal noise terms.
1 Introduction
The standard CDM model describes the evolution of the
universe in agreement with observations [
1
]. The fast
expansion at the early stages of the evolution can be explained in
terms of a scalar field (inflaton). A quantization of the scalar
field and gravitational perturbations leads to fluctuations
which can explain structure formation and the power
spectrum of density fluctuations in the universe [
2–7
]. The model
is introducing some new (dark) forms of matter and energy
which are not interacting with the inflaton. If we assume that
there are some interactions of the inflaton with the unknown
forms of matter then the wave equation for the inflaton is
transformed into a stochastic equation which in a flat
expanding metric (with the scale factor a and H = a−1∂t a) takes
the form
∂t2φ − a−2 φ + (3H + γ 2)∂t φ
where γ 2 is a friction related to the Gaussian noise
η(t )η(s) = δ(t − s)
according to the fluctuation-dissipation relation. Equation (1)
has been derived in [8] (see also [
9
]). The friction γ 2 is
proportional to temperature. The noise η comes from the thermal
(Gibbs) distribution of the initial positions and velocities of
the particles of the environment (in general, the environment
(1)
(2)
may consist of any degrees of freedom which are
unobservable and averaged in a description of an interaction with φ).
Equation (1) is a basis of the warm inflation [
10
]. In such a
model the resonant reheating is unnecessary as the
temperature during inflation does not fall to zero owing to the
creation of radiation as a result of the decay of the inflaton. The
quantum fluctuations of the inflaton and gravitational
perturbations are usually described [
3,7
] in a linear approximation.
Starobinsky [
4,11
] (see also [
12
]) discovered that quantum
fields at large time in an expanding universe behave like a
classical diffusion process. Then, the high momentum part
of the quantum field can be treated as an additional
(quantum) noise in the inflaton wave equation. Such a treatment of
quantum fluctuations during inflation goes beyond a linear
approximation. The quantum noise has been widely
studied in Refs. [
13–23
]. The Fokker–Planck equation for the
probability distribution of the inflaton has been explored in
detail. In principle, the Fokker–Planck equation contains all
the information about the probability distribution. In
particular, the power spectrum of fluctuations could be calculated
as discussed in [
21,22
] (see also Appendix B here).
However, in [23] (following [
11
], see also [
24,25
]) an alternative
method has been proposed for a calculation of the power
spectrum of the quantum noise based on fluctuations of the
e-folds. In this paper we extend the method to the calculation
of the power spectrum of the system which contains both the
quantum noise and the thermal noise. The plan of this paper
is the following. In Sect. 2 we discuss the stochastic
equation with the quantum and thermal noises in the slow roll
approximation. In Sect. 3 following Refs. [
23,26
] we obtain
general formulas for the expectation values of e-folds and the
fluctuations of e-folds (spectral function). Then, in Sect. 4 we
discuss approximations leading to some explicit formulas for
the spectral function and the spectral index. In Appendices
A and B we discuss fluctuations in soluble models and the
relation between Ito and Stratonovitch stochastic equations.
2 Slow roll stochastic equations
We consider two sources of noise in Eq. (1) the thermal noise
η and the quantum noise ηS [
4,11,12
]. The quantum noise
comes from the large momentum part (above the Hubble
horizon) of the scalar field. It can be considered as a part of
the quantum inflaton equation. The thermal noise results from
an approximation of the interaction with an environment by a
Markov process. In Einstein equations the environment could
be represented as dark matter or dark energy if we properly
choose the environmental interactions [
27–29
]. We consider
Eq. (1) in the slow-roll approximation
(3H + γ 2) ◦ dφ = −V dt − 23 γ 2 H φdt + γ a− 23 ◦ d B
3 5
+ 2π H 2 ◦ d W
In Eq. (3) we write η = ∂t B, ηS = ∂t W and assume that W
and B are independent Gaussian variables. We use the
notation ◦d W (after [
30
]) for the Stratonvitch interpretation of the
stochastic differential and the conventional notation of the
differential in the Ito interpretation. The difference between
Ito and Stratonovitch integrals consists in a different discrete
time approximations of the Riemann sums approximating
the integral. The Stratonovitch integral f ◦ d W treats time
approximation of f and W (in the Riemann sum) in a
symmetric way (so called middle point approximation) whereas
in the Ito integral the time in d W is later than in f . For an
integral with a differentiable function W the various discrete
approximations would lead to the same result. However, W
is not differentiable. We discuss both interpretations of the
stochastic differential for an easy comparison with literature
on the subject. The Ito stochastic differential equation can be
expressed by the Stratonovitch equation using the rule [
30
]
f ◦ d W = f d W + 21 d f d W . So, both equations differ by
a correction term. The Stratonovitch form is convenient for
calculations because it preserves the standard rules of
differentiation (the Leibniz rule) [
30
]. It must be checked in
mathematical models which form of the stochastic equation
better describes physical processes.
In order to simplify further discussion we assume that
3H >> γ 2 and V >> 23 γ 2 H φ. The friction term γ 2∂t φ
is usually related to the decay of the inflaton into radiation
[
31
]. The omission of the γ 2 terms on the lhs of Eq. (1) means
a negligible density of radiation (which can be true during
inflation [
32
]). Now, the stochastic equation (3) reads
1 γ 3
dφ = − 3H V dt + 3 a− 2 H −1 ◦ d B(t )
1 3
+ 2π H 2 ◦ d W (t ).
The Starobinsky [
11,12
] slow-roll (quantum) system
corresponds to the limit γ → 0 of Eq. (4). In order to obtain
an agreement with the inhomogeneous inflaton and gravity
perturbations (most easily treated in the uniform curvature
gauge [
33,34
]) we must change the world time t into the
efolding time ν [
21–23
] (usually denoted by N ; we change
notation for typographical reasons) describing the change of
the scale factor
0
t
ν =
H ds = ln
a
a0
Now, the diffusion (small roll) system reads
(4)
+∂φ (3H )−1V P.
If in the e-folding time we treat a as depending on φ (not on
ν), then we obtain a stationary form of the Fokker–Planck
equation which for the Ito version is
(3)
together with the differential form of the Friedman equation
(taking a derivative in the Friedman equation can allow to
treat the environment as a dark energy [
27
])
d ln(H ) = −4π G(∂ν φ)2dν.
We can insert in Eq. (4) either a(φ) as a function of φ or
a = a0 exp(ν) (in such a case we obtain a non-stationary
stochastic equation).
In the slow-roll approximation we can derive from Eq. (7)
in the no-noise limit
(5)
(6)
(7)
(8)
(9)
(10)
(11)
H =
We could take in Eq. (9) the noise into account by means of
perturbation methods (the relation between a(ν) and φ (ν)
will still be discussed in Appendices A and B).
The probability distribution of the solution of Eq. (4)
(Stratonovitch interpretation) satisfies the Fokker–Planck
equation [
26,30,35
]
∂t P = ∂φ
1 1 3 3
3 P + 8π 2 ∂φ H 2 ∂φ H 2 P
H a 2
In the Ito interpretation of Eq. (4)
γ 2 1
∂t P = ∂φ ∂φ 18H 2a3 P + 8π 2 ∂φ ∂φ H 3 P
∂ν P = γ182 ∂φ ∂φ a31H 3 P + 8π1 2 ∂φ ∂φ H 2 P + ∂φ (3H 2)−1V P
+∂φ (3H 2)−1V P.
and Stratonovitch version
∂ν P = 18 ∂φ
It can be seen from Eqs. (6) and (14) that quantum and
thermal fluctuations are of the same order if γ 2 e3ν H (ν)5.
The estimate of the dissipation strength γ is relevant for an
estimate of the power spectrum and the spectral index at the
end of Sect. 4.
Equation (14) does not depend on the a(φ) approximation.
H (φ) as a function of φ in Eqs. (10)–(14) can be obtained
from Eq. (8). The dependence of a on φ in Eqs. (10)–(13)
in the slow-roll approximation is determined by Eq. (9). Let
us consider simple examples. If V = gφn (chaotic inflation
[
36
]) then
→ +∞ then a → 0, if φ
→ −∞ then
= exp
− 4π Gn−1φ2 .
a(φ)
a0
a(φ)
a0
If V = g exp(λφ) then
8π G
= exp − λ
φ ,
Note that if φ
a → ∞.
For a flat potential
φ2
V = K + φ2
we have
a(φ)
a0
= exp
− 2π Gφ2 − πKG φ4 .
For “natural inflation” [37]
V = g(1 − cos φ).
Then,
a(φ) = a0 exp 8π G ln 2 cos2( φ
2
The noise corrected a(φ) relation could in principle be
derived from the solution of Eq. (6) but in general this is
possible only on a perturbative level (see Appendix A).
3 Expectation value of e-folds
We treat ν (in Eq. (5)) as a random time (because a is random).
Let us consider a differential of a function of the stochastic
process (6) in the e-folding time in the Stratonovitch sense
d f = ∂φ f ◦ dφ
1
= ∂φ f ◦ − 3H 2 V dν +
= ∂φ f
1
− 3H 2 V dν +
γ 3 d B(ν) + 21π H d W (ν)
3(a H ) 2
γ
3 d B(ν) + 21π H d W (ν)
3(a H ) 2
1
γ
3 ∂φ
+ 2 3(a H ) 2
γ 1
3 ∂φ f + 8π 2 ∂φ H ∂φ H f dν.
3(a H ) 2
For the Ito stochastic equation
d f = ∂φ f ◦ dφ
1
= ∂φ f ◦ − 3H 2 V dν +
γ 3 d B(ν) + 21π H d W (ν)
3(a H ) 2
= ∂φ f
1
− 3H 2 V dν +
γ 3 d B(ν) + 21π H d W (ν)
3(a H ) 2
1
γ
+ 2 3(a H ) 2
3
2
1
∂φ ∂φ f + 8π 2 H 2∂φ ∂φ f dν.
In the rest of this section we follow Refs. [
23,26,38
]. In the
Stratonovitch case we choose a function f
1 1 γ
−∂φ fS 3H 2 V + 18 (a H ) 23 ∂φ
γ
3 ∂φ fS
(a H ) 2
1
+ 8π 2 ∂φ H ∂φ H fS = −1
and in the Ito case
1 1 γ
−∂φ f I 3H 2 V + 18 (a H ) 23
1
+ 8π 2 H 2∂φ ∂φ f I = −1.
2
∂φ ∂φ f I
Then, integrating d f between ν = 0 and ν corresponding
to the values φ (0) = φin and φ (ν) = φ we obtain (the
expectation value of the Ito integral is equal to zero)
ν =
f (φin) − f (φ) .
Let ∂φ f = u. Equations (24) and (25) for u are of the form
(21)
∂φ u + Q(φ)u = −r,
(22)
(23)
(24)
(25)
(26)
(27)
where
1
Q = − 3H 2 V
and (Ito interpretation)
r =
8π 2 + 18a3 H 3
−1
dψr (ψ ) exp
Q(X )d X .
φ
φ
φ1
u(φ) = −
dψr (ψ ) exp
−
φ∗ ψ
where φ∗ is chosen to satisfy proper boundary conditions.
Then (an analog of the formula derived by Starobinsky
and Vennin [
23
])
φ
Q(X )d X ,
f (φ) = −
dφ
dψr (ψ ) exp −
Q(X )d X .
φ
ψ
φ
φ∗
Q
−
φ
ψ
u(φ∗)
φ
ψ
Q
This solution satisfies f (φ1) = 0 and φ∗ is chosen so that
f (φ2) = 0. Then, according to Eq. (26)(setting φ = φ2 to
get f (φ2) = 0) we have (an approximate formula for a mean
value of e-folds appeared already in [
11
])
ν = −
φin
φ1
ν is the umber of e-folds between φin and φend = φ1. We
have to determine φ∗ from the condition f (φ2) = 0.
Another method is considered in [
26
]. There, the solution
of Eq. (27) is written in the form
u(φ) = exp
−
Q(X )d X u(φ∗)
−
dψr (ψ ) exp −
Q(X )d X
(33)
Then, the boundary conditions are expressed by u(φ∗) and
φ∗. Integrating Eq. (33)
f (φ) =
dφ exp −
Q
u(φ∗)
φ
φ∗
dφ exp −
dφ
dψr (ψ ) exp
(35)
= − ∂φ f I 2
9(a H )3 + 4π 2
φ∗, φ1, φ2 are arbitrary but a useful choice is φ1 = φend ,
φ2 = ∞ and φ∗ = φin. So, we calculate the first hitting of
a boundary of an interval [φend , ∞] by the process starting
from φin (we know that φ = ∞ cannot be achieved, so there
remains φend ). There may be some problems with
integrability in Eq. (36) with some potentials in an infinite interval
[
24,25
] (if there is no thermal noise). Then, Q −V −1
and the integrability may fail if either V = 0 or V does not
grow fast enough.
From the formula (23) we have
ν = f I (φin) +
∂φ f I
1
+ 2π H d W (ν) .
0
φ
γ
3 d B(ν)
3(a H ) 2
Taking the square and then the expectation value of Eq. (37)
we obtain
ν2
≡ ν 2 + (δν)2
ν
= ν 2 +
0
dν (∂φ f I )2
(an analogous formula holds true for fS). Assume that we
find a function FS such that
in the Stratonovitch case and FI
1 1 γ
−∂φ FI 3H 2 V + 18 (a H ) 23
1 1 γ
−∂φ FS 3H 2 V + 18 (a H ) 23 ∂φ
γ
3 ∂φ FS
(a H ) 2
Solving for u(φ∗) gives the formula for ν
ν =
dφ exp
−
×
−
×
φin
φ1
φ2
4 e-folds, their fluctuations and the power spectrum
The general integral formulae in Sect. 3 do not allow an
explicit calculation of the functions F and f needed for a
computation of the e-folds and their fluctuations. We need
a perturbative approach. We write Eq. (27) as an iterative
perturbation expansion in Q1 starting with u(1) = − Qr
u(n+1) = − Q1 ∂φ u(n).
Then, the zeroth order approximation in Eq. (27) corresponds
to setting Q1 ∂φ u = 0. Hence,
u(1) = f = −r Q−1 = 8π G V (V )−1.
In this approximation we have derived the “classical” formula
(9) for e-folds. We have
∂φ u(1) = 8π G 1 − 2η
with
1
= 16π G
1 V
η = 8π G V .
V
V
2
Hence, ∂φ u(1) = 0 in general. We get ∂φ u(1) = 0 for the
exponential potential. For a power-law potential φn we have
∂φ u(1) = 8πnG . u(1) is independent of γ but u(2) (44) depends
on γ as Q does. If the thermal noise is absent then the
expansion (44) is an expansion in G (i.e., in the inverse of the
Planck mass). Next, we need an approximation for the
solution of Eq. (42). Applying again the expansion (44) in Q1 (in
the lowest order Q1 ∂φU
0) we obtain
in the Ito case. Then, calculating d F in the same way as we
did for d f in Eqs. (22) and (23) using Eqs. (38)–(40) and
taking the expectation value we find
8π 2 + 18a3 H 3
.
(49)
(V )−1∂φU (1).
(50)
(51)
(52)
(53)
The power spectrum P U u−1 at γ = 0 derived from
Eqs. (45) and (49) coincides with the standard formula [
3–
7,39–42
]. It can be obtained from the general formulae of
Sect. 3 which involve Q. These formulae for e-foldings
in the quantum case (cold inflation) have been discussed by
Starobinsky and Vennin [
23
]. If γ = 0 then
φ 3 1 3 1
ψ Q = − 8G2 V (φ) + 8G2 V (ψ ) .
In order to calculate the integrals (30)–(36) they perform the
Taylor expansion of V (1ψ) around φ
1 1 1
V (ψ ) = V (φ) + ∂φ V (φ) (ψ − φ) + · · ·
Changing variables
ψ − φ = G2 X
and expanding the exponential in G2 we derive the
perturbation expansion (44).
Next, let us consider the thermal noise using the method
of Starobinsky and Vennin [
4,23,38
] (which is equivalent to
1
the expansion (44) in γ 2G− 2 and in G). Then, in Eq. (28)
(without the quantum noise)
V √V a3
6
Q = − γ 2
(ψ ) =
(φ) + ∂φ (φ)(ψ − φ) + · · ·
Then, in the integral (31) we have (neglecting ∂φa G and
the second term in Eq. (54) being of higher order in G)
ν =
dφr (φ)
dψ exp(∂φ (φ)ψ )
By means of integral formulae of Sect. 3 as well as with the
perturbation expansion (44) we obtain in the first order the
same formula for e-folds as we could get in the calculation
without noise (showing that the stochastic method of
reaching the boundary has the correct no noise limit; the stochastic
formula will still be discussed in Appendix A). The
calculation of δν2 with the thermal noise on the basis of Eq. (41)
involves calculation of the integral
F
2
φ
dψ ( f (ψ ))2 exp −
φ
ψ
Q
with Q of Eq. (52). The Taylor expansion (56) in the integral
(58) gives
1 3
U = F = −γ 2(8π G)2(24π G)− 2 V 2 (V )−3a−3.
The power spectrum can be defined by fluctuations of the
e-folds
P =
d
d ν
Hence
d (δν)2
d ν
,
= −( f )−1 d .
dφ
U
P = F ( f )−1 ≡ u
where (δν)2 is defined in Eq. (38). We have
(evaluated at the horizon crossing k = a H [
43
]) where f
is defined in Eqs. (24) and (25) and F in Eqs. (39) and (40)
(calculated from Eq. (42)). It has been shown [
23,38
] that
the formula (62) in the expansion (51) (no thermal noise)
coincides with the standard one for the cold inflation [
3–
7,33,39–43
] because we obtain from Eqs. (41), (42), (45)
and (51)
2G2
8π G V 3(V )−2.
Pq =
3
It is difficult to calculate P analytically for general quantum
and thermal noise using the formulae of Sect. 3. We calculate
the power spectrum with no quantum noise (solely thermal
noise) applying for F the same approximation which we
used in Eq. (62) (for ν , i.e. for f ). Then, from Eq. (63)
(63)
Pth = 8π Gγ 2(24π G)− 21 a−3V 21 (V )−2.
(64)
(56)
(57)
(58)
(59)
(60)
(61)
(62)
From the Q1 expansion using Eqs. (45), (49) and (62) we
obtain
P = Pth + Pq
(this simple additivity holds true only in the lowest order of
the Q1 expansion as can be seen from Eq. (50)). The spectral
index nS can be calculated as a derivative (61) over ν of
ln P. Then
nS − 1 = −( f )−1 d ln P = −F ( f F )−1 + f ( f )−2
dφ
We obtain from Eq. (64) (the spectral index for warm inflation
is calculated in [
44,45
] but under different assumptions)
nth
S − 1 = −3 −
+ 2η.
nqS − 1 = −6 + 2η
For the quantum stochastic inflation the formula (63) gives
(in agreement with [
39–43
]). On the basis of the Q1 expansion
using Eqs. (66) and (65) we obtain for the spectral index of
the scalar field in thermal and quantum noises
nS − 1 = (Pq + Pth )−1 Pq (nqS − 1) + (ntSh − 1)Pth .
(65)
(66)
(67)
(68)
(69)
For a small γ from Eq. (69) and Eqs. (63) and (64)
nS − 1 = (nqS − 1) +
32π 2γ 2
9
q
H −5a−3(ntSh − nS) .
5 Summary
The method of a description of inflation in terms of the
fluctuations of e-folds (called δ N method) has been suggested
long time ago [
4,41,46
], developed recently and applied to
detailed estimates of inflation parameters [
23–25,47–51
]. In
this paper we have extended this formalism to include a
thermal noise. The thermal noise modifies the results on power
spectrum. In principle, the method allows to calculate the
inflationary parameters non-perturbatively for a larger class
of potentials. In the lowest order of a perturbative expansion
we have obtained a formula for the power spectrum which
is just a sum of the density of thermal (proportional to γ 2)
and quantum fluctuations. The spectral index is an average
of spectral indices of thermal and quantum fluctuations with
the corresponding power spectra. The correction to the power
spectrum is proportional to γ 2 (which is small for low
temperature in the warm inflation models). The spectral index
depends on the inflaton potential. It is measurable in
observations [1]. Its value can give some information on the inflaton
potential as well as on the friction in inflaton wave equation.
αλ
8π G
ν
0
ν
exp
X0
αλ
8π G
(ν − s) d B(s).
Then (in the decomposition into classical and stochastic
parts)
3λ2 3
− 32π G (ν − s) − 2 s d B(s).
Acknowledgements The author thanks the anonymous referees whose
comments contributed to a substantial improvement of the initial version
of this paper.
Open Access This article is distributed under the terms of the Creative
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Appendix A
It is instructive to compare approximations applied in Sects.
3–5 with exact solutions (some solutions of stochastic
equations with quantum noise are discussed in [
16–19
]).
For the exponential potential (see [
34,52
]) V = g exp(λφ)
in the e-folding time the stochastic equation (6) reads (with
a(φ) derived in Eq. (16))
λ 1
dφ = − 8π G dν + 2π
8π Gg
3
exp
λφ
◦ d W (ν)
exp(αφ) ◦ d B(ν),
(70)
(73)
(74)
(75)
(76)
(78)
(79)
(80)
X ≡ Xcl + Xst = exp
αγ
− 3
3
8π Gg
exp
ν Y0
,
For the exponential interaction in the slow-roll
approximation
H
dφ
dt
2
= (8π G)2λ−2.
where
α =
γ
+ 3
exp
λφ
◦ d W (ν)
γ
+ 3
exp
In the Stratonovitch interpretation these equations can be
solved exactly if either the quantum noise or the thermal
noise are absent. No exact solution exists if the Stratonovitch
differential in Eqs. (70) and (71) is replaced by the Ito
differential. In such a case in order to approach the solution we
would have to use the relation [
30
] (with a certain
parameter σ ) exp(σ φ)d W = exp(σ φ) ◦ d W − 21 σ dφd W exp(σ φ).
The resulting Stratonovitch equation (which can be treated
as an ordinary differential equation) would not be linear. If
quantum noise is absent then we set
X = exp(−αφ).
Hence, fluctuations of δρρ are proportional to fluctuations of
φ. Fluctuations of φ in Eqs. (73), (75) and (77) can be
calculated in a power series expansion in the noise (see similar
calculations in [
53
]). So, in the case of the thermal noise
(ν − s) d W (s). (77)
We can calculate the power spectrum from the formula for
the energy density fluctuations δρ
λαs
− 8π G
d B(s)
φ − φcl = β
ν = νin − 8πλG φ. (84)
Hence, at the lowest order fluctuations of φ are proportional
to fluctuations of ν. Now, we can calculate the spectral index
as ∂ν ln (δφ)2 with the result
Stratonovitch form of stochastic equations. On the other hand
correlation functions are easier to calculate with the Ito
integrals (in particular, an expectation value of the Ito integral is
zero). One can relate both integrals according to the rule
1
f ◦ d W = f d W + 2 d f d W,
where after calculation of d f as a function of W we use the
rule d W d W = dν.
Let
1
∂ν φcl = − 3 H (φcl )−2V (φcl ),
where H is related to V by Eq. (8). Let δ = φ − φc. First,
consider the Stratonovitch version of Eq. (6). We write φ =
φc +δ, use Eq. (88) and expand Eq. (6) till the second order in
δ. Then, integrating, taking the expectation value and using
f d W = 0 we obtain
H
H
H
H
(87)
(88)
(89)
(90)
(91)
(92)
nth 3λ2
S − 1 = −3 + 16π G
for thermal noise and
nqS − 1 = − 8πλ2G
for the Starobinsky (quantum) noise in agreement with Eqs.
(67) and (68). In all cases (73)–(77) we obtain stochastic
corrections to the classical formula (9). As an example from
Eq. (77)
exp
1
− 2 λφ
1
= exp(− 2 λφ0) exp
exp
1
− 2 λφcl
Hence, in the approximation exp( f ) exp( f ) we obtain
the classic formula ν = −8π G φ . From Eq. (77) we could
obtain further relations between correlation functions of φ
and ν.
Appendix B
Let us repeat the derivation of the fluctuation equations
(without the thermal noise) of Refs. [
21–23
]with some care
concerning the rules of the stochastic calculus [
30,54
]. We
discuss here also the difference between the Ito and Stratonovith
versions. The conventional rules of the differential
calculus (in particular, the Leibniz rule) are satisfied in the
G H 2.
− 2π
δ
δ2
δ2
δ2 ,
δ2
δ =
ln
H
+ H
H
H
1 H
2 H
G
− 2π (H H )
H
H
G
− 4π (H H )
The equation for fluctuations reads
δ2
=
2 ln
G
− π
(H 2)
H
H
3G
− π
G H 3
H 2 δ − π H .
In these equations we have replaced the variable ν by φcl on
the basis of Eq. (88). The “prime” denotes a differentiation
with respect to φcl .
In the Ito interpretation of Eq. (6)
δ = ln
δ2
=
2 ln
H
H
1 H
δ + 2 H
H
H
G
− 2π
(H 2)
H
H
2G
− π
G H 3
H 2 δ − π H .
Equation (92) is different from Eq. (38) of [
21
] and Eq. (A.22)
of [
23
] as the terms G H 2 δ and G(H 2) are absent there.
We can solve both Stratonovitch (90)–(89) and Ito (91)–(92)
equations in a perturbation expansion in m −pl2 = 8π G. We
expand the solution of Eqs. (91) and (92) around the one of
[
21,23
] (with δ = 0 and no extra terms) then we obtain that
in such an expansion δ G2. Hence, the term G H 2 δ in
Eq. (92) will be of order G4. As δ2 G2 another extra term
in Eq. (92)G(H 2) δ2 G4. Hence, in comparison with
Refs. [
21,23
] the extra terms are of higher order in G (at this
order, our starting point, the equations for stochastic inflation
would also need a modification). The same argument applies
to the Stratonovitch Eq. (90). It is different from equations
of Refs. [
21,23
] by terms of order G4. For this reason till
the order G4 we have the same conclusions concerning the
solution of Eqs. (90) or (92) (these equations determine the
fluctuations and power spectra).
We can repeat the calculations of fluctuations for the
thermal noise rederiving the formula (64). Then, at the lowest
order in γ 2 we get the additivity of fluctuations (65) and as
a consequence the formula (69) for the spectral index.
Let us consider the particular case of the Stratonovitch
stochastic equation (6) for V = 21 m2φ2 with no thermal
noise
dν
dφ = − 4π Gφ + qφ ◦ d W (ν),
where q =
G3mπ2 . The Ito version of Eq. (6) is
1 q2φdν
dν dν
dφ = − 4π Gφ + qφd W (ν) = − 4π Gφ − 2
+qφ ◦ d W (ν)
Equation (93) has the solution
φν2 = φ02 exp(2q W (ν))
ν
1
− 2π G
0
φν2 = φ02 exp(−q2ν + 2q W (ν))
ν
1
− 2π G 0
ds exp 2q W (ν) − 2q W (s)
where φ0 is the initial value. The solution of Eq. (94) reads
We can express correlation functions of φ2 in terms of
correlations of ν. In particular, for the Stratonovitch version
φν2 =
φ02 − q−2
and for the Ito version
φν2 =
φ02 − q−2
1
4π G
1
2π G
exp(2q2ν) + q−2
exp(q2ν) + q−2
1
4π G
1
2π G
.
It can be seen that from the requirement of positivity of φν2
we get some bounds on the initial values φ0 and
expectation values of ν. In particular, the relation (15) φν2 =
1
const − 2π G ν holds true only in the lowest order in q. The
fluctuation equations for the model 21 m2φ2 discussed in [23,
Eq. (A.33)] follow from Eqs. (90) or (92). The exact solution
(95) and (96) does not tell us more than Eqs. (90)–(92) (with
H φ) concerning quadratic fluctuations. However, using
the solutions (95) and (96) we could get explicitly the higher
order fluctuations of φ2.
(93)
(94)
(95)
(96)
(97)
(98)
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